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AN EVALUATION OF THE ODDS RATIO AS
AN ESTIMATOR OF VACCINE EFFICACY
A T tirsis siihiiiit ted in coriforiiiity wit ti t lie requireriirrits
for the Degree of Master of Science
Graduate Department of C'oiiiniunity Healtli
University of Toronto
@ Copyright by hlicliael S. Slanrio. 1997
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A N EVALUATION OF THE ODDS RATIO AS AN ESTMATOR OF
VACCINE EFFICACY
l las ter of Science - 1997'
lliciiael S. Manno
Graduate Depart nierit of C1oiiiniuriit!- Healtli
1 *iiivrrsi ty of Toronto. C'anada
Abstract
Tlie ret rospect ive case-cont rol s t udy bas ofteri been soiiglit as a q u i c h r . Iess rspeiisive
alteriiative to t lie prospective coliort stiidy in t lie evaluat ioii of vaccine effica(->-. I- i i l i k r
the prospective stiitly. wliicli coinpares at tack rates or siisceptihility in the vacriiiatcil
and uii~acciiiatrtl. t lie cap-cor1 trol s t i l d~ . cuiii pares vacciiiat ion odds i ii disrasrd aiif 1
rion-cliseased. \Vtien t lie cliseasr is rare tliis poses lit t le probleiii. But. ivlieii tlisrasr
i tirideiicr is liigli or vacciiir efficacy is loiv. t lie odtls ratio terids to overest iiiiatr \.ai.riiit\
etficar>-. I'sing tlie niotlels of wcciiie action iritrodiirrd b>- Siiiitli ci al. ( 19d-l) NI.
rsatiiirir t lit* odds ratio iiiider two sariipliiig sclieiiirs for select in; cotit rols. Epidcitiir
siiiiiilat ions of uniforiiily niising aiid st rat ifiecl populatioiis are iisrd to esatii i i i i~ t lit.
rstiriiators wlirti varcitiat ion coverage is everily <List ributeci or cliisteretl t.o ~ar>- i i ig
clrgrers. FinaIl>-. rrsiilts froiii botli prospet-tive aiid retrosprctive rfficary trials uf
Hm rirophilirs i~tjliir nznc I> vat.c.iiies are discussrd i r i liglit of t lie tititliiigs.
ACKNOWLEDGEMENTS
11)- s~ipervisor Paul C'orey for tiis giiitlaiice. frieridsliip anrl. niost of d l .
i t i niy abi l i t~ . . Tliatiks Paul. IOii 're tauglit me s o itiucli iiiurr tliaii jiist
his c-orilidetiw
: statistic-S.
11). pareiits for thei r love ancl support (sorry a b o u t al1 tlie late iiiglits).
.\Iy b is sis' .\rina for t h e words of eticoiirageriient.
Daiiiiy Lupez for tiis fririicisliip.
Professors i\lirliael E s r o t ~ a r . Jeffrey Rosent hal. Rot, Ti hstiirniii. .Joliii Hsieti aiid I i r i ~ I i
I i iiiglit for t lie rrnding of t [lis t hesis a n d t lie lielpful supgrst ioiis.
T tir BI.-G.C; for a11 t lie fiin t iriies ( aiici coffee t~reaks!) . especially: ( 'arriicri --So art- yoii
goriria cal1 lier?" hIak. Rafal **\k-tiy tlori't j.oii learri eniacs:>" Iiiist ra. C'elia C; rerii wootl.
Firtis -.Reallj-'.'" Xir . Jaiwj* Sliiii. aiid Aiiriie Diipiiis ( for t tir (wrriprtitioii ).
Gillian Hawker arid ek-erybotlq- a t \$'CH.
Ttioriias Bacon for lielpiiig nie "clear ttic fog".
AI1 creators/protlurrrs/~~eiitiors of food. Iwverages. rritertaiiiitieiit and ar1j.t Iiiiig clse
t h I've co~is i in i rd ici tlie last 6 niontlis wliile 1 was piilling in- liair ou t writiiig tliis
ttiesis - C'HEERS !!!
.\nuone else I've forpotten (sorry!)
. . . . . . . . . . . . . 1.1.1 LTaccine Efficacy basrd on At tark Rates Z!)
3.2 The Odds Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :{ 1
. . . . . . . . . . . . . . . . . . . . . . 1.2. 1 I-iiniatched Estimators :I 1
. . . . . . . . . . . . . . . . . . . . . . . 3 2 . 2 llatchecl Estinintors :IS
3.1 Frailty Selrctio~i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -CU
3.4 O t her Sources of Bias to C'orisider . . . . . . . . . . . . . . . . . . . . 41
4 Simulation Study 4'7
. . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Siniillatiori 1Ioclel 47
. . . . . . . . . . . . . . . . . . . 4 . 1.1 The lrniform Mixirig lfodel 4S
. . . . . . . . . . . . . . . . . 4 . 1 . The Son-I..nifori.~i hIisirig lIodel 49
. . . . . . . . . . . . . . . . . . . . . . . . 4.2 Slodelling Laccine Action 51
4 . 3 \.k.\‘liy Siriiiilat ion:> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4 Results of the Siiiiulatiori Çtuclj* . . . . . . . . . . . . . . . . . . . . . .; 2
. . . . . . . . . . . . . . . . . 4.4.1 I.'iiiforrii ( Hornogeneoiis) S'lising 53
4 . 4 . Son- l'ni forrii ( Heterogerieous) Slisirig . . . . . . . . . . . . . . (j!)
5 A Look At Some Case-Control Studies of Hib 92
. . . . . . . . . . . . . . . . . . . . . . l Hnrir tophi l i i s i r f l a r r c z n t Type II !E
5.2 PRP-Dl'ac-ciric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9:j
. . . . . . . . . . . . . . . . . . . . . . . . . 5.3 1.. E S t t i d ~ of cjwiss Infarits 94
. . . . . . . . . . . . . . . . . . . . . 5.3.1 AgeGroup: 6-17 hfoiiths 9.5
. . . . . . . . . . . . . . . . . . . . . 5 . 3.2 .\gr Croup: lS+ Slontlis 97
. . . . . . . . . . . . . . . . . 5.4 Otlier Efficacy Trials for PRPIPRP-D I O 1
. . . . . . . . . . . . . . . . 5 . . 1.1 Effiracy Stiitlies of PRP-D vaccine 101
.i..L.? Efficarx S t ticlies of P RP vaccine . . . . . . . . . . . . . . . . . 10-1
6 Discussion 107
A Derivation of ~ - h * [ \ T ( . j ) ] 109
. . . . . . . . . . . . . . . . . . . . . . .A 1 Derivation due to Becker ( 1982) 109
. . . . . . . . . . . . . . . . . . . A. 2 Alternate Derivation of c -~T*[v ~ ( , j ) ] 1 1 1
B Standard Deviations of I -E Measures from Simulations 113
C Computational Details for I,'E Estimators in Simulation Study 120
Bibliography 122
List of Tables
Filial est iniates for varioiis nieasilres of I - E froni siniiilatioris ~ v i t l i i i ~ i i -
formly iriising population (c '~Is , , '1. hfodel: Leakj- ( i ). triie I 'E = !)O'i;. 517
Final estiniates for varioiis nieasiires of I -E froni simulations witli rir i i -
Eornily riiising population (r-F I S.,) . hlodrl: ;\II/Sotliing ( 2 ) . triita
I ' E = W % . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Final estinlates for various rneasilres of L'E froni siniiilatioris iv i t l i i i r i i -
fornilx niisiiig populatioii (L'F I sLtr). Slodel: Leaky ( 1 ). triie I 'E = 70'7;. 57'
Final tastirriates for i-arioirs riieasiires of 1,-E froni sirriiilatioris witli i i r i i -
Firiid tii~ari estiriiatetl attac-k rates frorii siniiilatioris witli iiriiforriily
. . . . . . . . . . . . . . . . . . riiisitig popiilatiori. Troc I-E = 90%.
Fi rial rriean est iriiated at t ack rates froni siniulat ions vit l i 1-iriifortii ly
. . . . . . . . . . . . . . . . . . riiising population. True I -E = 70%.
Firial estirmates for various riieasiires of I - E frorti sinitilatioris witti non-
. . . . iiriiforrii niixirig ((Fe f s,,). Ilodel: Lealiy ( 1 ) . triie I ,*E = 90%.
Firial estiriiates for various tiieasures of L*C fror-11 siriiulatiu~is witli ~iuri-
uriiforiii iiiisirig (FE k - s tv r ) - Slotlel: ;\lI/Notliirig ( 2 ) . trur L-E = 90%.
Final est iriiates for \sarioils iiieasiires of I*- E frorii sirriulatioris k v i t h riori-
. . . . uiiiforiii tiiisitip ( t ic f n,,). Slodel: Leaky ( 1 ). true I-*E = 70%.
Firial estirnates for various nieasures of F E froni siniulatioris witli riori-
tiriiforrii riiising ( FC k s,,). !dodel: All/Notliing (2). trtir 1,-E = 70%.
vii
iiiixirig. Triir I - E = 90%. . . . . . . . . . . . . . . . .
Firial riieari est iriiatecl at t ack rates frorn sirtiiilatioris IV
iriixing. True I - E = 70%. . . . . . . . . . . . . . . . .
Final niean stiriiated attack rates frorti sirniilatiotis witli riori-iiriiform
1
hivar ia te rrsults for 6-17 nioritli age group . . . . .
Estiniates of l - E ( 0 R ) froni iirirnatched 2 x 2 x k coiitiiigrilcy tahlr
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . arialysis
Estiriiates of I ' E frorii iiiatctird ad!-sis iisirig îoriditioiinl lusistic. ri.-
gressiori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Estiitiatrs of V E frorii iiiitiiatrlietl arialysis iisiiig iiiicoriditioiial logistir
Estiniates of \ ' E ( O R ) froni itriniatclied 2 x 2 x k coiitiiigeiic~- tal~le
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . arialysis
. . . . . . . . . . . . . . 1-nivariate results for 1 S+ iiioiit li age groiip
Estiiiiatrs of I -fi' froiii riiatclirtl arialysis iisiiig (-oiiclitioiial logist i u r r -
. . . grrssioti. ('uiivrrgeiice \vas poor so t lie est iiiiatt3s artb iiiirrliahlv.
Estiriiates of I * E frotn tiiirtiatrtid arialysis iisiiis iiricoritlitiorial Iogistii-
regession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Table of iarcinat ion b>- disease st at us st rat if ed by dq-(.are at trtitlaiii-v
Resiilts of PRP-D Efficacy trial for Fiiiriisli Infants . . . . . . . . . .
Resiilts of PRP-D Efficacy trial for Sativr :\laskari Infniits . . . . . .
Resrilts of \arioiis (.'ase-control Efficacy Trials for PRP-D i r i t lit. I-rii tecl
States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rrsiilts of Variuiis C'asr-cuiitrol Efficaq- Trials for PRP i n tlie I.*tiitrd
. . . . . . . . . . . . . States
Final estinlates of standard der
sirtiiilat ions witli tiriifornily niix
. . . . Leakv. triie I V E = 90%.
iatioris for various nieasiires of I.- E froiii
ing population ( ~ ( P c ) f s,(,.,) '). .\lotl~l:
B.? I-niforiii .\fixing populatiori. 4Iodel: All/Sotliiiig. t rur I ' E = !NI'?;
( * ( c e ) & s,~,,, ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
B . I-riiforin .\Iisiiig population. llodel: Leaky. triir L'E = 70% ( a ( t * ~ ) f .. ,,,,,) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I l ,?
B.4 I-iiiforrii Ifisiiig Iludrl: All/Sotliiiig. t riitl I ' E = T O T
( 6 ( l * ~ ) i . . ,,,,,,) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5
B..'> Final estiriiates of staridard deviations for various riieasures of I ' E frorti
siniulat ions wit 11 non-uni forni niisirig (if( i * ~ ) I A,(.,) ). .\loclt.l: Lrakj-.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . true I-E=c)O%. l l ( i
B.6 Final estirnates of standard errors for \*arioiis nieasures of 1'E fro~ii sirii-
iilatioiis witli non-iiiiiforrii niisitig (ir( r * e ) f s , ~ , . , ~ ) . Iloclrl: .ilIl/Sotliiiig.
t r w 1 -E = 90%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1
B.7 Firial estimates of staridarc1 errors for variuiis nitmures of I 'E froiii
siniiilat ioiis vit li iiori-iiiiiforni iiiisiiig (a( tic ) ir .sfl( Lrv ) - 1 t : Lraky.
triie C-E =70%, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11s
B . Final est iriiates of standard errors for various nieasures of I - E frorti si iii-
iiiat ioiis witli non-iiiiiforrii iiiisirig (a( c.6 )f s,~,.,) ). .\Iodel: ;\ll/Sotliiiig.
t r i i c I - E = iOY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 9
List of Figures
C'oniparisoii of Epideniiî ciirves . . . . . . . . . . . . . . . . . . . . -
SIeaii niraslires over t iriie for siniulatioiis witli iiriiforiii i t i i s i i ia fur al1
coverage lrl-els: po = 0.000S. True 1'E = 90%. Leaky riiodel ( 1 ) . . .
hleari iiieasurrs over tinie for simulations witli uiiiforrii riiisiiig for al1
corr rasc levels: po = 0.00 10. Triie C-E = 90%. Leaky niodel ( I ) . . .
Sfeaii nieasures uver t ime for sinidations with iiiiiforiii riiisiiig for al1
cuverage Ievels: pu = 0.0005. True 1-E = 90%. All/Sothirig niodrl (2 )
S:eari riieasiirrs 01-er t iiiie for siiiiulatioris wit li iiiiiforiri i i i i s i rig fur al l
i-ovrragr lei-ris: II,, = 0.0010. Triie I * E = 90%. All/Sot liirig iiiodrl ( 2 )
Sleari riieasures oierr tiiiir for siriiulatioiis u-it li iiiiiforiii riiisirig fur al1
coi-erage Irvels: po = 0.000S. Triie I -E = 70%. Leaky riiotlrl ( 1 ) . . .
Slean iiieasiires over tiriie for siiiiulatioiis wit ti iiniforiii iriisirig fur al1
rovrragr Ievels: po = 0.00 10. True I* E = 70%). Lraky iiioclel ( I ) . . .
Sleari ilirasures over tiiiir for sini~ilat~ioris witli iiiiiforiii niisiiig fur al1
roverage Irirls: pu = 0.000S. True 1-E = 70%. .-!Il/Sotliiiig iiiodrl ( 2 1 Sleari rii'asiires 01-er tiiiie for siiiiulatioiis iv i t l i uiiiforiii riiisiiig for al1
coverage levels: po = 0.00 10. True I - E = 70%. All/?;ot lii~ig riiodel ( 2 )
C'oniparison of rriean nieasures over t inie for siniulat ions wit li iiori-
uni foriii niixing under various levels of varcinat ion clustering: True
I 'E = 90%. Leaky mode1 ( 1 ). vaccination coverage is -IO% for cliilciren
and 20% for aclults. . . . . . . . . . . . . . . . . . . . . . . . . . . .
C'orriparisoti of nieari nieasures over t inie for siniuiatioris with riori-
uni forni niixing iinder various levels of varci n a t ion cl ust eriiig: Triir
C-E = 90%. Leaky riioclel ( 1 ). vaccination coi7eragr is 60% for rliildrrri
. . . . . . . . . . . . . . . . . . . . . . . . . . . ancl 30% for aclults.
('oniparisori uf nirari nicasures ovrr t iriie for siniiilatiorls wit li riori-
iiriiforrii niisirig tiridet- varioiis Ievrls of varcirint ion i-liistrriii,q Triir
I' E = !IO%. Leakj. niodel ( 1 ). 1-acciriatioii ro i - r rage is SU% for cllililrtw
. . . . . . . . . . . . . . . . . . . . . . . . . . . and 40% for adults .
C'oniparison of niean nieasures over t inie for siniuiat ions wi t li noii-
iiiiiforni niisiiig under various levels of vaccination clusteririg: Triii
I ' E = 90%. AIl/Notliing niocle1 ( 2 ) . varririatioii roverage is 4OY for
. . . . . . . . . . . . . . . . . . . . . . c h i l d r ~ r i a n d 20% for atliilts.
C'orriparisori of niean nieastires u i w t inie for siniiilat ioiis ivit li iiuri-
uni forni riiisiiig iiiitier ~ a r i o i i s levels of vacciiiat ioii t-liisteriiig: Triie
1-E = 90%. .AII/Notfiing niodel (2) . vaccinatioii roi-erage is GO'% for
. . . . . . . . . . . . . . . . . . . . . . cfiilclre~i and 30% for adults .
C'oniparison of nieari riieasiires over tinie for siiriiilatioiis with rioii-
iiriiforiii iriisirig iincler varioiis levels of varciriat iori cliistt+rig: Triw
I 'E = 90%. ;\ll/Totl!iiig nioclrl ( 2 ) . vacciriatioii covtbragr is $O% for
. . . . . . . . . . . . . . . . . . . . . . c-fiiIdrt.ri arifl -KI3 for d i i l ts.
C'oiiiparisoii of nieari irieasures over tirne for sitniilatioiis w i th tiori-
iiiiiforrii ni ising under varioiis levels of vacciiiatiori clustering: Triie
I V E = 70%. Leaky niotfel ( 1 ). vacciiiatioii coverage is 40% for rliiltlreri
. . . . . . . . . . . . . . . . . . . . . . . . . . . a n d 20% for adults .
Coniparisoii of iiieati iiirasiires over t iriie for siriiiilatioris wit li iioo-
iiiiifor~ii riiixing iiiider various levels of varciriatioii clitstrriiig: Triie
I -E = 70%. Leaky mode1 ( I ) . vaccination coverage is 60%. for cliiltlreii
arici 30% for adults. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Coin parisoii of niraii ilirasures over t irtir for sirtitilat ions \r i t li
uni forrii niisiiig iincler varioiis Ievels of vaccination cliisterir~g: - I'E = i 05%. Le&!- iiiotiel ( 1 ). t-acciiiatioii çoveragr is SU% for rlii
. . . . . . . . . . . . . . . . . . . . . . . . ancl 10% for aciults.
C'oniparisoii of nieaii iiieasiires over tinie for sitiiulat ions tvit 11
S . .
11011-
uiiiforni iiiisiiig under various levels of vacririatiori rliistrriiig: Trirr
I -E = 705. .Ail/Notliing riiode1 (2) . varîiiiatioii coverage is 40% for
. . . . . . . . . . . . . . . . . . . . . . children aiid 20% for adul ts .
C'oinparison of niean irieasures ovr r tiiiir for siniulatioiis with rioil-
iinifortii rtiixing under variolis levels of vaccinat ioii cliistrririg: Triir - I'E = 10%. All/Sotliing niotlel ( 2 ) . L-aCciriatioii covrra,qe is Ml'% fur
. . . . . . . . . . . . . . . . . . . . . . cliilclreri a n d 30'3 for aclults.
C'oniparison of rtieaii rtieasures oi.er tiiiie for si~iiiiiatioiis \ r i th iioii-
tiriiforni i i i i s i i i ~ uiitler various levels of vacciiiat ion rlusteririg: Tri ir
I ' E = 70%. .\ll/Yotliiiig nioclel (2 ) . vaccinatiori (.-O\-erage is SO% for
. . . . . . . . . . . . . . . . . . . . . . rtiildreri arid 40% for acliilts.
Sirriii1att.d cListrihiitioris of I - E(.4 R ) foi. c-oliurts of 1000 aticl at tar-k r;rttl
uf 10% iii the iiiivacriiintrd. 5000 siiiiiilatioiiu t w r r (lotit. iisiiig si inplt~
biriorriial rarido~ii \*ariabltls. . . . . . . . . . . . . . . . . . . . . . . . .
xii
Chapter 1
Introduction
1.1 Background
siicli as nieasles. pertuçsis. ancl influeriza. In orcler for a vact-irw to Iw lic-etistvl Li-
lise in C'aiiatla or t he r t i i t rd States. its safety and efficacy iiiiist be proi-rii. Epiilriiii-
ologic stiidies siicli as raiidornized coiitrolled trials liai-e generally bwii wccptrd as
t lir Iwst tiiet liod for deteririitiiiig t lie protrctive r f f i c a c ~ . of a varcine. S t iid>- aiil~jtv-t s
artJ raii(loiiiizrc1 to oilt. of ttro groiips - taccinateci aiid iioii-iacciiiatrtl - aiiil ilist.ns<~
i iiricltwcr is siil)seqiieiit ly roiiiparetl in t lie tii-o groiips.
Tliere are alterriati\-e riiethods of assessirig vaccine etfectivt.riess otir of ivtiicli is
t tiruiigli serologiî stutlics. Serolugic stiidies are a niore iriciirect iiirasiirr uf i b t f ( ~ -
tivriiess since theu geiierally nieasure sorrie surrogate indicator of cliiiical protectiuii.
Tlierr are two types of serologic stuclies - srroconc~rsiorr and s r roprrra1fnc-r. I i i se-
roconversion studies. a blood saniple is obtained twice - before racciiiatioii. aiicl after
taccination a t a tiiiit. wlien a certain ariiourit of aiitibodies caii s ~ i p p o s ~ d l y Iw de-
trctrtl. The serocoiirersio~i rate is t lien corriputed as t tie proport ion of siibjeets wlio
rlcvelop aotibodies. Iii seroprevaleiice stiiclies. a blood speciirirti is obtaiiircl aiid tlitl
atitibodj. levels a r e tlicn compared wit h the su bject 's dispase and vaccinatioii liistory.
The proportion of vaccinated persons witli detectable antibody response I>qoiicl a
certain protective level is then taken t o be t he niinimum proportion proterteci. Two
iiiajor clisadvaritases of serologic studies is tiiat t hej- require extensive laboratur~. work
aiid rio not assess the impact (both direct aiid indirect) of vaccination at tlie (-oriiiiiii-
iiity and/or population level. For tliis reason. epidmiiologic stiidies are carriscl oiit
to adclress t hese short coniings.
Epirlriiiiologic st uclies nieastire t lie relative risk of diseasr anioiig t lie rare-iiiatrv l
corn parer1 wi t ii t he iiiiva(-ciriatecl. In gerit.ral. ariy st r i{f'- n-liich is c-aliahltb of (lc-terriii 11-
iiig relative risk cari hr used to calculatr vaccine rfficac~-. Siiice 19 15. the pruttv-t i \ -P
effcct of a vacci tic lias been conii~iorily evaluatect froni ratidoiiiizecl trials (or w-eii
uotl~reaks) L>j- iiieaiis of t lie folloivirig foriiiula:
wliert- I E is the vaccirit- effiracy. .AR1 is tlie nt-tack rate1 i i i tlir L-a(-(-iiiatrd guiil).
.-\Ru is the a t ta îk ratr iii tlir ~iiivarcitiateci groiip. aiici R R is the relative attack rate..
The qiiaiitit>- 1-E is cletirir(l as tlie perrriitagt. rrcliictiuii i r i tliseasr i i i r - i c l r i i w < I i i c l
tu varciiiatiori and lias range ovrr the iriterval ( -x 11. I f tlic racciiir i i i qiiestiuii
otkrs soriie protectioii then -4 RI ivill be l e s than -4 Ro and \,-E will falI betweeii O
aiid 1. -4 L-accine rfficacy value of 1 nicaiis t h a t t lie vaccine is coiiiplrtrly siiccrssful
i i i eradicat iiig clisease. Laliirs of vacciiie efficacy ~vliicli are iirgat ive. iiirari t liat t lir
t-a(-ritir is tosic- or i~iirtiiiiiosii~~presive. S I ~ C ~ I c0111d Iw the ritse i f a va(-c-iiir is itiatlc
froiii a wilcl virus ivliicli lias i i o t berii su<-c-essfrill~. iriac-tivatetl.
Tlie clefiIiitiu~i of varcirie efficacy gi\.eri abovr (-ail orily I>e rstiriiatrcl i r i a i-oiii-
parati1.e cohort stiicly rvliirli iiivolves allocating one group of siil~jt~cts to a vacciiir
group aritl ailotlier group of siibjects to a control group ancl followiiig tlie coliort.
iliroiigli tinie. While trials of ttiis nature are superior to any otlier design tliry are
tinie-consuniirig anci costly. It is for t Iiis reasons t h a t the case-control stucly is soiiglit.
as an alternative. especially i f prior evidence of efficac:. already esists.
'The attnck ratr is sirnply the proportion o f sirbjects who betonie infectecl over t.hc course of tlir rpicieniic. The use of "rate". as opposed t o "proportion". in t h e espression is actiially qiiite appropriate since it is a nieasure of incidence for a girven tnlrnral of Irrrte.
1.2 Case-Control Studies
('ase-control stticlies have breii regarded as important alternatives to coniparativr
rrliiiical trials. If the efficacy of a iaccine has beeri establislied froni prr l-bus studies
and t lie vaccirir lias alrad!- beeti liceiisrd for use tlicri it ni- b r d i i i t r r~s t tu rvaliiatc
its rffir-acj. iri otlirr s t rata. C'ase-cuiit r d stuclies ha[-e tlie ad\;ziitagr of l~eiiig I ~ s s
txspr~isive. less t iiiir-rorisuriiiiig aiicl ret rospect ive in nature. Drtailrtl iiitrrvirws of a
rdatively sniall niiiiiber of cases and controls can t>e coriiliicteil. Herice. it becoriies
possible to ascertain and adjust for a fair number of possibly coiifoiiritliiig factors. So.
t tir qiiestion t lien is: ho ri? accrirntf are case-cont rol stiidies in assessirig t lie protectil-r
eficacy of a \.accine:>
To hegiii. al1 the biases iiiliereiit \vit11 iisiiig the ocicls ratio as ai] rstiiiiatt> uf
prospec-tivr rt'lativt. risk rsist ii i vaccine studies. (.'ase-c-oiitrol ~tir(lit5s (Io riot alIow
direct estiiiiat ioii of at tack rates but rat lier approsiiiiatr tliti ratio of at ta& rat r s
wit li t lie oclds ratio. [ r i otlier words. i f Ive allow D to deriote ( l i s tue arid I tu cleiic)tt>
vaçciriatiori. then the cpiatitity
is h r i i i ~ rstiiiiated II!. t lie ret respective otlcls ratio:
disrase. P( D). is stiiall t lien tliis poses lit tle probleni since it is generally accrpterl
tliat uiider tlie rare disease assuniption t h e otlds ratio does a gooci joh of estiniat-
iiig relative risk. However. i f the disease is coiiinion (as in the case of iiieasles aiid
utlier siniilar iiifectious iliseases prevalerit in scliool-aged cliiltlreii) tlieii t h odds ratio
hrconirs iiirreasiiigly biased. Tlie reasuii is tliat tlir coiitrol group lieroiiirs ovrrpopti-
as opposecl to the population of risk'. The resulting odcls ratio is lower îoiisrqiieiitly
t raiislat ing into an art i ficially liigli vaccine efficacy - - - - - - - - - - - - - ' Later it will be shown that this is an exarnple of frailty selection
Tliere are otlier statistical issues as ivell. Tlie previous paragrapli clws riot gu
o i w any rirw t,erritory since it is iisiml1~- the bcasis for esplairiirig t h e "rare cliseasr . .
irssiiiiiptiuii" iii iiiaiiy i~itroriiictory tests oii rpideriiiology T'lie a h v e s(.eiiiirlu 1s
a ruiiiriiori probleni i n rase-mritrol studies wtirre ~.uriirrlntirr-iri<-idf rr c r snmpling is
eiiiployed. rncler cuitiulative-incitle~ice sarnpling. rases are srlectecl frorii tliosr wlio
c le~dop disease in t lie population w.er some period of tirtie ( to. t wliile <-ont rols are
sariiplecl froni t liose wlio are diseasr- free at t ,. h d e r t liis drsigri. t lie rlisensr itliist
t)e roiisiclered rare i f orle wislies to estiriiate the relatii-e risk iisiiig t lit. otltls ratio.
I f roiitrols art. rtiatrlietl for tiriir as i r i d ~ a s i t y sarriplirig tlirii tlirrr is t*orisi&d>lt*
iitiproveiiient iii t tie otlcls ratio as an estiniator of relatiïr risk. Eac-li coritrul. iii rliis
rase. is niore representative of the populatiori at risk hecaiise lie/slie \vas saiiipltd n l
/ / i f t inir thr cnsc d f c r f o p r d d is fasc . Tliese. alorig witli otlier issiies rrlrvarit to va(-riiir
st iidies have to Le aclclressecl.
L,-accirie stiirlies iri particiilar deserve special attention heraiisc we are dralirig n.i t l i
ail acii t i ii frct ious disease. Tlic data gerierat rd frorii ail iii fect ioiis riisrast. pruwsa
iisiially raiiiiot Ile t r e a t d i i i tlie sanie w+- as a t>-pical clirii(-al trial or coliurt stii(l>- fu r
il 11011- i i i f t ' r t i~~is rlisc*asc brraiisr statistical iiiclepeiicl~iictx iisiially i-oiiirs iiito qiitbst ii~ii.
Tlie iiiiiiil)rr of people becoiiiing iiifrcted is cieperitleiit oii tlie riiiiiil>er aIreiid>- i r i f t l i - tv i l
so vacciiiation of a portion of the populatioii iiicloces botli dir-rct aiitl ii>dirrr*t rfftlcts.
Ttie tlirect effect is to protect tliose w h o are vacciriated. Ail iiidirec-t effect occiirs
hecaiise t lie infect ious groiip in t lie popiilat ioii is effectivrl>- sriiall~r aiid t ratisriiissioii
of the disrast~ is lrss effiîieiit. If the riiirii1)er i r i ttiis grotip is brlow a rcrtaiii tlirc~sliol<l
t heii t lie infect ioiis proress is so inefficirrit t liat it caii iio loiigelr I)c sixstai~i(vl. Siii-li
hrrd imnlunity. as it is roiiinionly referred to by infectioiis <listmr rpideiiiiologists.
is wliy full vacîiiiatioii coverage is iiot uecessary to eradicate disease. Bot li of t lirsr
effects tiertl be taketi iiito accouiit in orcler to properly assess the acciiracy of aiij.
estiriiator of vaccine efficacy.
Tlie priniary objective of tliis work ivill be to evaluate the acciirac!. of tlie udcls
ratio as ari rstiniator of vaccine efftcacy wlieri roiiiparecl to otlirr niorr relial>le aiid
iveIl-establis~ied estimators wliicli are based on attack rates. Oiir arialysis of thcl
problerti will rtiake ilse of several tools. I 'siiig two differrnt iiiodrls of vavriiir actioii
establistied bj- Sniit 11 cf al. ( 1984) and stocliastic siniulatioris baseci ori a tvt.11-kiio~-ri
rpirleniic iiiodel. we will examine the otltls ratio for bot li iiiatclird aiid itiiriiat(-litvl
est irriators tmder different sanipling schenies.
1.3 Why Vaccines Fail
SIost of the matheniatical researcti i r i tliis area lias iiltiriiately beeri iiitliiencxd b>- tilt.
iiiabilit~. of vaccines to pro\-ide "periect" protection. \\,*hile nian?. racciries have bwii
kriown tu be as rfficacious as 9.5 per cent or liiglier. tliere is still an iiriavoidal>lr clrriitwt
of Iiuniari rrror a1ii.l iiiiprerlictability. Tliere are nian!. possible reasom \vli!- [.a[-riiirs
fai 1 to pro~.idc protect i ve iiiiiiiiirii tu. The terni -vacci rie fai luri.". wliile coiiiriiuiily iisrcl
in t tir litrratiirr is riiisleadirig since it iriiplies tliat t lie vaccinr itself is t lie c\ilprit. .\Iost
of tlw reasuiis for ~ w < - i r i r fai turr are due t.o errors i r i liaritltitig aiiti in ;drtiiiiistratioti.
As a resiilt. Fine aiid Zell ( 1994) prefer t lit. niore acciiratr trriii "vacciiiat iuii fail tir-<B.. .
Slost of tlir caiises for failiire bave iiot k e n stiitlird in great detail. Early qy uf
L-ai.ciiiation Iias roninlonly beeii cited as a poteiitial reasoii fur failiirc d i i t b tu t lit.
iriterfer~ric-e of iiiatrrrial a1it.i bodies. O t lier poteritial caiises iricliick ( Fi rie k Zell
( 1994) ):
Handling factors: Past rspirat ioii date. clisiiiirrtaiit i ii suri iigt.. iisr of i i i i -
proptar diliieiit
Administration factors: Iriatlaquate ariioiirit injecteri. incorrect iriject ioi
Host factors: C;enetirall~/pliysiologicall~- iiot responsiv~. waiiirig i i t i i i i i i r i
i site
it>-
be tiiglier t han tliey stiould. Strucliiner F I al. ( 1990) carry out a siriiulatioii st iidy of
riialaria vaccines to explore t lie beliaviour of nieasiires of associat ioii in case-coiit rol
st ildies. Tliey fincl t liat i c i iriteract ing populations. riiatclieti oclds ratios haset1 on risk
srt sairipliiig yield iinbiased estiriiates of tlie liazard ratio. (-iiriiatclitd rstiiiiaturs. oir
t lie ot lier liaiid. were pronr to hias niid were especia1l~- sriisit i v ~ to stvtsuiialit.>-. Tlitir
liypot liet ical populations. liowever. are iiiore tlynariiic in st riict cire. H~iicr. t lie wii-
ilitions emulatetl are niore endemic as opposed to epideniic. C;reetilaiitl aiiri Frrric-lis
( 19SS) responcl to soriie of the issues raised by Sinit ti €1 al. ( 1984). \Vit11 rrsprct tu
case-control stciclies tliey find t liat in orcler t« propt-rlj- estiiiiatr relative risk wit li t lie
ret rospect ive orltls ratio. one slioiiltl select coiit rols \vit liout regard for t hrir rlisrilw
st nt lis wlieri t lie disrase is r-oriiriion. Tliry also stress t lie i rii purtaiir-a uf sc*lrrt i 11%
r-ont rols t hat were at risk of coiit rart iiig t lie (disease cliiriiig the prriod of olxwrvntiuii.
Alost of [iteratitre uii the topir of i.arcirie efficacy fociisrs oii estiiiiators ivliicli artx
Imsed or1 attack rates. Firie aiid Zell ( lc)W j look at factors associateci witli ttit. failiire
of vacciiiat ion. tri particiilar t liey look at the estent to wliicli iiurirancloni distri biitioii
of tliese failiires ancl selectiori of srver oiitlireaks for iiivest igatioii ii if l i i r i i i -ca rstiiiiat es
of varcirie dficac~.. Haber t-t al. ( 199 la) estetid t tie t~vo ~riectiaiiis~iis uf va(-cirie a r t ioii
irit rodiicrtl by Srnit li r t al. ( 19S-L) Ijy iiicorporatirig an iiiidrrl>-irig tl~iiairiic rpi<lriiiii-
niodrl of ari irifectioiis cliseasr. Tlicy slio~v analytically ttiat iirirltlr t tir iriorlrl u.litw t Iic
vaccine rediires tlie probabilitj- of irifectioii giveri esposurt.. 1 ' E hasrd uii tlitb relatii-v
attack rates uii(1erestiniates the protective effect. The- t lien iiitrodiire niid drrii-r a
tiew est iniator wtiicli niore arcuratelv estiniates the vacciiie's protect i ~ e effect.. This
est iiiiator. ivliicli wi Il I>e esplored i n more clet ail. is gi \r i1 tiy
I 'iitler t tir iiiocld i II wliicli t lie iaccine iiiecliaiiisiii proviclrs i-orii plete i ri1 ii i i i i i i t j - t o a
r-ertaiii proportion wliile les\-ing the otlier proportion witli no protrctioii at dl. t lit.
I ' E based on relative attack rates will be unbiased. Siniilar resillts are ohtaiiietl hy
Svensson ( lY9 1 ) who anal>-sed effects of vaccines in stratifieci popiilatioiis. Loiigirii f t
al. ( 1993a) and Haber ct al. ( 199 1 b) explore furtlier the beliavioiir of the est iniators
iritrocliiccd by Haber € 1 al. ( 199 la) undrr iiiore coniples sitiiations. . \ Io r~ sperifirallj+.
t lieu look at t tir impact of indirect effects of vacciiiat ion aiid iiit roduc-r siiriiiiiarJ-
nieasures of 1- E to incorporate population Iieterogprity. Iii acldit ion. iiirtliods of
ronipiiting vaccine efficacy ivlien seconclary attack rates from tioiiseliold ancl t-uriiiiiti-
nity data are available are presented. Halloran r t al. ( 1992) specify t h r w acidiiioiial
txpes of \xcci~ie action ivhich are tvariat ions of t lie tivo basic tj-pes and drri\-r si1iiiiiiar~-
iiieasores of i-arcirie effificaç~ under t liese iiiodrls.
Becker ( 1993) presents a coiiriting process mode1 for epitleniic processes basrd <,ri
the trsdit ional s IRQype niodel of infectious disease aiid deniotistrates lioii. it cari
b r iisecl to derive the IVE uritler tlie leakx niodel. Lorigiiii ri al. ( 19931)) rsteii<l
tliis niocle1 to a population tliat is stratifieil by vaccine-relatecl factors aiid i i s r tlir
inartingale-basrd nirtliotls of Becker to ilerivr siiirirriary riirasures of I ' E alorig \vit li
t heir variances iintler t liis tieterogerieity.
3 ~ h e SIR model. which is short for 'Susceptible-InfectioiisRernoved". will be discussed i n gr~atrr detail in Section 2.4
Chapter 2
The Measurement of Vaccine
Efficacy
2.1 Models of Vaccine Action
kaci-irws agaiiist iiik(-tioiis diseases are i i i t r i idd tu lorvrr ari iii(lii.i<liiiil's siisi-rbptiI>ilit~-
ti> iiilrct ion. Howe\-rr. it is iiiiporta~it to Iw aivare of t t i r iiiiclt-rl>-irig iiitacliaiiisiii i ) r t i i i i ( l
t l i r vawiiie's nctioii. Tliere are several different esti~iiators of vacciiir rfficac-y a\-aila1)lr
aiid t l i 4 r use ancl biological iriterpretation will depend un ~vliat kiiicl of protwtioii t l i t -
vaccine provides. For our purposes. two nioclt.1~ of protest ion (or actioii ) rvill I ) t b
prtwiited arid ~ s p l o r e d . Later. it will be sliowii t tiat t liese tivo tiiudrls are art iiall~.
special cases uf a riiore geiicral iiiorlrl.
2.1.1 Mode1 1: The Leaky Vaccine
Suppose tliat in a sniall interval of tiiiie ( t . t + d t ] the probabilitj- of a siisrrptil>lr.
ri~ivacciiiated persoti becoiiiirig i tifected is giïen by ho( t )dt. In essence. t lie ptmoii's
siisceptibility at tiiiie t is nieasored bj. h o ( t ) . I f vacciiiatioii reduces t l i i siisrrptil>ilit~
11'- soiiie coristant proportion to say h l ( t ) = k h o ( t ) . wliere O < A. < 1. tlieri the
~ a î r i i i a is saitl to follow the l d y cnccir~r iiiodrl. Iritiiitivel~. we (-an tliiiik of tlie
the iiiiiiiuiie sj-sterii as Iiaving -gaps" t lirough vr*liicli infect ious agents caii tariter ancl
tliat tlie vaccine serves to rediicr the size of these gaps. Tliereforr. t tir prol>al>ility o f
gett itig tlirougli t he gaps tirrrrasrs \vit li varciiiatioii. It does iiot. liowe\-rr. rrd1ir.r t tir
probabilitj. of infection to zero. Vaccines beliet.et1 to beliave acrortliiig tu tliis iiiuclrl
are typicallu killed-virus vaccines or vaccines t liat protect agaiiist hactt~rial agents
(Srititli e t al . ( 19S4)).
1-iidrr tliis riiotlel. vaccinatiori reriders a proportion. say n ( O < n < 1 ) of ttir vami-
iiattd group conipletrly ininiitne and leaves t lie reniairiing proport ion 1 -ci iiiiçliaiigt-ti.
Tlir rai-i.iiiatrt1 groiip t lierrforr corisists of two iiriiderit ifiablr s t ra ta n i t l i silswpt i hil-
ity: t
Tliis niodel clescri bes a varrine wliicli is bot li liiglily poterit j-et r s t rriiirlj- sriisit ii.e
t.o liaiidliiig. Vacriiirs t liat bellai-e i r i t liis fasliioii artx pririiaril~. live- i-irirs i.a(.i-i 11t.s
(Smith c t ai. ( 19S-l)).
2.1.3 The General Model
Tlir two niodrls presmtr(l a r r nctiially sprcial casrs uf a mure gpiirral iitodcll uf v;w-i-
iiat ion Iieterogeiieit~.. Xssiinir ~vrrybocly i r i t lie population lias suscept i hili ty f i i r i ( - t , i i~r i
h o ( t ) M o r e vacriiiatioii. Aft.er varcinatitig a fraction 1 of the popiilatioii. a propur-
tiori o of t liose vacciiiated liave siiscrptibility h l ( t ) = r(sl h o ( ! ) and the rtmiairiiiig 1 - n
have susreptibility h( t ) = kz h a ( t ). wliere ki 5 k.2 5 1 ( kl a11d k2 cotistaiit ). I i i otlirr
wortls. tlie siiscepti bility for a vacriiiated iriclividual looks li ke
. , ( h 2 ( t ) witti probability I - o
h[otlel i is a special case wliereby lzi = k2 = k while Mode1 2 is eqiiivelriit to settitig
Li = O and k2 = 1. Halloran at al. (1992) use tliis franiework to creatr ail atlditiorial
2 niodels of vacririe ac t ion ~vliicti a re siniply differerit combinat ions of I i r t r r o ~ r i i i t it?i
across t lie vacci tiatetl st ra t a. Tliey are:
For our purposes. only t h e leaky and al l /not liing niodel will be cotisidrretl siiicr t liry
represeiit opposi te e s t ren ies and mil1 b r sufficient for studyirig bias i i i t l i r odcls ratio.
.-\]su. t lie!. a r e t lie easiest t o interpret s i n î e t lie' involve t lie fewest para inr t r rs .
2.2 Estimating Vaccine Efficacy
Iii tliis section. ire will ckscribe the ciifferent nieasiires of I - E ttiat arc- l~asecl on
saiiiple and popiiiation a t t ack rates and transniission rates. U e assiiiiir tliat t l iv
s t i i d ~ . des igi~ is s imple iii tliat we have r r l vacciriated and 120 u r i v a r r i i i a t d iiidividiials
a t t lie Iwgi iiiiiiig of o u r s t udy. ;\ssiiiiir t liat vacciiiat ioii r d u c r s t lir siiscrpt ibili t>-
a-[-urrlitig tu o n e of t h e t ~ v o ~ i iodr l s oiitliri<-d a l ~ o v e . \L-e cldirit? the a t t ack rate i i i
wliere ,Y, is tlie raricio~n variable for tlie riuriiber witli vacci~iatiori s tat i ls r * ~ 1 1 0 I w c o ~ ~ i r
irifectecl tluriiig t h e course of a n epideniir o r outbreak.
2.2.1 Vaccine Efficacy Based on Attack Rates
The iiieasure of l,-E bwed on tlie -4 R's is given b ~ .
LVe rail estiniate I'E(.-1 R ) froiii oiitbreak data usiiig the rstiniatrs for .-I Ri aiii.1 .-i Ru.
proportioris aiid staiiclard resiilts froiii niasiniiirii likrliliood tlirwry tell 11s t l i i r t
ivlirrr Ei.4 R ] = .-\ R , and .\-(p. a') iiidicates a rioriilal tlistri but ion wit li riiraii p aiil1
variarice r~'. So. using the delta nietliotl we fintl the approsiiiiate variance for t l i ~
q~iaiitit>- log( 1 - 1. E ( . S ) ) . tlrriotetl by a'( .S) to l x
Lrttirig RR(.-I RI = .-1 RI/.-& t l ien a 100( 1 - a ) % c.orifidrric-e i r i t t n d f o r - I ' E(.+IH) is
I f iid(li t ioiial i-ovariates art- avai lablr for arialysis. we caii iisc t tir logist ic iiiuclrl
P ( D = I I \ ' = Y . 0 ) log 1 = ho + b , ~ + b f r
P ( D = O 1 \ - = &*.X)
. ~vlirrr bu. hl and b2 are the inasirntini likeliliootl estirnates (1ILE.s) of bo. 6, aiirl b-.
respect i vely.
For niore cletailed data. with iiiforriiatiori available at the hortseliolcl le\-el. aii
aclclitioiial sstiiiiator of 1-E is available based o n secoridary attack rates (.<.A R's). TIiv
.zfcondnry nttnck rat€ is tlefined as the probability that a susceptible brcoiiies irifrctcrl
hj- ail irilective in the sanie Iioiisehold over the rourse of the latter's irifrrtioirs prriml.
This rtieasirre is defineci as
Several niet liocls are availablr for est iitiat ing secondary at tack rates. Orir wlii(-11 lias
been stiown to work quite well is the mettiod of Longini and lioopniari ( Lougiiii
r t d. ( 1993a)) wliicli is basecl oii a probability niociel of infection traiisiiiissioii i i i
Iioiiselioltls. Tlie 11 LE of .>'.-IR is obtaiiied by iiiaximizing t tir likcliliood
is tlir probability tliat j persoiis in a Iiouseliold of size s I>eîoiiie iiifected. ( 'P I is t l i r
coitiitiiiiii t j - probability of iiilection *. and r i , , is t lie iiiii i i ber of liotiwliolrls uf sizr s
\vitIl j iiifwtii.es.
For t lie purpusr of t liis ~vork. I -fi'( .\'.-I H) will iiot lw c-utisidtw(i flirt lit^ siiicx) rase-
(-uiit rol st d i e s are ufteri riot desigiied to anal'-se data a t the Iioiise!iold Ir\-rl. I i i
arltlitiori. Halloran r t al. (1991) have sugestecl tliat itiatcliirig of rases aiid coiitrols
\vit liin Iii)iiseliolrls (lors riot riecessariiy control for t lie amo urct of esposiirr to infect ion.
l lat cliirig at t lie lioiisrliolcl lrvel is based oii t lie same assiiiiiptioiis al~oiit cliiateri iig
uf tlisease traiisiiiissioii iised i i i t lie estittiatioii of .S.-\ R's. But i i i practiîe. (-ont ruls are
quite ofteri sani pletl frorti t lie same rieigliboi~rhoocl or îlassrooiii rat lier t Iiaii t l i v saiiics
liouseliold.
2.2.2 Vaccine Efficacy Based on Transniission Rates
This second est i iiiator. wliicli seenis t O draw sonie inspiration froiii t lie Iiazard ratio i ii
siirvival aiialysis. is tlerived froni the well kiiown susceptible-iiifectetl-reniovecl ( S I R )
' C ' P I is the probability ttiat a siisceptible becomes infectrd from oriisidr the housel~old ovrr t . 1 1 ~
entirc cpideniic period (Haber P I nl . (1991 b ) )
differrntial equation mode1 ( ivtiich will be presented later in t his chapter). Tlir irioi1t.l
assiimes tliat t lie population iiritler st iidy is closetl (except for t lie iritrodiict ion of oiir
or iriore initial iiifrctivrs). mises randonilj-. and t liat vaccination (lecreasrs t lie prol)-
alrili ty t liat a ~acc~ii ia ted persori berorries i rifectrtl. .A siisrep t i blr persoii riia~- l)tm)iiir
iiifectecl alter wrriing i n r o ruiitact with otlirr infecti\-es. Tlir prrsoii t l i r i i I~tv-oiiirs
irifertetl for soiiie short prriotl of tirne ancl fiiially is s r r ~ ~ i o v r d ' (iisiially t1iroiio;li iiat-
iiral iniitiuriity) ancl no longer p l o s atiy part in the infection proress. Let .ju aniiil
.jl ( d l < .jo) deriote the transmission rates to an unvaccinatecl and vacciiiatetl SIIS-
rrptible. respect ively. Ttie t raitsniissiorc r-atr is the probability t liat a11 uiivai-ciriattvl
iiitlivitliial becortirs infectecl git-en contact witii an infect ive in tlir iiiten-al ( t . t + d t ] .
For ILIIL-acciriated individrials. tliis is giveii by .&dt. Also. let Ï hr tlir a r - r r q r Irtigtli
o l tlir. iilfrrtiuiis prriucl aiid Itht .Y lx t l i r total ri~irtibtlr 01 i t i lwt td itidividi~als i i l tlw . .
p l a t 1 . .-\ri-ordirig to Halwr nt al. ( 199 l a ) fiiial est iiiiatrs of t lie traiisiiiissiuii
rates are given t>>-
L .j" = --
1 log(1 - .-IRo) and J1 = -- log ( 1 - .4R1 j .
r .Y 7 .Y
T\ie ttieastire of vaccine vficac>. l~ast-cl ori the transriiissiori rates. d r~ io t r r l 1,~- I - E ( .j )
is tlierefore siveii 1,'-
This riieasiire of L'E lias a clifferent biological interpretation froiii 1, ' E(.4 K). Iiisttwl
of iiieasuritig the percent retliiction in disease incidence d u e to vacriiiatioii. \ ' E ( . i )
rrpreseiits t lie ptarceritage reductioii iti susct.ptibility. \ - E( . j ) is founci risiiig t lw es-
t iniatcd at tack ratcs. Ttie variarice of C'E( 3 ) is derivecl usitig rriart i n p l r riirt liocls
( t o he iritrodi~ct.rl in Sertion 2.4.2) arid according to Hahrr a / al. ( 199 1 l>) is gii-cri I)y
6 2 ( ,j) w\.litv
Tlieri. a 100( 1 - a)% confidence interval for I.. E ( 3 ) is
2.2.3 Vaccine efficacy based on the Odds Ratio
Tlie use of t lie oclds ratio froni case-control studies to assess vacciiir rfficac~ d u r s
riot go O\-er any riew territory Al1 of the esisting mettiodology cari b r foiirid i r i staii-
dard texts on case-control studies such as those by Sclilesselman ( lW2) aricl Breslow
aiid Dai- ( 19Sü). h ï t 11 case-çorit rol stuclies. disease a t tack rates caniiot be est iriiatcd
rlirrctl- I~iit wr cari approsiniate the ratio of attack rates iisiiig the otlds ratio. Esti-
rnat iciri in t his sc~iiario reqiiires t hat we know t lie vat-ririat ion histories o f cases ari(l
coiitrols. \'accirie efficacy. wliicli we'll denote by I .E(OR) . is tlien obtained 1)'. tlir
simple forniiila:
C-E(OR) = 1 - OR (2.14)
Tlie typiral l+.oiit for conipnting i-aîcirie rfficacy ir i aii iiiiitiatrlird c-asv-writrol stiid>-
is giveii by the followirig table:
Vacci riated C'ases C'ont rols Tot al
- Tlir est iiiiatrd odds ratio. O R is siiii pli- t lie cross-protlurt ratio rcd/bc (assu~iiiri,q al1 - i-alues are iioii-zero) in wliicli case Cs-E(OR) = 1 - nd/bc. In the event tliat tlirrr is
ari eriipty ce11 tlien ive siniply acld ? to cadi ce11 ancl proceed in the saine \vay' (( 'us - (1970)). Tlie estiiiiated variaxice of tlir log(OR) = log(1 - I F E ( O R ) ) is drrivrd iisirig
the delta iiietliotl aiid is giveii I>y
'.AS pointed out l iy a reviewer. tliis can be assunie a flat prior for each disease category.
I f we have ot lier corariates available i r i t lie arialysis t heii ive (-an iisr [ogist i c - rrgrrsuiori
riiodel to coriipiite vaccine rfficacy bi- assirniiiig t lie inodt.1:
ivhere I* is vaccination status. x is a vector of covariates (one of wliicli is iiisease
statiis). and bi is a vector of regession coefficients. Tlie ~stiniatrcl vaccina rtficary
is si tiiplj- 1 - r ~ p ( b . ~ , , ) wliere I I , ~ ; , is the SILE of the ïoeffiririit for disrasr s ta t 11s.
Iiiferericr proceeds as iisual iisiiig standard l i kelilinotl t lirory.
In a niatctied case-coiitrol studj-. arialysis is sitiiilar Liotvrver tlir layolit is sliglitlj-
ciifferent. In this case, we are only ititerestecl in discordant pairs. ttiat is. pairs c-un-
sistirig of a vaccinatecl case ancl an iirivacciriated control. or of an ilrivai-ciiiatrd r - ~ S P
niatclietl ivitli a vaccinatecl cout rol. .A typical table in tliis design is as follow:
C'ont rols
C'ases Vacciriatecl I -~i\acciriatecl
Tlir estiiiiated odds ratio is ÔR = . s / t . providecl of course tliat t > O. Larriiir rfficiicu
is siriiplj. 1- E(ÔR) = 1 - ( s / t ). The estimatecl variance of = log( 1 - \ - E( ( )H) )
The est iiiiatecl \.ariailce of 1- E ( 0 R ) ( iiiatclied or iiiiniatclird) is ol~taiiir(1 iisirig t lit^
delta niet hod aiid by iiot ing tliat
C'onfideiice iiitrrvals ancl tests of hypot liesis ran t lien l x const riicted for t lir odils
ratio iisiiig t lie al~ove results. Coiifidence intervals for 1. E reqtiire sliglit iiiocli firatioii
siiice tlir endpoints for the OR confidence interval iiiust iiot only b r subtrai-trti frotii
one but have to be rrversed to be nieaningfiil.
Multiple coiitrols require niore extensive coniputatiori. In general. i f ive have a
vector of COL-ariates for each subjert. x = ( r i . . . . . s,). wliere oiir of tlir c-ovariatrs.
sa- r,.,;. (-orrrsporitis to varririatiori statiis and case i ( i = 1.. . . . .\', ) l i a s h , ii~atclirii
c u i t rols t.lit-11 vaccirie efficac~. is estiriiated Ily 1 -espf b,,,) whrre h,.,,, is the appropriatt.
eiitry frorii the vector b wliicli masiniizes the partial likeliliood
The likelihootl stiown liere is actiially a special forni of the partial likdiliuod iisrcl
i n proportional liazarcl regessiori witli a special assiiniption for Iiatidliiig ties. Tlir
(tetails 011 tliis proredure. whicli is apt ly titled coiiditimal logistic rrgressioii. i-aii Iw
foiirirl iri Breslotv arirl Da'. ( 1 O N ) ) .
2.3 Vaccine Efficacy in Stratified Populations
= \ I l of the estirnators clisciissed so far rail be applied to populatioris ttiat are stratifir~l
or Iietero~rneoiis \vit li resprrt to soiiie host-relatecl factor siicli as asr. .;lis(-rpt ibi1it.y
or ot lier factors l i kr i.acciiiatiori (.overage. \\-e cati partit ion t lie popiilat ion itito 6'
strata aiid siniply adfi ailotlier subscript g t o the quaritities alrrarlj* prtwwtrcl tu
011 tain st rat urii-spwi fir nieasures. Tlierefore. -4 R., for esaniplr. woiild rrprestwt t lir
attack rate for tliose with vacciriatioii status L. in stratüm 9.
Tests for liotiiogriieity across t lie st rata and suiiiniary riieasures for \ ' E lia\-r I x w i
proposed l>y Haber nt al. ( 190 111) iisiiig inverse st rata-sprrific i-ariaticrs as u+t.igli te.
I f we Irt rrrcnsccrc rrfrr to orle of .AR. .j or O R tlieii t lie wriglit for strattiiii 9 is gii-cii
I)y I C ~ ( ~ H F ( L P ~ ~ . F ~ ) = [ \ ' « I I ( T I L F ~ P L I I . F ~ ) ] - ' = [g;( I I L E ( I I ; I I I - C ~ )]-' . \\C t ~ I C I ~ &fine:
O ) = bg[I - LPE(.4 R I ) ]
O ) = 1 - LvE(3,.,)
o,(O R ) = log[l - C* E(O R, ) ] .
A test of lioiriogeiieity across tlie C; strata. i r i otlier tt*ords a test of the li~-potlirsis:
cari be carried out 11'- iisirig t h e test statistic:
of frredoiii. If LW do [lot rc~jrrt the iiull liypotliesis tlitw o. is i~srd to cumpiite tlir
suriiriiary estiniate for I-E. The variance of tliis siininiary nieas~ire is giveti \)J-
('oiifitlrricr intervals for the siinirriar>- b P E cari tlien br conipiited cliiite easily usin:, tlir
espression for tlie silrriiiiarj- C * E itself and tlie estiniated variance. I f t . 1 1 ~ nul1 Ii~*potii-
rsis is rrjectrtl tlir~i a siiriiriiary iiieasiirr of I ' E is ohtaiiieil iveiglit iiis racli st rat i i i i i
\ ' E b ~ . the proport ion of the vacciiiatetl popuiatioii i i i the stratiiiii. Aiialagotis tests
aiid nieasiires sudi as t lie Breslorv- Day test for Honiogeiieit~- aiid tlir Slaiitel- Hiier1szt.l
sil~iiiiiar~. iiieasiire are also atailable for t lie oclds ratio (See Breslow ancl Da!'. i..oI. 1
( 10SO)).
2.4 Epidemic Models
T1ie basis for iiiucli uf ivliat lias brrn disciissrd woiild iiut liai-r Iwrii possiblr i\.itliuiit.
mat tirniatical iiiodels of epideniics. The stucly of vaccine efficary iiiliereiit ly rrqciirrs
the stiidy of the spread of infectious disease. Iii tliis sectioii. tliree iiiatlieiiiati(-al
iiiodels wiIl be presented. one of wliich will serve as the basis for a siiiiiilatiori iiiodel
r i a p t e 4. The firiiclanirmtal ba i s for al1 of tliese riiodels is tliat the sprratl of
disease iiit.olïes the interaction of two populations - s~isceptiblrs autl iiifectivtbs-aiitl
t liat t lie cliseasr propagates tliroiigli dirfct contact.
2.4.1 The Kermack-McKendrick Mode1
Tlie riiorlel due to W.O. Kerriiack. a cliriiiist. aiid .-\.Ci. .\.I(-lir~lclri(*k. a p tiysiciari.
dra~vs its inspiratioii fro~ii t lie science of clieiiiical reaçt iuris. 111 t lie siitipl~st s-e~iario.
assiinie t liat t lie disease îonfers periiianerit irtiniuriity un an- iritliviclual ivlio tias
just recoverecl aritl t liat t lie iiiriibatiori/lateiit period is of negligiblr durat ioii. Tlie
population caii he tlivided into tliree classes of iiidividiials: tlie sus(-petible rlass (5).
tlir iiifertetl rlass (1) aiiJ tlie renioved class ( R ) . The spreatl of iliseasr is pvrriiwi
by t lie folloiviiig assiimpt ions.
1. Tlir popiilatiuii reiiiaiiis nt n fiseri lei-el T L for tlie eritirc ul)srrvatiuii ~wrioil.
So. everits siicli as hirtli. cleath. infection froni ottirr rliseasrs. cniigratiuii aiicl
i i~ii~iigrat iori are igrioretf.
2. Tlie rate of cliaii~e of susceptibles is proportional to the proiluct of tlir iiiiiiil~rr
ot' siiscepti bles aiici t lie riuniber of infect ives.
3. Inclivitliials are renioved fro~ii the infective class at a rate proportioiial t u t l i t b
.\il iridiviiltial's prugress tliroiigli tlir cpiclwiic ran he iIcpii.trd I)).:
tIeiirt-. iiiociels of t liis type are ofteii callecl SIR ~iiotlrls. Letting .\'(! 1:'. t ( ! ). aild R(t )
ileiiote the niiiiibrr of iiidivitiuals in t lie siisceptil~le. infectiw nid rritiuvd r lass at,
tirtir. t . t 2 O tlie iiioclel rail br espressecl with the followirig systeiii of ditfrrriitial
cqiat ions
3\Vhile ttiere are sirnilarities. the S ( t ) used here should not br. confused w i t l i tlie sirrvivor functiori S(f) in ('hapter 3.
wtiere 3. is tlir infection rate and -, is ttie rernoval rate ( 3 . -, > O ) . Tlirre is the
aclditiorial constrairit rc = > ' ( t ) + I(t ) + R ( t ) and starting i.aliies .\'(O) = ri. I ( O ) = O f .
( O f represeiits a sniall numberj RIO) = O. This iriodrl lias a iiii:e intt~rl>rrt;rtiu~i:
if t lie population is t liorotighiy riiisirig and the rate at ~vliicli a stiscrpt il)le riitrrs
t lie infecti1.e state is proportional to the niiniber of irifectious contacts ( iii uiir c-asr
. i I ( t ) ) . ttien the total nilmber of susceptibles who becorne infectious is tlir siiiii uf
tliis rate over al1 the silscrptibles - 12:' 3 I ( t ) = 3 S ( t ) I ( t ) . This systriii of rqiiatioiis
rtapreseiits tlie contiriuous t iiiie version of t lie lierriiack-SIcIieiidrick iiiodel. I t also
rese~~ihles t h e lait? of mas.$ action in cliernistr-- wliereby ttie rate of tivo rlirtiiic-als
rract ing is proportional to t lie prodiirt of t lieir coiiceiitratioiis.
2 A.2 The Cont inuous Martingale Mode1
The coritiriuoiis inartingale nioclel. proposed Becker ( 19S.L. 19SR). is css<vitiall~- a
stocliast ic - (zrialogiie of t lie Iierrnack-LIcKendrick nioclel. It retains al1 t tie esserit ial
ctiarasteristics biit expresses it as a probabilistir process as opposeri tu a cIrtrriiiiiiisti(-
orle. -4s Iwfore. Ive coiisicier a closed coiriniuiiity of intlivicliials. At, ti~iit. f = O a
riiiriihrr ri of iiidividiials are siisceptible uliilr soriir sriiall iiiiiiibrr uf irift)(-ti\.rs a r e
irit roilricrrl iiito t lir popiilat ion. Tliis procesr oprrates iiiider tlir saiiit* assiiriipt ioiis
oict liiied above. For rori\+enience. let .V(t ) be t lie total iiiinibrr of transit ioiis froiii t , l i ( s
siisceptible statr to the infectioiis state i n tlie interval (0. t ] . Tlie Iiist,or!- of tlir prucrss
(soiiirtiiiies calird tlir filtration) is giveii by 'H, = a { . s l ( u ) . Q u ) . R ( ~ L ) : O 5 i i 5 t }.
Tlie progression of t lie epitleniic process is deterniined bj- t lie followiiig t rarisit ioii
probabilities:
wliere o ( h ) satisfies l i ~ n ~ - ~ = O and the notation dS(L) is ii i terpretd as S ( t ) -
' ; ' ( t-) . Statistiral iiifrreiice for tlie mode1 parameters .3 ancl -, is possible i f i v r drtiiie
tlie processes I1.y aacl -LIR ivliicli t ak r tlie foriii
Tlie processes .ilv a n d .\IR a r e zero niean niartingales sirice tliey t akr on ttir foriii ob-
scrccd niin u s . . E I P F C ~ C ~ " . For t hose not familiar wi t ii t h e t lieory of coiiiit i ~ i g prorrbssrs.
a iiiart,irigale is siriiply a randotii process wtirreby its espectetl value nt s r i > - futur<.
poiiit in t inie is rqrial to i t s ciirrent value. In otlier ivords. i f .ll = { l l ( l ) : t 2 O } is a
iiiartiiigale witli respect t o filtration 'H then E(. l I ( t + u ) 1 %t) = . I l ( t ) for al1 ii 2 0.
Estiniation of .j ancl -, proceeds qiiite easily i f ive knoiv tlie valiies of . s * ( t ) . I ( / )
ancl R ( t ) a t al1 possible t i n i r points. First we rlioose a n appropria te stopping tiiiir
sucli as T;, = iiif { t : I ( t ) = O}. Tlien equating I lv aritl -\IR to tlirir riieaii. iiaiiit>ly O.
a t stoppirig tirtir To. ive $et
[t also tirrris oiit tliat .j arirl 5 are riiaxiriiii~ri likeli liood estiiiiators. Tliew est.iiiiators
are iiot useful siticr ttiey rrqiiire full uhsrrvatioii of tlir rpicleiriii-. This prvseiits a
I>rol~leni since iiiost ofteii. oril' final size da ta is availal>le for out breaks atitl rpic lrrtii(-S.
tii'es a re relrioi.ec1 is cotistant. irriplj-i~ig an infectious period ivlii(-li is expoiicwtially
tlistri biited. Ari iiriprovenient coulcl I>e niacle bu i nt rodiiciiig several irifert iot 1s periocls
eacli ~vi t l i t lie saiiie rate. For esaniple. i f ive introduced a riiulti-state i~ifwtioiis pr-
riotI ivitli a total of J infectious states. say I I . . . . . I J . each witti t h e saine trarisitioii
ftirictioii. tlieii t l ir iiifert ious period ivoiilcl have a ganii i ia dist ributioii a i t l i s l iapr
paraiiieter J and r a t e paraii ieter 7 .
Derivation of IV E ( . j )
Slartingale niet,liods. as it so happens. are well sititecl t o tlie task of cleriving relative
siiscept ibili tu. Tlie derivat ion liere was first introducetl bu Becker ( 1982) and lias I~eeii
rstended by Longirii rt al. ( 1093b) to situatioris iiivolvitig iiiorr tliaii ~ L W strata. Tt.
above niartingales caii Iw riiaiiipiilated i t i sucli a way as to iiiake the tiiial rsprrssiuiis
deperideiit orily or1 observable qciaiit it ies. -4s Lefore. if we partition t lie popiilat ioii
irito vaccinatecl st rata (su bscript 1 ) and iirivacciriated st rata (subsc-ri pt O ) arid itssiirtic
that vaccination offers &Ioc!el 1 tvpe laccine action ive can rr-forniiilate tlir nio(lz1
hasecl upoii the folloiviiig transition probabilit ies:
ivliere I ( t ) = I,(t )+ I l ( t ) and R ( t ) = Ro(t)+ R l ( t ) . The probahility that iio traiisitioiis
uccur ici ( t . t + d t ] is siniplj- the suni of the above traiisitioii probabilities sul)tra(.trtl
froni 1. Lettins .\;(t) aiid ( t ) he t tir nriniber of itiivacriiiatecl and \-acciiiattvl
siisceptiblrs ivtio enter the inf~rtioils state i i i (0. t ] Ive can reforiiiiilatc tlir i~iodrl as:
I~~iplir i t in t liis riioclcl is t lie assuiription that infect ives of eit lier i-acciiiatrc! or iiiivw-
ciriatd st atiis are ecluallx inkctioiis. Based on the sarne notatioti as Ixforr~ WC (-an
forrtiiilate t lie riiart ingale eqiiat ions
( t ) = Y ( ) - , . , ( l - ) f ( - ) l l c = O. 1. 1' Sow. define the preclictible process B , ( i ) wtiere
By iiitegrating R , ( t ) witli respect to the martingale .blL,l,(t) we se t the riiartiiigalrs
Iri order to evaluate t h e integrais it is lielpful to notice tliat
siricr xi=, J: = log( O/n) (artlially. a b r t t r r approsiiiiatioii is givrii 1)'. lug[( IJ+
1 ; ) / ( c l - a ) ] .. but we'll stick with tliis one). Substituting tliis approximation iiito (XjT)
followiiig estiniatiiig eqitat ion:
Fi~ially. ive get \ - E( .i) 1)- takiiig 1 - O. Siiriilar iiiaiiipulation aiid iisv of the rtktiuii-
sli i p 1
I-nr[ . l l , ( t)] = E [b ~ l ( u ) d \ - , ( u ) ] . c = O. 1. (2.43)
caii be risecl to sliorv t liat tlie variance of 1 E ( 3 ) is estimatecl hy:
- - ( T - ) + P . 4 R~/.$, ,(T-)
\ 'n r-[l,- E ( .>)] = - 2 (2.44 ) []Og( L 1 - . \ R ~ ) ]
'Without getting too technical, the Optional Sampling Theorem s ta tes that if a rriartingalr . \ l ( r ) (with respect to 'H t ) is sufficiently bounded. and T is a random stopping tinie siich that P(T < x) = 1. then E [ M ( T ) ] = .Ci(O). Frorn another viewpoint, this theorern just states that yoii cannot beat a fair garrie. A martingale is an example of a process where no arbitrage esists.
For t lie tlrtails of t lie derivat ioii see .\ppenclis A. Tliis expression s tioiild look fariiiliar
l>y rerogniziiig tliat S , ( T - ) is just i l , ( 1 - -4 R,). Ttie results Iirrr are basrd or1 fi~ial
attack rates froni stoppetl rpidemics. We could also estiniate t lit. relative siisi-rptil>ility
a t sonir t,iriie t before the end of the epideniic. Tliis estiiiiate woultl Iw b a s r d uii t tic
attack rates iip to t inir t given by
Estimation of LVE(.j) woulrl tlien proceed as iisual but the notatiuii woiilrl Iir iiioilitird
to LmEJ(t) to point out the tinie-dependeiice.
2.4.3 The Reed-Frost Mode1
Ttie Rred- Frost cliaiti- hinomial prorrss represtwts t l i ~ tliscrete-t iiiit* aiialogiie uf t lie
r-lassic .\'IR riioclel. .As i n the previoiis tivo niodels tlit. poptilatioii is rlusrtl ancl i i i i s t ~ h
i i i \\é denote tlie iiii~iibers i r i t lie sriscrptible aiid iiifectivr class I)y .\", aiiil
II. resprctively wliere I E {O. 1.7. . . . }. . \ S S U I I ~ ~ tliat initially (.\'(l. Io) = ( T L . r r t ) whert~
r > r . .At rvery tiiiie point I . a susceptible coniirig into cotitact witli a siriglr
iiifrctit-r cari br infectrd witli probal~ilit>. p. O < 11 < 1. r\ssiirtiiiig tliat al1 <-untacts
tvitli iiifectivrs arc itidepriiiiriit. tlirti givrii (.qt-i. II-, ). lias a hirioiiiial clist i i h i i t ioii
tvitti freqiiency fiinctiori
~vliere 8, = ( 1 - p)'l-l. Tlie nimber of infectives at t is simpl!: It = .5't-1 - .\",. Aiiy
disrrrte distribution can Iie specified for t lie lengtli of t lie infertious period. t liiis
giving the espt.riiiienter the Resibilitj- t o properly mode1 t lie disease of interest.
\\*.liile iiot iiiiiiiediately obvioiis. this niodel is actually an excellent cliscrett~-tiiiir
apprusiiiintiori to tlie contiiiuoiis riiartirigale niodel. Ln order to see tliis ~ v r cati r l r r l i i w
frorii t. lie previoiis sert ion t liat iii t lie iiiart ingale riiode!
The equitalent of - d S ( t ) = I ; ( t - ) - S ( t ) i n the Reed- Frost rtiotlel is .5't-i - s',. Su.
takirig espectatiotis ~ i v e s us
wliere p z .j dt (sirire At = 1 ). Tlir fart t liat t liis iiiucle! is esserit ially a Ijraili-liiiig
hirioniial process is appealing hecause i t provitles a simple yet po~wrf i i l alguri t l i i i i
for siniulat irig epideiiiic procrsses. Siiiiiilat i rig hitioiiiial rniidoiii variahles rrqiiirtns
grrierating onlj- uniform rantloiii variates. Tlierr is also the adclrrl ad\-aiitagr of rasily
incorpurat i tig popiilat ion s t ruct iirt. aiid Lieterogeiiei t): Motlellirig bot li t l i t . Irak? a i i l
tlie al1 or notl i ing vaccine a.ctioii is achieved quite easily 1)'. adjustirig the p parariirter
for tlie differetit strata. For t liese reasotis, the Reecl-Frost iiioclel \vil1 seri-tl as t.ltr-n
hasis for uiir siriiiilat iori mudel i i i C'tiapter 4.
Chapter 3
The Odds Ratio and Direct Effects
I n tliis cliapter. tlie airii will be to rsaniiiie the dirrct rflcct of vacciiiatiuri arid its
iiiipart oii iiieasiirn; of relative risk. Tlir direct effrct of aiiy kiiid of iiitrrvrritioii
p q r a i i i atlriiiriistrrrrl to a grotip of ititlividiials is tlir ditferericr Iietwerri tlir oiitcoiiw
i r i t lie iricli\-irlual u'it ti t lie i nterverit ion aricl ~vliat t lit. oiitcoitie nurild liai-e l~rwi \rit t~otlt
tlie iiitrrvrritiori. ail otlirr tliirigs beiiig eqiml. .An esariiple of a direct etf'rct ( ivl i i t - l i
will serve as a hasis for tliis ctiapter) is the redtiction in the probability u l Iwc-uiiiiri,q
iiifected t liat resitl ts froiti beiiig vacciiiated. [ n d i r ~ c t eflccts. or) t lie ot lier liaiid. rrsiil t
froiii sit uatioiis iii wliicli t liere are depeiitlerit liappetiiiigs. .A prrfrct tsuiiplc uf
aticli a sitiiat ion is the t raiisniissioti dxnaniir of irifectioiis disrases - t tir riiiiiilirr of
tlir itiiiiititiologic stattis of a popiilatio~i of liosts relati\.e to t h e disensr. Tliis is (-alIrcl
I r f rd ininz uriity and for reasoris of niatheniatical tractibility will iiot be arcoiiiitcd
for liere. For our purposes. the study of direct effects on tlie odds ratio will Iw
accoriiplislied usiiig traditional niodels of siirvival analusis.
3.1 Failure Time Mode1
Let T be a noiiiiegat ive randoni variable represe~it iiig t lie tiiiie i~iit i l a persoii I~rroiiirs
iiilected frotn the disease of interest. Suppose for now that we have a closecl roliort of
iiidividirals iri a randoriiized controlled trial and tliat. for the short terril. otlirr inocles
of erit ry and exit sucli as birt lis. deat hs. imniigrat ion. eiiiniigrat ion aiid reiiioval 1)'-
d i e r diseasrs cari I>r igriored. Me also assunie no luss to follo\v iip. At tiiiit. / = 0. a
Fraction / of tliis coliort is allocated to a 1-acririateci groiip wliile the retiiai~iiiig frai-tiori
1 -/ are gi\.eii sonit. placebo. Herice. a persori wlio is vncciriated lias vawiriatiuri statiis
\ = 1 i l a i . i ~ i v a c i ~ a t e l i r d i i l i a l l a I = O. .-\ssiiiiie tliat iiift~.tiuii hy tlit*
ilisease rriiioves the iiiclividiial froni the popiilatioii at risk. Let S(t 1 1- = r * ) deiiutr
the siirvivor fiitiçtion for a person with vacciriatiori statiis c*. c* = 0. 1 . Tliat is
S( t 1 I - = i?) = P(T > t 1 1- = I V ) (3 .1 )
= P([iifectiori free at tinie t 1 I - = L. f. ( 3 . 2 1
Siiiiilarl>-. let , \ ( i 1 I ' = t * ) deriote t lie liazarcl fuiictioii fur mi indi\-idi.iaI \vit l i \-n(-t-iiiii-
t iori stat 1.1s P . Tlicrefore
P ( t < T < t + I t [ T > t . L = L ' ) h ( t 1 I ' = 1 . 1 = lin1
At-O At P(Iiifecter1 i r i ( 1 . t + At] 1 .\t risk nt t . I' = i 1 1
= Iini At-O At
Tlie survil-or fiiiictiuii espressecl iri terrris of tlie liazard fiinctiori is t h r i i
Tlie final piecr of tliis iiiodel is given bu the probability of heirig \*acriiiatecl/iiriva.acrii~atr~l
at tiriie t = 0:
P ( = ) = ' 1 - ) - , *=0 .1 . (3.7)
Let h ( t ) deriote sonie t ilrie varying rate of esposiire to irifectiori iii t hr pop~ilat ion
at tirne t or a baseline force of irifertioii. For exaniple. h ( i ) coiilcl rrprpseiit soiiir
fiinction of t tie nuniber of infectious contacts in the envirorinient at f i . Natiirally. -- - -- - - --
'This is logical in ttieory but not feasible in practice. Becker ( 1989) has suggested a hnzard of the forni I i ( t ) = k l ( i ) . where k is constant and f ( t ) is the number of infectives at t . Sirice 1 ( t ) is a randoni and unknown quantity, i ts estimation is not often possible.
ive assunie that bot11 t h e vacci~iat rd aiid tiiivacciriatec~ culiurts are siibjtvrt to tyiial
iiiterisity of tliis uiiclerlying r a t e at ail tiriies. .As outiiiied iii t h e previoiis r l iaptrr . IW
assiiriie two niodels of t-arcine art ion. Ivnder SIotiel 1 ( Leaky). var-riiiatiori rrfliic-es t lit.
siiscrptibili t ~ - of an indiridiial 11j- a coristarit aniourit. L7nder SfodeI 2 ( All /Su t l i i r i~ ) .
t lie lacci ne coiifers IOOR protection for a proportion n of t lie vaccinated popiilat iuii.
wtiile t h e other 1 - a a r e no bet ter off. .Assimie for bo th niodels tliat the Iiazarri for
t h e unvaccinatecl is given
ivhere . jo is a constant represeritiiig either a transniission r a t e or a Itazard rate for aii
iinvacri riateti iiiciividual. If Ive assume lfoclel 1. t tien t h e liazard for t lie vacc-iilatc-d
group beconirs
wliere .jl < 3". This riioclel t herefore recluçrs to t lit. proport ioiial Iiazartls iiiorlel.
I..iider I locid 2 . tlie sitiiatioii is niore casil' espressecl i i i teriiis uf t l ir siin-ii-or fiiii(-tiuii.
Rtv-al1 tliat ivitti t l ir .-\II/Sot liiiig iaccitie. t lie va-ci riatecl (-oliurt i-orisists of t\vo s t ra tn
- one n-it Ii coiiipletr protection. t lie otlier wit ti no protectiori. First. for wiivriiieri(-<-
for r v = 0. 1. T h e sur\-ivaI fiiliction fur t he ~-açc-iriatcci t tieri Iw(-oriies
T h e overall or aggregate hazard for tlie vaccinated groiip is t hen
T h e iinvaccinated groiip s t i 11 retairis t lie sanie Iiazartl. So as before
x ( t 1 c- = O ) = . joh( t ) .
3.1.1 Vaccine Efficacy based on Attack Rates
Lai-cine efficacy for bot h t lie Leaky niodel and tlie ;\ll/ Nothirig mode1 Iia\.r di tfrrriit
l~iological interpr~tatioris. Recall that tlir I-'E Imsed or1 re1atii.e attac-k rates \vas
iriterpreted as t lie percent retiuct ion of clisease iilcidt.ti(-r at t ribiitablt. t u vac.i-iriat i w i
wiiile t h e I.*E hasetl on relative trarismissiori rates representrd tlie redit(-tiuri iii siis-
cepti bility. (*rider t h assuniptions of our nioclel ive have t tliat the espei.tril fiiial
attark rates at t . derioteri by -4 R,(t ). are:
111 adrlit ion. otit of iieccssity. I v e will riiakc use of the assiiriiprioris
and tliat EILI(.\-)] ;r g( E( .Y) ) for sonie differentiable functioii g. IÏitlrr t l i r proprr
wriditioris. al1 funct ions of raiidoiii qiiantities \vil1 converge iii probability to t 1 1 t h
quantities on die riglit liand side of tlie 2- sigii by Çlutsky's Tlieorcni.
50 i i d r r .\lt>dcl 1 . the rsprcted value of tlir \acciiir efficac?- I>nsrrl oii n t t a d ratos
tferiote~l II>- I ' E . 4 K ( 1 ) is
Tlic rspeî ted varciiir rffirary basrd on traiisiiiissioti rates is gicrri h>.
wliich in ttiis case gives LIS the Iiazard ratio. Tlirrefore. imder hlodrl 1 t l i r Iiazard
ratio yields a consistent tiiiie-invariant estiniate of \acciiie rffifirac~. based o ~ i rt.lativr
siiscept i bilitj- wliile the L-acririe efficac~- based or1 relative at tack rates is a fiiiict iuii ul
the follow-up tirrie t .
I-rider Mode1 2 ive get a differeiit resiilt. For \ E . I R ( t ) ive have
Su. tlie I ' E I ~ a s d oii relative attack rates sliould yield a n iinbiased and tiiiir-itivariaiit
rstiiiiate of n. the proportion of successfiil vacciiiees. I f o n tlie otlier haricl. ire iisr
t l i t x relative siisceptibility t lirri wr have
i f ive use the soniewliat arialagous iiieasure of tlie Iiazard ratio us i i i~ the -nggre,qatr'
Iiazard for t lie i-accinatrd sroop tlieri ive ;et
- a - (:W.y
o + ( 1 - a ) e s p ( - . & H ( f ) )
So. i i i Imtli instances wr Iiavr tliat rieitlier of' the two Irieasiires of relative suswptil>ility
abovr xield a coiisisteiit estiiiiateof ci. In fact. tliey'rr botli Iiiiictions of tlir follo\v-iil~
tiriie t . Oi-erall howeiw. the results foiiiicl liere are ronsistent witli ttiat of Smith r t
al. ( 111s-1) aiid Haber r t al . ( 199 1 a ) wlio denionst rated t hat: ( I I under t lie Leakj-
i-accine nioclel tlie ratio of transiiiission rates yields ail uiibiasrd est iiriatur of I ' E ( .i)
aucl. ( 2 ) iiricler tlie .All/Sotliiiig vaccine. the L'E basecl oii relative attack rates j-irlils
an i~iibiased estiiiiator of o. It's also iiiteresting to iiotice tliat. irrespective of tlir
iiiodel of vaccine action. IV E ( J ) will always esceed L-E(A R) wlieri the vacciiie offers
sonie protection. This is a simple conseqiience of the fact tliat for O < x < y < 1
C'oiisequeiitly. uiitler tlie Leaky iriotlel IF E(.4R) rvill cindrrrstirr~att t lie tri ir iiitXasiirr
of rfficacy I,'E(.j). Sirnilarly. under t lie .AH/ Xotliing nioclel. I ' E ( . I ) will ocrrf.~tiniatt
t \le t rile pro port ion of successfui vacci nees.
3.2 The Odds Ratio
3.2.1 Unmatched Estimators
I.-riiiiat clietl otltls ratios grnerally take t lie forrii
ivliere . -II aiid .Ao arc the noniber ( ~ r o p o r t i o n ) of esposed aiiti iiiirsposrd i-nsrs. rt.-
spect ively. arid BI aird Bo are t lie niiiiiher ( proport ion ) of rsposed aiid iiiirsposrd
(-ont rols. r rsprr t ivclj.. Iri oiir rase. t lie esposure variable is vncrinatioii s ta t lis. Thr
relatiorisliips tliat \vil1 br tlisciissecl ivill assunie a case-coiit ru1 st utlj- iii a pupiilat iuii
t liat is of tisrtl sizr. Llirrefore. t lie populatioii \vil1 be depleted II?. diseast iii(-irlvtii-v
biit iiot Iie repleiiislitd. Jiist like tlir rpicleiiiic itioclels preseiitril previoilsly. \vtl as-
sliiiie 110 gains or losscs via i~iiiiiigratiori. eiiiigration. tleath. hirtli or disrases otl1i.r
ttiali tlir oiir ~iri(lrr stiitly. Also. biases frorii censoriiig and loss to f o l l o ~ i i p ivi l l riot
be takeii iritu accoiiiit. Assuitie tliat the eritire popiilatiori cotisists of .V iiidi\-idrials
aiid tliat tlie prol~aliility relatioiisliips already preseritecl still Iiold. Herice. tlir iiiiiii-
hiar allorated to the racciiiatrtl group at tiiiie t = O is /-Y aiid tlir iiiiiiihrr i i i t l ie
uiii-acciiiated group is ( 1 - f ).le. Also assunie tliat t lie riiasirniirii folloiv-iip t iiiie is r .
For the first esatiiple. wr will assume cuntulatice-irtridt.rzrf sanipliicg (C;rrrnlaiid
Sc Tlioiiias ( 1SSZ) ). IIntler t liis sampling design. cont rols will be selected froiti t liosr
st i l1 free of disease at t iiiie T . Tlierefore. t lie espected proportiotis for t lie iiriiiiatclird
odds ratio rvill be:
fur rv = 0. 1. I-siiig Bayes' Tltroreiii Ive get tliat for an! t E [O. r ]
Su. after plusgiiig iii tlic appropriate espressioiis irito uiir foririula for ORr- IYP Iiavr
wliicti is wliat we woiild rspect e\-eii Iiatl the study beeii prospective iii ilcsigii. su . frorii our prosprct ive aiialysis in t lie prrvioiis expression IW rtv-wgriizr t lie qiiiiii t i t y
tu hr the ratio of attack rates b 1 tiiiie t . Tlir o\.erall qiiantitj- is ttirreforr 1)iasrd l,>-
the ratio S(t 1 1- = O)/S(t / \ * = 1 ) . .Assiiniing. as always. tlint the \.ar(.iiir coiifers
soine kint l of prutwtiori wr woold expert tliis ratio to br Iess tliati 1. Rrgartllrss c)f t l i r
iiiotlel of vacciiir action. WC iroiiltl cspect siicli aii odtls ratio to be l o w r t tiaii tlir ratio
of attack rates. .\s a resuIt. vaccine efficacy based on tliis odds ratio \vil1 hc. Iiiglier
tliaii vaccine efficacy Iiwecl on relatil-e attack rates. I f ive assunie the .\ll/Sotliiiig
vaccine. tlieri tlie estirnate for vaccine efficacu will not converge to o. Katlier. it will
lx. ronsisterit for
Q I V E ( O R r . ) = 1 - O R [ - = (ij.36)
a + ( 1 - n ) S o ( ~ )
So. oiilu i f survi~al is fairly liigli (or incidence is lorv) ici11 the abovr qiiaiitit~ provr
iisefiil. On t h e otlier harid. i f we assunie tlie Leaky vaccirie then Ive have that ttie
odds ratio I .E estirriates t.hr quaiititu
.-\gain. if iiicidmcr is low. iniplying ttiat the quaritities .jo H ( r ) aiid .il H ( 7 ) are (- lus^
to zero we can irifer iising the approximation edZ z 1 - r for sufficient ly srtiall s tliat
I f tlit% iiiridence is liigli. l .-E(O R c . ) will be liigher tlian I-*E(.-I R) (coriiparr
(3.1.3) to (3.38) + I , ' *E(ORr-) = ed1"(')l'Em4R(r) ) l , -E ( . j ) will also b r lowrr t l i i i i i
I - E ( O R [ - ) . To sep tliis. al1 wr nerd to do is take advantagr of tlir follon*iiig iiieqiia1it~-
ivtiicli caii I)e proveii usiiig rleiiirntary calculus:
f o r O < r < g < 1 .
\\-r cari also conipilte ari uriri~atch~cl est irriator where cont rois art1 mat (.lied for
t iiiie. Tliis desigii. ivliicll is often called dcns i ty àauiplircg ( C; rwnlaiid k TTlioriias
( 19S2)) or tirrie-matchrd sampling ( Lubin SL Gai1 ( 19S-4)). iiivolves srlt.ctirig a control
for t~acli case a t t lie occurreiicr titiie of racli case. Sou-. i i i the iriterval ( 1 . 1 + dt] t lie
rxpcc-trd ii~iiiibrr of cases witli vacciiiatioii statiis is giveii 1)'.
The total iiuiiiber of expected cases witli vaccinatioti status i1 over the iiitrri-al [U. r ]
The e s p e c t r d riiiriiber of coiit rols selected a i t l i \*accination statiis 1.' ici t lie ititrrval
( t . t + dt] is giveii LI>-
for t1 = 0. 1. Xotice t h a t t h e portion on t h e riglit is siiiiply t h e proportion of aiirvii-ors
iii \;?ct:ination groiip 1.. This leatls t o
k i n g t lie total espertecl riuniber of roiit rols with vacciriatioii s t a t i ls t 1 ovrr t lie i i i t . t v x I
[O. r ] . Ttie fitial esprctecl otlcls ratio is sitiiply
111 orclri- to app1~- tliis t o otir t w tiiwlels rri-al1 fur t tir Lraky ïw(-iiir t tiat
Tlir probahility of a11 icirlii-idiial beiiig ic i groiip L- giveii tliat Iie/slie Ilas avoidt>tl
irifertioii rip uiitil ticiie is
Oiir espressiolis for t h e etitries in t h e otlds rat io becorrie
O 2 4 6 8 10 Time
Figure 3.1 : Ttvu di frereri t ~(wiarios for c-iirve represents t-acriiiatetl ititli\.icluals.
O 5 1 0 1 5 20 Time
epitleiiiir ciirves. 111 hot li ~ > i ~ ~ i e l ~ tliv luwrr Tlir pariel o n t lie It3ft rrpri3sriits a si t iiat iori
w here bot li ~.accinatecl and iinvaccinated groiips have t lie sanie riirari t irtic of uiisi3t. In t h e riglit panel. \-accinated indit-iduals lia~ve a later nieari oriset tinie.
to \-arj- in riiagrii t ticle depending or1 t tir appearanrr of t lie epitlritiic (-iirïes of Ilut li
jroups. Tlir sliape of . j ( / ) will I)e priniaril>- Rat for Iiiglier valiies of t I ~ i i t fairly s t r r p . .
Fur siiiall valiirs of 1. Assii~iiiiig t lie \-acciiir uKrrs soriir proter-tioii. 1 1 t lit. sitiiatioii
resriiil)les tlir lrft paliel uf Figiire 3.1 tlieii tlir al>o\*r ratio \vil1 I)t. ruiijIiIy rqiial tti
r i t Biit if t lit. pro1)ability deiisitj- fuiictioii i r i t lie iiriva(-r-iiiatrd groiip placrs i~iust
uf i t s weiglit riear t lie begiriiiirig of t lie stutlj- iriterval (iriiplyirig t liat ii1i\ai~iiiatc~1
iiidividiials get irifected soorier 0x1 average) tliis will create a ratio tliat is greater tliaii
1 (riglit paiirl). This would produce a Iiigtier odds ratio aiid tlierefore a loiver I ' E .
By t lie sanie reasoniiig. if vaccinatecl individuais ( by sonie peciiliar reasoii) are Iwirig
irifrrtecl soorier. the opposite occitrs - tlie ratio of espectatiolis will br less tliari 1.
Tliis d k c t will sliririk t he odds ratio and coiiseqiieritly iiiflatr 1-E.
For t lie All/Sot liirig vaccine. recall t liat the rtiodel corisists of t lie qiiaiitit ies:
The ol-erall siirvival futictiori for the eritire populatiori is givrii b ~ .
.\gairi. iisiiig tiir aboïe relationsliips for . -L i . .-lo. Bi and Bo. ire have:
III a siniilar fastiiori let
Oiiïr agairi. ive cari tliiiik of . j ( t ) as being a u-eiglitrd average of t,lie si.is(.rptibilit it~s
o \ w dl t lie strata. Baser1 ou this riew riotat ion Ive have
The orl(ls ratio t heri lwcoriies. after sorrie ubvioits caricellat iori uf terrr~s
If w r tlieri di\-ide t tiroiigh by / , i ( t ) h(t )So(t ) dt and siibtract the resiilt from 1 ive grt
The qiiaritity .$ is jiist a rveiglited average of tlie survital lunctiori for tlir iiiivai-uiiiatrd
strata and. as a result. will be less t han 1. Tlie resultiiig estimate of i-a(-citir rfficary
\vil1 tlien he Iiiglier tlian it sliould. So overall. we see t hat in botli rases. tlir iisr
of unniatclied otlds ratios under either forni of sanipling pields L'Es tliat are tiot
cuiisistent witli eitlier I.- E( .i) or I' E(.4 R). ijcliile ive rvoiild expect th. qiiaritit ics
to closely est iniatc tlieir coiiriterparts. the rsact magnitude of the discrrpaiirirs \vil1
l.arj- rvit li t lie lengt li of follow-ilp. t lie severity of t lie epitleniic. aiid t lie effi(.acy of t lie
\arririe. Overall. i f the efficacy is liigh. or at least i f L-accitiatioii coveragr is liisli aii[l
tlie vaccirie offers rreii tiiodest protectioii. Ive woiilcl riot espect the hias tu b r vrry
3.2.2 Matched Estirnators
For tlir purposes uf rsariiinirig iiiatclietl udds ratios ive ivill i-oiisidrr aii aiial>-sis uf
iiiatclierl pairs where ive assume I : I niatcliing. Tliat is. ive will look nt odds ratios u l
-111 0 OR.!, = - -\[O 1
w l i t v .Illu represmts tlie tiiiniber of niatcliecl pairs consistiiig of a vacçiiiritrd case aiid
aii iiiivacciiiatrrl coiitrol aiid .\lut is tlie iiutiiber of pairs witli ail iirivacriziatrd rasta
aiid a vacciiiated coiitrol. Assurriirig the sanie failure tiiiie riiodel as I~eforr aiid tliilt
roritrols are sanipled at t lie tirrie each case \vas irifected t lieri we can tlieii foriiiiilate
espressioiis in a similar fasliion. If sanipling is rantloni i t i t lie interval ( t . l + r l t ] . t l i r i i
the espected riuiiiber of pairs iii this inter\*al where the rase is in racriiiation groi-ip
i: ancl t lie control is in vaccination group IL* is givcn by
So. t h e total espected niiniber of such pairs over the interval ( O . r ] is givrii by
Tlie (esprrretl) innt clied oclcls ratio <*an t lien I>e espressrd as
For the Leakj- ~acrir ie riiodel ive get:
arid
Forti.iriate1. t lie fi rial rspressioii for t lie odrls ratio rrclilires rio si iii pli tic-at iuri at al1
sirice the iritegrals cancel iniriiediatelu. Tlie tinal expression collapses to
The oclrls ratio t heii hecoiiies
For corii-enierice let
.-\a Iwforr. t lie quantity is siinpl>- jiist a weiglitrd average of Su(! 1 over ( O . r ] iisiiig
h ( t )&( t ),/S(t ) as a tveight. Our tinal expression for vaccine efficacj- t lirii Iwro~iirs
The value of will clearly fall between O and 1 tlius niaking I,-E(OR.\,) > o. Su.
to srininiarize. under !vIocIel 2 the value of I. 'E(ORVlr) iç not onlie tipivardly biastd. . . i t s a fiinctiori of the follow up tinie r as well. L-siiig a pairetl rstiniator iiiitler tinir-
itiatrlietl saniplitig in this sitiiatioti ttoes iiot seeni to provide an wtds ratio hastxi I .E
t tint converges t o 1,- E( -4 R) .
3.3 Frailty Selection
Tlir biases we'~.e heeri seeirig in t lie above est iiiiators and sanipling scliriiirs rspriiit
iri r tir case of the All/Sotliiiig itiorlel) is an esai~iple of wliat soiiie liai-e rt3frrrwl tu
Erailty selectio~i (Briinet c t al. ( 1993)). By frailty. ive riieari tliat t h e fractioiis of proplv
nt risk ir i rithPr va(-citiattd or iiiivarriiiatrd gruiips i-ary tvitli tiriic.. Tliis i-ariatiuii
causa a distortioil i i i t h e fraction wlio have beeri 1-acciriated for bot l i rrioclt4s. I r i tlie
.-\ ll/Sot l i i ng itiodcl i t causes an arldit ional tlistort ioii in a. t lir proport ioii of sii(:crssCiiI
va(-ci riees. At t irtic t. the proportion of suçcessful vaccinees woiild be
For 110th iiiodels. ttie proportion varciiiatecl still a t risk at titiie t . wotild bt3
Recall froni eariirr tliat Sniith t-t al. ( 19S-L) poirited out t h . iintler the .All/.\:otliiiig
inotlrl. cotitrols slioiild be selecteti froni al1 potential sul3jects. regardless of wlietlier
tliey have coiitractecl t lie disease or tiot. The ratiotiale for sanipling tliis way is to
siriiplificatiori oiider .\lotiel 2. I n wortls. ive are sanipling frorii botli prrvioiisly iiifrctrrl
aiid iiori-infected indiviciuals. For al1 three cases. the odcls ratio reduces to 1 - a.
I f applied to tlie Leaky iiioclrl. we arrive at equatioii (3.35). Tliereforr. iiirliidiiig
persoiis alreacly infected. even ttiough peciiliar froni a n rpideniiologic persprrtirr.
actiially yielcis an uiibiased estiniate for tlie relative attack rates. ('oiisrqiirrit,l~-. iri
the .All/Sot liiiig riiudel. i f the diseasr is rare. tlirri the (liifereiice i r i rstiiiintrs froiii
ritlirr sairipliiig sclieiiie will he tiiiniiiiial. Biit. as pre\-airriïr ur iiirif-lrricr in(-rrastar.
t lie (listurtioii iiiducrtl by tlie frailty will be severe a n d t lie t i v ~ estiriiates wi l l divergtb.
3.4 Other Sources of Bias to Consider
A s a coiit iriiiatioii of tliis sert ion. LW will fiirtlier look at odds ratio rstir~iators 1)'-
adopt i i i~ the iiiodel crrated by Liibiii k Ciail ( 1%-1). !-iicler tliis appruacli. n.r alniirloii
t l i r 'aggrrgatr' iiietliwl iisrd thris far ariti look siiiiplj- a t ocl(1s ratios fur a siriglt. (-asex-
i-ont 1-01 pair basrd on t lie iiiodel alrend!- creatrd. This \vil1 $ 1 - i x 11s niorta Hesil~ili t j .
iii esploriiig more elatmrate iriodels \vit tiout liai-iiig to tvorrh- almit si riipl
iiitrrpr~tiiig iiitegral espressioiis. It will also allow us to assrss hias froiii
furiiis of (:uiitrul sainpliiig aiid. at the saine tirrie. alloiv ils tu esaniirie otl
ivliicli iiitliierice siiscept ibility.
Tlir priiiiary txpe of sariipling for writrols esplorecl so far lias b r ~ i i (-all<d dt-nzi lg
snrrtpl i i ig . To rritrratr. oiice soiiieoiie I>ecoiiies a case a t tiiiir 1 . n ruritroi is c l i o s t~
froiii tlitl populatioii ivlio are stiil infection-frer. .Altrriiatii.ely. a coiitrol (-uiil(l 1x1
clioseii froii~ the popiilatioii at risk at tinie t - . tliat is. i r r ~ t n f d i a t r l ~ Lefore t . l-iirlrr
tliis ride. a case could encl iip being its own control. This is geiierally avoidrtl i i i
practicr because replication is considered to be uiiinforniative. 111 a siiiiilar fastiioti.
a prei-ioiisly selrcted coiitrol coiilcl he iiifectecl nt a later tiiiie poirit arid I>r st.1~~-trcl
agaiii for studj-. oril- tliis tiriir as a case. Furtlirr. a control roiilcl b r selrrtrd riiorr
tliaii once as a coiitrol if Iie/stie nianages to avoitl infectioii for a goocl part of tlir
outbreak. Tliis rarely liappens siiice the population of eligible controls is tisiially qttitr
large and sampling witti or wit hout replacenient nialies lit t le diffrrencr. B~i t . wl i i l~
surtir of tliesr srlrctioii rules iiiay serin awli~varcl in practiçe. allowiiig the iiidi\-icltial t o
reiiiaiii iii t lie ris k set despi te bri tig sain plrd is act iially fuiidaiiierit al tu t lie priii(.iplrs
of siirvival analysis.
Tu lwq,iri. let ORT.\* denote the oclcls ratio iii tlie 2 x 2 table wlirrt*l>?+ for iill
stibjects wlio becoiiie irifected at tinie t we take as controls a randoni sariiplr of dl
patients still at risk at tinie t . The otlds ratio is siniply
P( I+ = 1 1 Case at t ) / P ( I W = O 1 C'asr at 1 ) O R T . ~ = (:i .M)
P ( \ - = 1 / C'oritrol at t ) / P ( 1 - = O / C'oiitrol nt t j
I r i order to foriiiiilate aii esprrssioii for tliis. Ive nrrd a frw prol~al~ilitit-as. First.. t l i ~
probability of beinj infecteci i r i ( 1 . t + d l ] given vaccinatiuri statiis r* is givt-ii
Tlir iiiicoiiditioiial probability of failiire i i i (1. t + d t ] is
S ( t 1 I , ' = f!) , \ (t 1 I ' = L ' ) P ( I - = i l )
P ( 1 7 = r v / Case at 1 ) = d t )
Tlie probability of being sanipled as a control. P( 1 ' = r 1 C'oiitrul at 1 ) is jiist
iiiiiltiplietl bj- tlie probability of heiiip sanipletl at t . So. accordiiig tu L i i h i t i k (;ail
( 19S4). assuiiiiiig the saiiipliiig probability is eqiial for vacciiiatecl aiid iiori-i*ar(.iiiatrcl
individuals. t h e odds ratio becornes
h o t lier coiiiiiioii source of bias in case-cont rol studies is t lie addecl reqiiireiiietit ttiat
n control reniain disease-free for the riitire follow-up period. Tliesr type of coiitrols
are referreti to 1)- Lubin k Gai1 (19S-1) as 'pure coiitrols'. Pure rotitrols are uftrii
selerted iii coliort-type st iidies even t liough a niore appropriate cont rol groiip iroiild
br ttie populatioii at risk. If. as before. we assume that the iiiasiriiuiii possible fullo\v
iip tinie for a siibject is r . tlien tlie odtls ratio based oii piire coiitrul saiiipliiig is
P(1- = 1 / C'ase at t j l P ( 1 ' = O 1 Case at t ) O R p , = (3 .9 - i )
P I \ - = 1 1 ('ontrol a t r ) / P ( L V = O 1 Coritrol a t 7 ) -
Basrd oii our previous formulas. the probability of tiaving vacciiiatioii stat 11s I ? aiilors
tlie çoritrol group for sornebody selected at time t < T wt:o stavs irifertiori fret. i i i i t i l
r t i i t l t iplietl 11y t lie sainpli tis probabili tu. After soiiie siiii pli firat ion ive get
Applyiiig botli of tliese rspressions to our niodels of vaccine actioii aires lis siiiiilar
rcsitlts to before. 1-licier liociel 1
irliilp iisirig piire roiit roli leatls t o
Once again. tlie tiriie iiiatched odds ratio does ir-el1 in estii~iatirig t lie rrlat iw siisi-rp-
tibility. Tlir oclds ratio basecl on pure control sanipling gives us a qtiant.ity t liat is
lower. The estent of tlie untlerestimatioii actually depends upon the differrtire br-
t wecw t lie t inie t lie case occurred ancl t lie end of t lie study. Tlie larger t lie gap bet ivreti
tlirse tivo rverits. the larger the bias. Tlie direction of ttir bias wit li resprct tu I* E is
corist-qiieiitly positive aiid overall we find tliat LSE(ORpr) > I -E(ORTJr) = \ 'E ( . j ) .
I'iicler l lodrl 2 the resiilts are parallel to previo:is fiiidiiigs. Botli sariipliiig scliriiira
givr estiniates that
.All/Not hi ng iiiodel
are Iiiglier tlian the true value of o. Applyiiig tlie foriiiiilas to tlic
yields
for t inie-niatciiing of coiitrols wliile for pure controls wr lia\-e
Botli of tliese quantities are easily seen to eaceed 0 ivitli 1 E(O Rpc. ) cloirig progp3s-
sively n-orse as would he erprrterl. .-\ rareful look also re~-eals t liat I/-E(OHT.\I ) is
act iiallj- t lie es perted proport ion of successful varciiires iii t lie vacri~iat ei 1 groiip at t
ivl i i l r siitiilar1~-. I- E ( 0 Rrc- ) is t lie espectetl proportion of sucrssfiil i*a(.riiitrs at t i i r i t .
r . Again. we're ol~serïing Irailty srlectioii at ivork. Tlir vacciiiatecl group is hrt-oiii-
iiig r~irictied with siircrssfiil vacririers and causing tlir proportion n( t ) to grailiiall?
i ncrease (II-es t i me.
Atiottier obviotis soiirce of bias in st udies of vaccine efficacj. is diflcrc rrtinl .suscrp-
tibilily. For any vacrine ttiat rloes ~ i o t provide perfect protection it is iiot cliftiriilt to
roiiipretien<i t liat if inclividiials vacci nated or uti1-acçinatrtl ) systrriiat iral1~- cliatigtb
t tieir siisrrpt ibili ty to disrase berairse of t lieir varciiiat ion stat ils t l i r i i t ~ s t iiiiat.t.s of
vacrint1 efficacy will riot go iiiiaffrctetl. One possible sceiiario is tliat of a 1.ac.i-iiiatt.il
iiicli\.icliial ivlio irirreases liis/lit.r riuniber of contacts ivi t li infect ious liosts brcaiist> o f
a Iirigliteiid serise of protection. Ttirre are certaitily more factors at work. [f tlie va<.-
(mine proterts accurdiiig :O a Irak' txpr irioclel ancl tlie vacciriee is atvart3 of tliis. tlitw
t1\-cii a rerliiced cliaiice of infection nia>- deter siich l~eliaviour. 0 1 i tlita otlier liatiil.
i f t lie ~acc ine g i \ w .-lll/Sot liiiig protcct ion t lien t lie ~acri i iee ni-. ivisii to kiicnv i f
t lie vacci iiat ioii nas acliiiinist errtl siiccrsfiilly ancl ri~odif~. t lieir bella\-ioiir a(-(-orrlirig1~-
( Halloran r-! al. ( l W 4 ) ).
Tu setx tliis. we have to reforriiulate our liazard. .Assiirrie tliat oiir liazarrl iioiv takvs
t lit. foriii siiggested by Halloran r t al. ( 1994 ):
P( [nfectio~i 1 Contact ) x Contact Rate x Prevalence.
So. iiiattir~iiatically oiir Iiazarcl Iooks like
wliere .j is tlie traiisiiiissioti probability per contact witli an infective. ~ ( t ) is the rate
at whicli contacts orcur. and p ( t ) is a nieasure of overall prevaleiic~ for tlie coriiiiiiliiitj:
Erteiidiiig. t liis tu t lie Leaky ~acci i i t . niodel. t lie Iiazard for prrsuti wit li va(-ciiiat ion
stattis r v is theri
,\(t 1 \,' = c ) = .je R u ( f ) p ( t ) IV = 0. 1 ( : L l w )
.As Iwforr. .jl < .j0. so t he vaccine still serves t o reduce tlie probability uf iiifrctioii
git-ri1 contact . This is not i i r î e s s a r i l~ t h e case for ~ ~ ( t ) ancl ~ ~ ( t 1 which is n w r r
a fiiiict ion of t tir person's beliat-iour. Tlie resultiiig odds ratio using t irtie-itiatclir~l
t-oritrols woiilcl the11 l>e
Su. t lie bias ~ v i I l clepencl ent irely or1 t lie coiit a r t paraineters. II 1-acciriatrd siibjt*rt s
are iiiaking niore contacts tlien 1*E( .9) will be lower tliaii espected. If iiiivacriiiattvl
siil)jects a rc itiakiiig more contacts t lien t he resul t irig I* E( J) \vil] appear rtiore r f f r c t ivv
tliaii it actrially is.
I'ricler t lie A l l / S o t liiiig ~acc ine . t tic rrorresponding stirvi val fiiiict ioiis arc
for the vacciiiated aiid
for t lie iinvacciiiated. For rotiieriieiice. let .\,(t ) = Ji & K , ( u ) p ( u ) du. 1 *iidc~r thesta
c'ire-ii~~istatices. a case-cuit rol aiialysis will lead t o eçtiniatiori of t lie qiiatitity
Tlie r e s d t s are siitiilar to beiore in tliat i f K~ ( t ) = x o ( t ) . t lie expressioii ïollapsrs
to the proportion of successfui vaccinees at t inie 1. Not surprisiiigly. i f varciiiatrd
people iiiake niore contacts. t liis proportion will be uiiderestiniated. If t lie o p p s i t e
ocîiirs aiid uiivacciiiated indiiidilals iiiake niore iiifectioiis contacts. t l ir proportioii
will iricrease. Tl ie uiifortiiiiatr part of tliis little eserr ise is t liat wliile it tloes Iiigliliglit
t lie iiot ioii of different ial siisceptil>ility/cxposure. it is of liniited ilse siiire t tir ~ iu i i i l ~e r
of contacts iiiade by an!- given person is usiially iiot observable.
C\;e r an gerieralize ttie above resiilt t o aiiy sce~iario ~vlierr cases aiid coiitrols arts
siibject t o clifferent infectious 'cliniates'. If. for exaniple. cases are sui>jrct tu irifrrtioiis
force h ( l ) and controls are sulijert to a dif€erent force. sa5 p ( t ) . tlirn rveii rriat(.lii~ig for
factors sucli as age or ses will be liriiitrtl i i i retlucing hias. Tlie hias \vil[ be 5rratt.r i f
vacrinat ion coverage aiiiorig t lie controls is cliffermt. Socli a sit iiatioii i-ail Licriir qui t e
rasily if coiit rols a re sanipled Ironi areas of differeiit socioeconoitiic s ta t ils. Poorer
areas iiiay. for esairiple. liavr 1iight.r prrvaleiice. lower vacrination covrrage t.aii(l loivrr
uvrrall tiygierir. hlatching to accoi~rit for t tiese variables is iniportatlt i f oiir is 1.0
obtairi a t rue [neasrire of a vaccine's effect iveriess.
To see ttiis. assume the hazarcis already defined and assume tlie probahility of a
casr liaviiig vaccination status L~ is also as belore. Now let tlie vacciiiation probat~ility
fur ïotitrols lw P(1- = I * ) = ou( i - O ) ' - " . For cuiiveriiencr. defiiie .1I(t) = Ji ,u(il) ilu.
I f WC rt.-aclopt t lie notation froiti section 3.1 and assume ttit. Leakj- iiiodrl tlirii t lie
t, iriie-rriatclied oclcis ratio beconies
I-iicler the .-\ll/Sotliing vaccine t he odds ratio estiiiiates
Iii hot l i cascs. t lie trrnis itivolve t lie odds ratio o l beirig 1-arc-iiiatrcl i i i t lie ras ta pop-
iilatioii relative to tlie coritrol poptilatioii. / ( 1 - o)/o(l - /). Tlir i~ t lds ratio tirider
t lie .-\Il/ r o t liirig iiioclel inr*ol\.es the proportion rvho ivere itifectrd iip to t i i i i r / iii t l i t )
populnt ion frorti wliicli t lie coiit rols were selectetl.
Chapter 4
Simulation Study
III urcler to test t lie robiist riess of t lit. varioils odrls-rat io tmsed t~s t itiiiitors iirirlt~r c-ir-
ciiiiistaiicrs \rlierr iiidirert effects are also preseiit. sturliastic siiiiiilatiuiis r v t w <-arrit3il
oiit iiicorporat ing t lie t wo rriotlels of vaccine action presented. I'sirig a li>-pot liet ira1
populatiori of individiials. several sets of siniulatioris were riiri for ~arioi is lev& uf
rfficacj-. \acciiiatioi~ coverage. aiid vaccination clusteriiig. Tlie goal of the siriiula-
t ioii st iitly Lvas to assess t lie i iifliience of factors wliicli iiiducetl i i i r l i rect r f f t v - t a ailcl
alfertecl O\-rrall in(-irlrricr. Tlir first set of riiiis iiio&llrd a popiilatiuri tliat i i i i s t v l
iiriiforiiilj- wliilc t lit1 st3roiitl set ~01isisted of a IIIOI-C rralistic p ~ p 1 1 l a t i ~ 1 1 (-Iiara(-teri~t~tI
hy iiuii-iiriiforrii riiisirig and i~iiiltiplt. le\.els of siisrcpt,ihility. In rarli case. iiiriiiis ;r i i ( l
stati tlard errors were coriiputecl for a11 t lie available est i iiiators of I " E prrsciitrrl t liii.;
far. hIeari attack rates were coiiiputed as well. But. Liefore tliscussirig the resiilts.
LW will Iay clown t lie t lieoretical and structural frariiework for t h c probal~ilist i i riiode1
iisetl to gcnrrate t lie data.
4.1 The SimulationModel
Tlie riiodel involvecl in t he creation of the simulated epidemics is t lie Reed- Frost iiiutlel
introcluced at the end of Chapter 2. The mode1 is Markov in nature. rneanirig tliat tlir
future state of tlie epideniic is deterniined by the current state. In lact. tlie Red-Frost
tiiodel beloiigs to specific class of .\larkov rhains known as brnnrhing procrnsrs. \i;v
cati t liitik of eacli infect ive as being a -pareiitn aiid eacli suscepti1)le t liat t lie iiifri-ti\.r
roiiies iiito coiitart ivitli as a potential -of-Fspringn. Froiii tlie tlieory of hratir-liitig
processes (Resiiick (1992)). we kiiow tliat if tlie average iiuiiiber of offspriiig is lrss
tliari 1 tlieii the brancliing process dies out or goes extirict. If the opposite is triir.
t liat is. tlic niean riilniber of offspriiig euceeds 1 t heii t he popiilatiuii grorvs \vit lioiit
~ i . iii epiderriic tlirorj-. t liis -explosion" of offspriiig ( o r iiiurt. apprupriatrly. i - i w r ~ )
is rvferrecl to as ail cpidemic.
4.1.1 The Uniform Mixing Model
Tlie assiiiiiptioiis riiacle in tliis riiodel are actiially few and siiiiplt.. I t i a closel. frer1~-
iiiisi tig populatiori of susceptible intlivicli~als. orir o r more iriferti\.es is irit rudiicwl.
Tlir itifrctiuii tlieii spreatls froiii iiifertires to otlirrs bj. **ntleqiiatr wiitai-t". On[-(1 a
persoii beconies i r i frct etl. tie/slie passes t hrougti a latent period followcl II!. ail i t i f w -
t ioris prriod wlierr t lie persori lias t, lie poteiit ial to iiifert ot iiers. Eacli siiscrp t iblr's
prolmhili t ~ - of rorit ract iiig disease giïeti contact ivit h an iiifert i\-e. ivliicli ~ v ~ ' l I cleiiot(b
h ~ . p w\.lierr
p = P(hfection 1 Contact ).
is ronstaiit for al1 itirli\-itloals in a ririiforiii1~- riiising poI>iilatio~i. Tlir paraiiirt(br p caii
aiso hr tlioiiglit of as a iiieasiirr of roiitagioiisiirss.
For a typical siist-rptil>le i r i oiir f ctitioiis popiilatioii. his/lirr progrras t liroii,qli t l i t s
cpitleiiiic caii Le clepicted bx:
IViilike the niotlels preseiited earlier. a latent period was iiitrodiicecl to pro\-ide
ail atlilitioiial elriiieiit of realisiii. So. once a person beconies infertecl. lir/slitb passes
t liroiigti a Iatrrit prriod of soine raiidoiii durat ion prior t u bccoiiiiiig irifrct ioiis. Diiriiig
t tic latent period. t lie iiifrctioii tlerelops strict ly iiiteriially aiid the persori does riot
ciiiit ariy iiifectious niaterial. Hence. the individual is in fected but r~ot in ffctious. I f
ive let:
e.
< C I . *
IL:,'( t ). ~ " ! ( t ) be tlie nuniber of latent iridividiials. iiifecti\.es niitl reniovrd irifrrtives.
resprrtivelu. for the sariir collection of labels. Si~iiilarlc for a sus(-rptiblr i t i a g gruiip
cr let:
p ( c L ) l Lv = within-tio~.iset~Uld probability of infrctiori givrn <-ontai-t
p:" = I w t iveen-lioiisehoId probability of infection giveii coiitart
p ! L ) = I~rt~veeri-tieiglibo~irliood probabilit,~ of infect iori giveti I-oiitai-t
For coiivenience let = p y ) j. 111 addition. we can rietirie:
the total niiriibrr of infectires in neighbourhood rc and fitially
The martingale analogue for tliis iiiodel caii bc easily o b t a i n d bj- iisirig t l i e approsi-
iiiation ( 1 - z I - r1.r (for siiiall a ) . espaiiditig o ~ i t the trrtiis aiid (lisrardiiigi al1
rioii-li ripar ternis i n p. FinaIl!: replariiig t lie p's b ~ . .i dt aiid siiiiplif~iiig yirlds
wliere X and 7 are the inverses of the niean latent and iiifcctious periods. respectivel~. --
' .-\ctiialIy. WP are su bstitiiting these terms with o(dt ).
4.2 Modelling Vaccine Action
In al1 sceriarios discussecl so far. iniplenienting ttie Leaky (Ilotlel 1 ) arid .\II/Suttii~ig
iiiotlel ( Slotlel 2 ) for our siiiiulation niodel is actiirved bj. a siiiiplr t raiisforniat iuii o f
t lia escape probal>ilitv Q. I f we preset the desired I * E to sonie value I . E . s r t pu = p.
and pl = ( 1 - I ~ E ) ~ ~ thcn a vaccinated individual's escape probability for t l i r Lraky
riiode1 is giveii by Q ( ' ) = Q ( t . pl ). I-nder the .-\II/Sot liinp riiotlel t lit. vaci-iriatrrl
group corisists of t ivo s t rata - siiccessfirl aricl tinsuccessful vacciiiees. .A va(-ri riee Iris>+
receive coniplete protection froni disease ivitli probabili t~. ce or rereivt. no protrctioii
ti*tiatsorver tvitli probabilitj- 1 - ve. Tlie escape probability for 1-acciiie actioii of tliis
t j-pr is $ive11 by
for siicressfiil vacri nees Q(') =
Q( !. po ) for i~risiicressful \-accinres
Siniilar1~-. i r i t lir noii-iiiiiform iiiixiiig case. tlir escape probal>ilitirs art8 Q1') =
Qj1,, ( t . p(ln' ) in t lie Leaky iiiodrl aiid
QW = 1 for successfirI vacci riees
Q , ~ , ~ ( t . &)) for u r i s~ i~ res s f~ i l I - ~ C C ~ I I ~ C S
( , I ) for tlie All/Sotliirig rtiotlels ivliere = ( 1 - c-t)h .
4.3 Why Simulation?
TIitl iiori-liriearit>. of itiost rpideniic rtiotlels conipountletl II?- tlieir typically l larkov
forni are probabiy t h e two iiiaiii reasons tvliy iriost rpidrriiic iiiociels do iiot leiid
tlieiiisrlvrs to compact. closed solutioiis. Stocliastic sinlillation offers a frasil>lr aiid
tlcsiblr solution. ~vitl i tlie onl). real disadvaritape k i n g tlie aniouiit of co~iipiitr~r tiiiie
irit.olved.
As ail esatiiple. t h e simple t<erinack-SIcIïendrick riiodel in its original forrti lias
no esplicit solution. Hotvever. If tlie epideniic is not too large (see Braiin ( lOS3) for
a niore cletailecl treatnient ) an approsiniate solution for d R / d t is available coiisistiiig
of a coniplicated espressioii involviiig serh and tanli-' fiirictioiis.
The Reecl- Frost niorlrl. clespite being a rlisrrete-t inie riiotlel uffers ver!- lit t le uvrr
the Kerinack-LIcKendrick riiode1 in terms of analytic sirriplicit!.. .\ssiiriiirig ttit. siin-
plest possible forni of t lie Reed- Frost - a single liornogenroiisl~* riiisirig popiilat iuii. iiu
latent period and an infectious period of unit lengtli - Lefevre and Picard ( 1990) liavr
slio~vti the t hat the frequency funct ion for the final size of the epiclriiiic is ,qivrti hy
~1 ! ~ ( 4 =
(ri-k)(rnf k ) (I ( 1 1 q ) . X. = O.. . . . rt
( 7 1 - k)!
\\.liere n is the initial riurrher of susceptibles. r r j is the initial iiiiniber of iiifrctives.
q = 1 - p and q'" = {qr'\ qmf l . . . .}. The fiinctiori G B ( - 1 * ) is kiiowil as ail .\lwl-
C;oritctiaroff polynoniial (Lrferre and Picard (1990)) aiid is tlefirierl as
wliere U iç the set of real nunibers { i i o . u ~ . 1 1 2 . . . . } . This should gi\-e tlir rra<
soriic idea of t lie lei-el of con~plexi ty involvecl. lliilti-dirrieiisiorial arialogues fur r i i d t i-
p o p ~ ~ l a t ioii R e d - Frost niodels are also t leri~ed biit t lie resiilts. IV liicli (-uiiil)iiir Iiigti
(Iiiiierisional fiirirtiotis aiid îoniples reciirsioii. are estrvriirl~. roiiipliratd.
4.4 Results of the Simulation Study
For t w - l i iiidivitliial set of rontlitioiis n-e rail LOO siniiilatioiis. Typirally. 100 riiris
nia>- seeIii iiiodrst for çiiiiillatioli pi-irposes. Brit. ive stiick ~vi th tliis anioiiiit for tuTi,
rcasons:
1. Tlir 'rIarkov nature of t lie iiiodel riiatle it fairlj- nieiiiory iiittwsi\.r. This i ras
especially true i t i tlie more realistic niodel (whicii tiirned out tu Iw iiiorc of ail
accoiiiiti~ig probleiii tliaii a prograriiniing problriii)
2. I t kvas found that ttiere were virtually no differences in tlir niagtiitiidr of tlic
estirriates atid/or tlieir hias 1 1 ~ doubling tliis nuniber to 200 or eveii goiiig as
liigli as 500.
For al1 siniiilatioiis. the epideiiiic was allowrtl to riiri for a itiasiiiirini of '15 t i t t i r
iiiiits. Tlie latent aritl iiifrctious periocls ivere botti of randorii duratiori witl i a [>ri.-
specifird probability distribution. Tlie latent period ranged froni 1 to 2 tinie periuds
~vitli a niean of 1.5 wliile the infectious period ranged from 1 to 3 t iiiie periocls witti
a nirati of 2. Fiiiai estiinates of I,-E(.4R). EWE(.3). I-- E ( O R I - ) aiid ImE(ORI I ) i v r r o
roniputed froiti tlie final totals nt tlie end of racli riiri and averagrd ovrr t l ir 100 riiiis
(the exact foriiiulae 1-ised are clesrribed iii Apperidis C'). Interitiecliate t-aliirs ivrrt3 also
coiiiputed for al1 tinie poirits aiid plottecl to see ttieir evoliitioii orer tlir cuiirsr of t l i r
rpidemic. Tivo versions of the otlds ratio est iniators were coiiipiitetl - the first I)asr(l
on time-niatched sanipliiig aiid the otlier based on cuniiilative-iiicidence sanipliiig. Al l
values of FR and .se(O R ) mere calculated wit li an adjiist merit of $ adtled to al1 c<~lls
to eliniiriatc undefiiied values of I.- E i i i ttie event of scarcity (Cox ( 1970)). Staridaril
tkrrurs of al1 t lie avai labk I , 'E nieasrires were calculated as lvell. For t hose i ~ i t e r ~ s t d .
iiitBaii sta~iclard rrrors of al1 estiitiates of 1-E basrd oii the foriiiiilas froiii ('liiipti-br 2
artb ai-ailahle iii Appeticlis B. As ail addi t iorial arialytic aid. itirari at tack rates ( Iwt li
orrrall aiitl st rat iirii-specifi c ) werr coiti putecl froni t lie final totals.
4.4.1 Uniforrn (Homogeneous) Mixing
Tlir pupulatiori for tliis set of sirtii~latioiis cotisistecl of a (-lost~l roiiirtiiiiiity o f 2000
lioriiogeiieoiisly iiiisiiig iiirlik-iduals. For hot li rtiodels of vaccititl act ion we rail t. lit.
siiiiiilatioris for al1 possihlr roriil>iiiat.ioiis of:
pu = 0.0004. 0.0006. 0.0008. arid 0.00 10
a C-E values of 7'0% and 90%#
O[-erall. the resirlts are ronsisterit ~vitti the literatiire. Ttiat is. estiiiiators for
wliicli ttie iiii<lrrlyin,a rtiotlrl assuniptioiis were satisfietl proved to be uribiaserl. 1.-licier
1lorlt.l 1. b + ~ ( . j ) was coiisistently oii target while bVE(.4R) Iiatl a ~~~~~nwanl hias.
I'nder hlodel 2 . t lit. direct opposite was t lie case - I.* E(.;IR) mas alwqvs oii tlie iiiark
for û tvliile I V E ( .i) macle the vaccinatiori appear more rffertik-e t liai1 it r~a l ly \vas.
The extent of the ditference between the two appearecl to deprnd or1 tlirrr fartors:
tlir tlegree of vaccine coverage. azid the niagziit ude of t lie rfficacy i tself aiid t lie i-oiita-
gioiisriess of t lie disrase ( rde r t ed by t lie paranieter po ). I r i otlirr wortls. aiiy (-1iaiio;t.
t liat atfected t lie overall i ncideiice of t lie disease aiid cu~isc~t~iirii t ly iii(liicrd ail i r i ( l i -
rwt cliange i t i tlie aniuunt of iiifectioiis niaterial circiilating Iiail a rlirrtt effet-t i r i t lits
riiagriitiide of t lie Sap bettvem I.- E(.Z?) and I- ~ ( . j ) . .An exairiiriatioii of Figiirrs -1.1
tlirougli -1.S. wliicli plot the iiiean 1'E's of a11 the estiniators corisidered for al1 t l i r r r
wvrrage le~.els. show tliis relatioiisliip qiiite clearl~.. Wlieii botti vacriiie rfficacy a i d
~.acrinatiori roverasr are liigh. \,,-E(.-\ R ) and I - E ( . j ) are virtiially icleiitii-Ir rrgar(1lt~ss
uf t lie iiiodrl of vacri iie action ( Figiires -1.1 aritl 4.3. 70% cm-eraje). If l v r look at t l i c b
oppositr sitiiatioii ivlier~ tlie varcirir is iiiferior. tlie cuverase is loiv ari(l t l i r ilisrnw is
Iiiglily roiitapioits (Figiires 4.6 arid 4.S. 30% c~vrritge) tlirii the clis(-rrpa~icy is large.
Tlir t~vo estiiiiators diverge iii a ver). predictable fasiiioii as wc ctiaiige ail- factor tliat
direct ly or iriclirect 1 - increases incicleiice.
Tlie ocltls ratio-hasec1 estimators t.sliibitec1 siriiilar corisisttwcy biit in ari opposite
dirtlctioii. I--ridt.r al1 circiinistaiicrs. tlir odds ratiu gave a I - E i-aliir t liat \vas qua1
to or liiglier t lia11 t lie triir itioclel 1- E. Iii fact. odds ratios ririplo~-iri tiiiir-rtiatc-liwl
iiiatclird coiitrols provrd to be iitibiasrtl estirnators of \ - E ( . $ ) rrgardlrss uf ivlirtlitv-
tliry lrrre basecl oii riiatclied pairs or iiot. Coiisequeiitl~~. iiiitler the Lraky itio(1t~l. t lit.>-
wrtx jiist as acrurate as tlieir prospecti\*e analogue I * E ( . j ) . Biit. i f the oiids ratiu \vas
Imrd oti coiitrols wlio reitiaiiied tlisease-free for the eiitire tyitleriiic (ci~iiiiilative-
iiiciderice saiiipliiig). tlieii 1 - E ( 0 R ) ivas consistently liigher tlian tlir t r w itiorlel I'E.
The riiagnitutle of the bias in / ? E ( O R ) was. again. a functioii of: va(-riiiatioii
CO\-rrage. t lie t rile niotlel efficacu. and t lie contagioiisriess of t tir disrase. To reprüt
tlir esercise. an esaniiriatioii of Figure 4.1 and Figure 4.3. aloiig witli Tables -1.1
aiicl 4.7 reveals t liat iiiitlrr Iiigh rovrrage. liigli rfficacy and lowrr c.oritagiotisiirss.
al1 est iiriators. iiicludirig t lie cutiiulat ive-incidence OR. are qui te acciirate. Tlir rare
disease assu~iiptiori is obviously at work wlien al1 tliese factors are conibinetl. Tlir
only exception to this observation was the most extreme case - tlie highfst coverage.
Table 4.1: FiiiaI estimates for varioius nieasures of I e E frotii siiiiritatiotis witii i r i i i -
formly inixing population ( F e i s,, '). Slotlel: Leaky ( L ). triir 1-E = 90%.
Model: Leaky
Trtie VE=90% Time-5Iatched C'uniiilat ive-1 ticideric~
Fraction
PO vat. I-E(.=IIR) I - E ( ~ ) I*E(ÔR[-) I+E(ÔR.~~) I - E ( ~ & - ) I-E(~%.,,)
the hightsf efficacj* and the lori~-.d contagiousness. I-nt1t.r al1 t lirer of t tirsr. coiidit ioiis
c-o~iil~iiied t here \vert> so ft~w rverits t ha t al1 the effiracj- val iies irttrr I rnst able. Frirt lier
p r o ~ f of t liis cari I>e seeii froiii tlie large staiitlard error values.
But. i f rvc look at a sitiiatioii wliere tlie diseasr is liiglily coiitagioiis aiid I~otli
tlie efficacy and the corerage are low. then (as seeii in Fizure 4.6 ancl Figiire 4.S)
I ' E ( 0 R ) under cumulative-incidence sainpling ceases to be ail unbiased estiiiiator
of tlie triie efficac-. In fact. Crider Mode1 2 . this odds ratio drifts as Iiigh as <)SR iiiider tliese cirruiiistances wlie~i the t rue I1.E is actually 70%. The tiriie-iiiatrlied
odds ratio. oii t h e otlier liand rernained iinbiased for 1. E( . i ) . ei-en iiiicler siic-li liigli
iiicideiict. conditions. .As a result. it kvas accurate for al1 1Iodel 1 scenarios I ~ u t iiot
for .\Ioclel 2 . A s ~voultl be expected uiider Model 2 . the tirne-niatchd odds ratio
Table 4.2: Final estiniates for variotis nieasures of I rE froni siriiiilatiiiiis witti iii i i-
foriiily tiiixirig popiiiatiori ((ce I .sw ). .\lodel: .\Il/ Nottiirig (2). triie L* E = 90%;.
uvtwst iriiateri t lie t riie r~iuclel 1,-E. But t lie esterit of t lie overest iniatioii \vas riot as
i i i i i d i as t Iiat seeii for t Iir odcls ratio iiiitlrr riiiiiiilat i ve-iiii-idcri(x1 saiii pli iig.
So. ivliat is tlir rrasoii for t lie odds ratio hc41ig s~irli a (*orisistriitly iiiHatd rsti-
riiator of I , -E. It was alreatlj- establislied i n C'liaptrr 3 tliat:
1. I -rider .\lotle1 1 ( Leaky) \-accirie action. 1,- E ( j ) pro\-ides ari uribiasetl (coiisistriit )
aiid tiiiie-invariant estixnator of the reduction iii siisceptibility dile to vacciiia-
t ion.
2. I-rider Slotlel 2 ('\ll/Sotliirig) vacciiir actioii. E proricles nii uribiase(l
(roirsisteiit ) aiicl tiiiw-iii\.ariaiit estiiiiator of t lie prupurt ioii of sii<-(*rssfiill>- i i i i -
niurie vaccinees.
Table 4.3: Firial estinlates for varioirs nieasures of I.-E frorti sirtitilatiotis with irrii- forriily rtiising population ( 1 % k n,,). Model: Leaky ( 1 ). triir L m E = 70%.
Model: Leaky
Trur C'E=CO% Tirrie-!vIatched C.'timiilative-Inciclence
- Recall tliat \ - E ( . j ) is a ratio of rates wl~ile I-E(.-I R ) is a ratio of prol~al>ilitics. Tlir
tliscrqm.iic~- ivotild seerti to iritlicate tliat t hr drriomiiiator heirip usetl to est iiiiatr
tlir vacriiiatioii rates iiiay Iiold a cliie. Tlir tirne-niatcliiiig tliat iras ~ i s p d to s ~ l t ~ . t
controls for the first ocitls ratio is actually just aii esample of risk-set sariipliii i i i
siirvival aiialysis. So. evrrj- tiiiie ail infection ocriirs. a coiitrol is selectrd ~ l i o is
represeiitative of the popiilatiori at risk a t the t i m ~ of I l i f infcdiorc. Tlir udds ratio
uricler tliese cireunistances is essriitially a pooletl siini of the iridividual odds ratios
for eacli failure tirrie and is ttiereforc unbiased for the hazard ratio. Sirice Moclel 1
is really just a proportional liazards niodel for infectious diseases. the resiilt slioiild
corne as no surprise.
In the case of hlodel 2 liowever. we are dealing witli a Iiazartl of a differriit sort.
Rrcall. tliat the raccinated groiip consists of tivo unidentifiable strata - those wlio are
Table 4.4: Final estimates for various measures of C-E froni siniulatioris \vit li iirii- fornily mising popiilatiori (Ke rt su, ). l lodel: All/Sotliirig (2 ) . triie 1'E = 70%.
Model: Ali/Nothing
True VE=70% Time-JIatched C'iiniiilativr-Incitirrice
Fraction
PO Vac. E ) L . E ( ~ ) I*E(OR[:) I'E(ÛR~~) l .E(OHr- ) I 'E(OH. \~ )
(-oiiiplrtrl~. in1 riiuiir (siiccessfiil vacrinees) aritl t iiosr wlio. I>eraiis<~ uf va(.(-iiie fai liiw.
have iiu protection nt al1 (iirisii(-cessful \*arriiirrs). Becaiisr tliere csists a lit-tt-ruatwcity
in siisrrpt ibility t tiat is riot ol>ser\,able. t lie overall liazarrl for tliis groiip is i iot wii-
starit over tiriir. -As a resiilt. the vaccination rates arnong rliseasetl aiicl non-diseaswl
cliaiige over the course of a n epitlrniic. .As the epideriiic progresses. tlie tioii-diseased
grotip will coritain a disproportionately liigli nuniber of successhil \-accitires. Bruriet
c t al. (1993) have referred to ttiis phenornenori as a frailty. Tlie odds ratio tiritler
ciiiiiiilative-iticitlence saiiipliiig etids u p estiniatitig tlie proportioii of successfiil vacci-
rires at the tinie tlir saiiipliiig was done. It is not Iianl to ser that i f t l i r epidriiiir is
srvrre. tliis proportion will be close to one. Ivrider low iticidetice. tliis qiiaiitity \vil1 Iw
dose to the true efficacy but rvill still exceed it. In the case of tlir tiiiie-inatrlie<l odds
ratio. t lie risk sets at later tinie points i n the epidemic also becoine over-represeiiteti
3 9
Tablc 4.5: Final rriean estirnateci attack rates frorii siiiiulatioiis witli uriiforrii1~- riiisiiig population. Trile L - E = 00%.
True VE=90% Leaky .Ul/Yot hing
Fraction - I Vaccinated -4 R1 -4 Ro -4 R I .4 RIl
wi t l i siic-wssfii 1 \.nrriiiet*s. Biit . because saiiipli rig is dorie t lirou,qlioiit t t i r ol~srrvat iuii
prrioil usiiig iiiteriiirdiatr risk sets as opposecl to the population statiis rislit at t l i r
eiid. tlir hias is reducetl 11' rarlier niore reprrsentative saiiiples. T1ic utltls ratio i i i
tliis case is still Iiiglier thari it should. but iiot as Iiigli as cu1iiiilati\-e-iiiric1c~tire saiii-
pliiig. A s ive saw Ironi C'liapter 3. the t iiiie-iiiatclied odds ratio act tially rstiitiatrs a
weighted average of tlie proportion of successful imniuiies over t lie eiitirr observation
period. As Sniit li E L a l . ( 19S-4) have suggested. t lie only way to alleviate tliis probleiti
is to incliide botli diseasecl and 11011-diseased inclivicluals in tlie cotitrol groiip. Tlir
resultiiig oclcls ratio tlieii collapses to an uiibiased estiriiate of I*'E(.-1R).
Table 4.6: Final nieari estiinatetl attack rates froiii siniulatioiis witli tiniforni1~- riiisiiig popdation. Truc 1- E = 70%.
VE(0R) - Time matched
I VE(0R) Pared - Time matched VE(OR} - Curn Incidence
Cumulawe Mean Cases - Vaccinated
Cumulavie Mean Cases - Unvaccrnated
Figiire 4.1 : lleaii riiras~ires over t iriie for siriiulat ions vit li iiiiiforiii iiiisiiig fur ;LI I rowrage levels: pu = 0.0005. Triie L E = 90%. Lcaliy iiiodel ( 1 )
VE(OR) - l m e malched
1 VE(0R) - Cum Incidence
--e
Cumulative Mean Cases - Vaccinatea
Cumulative Mean Cases - Unvacanated Cumulative AR'S - Unvaccinated
F i 4.2: llcaii nieasiires over tinie for siniulatiotis wit li uni foriii ~iiisiiig for i ~ l l
coverage levels: po = 0.00 10. True C- E = 90%. Lraky iiiodel ( 1 )
VE(0R) - Tïme matched \/E(OR) Paired - T h e matched VE(0 R) - Cum. Incidence
Curnulatnre Mean Cases - Vaccinated
-e
Cumulative AR's - Unvaccinated Cumuiatrve Mean &es - Unvaccinated
Figure 4.3: 1Iean riieasures m.er tinie for sini~ilations witli uniforni rnisiiig for al1 covrrage levels: po = 0.000S. Truc lr'E = 90%. .All/Not liitig iiiotlcl ( 2 )
Cumuiat~e Mean Cases - Unvaccinated
VE(0R) - Cum. Incidence
Cumulavve Mean Cases - Vaccinaîed
Figure 4.4: .\lean iiieasures over tinie for siiiiulatioiis witli uniforni niisiiig for al1 coverage Irvels: po = 0.0010. True 1-E = 90%. ;\lI/Sotliiiig tiioclel ( 2 )
VE(0R) Paired - Tirne matched VEIOR) - Cum Incidence
(='. I l -
Cumulatrve Mean Cases - Vaccinaled
- -e
Cumulative Mean Cases - Unvaccinated
* r +
Cumulative AR's - Unvacc~nated Cumuhtive AR's - Vaccinaled UY O
O O
= 3 O I
f o r - O 7
O
O O
Figure 4.5: Ileaii iiieasures over t inie for siniulations witli uni foriii iiiisi tig For al1
covrrage Ievels: po = 0.0008. True 1- E = 70%. Leaky niodel ( 1 )
VE(0R) - Cum incidence
Curnulatrve Mean Cases - Vaccinard
.@
Curnulatrve Mean Cases - Unvaccinated Curnuhtrve AR's - Vacci~ted
Figure -1.6: .\leaii iiieasiires over tiriie for sitiiiilatio~is with iiiiiforiii iiiisiiig fur al1 co\wagtX levrls: y0 = 0.00 10. Trite I - E = 70%. Leaky irioclrl ( 1 )
VE(0R) - Time matched
Curnulauve Mean Cases - Unvaccinated
VE(0R) Pared - Time matched O f
VE(0 R! - Cum. lncrdence
Cumulame Mean Cases - Vaccinated
F i e 4 : .\Iean riieasures oi-er tiriie for siniulatioris witli iiiiiforiii niisiiig for al1 (-overage levels: po = 0.000S. Triie I- E = 70%. .All/Sotliirig iiiodrl ( 2 )
VE(0R) - Time matched
0' VE(0R) Paired - The matched VE(0FI) - Cum. Incidence
- ~ e
Cumulatrve Mean Cases - Vaccinateci
Cumuhwe AR's - Vaccinated Cumulative AR's - Unvaccinareci Cumulative Mean Cases - Unvaccinated
ivitli uniforrii riiisiiig fur ail i r e 4 . : SIeaii nieasures over tinie for siiiiulations coiPrrage levrls: pu = 0.0010. True 1,'E = 70%. .All/Sotliirig riiotlel ( 2 )
4.4.2 Non-Uniform (Heterogeneous) Mixing
Tlie popiilat ion for t liis serorid set of simulations also ronsisted of a closed roiriiiiiirii t ~ -
of '1000 individuals. Ho~vever. in tliis case. the rotri~nuiiity lvas rliviclrd into 4 sriia1It.r
iieiglibourlioocls of -700 individuals. In eacli neigli bourliood t liere wrre 100 Iioiisrliolcls
niade itp of .? nieinbers - 2 adults (low susceptibility) and 3 rliildren (liigli sirscrpti-
bility). Ln tlie sirriulation. there are three levels of niising - at the coniriiiiiiity Ir\-el.
tlie neiglibourlioocl level and the lioilsehold level. Therefore. persons iiiis at diffrrriit
le\-rls in t lieir lioiiseliold ariti t lieir rieigli boiirliootl t han t lit.>. woiild wi t li persoiis froiii
ot lier tieiglibourliootis.
Tlir pararrietrr valiies choseri for th^ sinidation ivrre choseri not oiil~. to rrHrt-t t l i r
Iietrrogeneity in the mising but also to incorporate co~ifoiiiiclirig diie to the difFrrtwtin1
siisceptibility bet~veen adiilts and cliildren. .-\ furttier distortion was cwatecl b>- va(--
ciiiati~i$ twice as rtiany cliiltlren as adults. The paraineter valiies for t lie siriiiilat iuiis
iwre:
riutvs c-liildrt~ri atid (1 = 2 rienotes adiilts
0 Overall vaccination CO\-eragrs of:
Loiv
Moderate
Higli
a Degree of clustering: Sonr ( uni foriii ). lloderatr. Higli. Tlie iiidi\.iilual iirigli-
I>oiirliood co\.tm,se levels (giveri as a percentage) correspontlirig to t l i r Ir\-pl of
cliistrring are outlinecl iii the lollowing table:
Iudividuai Neighbourhood Coverage Levels (%)
C'accination
(l'overagr
High
l Seighbourhoocf
C'lustering 1 Stratiini 1 1 2 3 1
H igh .r\dults 80 O O O
C'hildren 100 60 O O
Sone Adults 30 30 30 :IO
C'hildreri 60 60 60 ti0
l lorierate Adiilts 60 tiO O 0
l r n 90 90 30 30
Higti Adults LOO 20 O O
('hiltiren 100 100 40 O
>one :idiilts -10 -10 10 40
C'hildren 80 SO SO SO
>[oderate .-!duits 80 8O O O
C'hilriren Ill0 100 60 i i O
The resiilts t liat lvere obserl-ed for t tiis seconrl set of siiiiiilat ions broiiglit t.u ligiit.
otlirr iritlirect rtferts whicli icere not iticorporated irito the previoiis iriodt4. Tlir iiiost.
apparent was the effect of clustering on t lie vacciriat ion coverage. .Aiso. to tlsaiiiiiir t h r
rffect of coiifoiiiitliiig prodiicecl 1 ) ~ - the differeiitial siisreptibilitj. l>etwerii adiilts aiiri
cliiltlr~ri. c r i ~ l i ( tiieasures of I . E. wliicli clirl iiot take iiito accourit age st rat ific-at ion.
ivere alsv calcrilateci.
I f ive oiily coiisider t lie factors esplored iii tlir prcvious set of siniiilatioris t lirti tlir
findings will be parallel. Tliat is. al1 factors tliat influence prevaleiice will iiifliieiicr I * E
estiiiiatecl iiiider tliis iiiodel i n the saine LVV. The results for b1odel 1 and 1'E = 90%
are displayeci in Table 4.7 and Figures 4.9 tlirougli 4. i 1. Tlie resiilts for .\lodel 2 aiid
I,'E = 90% are clisplayed iii Table -4.S aiid Figiires 4-12 tlirougli 4.14. Siriiilarl~.. for
I ' E = 70%. blodel 1 fi na1 resiilts are giveii i r i Table 4.9 aiiti Figures 4.1-5 t liruii.o;li 1 . 1 7
ivliile SIoclel 2 tinal resiilts are given i r i Table 4.10 aiid Figures 4. I X t o 4.20.
Even Distribution of Vaccination
I -11dt.r circrinistances rvhere t lie vaccine was distribiited evrnly ariiong t lir foitr iirigti-
Lmirlioocls. al1 t lie pooled est inintors denionst rated t lie sanie pat terri of hias iiiidrr
the iiiiiforiii iiiising niodel. In fact. eveii in the presetice of iiori-iiiiiforiii iiiisirig dur tu
lioiiseliolds aiid neiglibourlioods. 1- E( .>) \vas unliaseil iiritier SIotlel 1 wliilr. I ' E ( GR)
situations tliey Lvere acciirate to a tt.nt,h of a perrerit. The c-riide estiniators. ori t i l t>
otlirr liaiid. rieere sliglitlj- lower in riiagriitride tlian tlir pooled r s t i~ i i a t r s . By nssiii~iitig
t iiat tlie populatioii was honiogeneous. t lie result ing rrricle es t i~ i ia tes ivere hiascil [Iiit.
to rorifotincling. Recall tliat the vaccinatecl group contained tivice as niaiiy cliil(lrrii
as adul ts. Since ctiildren were niore susceptible. t lieir large riiiiii bers i 11 t lie i-art-iiintr(l
groiip iiicrrasetl t lie a g g r ~ g a t r siiscept ibili t ~ - ariiurig vawiiires aiid iiiaclr t lit* \ ï ~ ( - r i i i i .
a p p a r lrss i t i i pressi i-r.
Uneven Distribution of Vaccination
Tlitl r f w t of (list ri but itig t lie vaccine iiiieveii1~- aiiioiig t lie nieiiibers of a coriiiriiirii t.y
cati rreate a strorig hias in C-E if iiot accountecl for in the estiniatio~i. Soriieoiie liv-
ing iii a Iiravily vacciiiatrtl area. regartlless of wliet lier t liey tvere vacciiiated. w ~ i l i l
rrcrivr grrater irielirect protrct,ioii tliati sotiirotir liviiig oiitside s ~ i r l i arc3as. I i i urrlrr
to clriiiuiistrate. ~ v t . ran aii aclditioiial tivo sets of epi<letiii(- siiiirilatiutis n-lirr<~ \vtt i-tic--
ciiiatsd the saine aiiiouiits of cliildreii and adults but iiistead of varriiiatiiig rvriiiy
ncross t lie board WF. deiiberatelj. clustered the vacriiiatioii in 1 or 2 of t lie 4 iieiglil~oiir-
lioods. As cari be seen from al1 tlie available tables and figures. al1 estiniators Iiasrtl
on OR and trarisniission rates were found to overestiiriate tlie t r u e C-E iirider tliese
rircurnstances. T h e obrioiis exception kvas 1. E(.4 R) wliicli srverel? iiriderest i i~iatrd
tlie trrie I * E urider SIoclel 1. I n fact. a close esaniiriatiori of Tables 4.7. 4.S. 4.9 alid - -1.10 ret-eals tliat tlie estent to n-liicli I..E(.q R ) and C'E(J ) overc-stitiiate I- E iiicrrases
as vaccination coverage iiicreases. Tliis occurred because vacriiiated prrsoris rcwivrcl
addit ioiial iiiclirect protrrt ion due to t lieir lierd iiiiniunity wi t l i i r i t lie clusteriii,a iv l i i l r
iiiivarcinated persoiis. wtio niostlj* resided i r i areas witii little or no varîiiiatioti. re-
ceiveci lit t le iiidirert protectioii (ot lier t liaii t liat providecl nt t h r roiiiriiiiiiit~- Irvel )
.\Iat.liei~iat icallj-. t liis trarislates iiito liiglirr nt ta& rates for iiiivarciiiatr(1 popiilat ioiis
aritl loiver attack rates for vaccinaterl popiilatioris. -4s a resiilt. al1 nirasrlres wtiic-li arc
uiiable to corit roi for sucli differetit ial susceptibility will procliir~ inflatrrl est inintes of
efficacy. Tlie e\.idenee is in Tables 1 .11 and 4.12. For ail- giveri level of rovrragr. .-1 Ri
rierreases aiid .4 Ro i ncreases (overall ) as t lie cliisteri ng increases iii severi t>-. .As nri
twmiple. for t lie Leaky vaccine and adult /rliildren coverage rates of :IO% /GO%. (Tal>lr
-I. I 1 ). .-\Ri is O. 14S6 iiiider ilriilorni (-overase but decreases to 0.1 149 aiid 0.01) 13 iii tlit-
riioderate aiicl higli rliister sceiiarios. resprrtively. C'oiiverselry. -4 Rii iii(m3ast.s froiii
O.(i!)OO to 0.79'74 to O S 3 14 as cliistrririg is iricreasetl. I 'iicler Sloclrl 2. t lit. (-liaiigtns
i i i .-IRi aird .-\Hu as cliisteriiig iiicreased (for a givrn Ievel of 1-E) iverc iiot qiiitr as
rlrarnat ic siiice bot ti t lie o\.erall suscept ibility of t lie population aiid t lie iipper boiiiirl
on the iiuiiil~er of potcntial iiifrctives was lower tlian tlie Leaky 1Iodel. This reasuli-
iiig alsu esplaiiis wtil-. tiiider 1Ioclel 2. G-E(.4 R ) \vas iiot onlj- tlir lrast 1)iasril hiit.
reitiaiiirrl lairly ru lmr r ~ w i iiiiiler es t renitb clustrriii~.
Tlir tiiiit--iiiat<-lit-cl udds ratios stiowcl t l i r sanie trend iii hias aiiil agaiii iiiiiiii(-krrl
tlic I>elia\-ioiir of I ' E ( . j ) ver- rlosely ~iiakiiig it quite acriiratr iiridrr iclral Iluilrl I
c-oiirli tioris. Tlir oclds ratios baseci on curiiulative-iiiçitlriice tvcre. orict1 agaiii. t 1 1 t h wurst
offeiiciers i r i terrils of overest iniat ion. But. iirilike t lie other iiieasiires. t tir o\-vrc~sti~iia-
tioii ivas iiiore srvere for loiver coverage levels. Tlie clusteriiig of vacriiiet~s togrtlirr
wit li the frailty etfrcts tlescribed earlier producecl higlier t han rspectetl varri-iiiat ion
rates aiiioiig t lie riori-cfiseased and lower t!iaii rxpected vaccinatioii rates aiiiorig the
disr;~srtl. I I vacciiiatioii roverage decreased. the proportioi, of va(-ciiiatrcl i-asrs
&crcased whilr tlie proport ion of vaccinatecl coiit rols iiicreased. I l l e iirt r(1siilt \ras
a lower otlds ratio aiicl Lieiice. a Iiigtier I ' E ( O R). t f ive observe t lie trends i n Figurrs
-4.1.5. 4.16. 4.17 and Table 4.9 for b-E = 70% we notice that the odds ratio uiidcr
ciiniulat ive sani pling act ually cloes worse at loir. leïels of vacciiiat ioii coveragc aiitl
iitiprovrs as tl ir covrrage iiirreasrs. 'Vlieii t he vacciiiatioii is cliistrretl tlieii ttir dif-
ference in niagnitiide for I - E ( 0 R ) under al1 coverage levels appears to tiiiiiiiiisli. Tlir
sanie trend can be seen for t h e .All/.l:othing mode1 as ivell (Table 4.10).
One iiiterest ing plirnoriieiion was t lie -betitliiig-. beliavioiir over t iiiie uf \ - E( -4 H ) . Tlir effect is too subt le a t I V E = 90% but ver-- visible at I -E = 70%. Iriitinll~.. it
ivoiilti rise fairly steariily follorvetl by a rirciilar ~~~~~~~~~artl niotiuri tliat luokr(1 l ikr it
ivas -*wooping ~ i p - sliglitly towards t tir end. Tlir sliapr \vas iiiurr (Iraiiiatic i v l i r t i
t lierr kvas tiigti clustering aiitl loir rowrage. I.'iiclrr t lie Leaky vat-ciiir. t tir i-liistrri iig
produced fewer vaccinated cases so t lie clustered estimates of I -E(;1 R ) wrre Iiiglirr
aiid actiially closer to t lie trile efficaq. alt lioiigli. in Iiigh incideiiçe coiiditioiis. t l i q -
wrre al1 drastic iintlerestiniates as ive woiild expert. ITrider tilt1 .-lll/Sot tiirig rtio(lt1I
t l i r r r ivas a siriiilar sllape tiowerer tlie estiitiate always stayetl above the t ru r zstiriiattb
iit al1 t iriitls. \\'lien t lie ro\.erage was ION-. t lie est iiiiate rlrsçeritlrd closer I)rcaiisr t lie
TaIlle 4.7: Firial est irriates for varioiis tneasiires of I * E frorzi sirnulat ions \v i t 11 [lori- iiiiifor~ii ~iiisirig (L'F i s,, ). hIodel: Leaky ( 1 ). triie \ - E = 90%.
h/Iodel: Leaky (1). True VE=90%
Fraction C'ac. Time- Matciied C'iimulat ive- inc idwcr
PooIeci 80. l f 1.9 89.:3f L .O 88.Cf 1.3 $9.411-7 94.7f0.9 9-1.55 1.2
40% /$O% C'rude 83.2i2.6 $ 5 . : . 83..5f 2.9 5.5. If 4..5
Poolcd 6 . 4 2 . 89.3f 2.0 S8.'Lf 2.9 S . S9.8f 4.0 W.l*.ti.;>
\ 'acci t i~ Distribution hIotirrntely C'liisteretf
4 % ( ' r i itk d4.5k2.0 92.9IO.I) 92.'ïf. 1 .O 9 Ï . Ï fU.3
Poolrd - 2 9 4 . S f 0 .Ï 93.6i0.9 9 . . '3S.c)*U.2 9S.f kO.4
0 / ( r i 5 . 1 2 . 4 1.0 9 1 .$& 1 . 1 9tj.-IdS-.5
Pooled 1 ' 5 . 0 1 9-I.Sf 0.7 9:I-Ï*0.9 Y.ti.:{i1.0 9S.410.4 9S.3k0.5
40% /SU% ( ' rude S9.O& 1.6 92.SiO.9 9 1 .'LI 1.1 W . l f 1.1
Pooled 90.3&1.6 Y 6 . W 0.9 95. l f 0.9 9.5.Ïf 1.3 C)8.Uf 1.0 97.91 1.1
1,'accine Distribution Highly C'liistered --
20%/-10% C'riiric S . 1 94.5f 0.7 93.9f 0.9 9S.5f 0.2
Pooltd Sij.Of 2.0 0 . !Nefif 0.S 97.0*0.9 99..5&0. 1 9!).-Ifu.?
:lO%/GOCX C'riirlr SS.Ok 1 .C 94.Ïf O.7 9-1. I f 0.S 97.Ïf 0.3
Pooled 8 . 4 1 . 97.Of0.5 9.5.7fO.7 9Ï.ZkU.S 99.3f0.2 99.3k0.3
- IO%/YO% C'riicie 9 1 .Ïf :3.0 94.6*:3.0 9:1.6f 4.3 96. l f 2.2
Pooled 93.3f3.2 9 i A f 2 . ( j 96.7i53.33 K . 5 f 3 . 7 9X.Ïf 1.5 '3S.C)* 1 .Ï
Pooled VEIOR) - Tirne malched Cnide VE(0R) - Tirne rnarcned
Fi
Pooled VE(0RI - Cum Incidence
'ni
Crude VE(0R) - Cm Incidence Paaled VEIARI
-. Pooled VE(beta)
Figiirr 4.9: ('oiiiparisoti of iiieaii iiieasiires O\-rr tiiiir for siiiiiilatioiis witl i iioii-iiiliforiri iiiisitig I iiider varioiis let-els of vacriiiat ion rlusteri iig: Trtir C-E = 90%. Leakj- iiiotl<~l ( 1 ). vacciriatioti coverage is 40% for cliilclreri aiid 20% for adiilts.
Poaled VE(0R) - l m e matched VE(0R) Patred - Time matched
-,ri4
Paoled VE(0R) - Curn Incidence
-Fe
Crude VEIOR) - Cm Incidence Paaled VE(AFi)
Figiire 4.10: ('oriiparison of niean rrieasiires over t ime for siiiiiilat ioiis [vit 11 iioii- iiriiform riiisirig iinclrr ~arioiis Ievels of vacciiiatioii cl~isteritig: True \ ' E = 90%. Leaky rnoclel ( 1 ) . vaccination coverage is 60% for cliilciren and 30% for atlults.
Pooled VE(0R) - Time matched
Poaled VE(0R) - Cm Incidence
V E ( 0 R ) Paired - Tirne matched
Cnide VE(0R) - Cum Incidence
Pooled VE (beta)
Crude VE(0RI - Time matched 1
Pooled VE(AR1 0 1
Figure 4.1 1 : C'oriiparison of niaan nieasures over t inie for siiiiulations witli rioii-
iiriiforiii riiisiiig iiiider ~ar ious levels of vaccination clustering: True 1-E = 90%. Leak- riioclel ( 1 ) . vaccination COL-mage is 5'0% for cliildren and 40% for adiilts.
Table 4.S: Final est iniates for L-arious riieasures of C - E frorii sirnulat ioris wit l i rio11-
iiniforiii niising (rre k s , , ) . IIotlel: .illl/Nothing (2) . trur I S E = 90%. ppppp
Model: ALl/Nothing ( 2 ) . Triie VE=9O%
Fraction C'a c . Tirne-llatcheri ( ' i ir i i i i lat ive- Iricitlrticc~
- - - - -
Vaccine Eventy Distributeci
Pwled VE(0R) - Tirne matched VE(0R) Paired - Tirne matched Cnde VE(0R) - Tirne matched
1
- re
Pwled VE(0R) - Cm incidence
c e
Crude VE(0R) - Cm. Incidence
rt!
Pooled VE(AR)
Poaled VEjbeta)
Figure 4.12: C'ornparison of riieari nieasures over tinie for sitriiilatiotis witli iioii- iiniforiii niisirig under varioiis levels of vaccination rliisteririg: True I - E = 90%. AIl/Sothing inodel ('1). vaccination coverage is 40% for children aiid 20% for adillts.
Paoled VE(0R) - fime matched 0 1
-5
Pwled VE(0R) - Cum. Incidence
VE(0R) Parred - fime matched Li r 1
Crude VE(0R) - Tirne matched
*CL'
Crude VE(0R) - Curn Incidence
-*
Paoled VE(AF1)
Figure 4.13: C'oniparison of mean meastires over tinie for siriiillat ioris \vit li iiuri- iiiiiforiii 1iiixi11g under various levels of vaccination clustering: Trtie 1' E = 90%. All/Sotliiiig riiotlel (3). vaccination coverage is 60'3, for cliildren aiid 30% fur acliilts.
Pwled VE(OR! - Time malched VE(0R) Pared - Time matched Crude VE(0R) - Tirne marched
-. Pooled VEiAR)
-re
Paoled VE(0R) - Cum Incidence
-re
Crude VE(0R) - C m . Incidence
Cruae VE(AR}
Figiire 4.14: ('oriiparisori of iiiean nieasiires O\-er tinie for si~iiiilatioiis wit li riori- iiiiiforrii riiisirig urider \-arious leïels of vaccination clusteririg: Triie I *E = 90%. .-\Il/ Sot liiiig irioclel ( 2 ) . vacciriation coverage is 80% For chilclren ancl -10% for a h 1 t-S.
-Table 4.9: Firial estiniates for various rneasiires of LIE frorii siriiulat ioiis \rit li t i ù ~ i -
iiiiiforni iiiising ( tCe I .su, ). Model: Lraky ( 1 ). triir I ' E = 70%.
C'ac. Tinir- Matchecl C'uri i i i latÏvt~-Ir1ci~1~~1~~t~
P w l d VE(0R) - Tirne matched VE(0R) Patred - Time matched Cnide VE(0R) - Time matched
--5
Pooled VE(0R) -Cm. Incidence
-fie
Crude VE(0R) - Cum Incidence Pooled VE(AR)
of nieaii iiieasiires orri. tiriie for siiriiilatioiis witii I ~ W -
iiiiiforiii iiiising ilrider varioiis Irvrls of vacciiiatioii ïltistt~rilig: Tr~iv 1 - E = 70%. Lraky riioclel ( I ). vaccination coverage is 40% for cliildreri atirl 20% for atliilts.
Pooled VE(0R) - Time rnatched
-re
Poaled VE(0R) - Cm. Incidence
Cnide VEIAR! o r
VE(0R) Paired - Time matched
-pe
Cnide VE(0R) - Cum. Incidence
Cnde VE(0F) - Tirne marched
Pooled VE(AR)
::
Figiirc -1.16: C'oriiparisori of niean nieasures oi-er t i nie for siiiiulat ions \v i t l i i i m -
utiiforri~ riiising iiritler tarious le\-els of vaccitiatioii cliisteriiig: Triir I V E = TU?. Lraky iiiodel (1 ) . \.accinatioii coverage is 60% for cliildreri a n d 130% for adiilts.
P~oled VE(0A) - Time matched VE(0R) Pared - Trne matched Cnide VE(0R) - Tirne matched
- -e
Poaled VE(0R) - Cm. Incidence
' C O
Cnide VE(0R) - Curn. Incidence
Pwled VE(beta)
Figure 4-17: ('otnparisoii of nieaii rneasures over t iiiie for siriiilIat ions wit 11 riori- iiiiiforiii rriisiiig iiiidrr varioos levels of 1-accinatioii cliisteriiig: Triic 1 * E = 7 0 9 . Leakj- iiiotlrl ( 1 ). varriiiat ion coveragr is $O% for (-hildreri aii(l 40% for ndiil ts.
Table -L. 10: Final estiniates for various nieasures of I - E froni siriiiilatioris witl i tiori-
i n i i ( s ) Iloctel: All/.\:othiiig ( 2 ) . true I* E = 70%.
Model: AII/Nothiug ( 2 ) . Triie VE=70%,
Fraction C'ac .
Vaccine Evenly Distri btited
\.\crine Distribution hlorlerateiy C'lusterrtl
\,'ncciiie Dist r i l ~ i i t iori H ig hly (_'liistered
Pwled VE(0R) - The maiched
F e
Cnide VE(0R) - Cum Incidence
--e
Poofed VE(AA) Paaled VE(0R) - Cum Incidence O I
Figure 4.1s: Coinparison of triean nieasures over tinie for siitiulat ions \vit li 11011-
uriiforiii itiisiiig iirider varioiis levels of \-accinatioii rlustrriiig: Triit3 I'E = 70%. AII/Xotliing tiiodrl ('1). vacciiiatioii CO\-crage is 40% for cliiltlreii aiid 20% for n(1iilts.
Pooled VQOR) - rime matched VE(0RJ Pairad - Tirne matched Crude VE(0R) - fime matched
/--- z--- =---y-
4*c ' F _.*---
_*.*_*_.. - .---- -*.S.-
/[ /--- fi''
Et
Pooled VE(0R) - Cum incidence
--e
Poaled VE(AR) Crude VE(0R) - Cm Inciderce
-'
Pooled VE(beta) Crude VEIAR)
Figure 4.L'): C'oriiparisori of nieati rric-.açiir's ovrr tiriie for sirnirlatioris witli riori- iiriiforiii riiisiiig uiicl~r various levels of iacciiiat iori cliisteriiig: Triir I ' E = 70'3. AIl/Sotliing niodel ( 2 ) . ~accination coi-erage is 60% for cliiltlrr~i arid 90'7; for alti lts .
Pwled VEiOR) - Tme matched
0: / / -*-a =-= =-=
m C - O <&,
--**.- *.---.--- __..-*----
U1 ..&-- fo' 91
Prxlled VE(0R) - Cum Incidence O y 1
Crude VE(AR)
Crude VE(0R) - Curn Incidence
4:
Pwled VE(beta) 1
Pooled VE!AR)
Figure 4.20: ('oiiiparison of niean nieasures orer tinie for siniulatioiis ivitli iioii- iiiiiforrii iiiisi~ig iiiider varioiis levels of vacçiiiat ion clusteri 110: True ' E = 70% . .-\ll/Xot tiirig riiodel ( 2 ) . varcinatioii coverage is SOR for rliiltlreii aiid -10% for adiilts.
Table 4.1 1: Firial nieari stiniatecl attack rates frorri si~iiulatioris witli rioii-iriiiforrii
Sone
Fraction Caccinated Leaky C'accine
:WX/60% Adults 0.0880 0..5857 0.0.5:I6 O..?-L:J-I C'hilriren 0. 1688 0.8 116 0.01s 1 0.1170
Overall 0. 1486 0.6900 0.0719 0.6.5 12
-10'%/80% Adults 0.0 173 0.4877 0.0 1.73 0.45.54 C'hiltiren 0.064 1 0.9.511 0.0372 0.93.38
Overall 0.0524 0.6442 0.0317 0.61.5.3
Chapter 5
A Look At Some Case-Control
Studies of Hib
Tlic purpuse uf tliis cliapter will l x to coiiipare rrsiilts froiii a ~ v c r a l (-ast--(wiitrol
sti1dit.s clrsigii(4 to rstiiiiatr tlir protrrtive efficai-y of PRP aiid PHP-D i-awiiirs
;igairist diseasr raiistv.l 1)- H n r rrloplt i l u s irrjluc nzni type 1) ( Hib ). Aftrr suintB I)ric1f
lm-kgroririd oii Hib and ari atial>-sis of a i-ase-control stiidy of Stviss iiifaiits \vtb n-ill
luuk at rtlsiilts of ottier eficacy trials for P R P j P R P - D witli tlit. Iiopr t h t i t nia). slitvl
liglit oii the rcasoris for tlie \.ariuiis resiilts witnessed i r i case-coritrol trials to clatr.
Harntophilus inJ lurnra t is a patliogeiiic bacterial agent ~vtiicli attar-lis Iiiiiiiari Iiosts
tw-liisivel~: Tliere are 6 types of Haeniophi lus i n p u r rrznt-. iiariird a-f. T~*pe I > ( H i l )
is the rtiost patliogenic strairi affect ing yourig cliildren. Prior to the ilt~t.rlopiiiriit o f
rffectii-e vaccines. roiigtily '2.000 cases of Hih tlisease had beeii reported aiiriiiall>. i i i
('ariacla (Gold aiid .\lartell ( 1994)). Tlie niost coiiinioii disease prorliicctl I)y H i l ) ib
bnrt t~ial tiietiiiigi t is. Tlir tlisease also riiariifests itself i i i ot lier forriis sii(.li as I,i~i.-
trrciiiia/srpsis. ptiriiiiiotiia. epiglott i t is aiid septii- art lirit is to riaitir a ftaiv.
Tlie riiost iriiportaiit risk factor for Hi b tlisease is agr. C'liiltlrrii brtwerii t lit) agrs
of 6 arid 12 rtiorittis are at greatest risk. For infants uncler t h e age of 6 iiiuiitlis.
t lie (liscase is reiat ivelx rare. supposedly becatise of protertion conferrd II!- riiatrrtial
antiborties tlirougli brrast-feedirig. It's also siiggestrd tliat luw rsposiirr is partia1l~-
resporisiblr as well. Ariot lier iritrrest irig ctiaracterist ic of t lit. agt. risk is t liat rar-li
lorrii of Hill rliseasc lias a iiriiqtle agr of oçciirrerice. lleniiigitis. for rsariiple. t.ypi(-all>.
affects infants aged 6-9 riioiit lis. wliilr epiglot titis affects rliil(1rtw of aboiit 'L >.eili.s of
age ( \Carcl e l al. ( 1 094 ) ).
Otlier risk factors inclilde (i\,-ard rt al. ( 1994)):
1. Gender: Tlie eviderice is not o\-erwlieliiiiiig. Lut sonie st~itlies liavr s i iggrstd
t.liat I q - s are at sliglitly higlier risk (C'ochi r t al. ( 19%) for rsaiiiplt*).
2 . Race/Ethnicity: Sltich corit roi-ers‘ stlrroiiritls t liis topir. Suriie st iirlirs l i i t ~ t b
sliuwri tliat in(-idriice is sigiiificaiit 1'- liiglier in Black diildrrii aiid Satil-tb Anit~r-
icaiis - a t least double t liat of White cliildren. However. t lie Iiypot liesis t liiit
tliesr racial ciifferences are due to geiietic fairtors is iinsiibstaritiated sirirr i«riitx
have arguaci t liat coiifoiiiidirig due to socioecoiioniic stat us ancl croivtliiig pliry<vl
a sigiiiîirarit role i r i irifliiencing sticli f ritliiigs.
3. Daycare Attendance: llariy popiilatiori-hased st~idirs lia\-r foiirid tlint (-Ilil-
(lrrii ivlio at triid (laye-are arca at grrater risk ( l i i r to iiicrrasrcl t~rpusiirt3.
4. Socioeconornic Factors: C'roivdirij. large lioriseliold sizc>s. 1at-k uf a(-(-e1ss t.o
vacciiiat ioii and ot lier relatecl factors iiiflueiicril bj. iiicoiiir aiid /or ciil t iirtx affwt
risk.
5.2 PRP-D Vaccine
tIib roiijiigate vacciries are quite tiniqur froni ot her vacciiirs Iwraiise L liey are t lit*
first cori;ugnted vacciries licensed for use. BI conjiigated. L w iiieaii t liat t lie arit igtw is
clieiiiically liiiked t o a carrier. .A carrier is siniply â protrin wliirli provides a rr(-o.q-
nit iori si te for T-cells. Soiiietiriies a \-amine does not stiniiilate ail iiiiiiiiiiie respuiise
ori its owii so tlie carrier acts as a ~~iiiitldle niari". PRP-D vact-iiie (Coiiiiaiiglit Lal~u-
ratories Ltd. ). wliicli lias beeii PL-aluatecl rxtensivelu for its efficacy. co~isists of P K P 1
polysaccliaricle vaccine cotijugated witli diphilieria tosoitl. Tliree otlier iiiajor cuiijii-
sa te vaccines for Hib are also availablr. .-\II use tlie same P R P polj-saccllaride Imsr
(sonietimes callrtl a liapten). biit each one is coiijugatec~ witli its owri iiniqiir r-arrier
aiiti lias its oivri iiriique clieniical st rtwt iire.
P RP- D vac(*irir lias hem ssliowii to sigriificarit ly raise ant ilmly Ievrls i t i clii l i lrt~i
nrld acliilts. The rcsponse to PRP-D iii cliildren sretris to var!. s!.steiiiati(-ail?. u-itli
agr. Olcler cliilclreii generallj- esliibit liiglier levels of antibody pro(liirtioii t liaii in-
farits. everi wi t tiorit a second dose. Several case-cont rol s t udies i 11 t lie I - r i i ted States
were rarriecl o u t to assess efficacy in oltler children (> lS niontlis). Tlir resiilts \vr*r(-
favourable. \vit h protective efficaq. rstiiiiates rariging froiii 74% t« 96%. T ~ v o iiiajur
prospect ive trials were sr~bsrc~iieiit 11. înrriecl out to assrss protrrt ive efficac-u of F'K FI -
D iii yoiiiig infants and tlie rrsults w r e iiiixecl. .A large trial i i i Fiiila~itl fotiii(l t l i r
1-acciiie to bc at least (14% eficacious after 3 doses. By coritrast,. a large trial (wii-
~liiîtccl in Sative Alaskari infants failecl to slioiv acleqiiate efficacy aftcr 3 ilusea (4:I'X.
3 % ' 1 : ( 4 7 ) ) Tlie followiti:, tliscussioii will corisicier t tiesr alotig vit li ot lier rtfi-
(.a(.>- stiidirs uf PRP-D (ancl its pretlrîrssor P R P ) t u explaiii wliy siicli rriisrrl nbsiilts
oc(-iirrtl(l.
5.3 VE Study of Swiss Infants
A i-ase-coritrol s t~ idy tu deterniine the protective efficacy of PRP-D \vas corit1iictt.d i i i
S~vitzerlaiicl tliiriiig 1 !XU for irifaiits of age 6 mont hs or older. Berause of t lic rlifftw-iit
rrspotise to PRP-D for cliiltlren of differerit ages. the sariiple was separatrcl iiito two
Sroiips - 6- 1 ï riioi~t lis. aiitl 1 S iiiorit lis or older - and a separate arialysis was perfortiitd
on earti group. C'ontrols were obtained for racli rase frotii the satiie geograpliic area
aritl riiatcliecl for age. The nuniber of coiitrols varied per rase atirl rarigcd froiii O
(rare) O 4. Disease status aiid vacciiiatioii records were availabltl for al1 part icipaiits.
' Polyribosylribitol phosphate
-4 siibjrct was coiisidrreti rarciriatrrl if Iie/slir recrivrd 2 tlosrs of P R P - D at Iraest -1
iverks apart witli tlir firial ilose briiig at Ieast 2 wet~ks prior to oiiset of (lisrase. Otli<lr
covariates and/or poteritial corifounders t h were availahle includecl:
At terirlance in kiridergarten ( \ ies/So)
5.3.1 Age Group: 6-17 Months
.-\rial~.sis of tlie piiriger of the two age groups iailed to sliou- sigiiificarit prote<-tire
efiracy. Botli case aiicl curitrol groiips were siniilar tvitli respect to a p . and irieaii
1 1 1 r 0 s i l i i s ( T a 5.1 . The rase groiip liad a liiglier proportiori uf riialrs aritl
a Iiiglier proportiori of siibjects wtio atttrrirletl kiridergarten. Tlir proportioii at trii(liii,q
tlir niinibers n w e too sniall to Iw reliable. UÏtli respect to laccination aiid Disrasr
stat us t lie data broke don-ri as follows ( colunin perceiitages are i r i pareri t lieses ):
iacci tiat ecl (.'ases C'ont rols Total
Total 6 1 1-41 202
The estiriiatecl rfficacy frorii tliese riiiriibers alone was :19.1%l witli a 95% CI of (-42.6. 74.0).
C'oritrolliiig for otlier confoiinders ciid little to change the results witli the escept ioii of
daycare at teiitlaiice (Table 5.2) w liicli provided a lowcr and less reliable est iiiiate. ( 'oii-
trollirig for daycare yielded ari efficacj- \ d u e of ZS.23 witli a 9-53 C'1 of (-69.d. 69.6).
( 'oiidit iorial logistic regressiori iisirig age as t lie iriatcliiiig variahle [vas (loiir for
al1 possible coiiibiriations of t h e available covariates (see Table 5 . 3 ) . Berartsr ive
Table 5.1: I.'iiivariatr rt~siilts for 6- 17 itiorit ti age groiip
I C'ases ( 'ont rolç Tot ai (n=61) (n=141)
.\lean Age (s td . dev.) 1 1 1.4.5 (3-28) 1 1.3 1 (3.35)
Table1 5.2: Estirtiates of LVE(OR) frorii iiiiiiiatclied -1 x 2 x k r:oritirigtwcy table aiia1)-sis
( 11=202) 1 1 - 3 5 ( 3 . 3 2 )
Siiriiber A t tending D a ~ c a r e ( % ) Ses-SIales ( %A ) LIean Suriib~r of Sibiinns (std. dev . ) Siiiiibw Attendirig Kiridergarten ( % )
\vert. riiorr c-oric.crriec1 n-i t h est iiziat irig efficacy. i titeract ioris werr riut r.orisiclercv1. I r i
addi tioii. tlir sparseiiess of e\.eiits for soiiie of the predirtors iiiaclt- est iriiatioii of iiiti.r-
1/53 ( 1 3 9 ) 4/141 (2.84) 41/61 ( 6 7 2 ) 7Oj 140 (50.0)
1.07 (0.79) 1.07 (0.S1) 4/53 (7.55) l / 1-I 1 (0.7 1 )
Estinlated V E (9.5% C'orif. Iriterval)
:{9.1 (-42.6. 74.0) :{Tat i (-4:3.-!. 72.9) 3 . 2 (-69.8. GY.ti) 132.0 (-60.1. 11.1)
actions iiiipractical. Estiitiates of \ ' E raiigetl froni 27.4 to 39.4 biit the rrsiilts wrr(1
iiot sigiiiticaiit witli ail (.orifidence iiitervals raiigiiig t>-pically frutti -120'X r . 0 $O?(.
O t lier tiiorlels a[-liirvefl urily rilnrgirial si.qiiifiraiicr. l[o(lel 5 a i { l rtiodel S. tvliir-11 in-
(-liiderl Srs aiicl/or li iriilergartrti ivere t lie orily itiodrls ivi t l i P-\-aliir I ~ s s t liai1 1 O'Z .
The rstiiiiates of oclds ratios for soriie of the coivariates dicl hoivever I~riiig fort11 soiiitB
iriterestirig associations. &laies were approsiniatclx 1.0: to 2 tiriies ( p = O. 16) mortb
liliely to bp affected wliile attending kintlergarten made sonieoiw T to i: ( p = 0.0s)
t iities riiorc at risk. The iitiiiiber of siblings aiid claycare atteiidance dicl riot srriii tu
prot-idc an>- iisefiil iriforrziatio~i.
I'riro~iditional losistic regression follo~ved witli eclually disappoiriting rcs~ilts. Tlic
iiitt>rpretatioii of the predicturs \vas ctlifferrnt for tliis riiodel sirice t l i r oiitcotiir i r ,
tliis case ivas \acririatioii as opposed to clisrasr. Efficacy values rarigrd frorii 20% tu
39% wit h loiver corif clence liniits spannitig -90% to -43% aiitl iipper liniits Iioveritig
5/194 (2-5s) 1 1 1/20 1 ( - 3 . 5 2 )
1.01 (0.S-l)
5 / l9-I (2.5s)
Controliiiig For
Siblings Da~wrare .-4ttericlaiice
Ses
P-1-alue 0.250 0.266 0.430 0.3'lb
Table 5.3: Estiniates of I'E froni riiatchrd analysis iisirig coiitlit iuiial lugistic rrgrrs-
Estimated V E (9.5% C'onf. Interval)
29.1 ( - 144.9. Tg..?) 26.3 ( - 1.j4.9. 78-7) 2'3.1 !- I4.5.U. 79..5) :$O.? ( - 140.0, 79.7) 3S.5 ( - 1 1s-7. 82.7) 2.5-8 ( - 1.ïs.3. 78.7) 27.4 (- 149.6. 78.9) 36.6 ( - 127.7. $2.3) 30.4 ( - 13S.p5, 79.7) 3 8 3 ( - llS.0, $2.6) :i9.4 ( - 1 14.2, $2.9) 27.4 (-149.2. 73.9) 36.5 ( - 12S.2. $2.4) 37.1 ( - 124.3. 82.4) 39.3 ( - 1 14.0. 82s) :j7.1 ( - 124.9. S2.4)
Other C'ovariates: Risk Ratio Kirider-
Sex SibIings Daycare garten
1.9 17 1.000
0SX4 s..5 1')
1 - 9 4 0.9-1:j 1 -729 0.707 1 .Gd8 7.397
1.064 0.692 1.042 s.4 1 1
0.6.54 S .543 1.724 1.013 0.713 1 . 5 0.984 7.442 1-64 1 0 .GY6 7.4 19
1.0:3 1 0.tj68 ~ . - ! f j = !
1.6.51 0.976 0.686 7-49 1
aroiiiid 70% (Table 5.4). Overall efficacy iras iiiediorre at brst . .\lodrls wit li
iiiiiiil>ers of pretlictors did ~rorse tliari riiore parsinioniotis iiiodrls. To no siirprisr.
the riiodel \vit11 tlir Iii~liest estiriiate OF I - E \ras :lie oiir iii ivliicli tlip va(-ririatioii
pr~l)ahilit!- iras a fiirirtioii of disrase statiis orily ( p = 0.24).
5.3.2 Age Group: 18+ Moiitlis
Tlitl oldrr a s r groiip i i i the stuciy. made iip of cliildreri oltler tliaii 1 T iiioritlis of agtl
Iiad a saniple size of 299. of wliirli onlx 16 werr varcinateci. Out of the riitire groiip.
1 16 werc rorifiriiied as cases. Tlir case and corit rol groiips ivrre siriiilar wit li respect
to age witli the irieari age for the riitire saiiiple bcirig 40.75 iiioiitlis. Roiiglily 10Y uf
t lie groiip liarl at teridrd daj-care wi t l i cases Iiaviiig a riiargirial edge iii t liis r - a teqr~ .
(10.1% \-S. 9.:14<%). The case groiip tiad far riiore iiiales tliari tlir roiitrol g o i i p ( ? i .S< i :
iiiales vs. 39.2%) but Iiad fewer sibliiigs (0.97 as opposecl to 1.38). 1.-iilikr tlita Ci - 17
Table 5.4: Est iniates of VE froni uiiniatctied aiia1~-sis using utiçoiidit ioiial logist i i -
iiiorit li ase grou p. far inore ctiildreii iii t tiis group a t tencled kindrrgartrri ' Iiiit r h t .
Estiniateri C'E ('3.5% C'orif. Interval)
119.1 (-42-6. 74-0) 32.3 (-MIA. 71 3 ) 39-1 (-42.1. 74.0) 28.2 (-69.4, 69.6) 29.S (-69.0. 10.8) :{'L.I (-61.4. 71.4) 20.6 (-$9.9. 66.8) 22.3 (-89.2. 6s. 1 ) 2S.2 (-69.6. 69.6) 29.1 (-69.2. 70.8) 29.7 (-G9.2. 70.S) 20.d (-89.lj. 66.9) 22.4 (49 .1 . iiil. 1) 22.2 (-89.6. 68.1 ) 29.7 (-69.3. 70-S) 22.2 (-89.5, 68. L )
C'a(-ci riat ed C'ases Cuii t rols Tot ët l
0 t h ( 'ovariates: Risk Ratio Kitidrr-
iritercept Ses Siblings Daycarr gartrri
0.248 0.290 0.663 0.242 1 .O23 0.247 1.01.5 0.247 1 .334 0.267 0.658 1.083 0.287 0.671 1.138 0.231 0.612 1 -323 0.249 0 .993 1 .O69 0.2.5 1 O . C M 1 .:1-1-1 0.247 1 .OS 1 1 . X W 7 6 1.0.5 1 1.179 2 7 . 5 U.670 1.0-lI 1.291 0.286 0.672 I . 144 1.327 0.2.50 0.9SS 1 .O72 1 34.5 0.213 0.669 1.181 1.29.5 0.669
Tlie siiiall nuinber of vacci iiees preseii ted a probleiii for t lie ma t clierl aiialysis Iwraiist~
of tlir i.iii~isuall>. loir iiiiiiiher of discordant pairs. T h e iiiiiriatclietl aria1~-sis yieltlrrl
favuiirahle resiilts. ( 'ont iiigeiic~. table atialysis pro\.iderl stat ist ically sigiiificiiiit rst i -
riiatrs of I , 'E. \ \ Ï t lioiit coiit rolling for con fouiiders t lie rffiracy Kas ~ s . S % ( = 0-027)-
('otitrolling for ses and t h e nuniber of siblings dici not ctiaiige t h e results. But. con-
trolling for d-carr attendance gave an iriiprowd estimate of 88.9% (p = 0 . 0 2 ) .
'The owners of the data were not rlear a b o u t the meaning OF .-kindergartcn" in Switzerland or the age at :vhicli a child became enrolled
Table 5.3: Estiniates of LVE(O R ) froiii iiniiiatclied 2 x 2 x k c-oritiiigeii(-J- t.al)lr aiialysis
Tahle 5.6: l-riivariate results for IS+ nioritli age group
P-vaiiir
0.011 0.022 0.023 0.06s
- -
Est iinateci VE (9.5% Conl. Interrai) 1 Controiliiig For
lleail Siiiiiber of Sibliiigs (s td . dei-.) / O ( O ) 1.3S ( O N ) / 1-22 (0.4W)
7S.S (5.0. 95.3) $0.1 (2O.S. 9.5.3) SS.9 (26.6. 9S.3) 72.4 ( - 10.1.93.1 )
-4s siispec ted. mat ctied anal>-sis usi iig conditional logist ic regressiori \vas 1111 i ii for-
Siblings Da>-care .-\tteridaiice
Ses
To ta1
( 1i=?99) 40 . ïS (17.3S) 26/271(9..59)
llean -4ge (std. dev.) Siiiiiber A t tmdirip; Daycare ( X )
riiativr dur to lack of convergence. The results are sliowii iii Table 5.7. Effi(-acy
Cases C'ont rols (n= 116) ( n = lS3)
-41.16 (16.56) 4O..S.3 f 17.91) 9/S9(10.1) l ï / l S 2 ( 9 . 3 4 )
estimates irere 100% for al1 iiiodels but corifidvrice intervals jvere iiitiriite itirliratirig
t liat t lie l i k t 4 liood iras rit lier i-rry Hat aiitI/or Iiad ail as>-iiipt"ti2 aluiig t lie .j,.,,, asis
Kesiilts u l t lit1 iiiicoiidit ioiial losist ic re;rrssioii parallellrtl t liosr of t lit2 ( u i t iiigrii{-?
table aiialxsis (Tahle 3.8). Overall nioctel fit \vas good for al1 coiiibiiiatiuiis uf prtl-
dictors. Vaccine efficacy was estiniated to lx around SO% wlieri D q c a r e was iiot
iiiclutled iii the iiioclel and juiiiped u p to 88% wlieri Daycare Iras inclutlerl. Efficacy
iras statistically significant for 10 out of the 16 niociels and iiiarginal for tlie rriiiaiiiing
were iiiore likelj. to take precautioriarj- rtieczçures agaitist iricreasecl esposiire.
100
Table 5.7: Est iniates of I.- E froiii niatcbed atialysis using coridit ioiial lo~ i s t ic r r r r s - sion. C'onvergericr was pour so the estiiiiates a r e iirirriiable.
Ot her C'ovsriates: Risk Ratio Sex Siblings Daycare Kindergarten
2.403 0.559
0.604 0.498
2 . 5 0..57'1 2.20.5 0. .367 2.3 1 fj 0.3'74
O..? 17 0.342 0.5.5 1 0.*546
0.3 1 -4 O. 169 2.060 O . 0.40 1 2.114 0.576 O.-tO 1 '1 -.- *)os 0.2S9 0. 1 .5(j
0..32 1 0.2 14 O . 1-59 '1.071 0..5:11 0.20ij O. 144
Estirtiattvl C'E ('35% C'onf. Iriterval)
7S.S (5.0. 9-33) s0.5 ( 1 1.4. 95.7) S0..5 ( 10.9. 9.5.7) S Ï . 5 ( 1.3. 98.4)
13.2 (-20.7. 94.1) 82.1 ( 16.S. 96.1 ) 88.6 (9.4. 98.6) - -
( 9 . 3 ( - 12..5. 94.6) 88.3 (5.6, 98.6)
14.5 (-16.0. 94.4) 86.7 (-4.4. 98.3) S9.4 ( 13.7. 98.7) 76.5 (-8.4. 94.9) 8s. 1 (5.6. 98..5) 87.2 (-3.1. 98.4) 8d.6 (7.1. 98.6)
Otlirr C'ovariates: Risk Ratio tïinder-
Intrrcept Ses Siblings Lhycart. garttw
0.083 0.07 1 1.4.53 0.109 O.SU9 O .O60 fi. 1-12 O -092 0. (54 3 O. O93 1.4'7-1 0.S10 0.043 1-832 6.13 1 0.079 1.480 0.63.5 0 .O74 O .SC0 5.940 O . 11-1 O S 3 6 0.69s 0 .O69 5-64 t 0.557 0.0.57 1 .S.59 0.8.5 1 .5.939 0.097 1.498 0.840 0.6SS 0.054 1.787 .5..564 03.59 0.076 0.920 .ï ..5 73 0.58.5 0.060 1.801 0.909 .5.506 O..5C3 1
Table -5.9: Table of \.acriiiatiori hy disrase statiis stratifiecl !>y (loirare at:rii<laiiw
Did not attend daycare p a c ç i n a t r d C',zses C'ontrols Total 1
1 Total 80 16.5 2-1.3 1
Did attend daycare
1 Vacciilat~ci r a s ~ s Con t rols TO ta1
1 Total 9 1 1 26
5.4 Other Efficacy Trials for PRP/PRP-D
5.4.1 Efficacy Studies of PRP-D vaccine
Tivo Iargr cliiiiïal trials of t lw rffira(-~- of PRP-D ivrrt8 i-arrirrl uiit i r i Fiiilaii(l niid i i i
5atii.e :\laska~i i-liildreii. The Firiiiisli trial slioirrd tlir i-acc-itia t u b r liiglil~. rffi(-aciuiis
(9-1%. 95% C'I: (S:I.!)S)) w hile i ti riat i ve Alaskari rliildreii. the efficacJ- iras iiiiicli Ion-ri.
( 1 % . 5 % 1 : - 3 . 7 ) ) The trial in Fitilancl erirollecl over 1 16.000 infants witli
5S.000 in the treat~iierit arni and 56.000 in the control arrii. Tliose in tlie treatiiit3rit
arni received 4 doses of PRP-D at 3. 4. 6 and 14- 18 nioiitlis of age. .\Ierril~rrs uf bot l i
groups rcceil-ed cliplit Iieria-tetaiius-pertussis ( DTP ) vacciric. Tlie Alaskail trial iras
sriialler i r i size - 1054 infants receiveci closes of PRP-D at 2. 4 anrici Ij iiioritl~s o f age
ivliile aiiot ber 1 O U receivrd a placebo.
SIaiij- reasoiis have breii giveii for t lie large differriice in est iiiiates (Sliapiro ( 1 !)!)-!) ).
Table 3.10: Results of PRP-D Effiracy trial for Fiririisli Iiifarits
Episodes of Hib Diseasr/ $O. of S u b j e ~ t s
Dose .\ge I uccin € e s Controls Ef icac~ . 95% (.'1 Post I or 2 2-6 4/.5S.000 S/56.000 43% -9.5 to S:VX Post 3 6- 1 4/.58.000 3'7/.56.000 S9%, 10 to 06% Post 4 1 4 0/5S.000 27/.36.000 100% $2 tu 100% Post Ariy 2-24 S/.!iS.OOO TZ/.X.000 S9% IS to 9.53. Source: ijarti e t ai. ( 1994)
Table 5.1 1: Rrsiilts of PRP-D Efficacy trial for Nat.ive .-\laskari Irifarits
Episocles of Hib Discase/ So. of Subjects
Dose I iz rciri €es Controis Efficaci- !15%.Cq1 P O S ~ I :{/ 105-k 41 1 04s .),r , I c Tc -233 ~ C J S X Z Post 2 2/99 1 :3/996 3.5% -2SS to Sc)% Post 3 1/915 12/SS3 43 %. -43 to 7S% Post An>- 12/1034 191 1 O-ld 3 7 3 -29 to 69% Source: ii'ard e t al. ( 199-4 )
Fur uiie. tlir epidertiio1og~- of Hill disrasr i i i t lie tn.0 popiilat ioris was iiot rli(-> s;iiiitx.
Firiiiish infants I~eiiefit ted froni riiariy .~lusuriess~ irilierrnt in incfiist. rial stirict ics - brt -
ter socioeconoiiiic coiitlitioris. srnaller faiiiilies. less cro~vdiiig. etc. T11~ rlfrrts roiilcf
Iw srrii i r i t lie iriciderire of Hib irifectioiis for iiifarits < 1 year old. For Fiiiiiis ti irifaiits.
t.lic iiicitleiice for t his age grotip \vas 571 100.000 per yrar. III iiativr Alaskaii iii farits.
t liis iiiiiiil)t.r \vas iiii.ii:Ii tiigtier - about 1 ï O Z / 100.000 prr year. Ot lier possil>lc rrasuiis
for t tie lo\vt\r rffic-acj. iiicliicle (Shapiro ( 1994)):
The close sclietlule was iiot as strictlj- adliert~l to as in Firilaricl.
0 Sinaller saniple size - only 95% of eligible cliildren in Alaska participatecl. Sta-
t ist ical power was considerably lower.
0 Differwt \-accirit. lots.
Table 5.12: Results of Various C'ase-coritrol Efficacy Trials for PRP-D i r i the (-1iitrc1
s t ates
Inimunization Rates S t u d ~ , Cascs (?ont rols Efficacj- 9.5%. (.'I llinriesota 1 ( 1 ) 3 O / 1-70 (20%) 96% 6.5 to !)Il(% Los Angeles 3 ( 3 j 46/1-79 (29% ) S4% 36 to Y(i% (-'T/Pittsl>iirgli 1/39 ( 3 % ) :39/136 (2.5%) S4% 'IS to 9'7%
.\Lassacliiisetts 4 ( S ) 1 3 ( 1 ) S R 45 to 98% Source: LL7ard r t al. (1994)
D e r a r e is uiiroiii~iioii for irifants < 1 >-car of age ici Finlaiid.
Se\.eral rase-corit roi stritiics of PRP- D vaccine wcre coridticted in L-arioiis Iwat iotis
t liroiiglioiit t lie ( -iiited States for older iiifarits ( IS-.?) riioiit lis ) aiirl t lie3 rrsiilts wtmX
qtiitr fa\.oiiral>lr. The results are sliowii in Table 5.12. Tlie stiitlirs c-uiidiictril i i l
Slassarliiisrtts. ('orinect iciit jPit tsburgli and .\Iiiiiirsota werr al1 rrasoiiahl!- siiiiilar.
111 al1 tliree. cases were niatcliecl wit li (iip to) 4 controis wlio werr of t lir saiiir agr
aiid froni t lie sanie area or residenct.. The ot lier two liad differerit clt.sigis. The Los
.-\rigtdcs ('u~irity st i idj - iisrd iip t u 13 cont rols. ohtaiiit4 t liroiigli raiiJoiii-di~it (lialiig.
\vIio \vere in tlir sarile agr <-ategor>-. The ('DC' stiitl~, was a riiiilti-criitr~ stiirl>- ic.lii(.li
spaiiiird foiir itiajur arras. C'out rols fur t liis st iidy w r r iiiat rlir<l \v i t lii ii agv groiip
and foiind from a list of arquaiiitarices or frorii rhildreii in tlir saiiir dqi -are $roiil)
( Laiigliiiii r - t cd. ( 1992)).
Orerail. tliere appears to br fair corisisteiicy atiiotip the 5 case-roiitrol t~stiiiintîs
aiitl spec~ilatiori over possible tliffermces bj- putting t tirir stutly designs iiiider tlir
microscope is probably futile in tliis case. Obviously. no tivo trials are ttir saiiir
ancl soriie degrce of noise will be iiievitablr. The differe~ires roultl I>r siinpl>. < I I I ( * to
sariipliiig \.ariatioii. Tlie iriiportarit point is sinipl? t liat t lie vaccirie lias i-oiisistent lj.
pro\-en itself effective in tlwse populations aiid tlie results are statistically sig~iificarit.
But. wtieii the results are scattered to such a clegree that efficacy estiii~atcs rarigr
froiii very good to dowiiriglit tosic. further investigation is necessary. Sucli was tlir
Stucly Site Tinie Period Protective 05% ClotificIenc-e Efficacy ( 'Tc) Iriter\-al
( 'eriters for Disease ( 'out rol Iiaiser Pernianerite. No. C'al. C'eriters for Disease ('ont rol ~iassacliiiset ts hliri~iesota llinnesota L A C'oiinty. C'aliforiiia Soiirre: h a r d ~t al. ( 1994 )
case wi t li PR P I~ul~-saccliaridr vaccine.
5.4.2 Efficacy Studies of PRP vaccine
P K P \-awiiir \vas lirriised for iisr II? the FDA in 19S.5. Two large cliriiral trials \v(-rtl
carriril uiit to drterriiinr its protective rfficacy. Tlie first. in Sort11 ('ardiita \vas
iiicoiicliisive diie to a lack of statistiral piver. The secotid large trial \vas i i i Fiillaiid.
For rliildrrii 1S to 71 iiiontlis of agr. the t+firac~- \vas foiirid to b r 110% (95% ('1: (%i.
Tlie cotitroversy contitmed as a wliole sleiv of case-control studies iri t lie 1-iiitecl States
reported efficacies as liigh as SS% to as low as -58% (see Table 5.13). Tlie restilts of a -"
trial iii .\linnesota (-;>a%) convincecl niost pliysicians in tiiat part of the- country tliat
tlie \.arcirie lvas tosic and practicall~. put its iise to ari end. Virtiiallj- ail stiidirs iised
age-iiiatclied coritrols wlio were froni tlic same area of resideiicr or sanie tlaj-rare i-lass.
T tir oiily cscrptiori was tlie La-\ st u d ~ . wliicli used randorii-digit rlialiiig ( L ü i i ~ l i l i i i r i
al. ( 1 ) ) . So patterns lirikitig the estiniates to stitdy design or coiitrol s(4rctioii
0% 1 0% 20% 3OoA 40% 50% 60°h 70°A 80°h 9o0h
True Efficacv
Lliir.11 sprciilnt ion lias ber11 cloiie O\-er t liis rrratir art of resiilts. Drsiqi issiit~s
asidr. P R P \-a(-ciii~ lias (wrisisteiit 1- s l iu~r~ i tu 1-w a lrss i#ica(-ioiis iawi i i tb t liai1 its
wiijiigatecl sui-ressors. PRP-D \-ai-cirie. For rsaitiple. iiicliiws liiglirr let-rls OF aritilmlj-
tliaii P R P for al1 agcs. I i i fart. P R P lias failetl to rlicit Ixmster rcyuiisrs i i i i-liililrivi
9- 1.5 mo~itlis old ( \ \ à rd E I al. ( 199-1)). Tlir reason for tlit- large variation i i i 1 - E for
these stuclies could be siriiply due tu the nature of I 'E statistic itself. Rrrall tlint
al1 variance formulas for L,*E (011 tlieir original srale) are fuiictioiis of tlieir iiirari.
Distribiitioiis for ivliicli tliis is t rue are typically higlily skewed aiid tlir variaiicr t r~ids
t,o iiirrrase \vit11 tlie iiiraii. .\Icasiires of relative risk ( R R ) are perfect esaiiiplrs of
statistics t.liat bellave in this iiianiier. C'onseqiiently. a statistic like 1t:liir-Ir is jiist
1 - KR. is \.erj. volatilr at loiv effiracies. If the triir 1-E is Iiigli (close to 1 ) t . h t ~ i
sariipl~ estiniatrs will be cliisteretl fairly tiglit.11 aroiiiid the triie \-ali.1~. 0 1 1 tlic otlirr
liarid. i f t lie t rue 1- E is lorver. t tieri saniple est iniates wi1l be niore scat t e r d l~waiise of
tlir inlierriit nat urr of t lie statistir to esacerbatr siniplr noise (ur saitipliiig variatiuii)
at loiver efficacies. N,artl t-t al. ( 1 N S ) lias denioristrated t iiis risiils siiiiulat.ions o f
rase-control stuclies. For efficacy levels of 5O%, . t lie resiilts rangeci fruiii - 1 I %' r O
$ 6 . For a 0% Ievel of efficary. tlierr riras rveii grrater variation - - 110% tu 60%. For
tlie ciirioiis. rve carried out a siiiall siiiiii1atioii iisiiig siriipir L>itioiiiial raiidoiii r-ariablrs
for fictitious coliorts of size 1000. Tlie result is iii Figure 5.1. Srtialler coliorts ri-oiild
lie csperterl to have tvider ranges.
Chapter 6
Discussion
('asc-coritrol stuclies for the evaliiatiori of vaccine efficacy a re L-ery ~isefiil htv-aiise
r liry are ret rospect ive. relatively inespensive. aiici are quick t o iiiiplrriiriit. Tliry are
particiilarly trrll stiitrd for diseases witli low iiicidmrr a n d for s t i i t l~ . i r i~ rspoaiirix
\-ariahlrs. .-\a a rrsiilt . i-asz-cont roi s t iidies allow 11s to s t iiciy esist iris 1-ai-i-iiiib lis<.
( \\*art1 t 1 a /+ ( i 9M ) ).
However. c-ase-cotitrol stitdies a re far frorii "l~tillrt proof" ir.lien it coriitas t u stiicl~'
(lrsigiis aritl. as a result. tliere are iiiaiiy potential ireak spots wliere [lias cari 1 x 3 iiitro-
rIiicecl. For starters. case-coiit rol st uclies tend to I>e based on siiialler satiiplcs. Ii i rluiiig
so. tlir reliabilit>- of t lie saiiiple i c i reflectirig t he vaccination protile uf t l i t* pupiilatioii
vunies iiito qiiest ion. ('ase-ioiitrol s t iidies operattD or1 t lie ass~iii ipt iuii t liat \act-iiie iisr
is t l i t . ont. Sac-tur t liat cliffererit iates cases frorii corit rols. This ~voril(l II(% tiiir i f va(-c-iritx
lise \vas rverilj. distri1)iited ii i t lie popiilatiori but siirli is iiut t lie vase. L;2(-ciiit. i i s t x
is ofteii iiiflwncecl bj- a wliole ni>-riad of socioecononiic factors spaiiriiiig froiii stat 11s
t o ani labi l i ty to differerices in pliysiriaii practice (\jeard e t al. ( 1958)). Bias can also
be iiitrodiiced by iniproper selection of cases and controls. Cases are more proiie tu
tktectioii hias ancl iiiay over-represeiit cases i r i t he population. C'oiitrols. oii the otlitxr
liaiid. r r i i c s t br srle(-tetl so that tliey are. iii ail respects except for disrase. siiiiilar
t o rasses. ;\ ri>- coiifoiirithg factors t hat affect vacciiiat ioii o r t lie aiiioiirit of rspusiirr
t o diseasr iiiust lw equivaleiit bet~veeti t tie t IV» groups. I-il fort iiiiatel~.. roiifoiiiirlrrs
wliicli are ei tlier tiiiknown or tiarcl t o quantify are nuinerom so takiiig iiieasiires tu
ensiire tliat t lie are acljustrd for is riot feasiblr.
The sliortcoiiiiiigs of case-coiit rol st litlies clisc~issrd i n t lie prrcrdirig para,qrapli art.
far froni esliaustive. 111 fart. one coiild coriie tip witli an alniost eririless list of potrritial
biasrs or rksigii tiaws. I t is the opiriiori Iiere tliat it nia' be iirrrssary tu rrtliiiik t h i x
tv- we evaluate t,lie effect iveness of vat(-iiirs riit i rely. Tratli t ioiial st ai ist ic-al iiirt.liorls
wurk well iii rircuiiistarices where tlie assu~iiptiuiis are satistitd hiit tlir assiiiiipt ion of
-iric!epencient and ident icallJ- tlist ri biited" does iiot apply i r i epitleiiiics uf irift~rt iuiis
(liseases. Infect i o u ~ cliseases propagat e t lirougli contact . .As a roiisrqilrrii-tx. iilos t
of wliat goes 011 i r i an epicleriiic is iiriobservable. Drsrribirig the ~~~~~~~~~rritass of il
rarcirir \>y a ratio of ttvo ( possilily siisprt-t ) rates basecl oi i fiilal-sizr data is soiiirtvliat,
iiriset t lirig.
For t liesr aiicl riiany ot lier reasons. tliere are riiari!. (-liallriigrs t liat iiiiist I)r i i i txt
i r i tlie area of epitleiiiic iiioclellirig aiid the atialysis of epicleiiiic (lata. .-\ priniar~. goid
sliuiild I)e to fiiitl niocl~ls tliat aclequately describe the CI>-iianiirs of epirlrriiir (1at.a. -4
g e n t (Im.1 of resenrcli is ciorie esploriiig detrrniinist ic riiodels. Firitliiig L W ~ S tu a h l i t
statistiml iiietliods for t l i r piirpose of fittiiig silch iiiotlels to ilata aiid assessiiig guwl-
iiess uf fit rvoiild Iw a big strp. Qiiite oftrii. Iiowe~w. tlir rioci-li~itaar iiatiiro of siii.li
riioclels r i d e s t tierii clifficiilt tu deal \vit 11. Tratlitiurial statistic-al nitD! liorls art* ~ ( 4 1
rq~iipprd to (leal wit li liiiear iiiodels but leai-e riiiicli to I)e (tesirrd tvlitw t liis is iiot
the rasc. I i i liglit of tliis. Ive iiiay also iieed to rr-rvaltiate tlir tj-pr of ~1at.a w r (-olliv.t
aiici the niaitrier in which Ive rollect it. hIore eriipiiasis shoiilti l)c platwi ori c-oiit a(-:
patttarris arid t tir riiisiiig beliaviour of specific groups. K i t li rrsptbct tu t liC drsigii of
\-a(-ciiie rfficary stuclies. al1 riieasiires iiiiist Ilr takeii to eiisiire tliat vacïiiiattd arirl
iiiit-arriiiated groiips are equally esposeci to disease. 111 ad(litiori. ive rir~rd to soiiit4ioti.
iiiodel or adjiist for t lie rffert of r-ontart pat terris and lict erogeiiei ty i n est iriiat irig rffi-
cacy. The future for epideriiir niodelling and t lie assessnieiit of \-acciriatiori prograins
iiiaj- be i ~ i spatial models. Popiilations are iiot Iioriiogerieous aiid f ritliiig iiiodrls tvliicli
dorunierit the spread of infections over space as well as tiriie itinj- lioltl tlir krj- I;r
iindrrstaticliiig epicleiiiics.
Appendix A
Derivat ion of Var [V E ( P ) ]
A.1 Derivation due to Becker (1982)
Recall t hat t lie systrni of rqiiat ions for t lie niart ingale epicleniic riiodtll i I I oiir t wo
popiilat ioti scenariu iras:
111 order to proccctl ive evaluate tlie integral \vithiri the rspectatiori on tiir right
side of Eqiiatiori ( A.1) ancl use it as an estiiiiate of its uwri iiwaii. S o . after soiiir
siriiplifi cati011 we get:
Brraiisr t lie two niart iiigales are ort liogonal t tirrr is rio co\.ariarirr trrrii. Rrbcall t liat
witlioiit the Iogaritliniic approsiniatioii the es t inia te of 0 was gi\-tw II>.:
F i r i a l i)y siibstitiitirig t lie iiitrgral esprrssiotis wir li t lie approsiiiiat iutis alrt.atlj-
( h i \ - r d . lettitig -4 R , = . - iR.(T-) . and replacirig 5) n-itii 1 - \'&(.;) \rr arrive at
A.2 Alternate Derivation of Var[VE( f l ) ]
T i r s 1 1 r i 1 - Brckrr iisirig tlit- riiarti~igale nirtliods (-ail hi. rrpro(lii(-tvl 1))-
iisitig t lit. iiiet liud of stat ist ira1 difleretitials or tlir dr l tn rr~rthod (Set-. Agrcsti ( I!NO) ).
111 the iiriivariate case. the \.ariante of q(.Y). wliere .Y is a ranrloin r-ariahie tvitli riitsati
Froni C'tiaptcr 2 Ive learned that
So. appl~~i i ig equatioti ( A . 14) git-es
u - 3 II)
E I U
Li . - II) aJ
rc. w
5: II
e,
CJ > 5 d
QJ
2
Appendix B
Standard Deviations of V E
Measures from Simulations
Table B. 1: Final estimates of standard deviations for various riieastires of I - E frcj~ti
sirniilatioiis vit 11 iitiifortiil~ ~iiixiiig popiilatioii (a( l*e ) rt .$,( ..) '). I l : Lrnky. t r i i i *
r - E = 90%.
Model: Leaky
Truc VE=YO% Fraction
po Vaccinatrd 6(.G) 4) fl(o^Rt. ) m i 0% .tr n(O^Kr*) (u%\I 1
0Ll00-l 30% 4.7.5&1..54 4 .5211 -67 5.61f 2.86 6.17f2.91 .5.1612.01 .i.97*2.t;7
0.0004 .SOS S.1Yf 10.67 7.46f 10.49 t3 .77I31 .69 17.36f49.08 1 . 7 1 2 lS.S2&52.:1.1
0.0004 70% 13.;-t138.92 1 2.07f 36-88 33.2lif l ' ~ . j . ï - ~ 6 29.01 f lu2.1.5 ?7..1~~*,54..5:$
0.0006 30% 2.12f 0.25 1.61f 0.3 1.85f 0.39 2.1 1 k 0 . 4 1 2.5 10.:39 I .tii&U..t.ï
0.0006 .50% "..5Yf 0.74 'L.41f 0.77 2.9 1 I 1 . 0 3 . 5 i f 1.3'1 2.ti2* 1.1.5 3.27k 1.2!)
0.0006 70% 4.19&2.0.5 4.12J-2-06 .5.9713.7? 8.0.5f 5.0s 5.99i.5.2 ?if 6.22
Table B.3: I7 i i i f orm .\lixirig popiilatioii. Slodel: Lraky. tr i ir I 'E = 70% (<i( r r
.$r, t r r ) 1
1 Mixiel: Lerky
Tnie VE=loC/c Tixtie-llatciied C'iutiulrit ive lricideiv-e Fraction
po Vaccinated r(.z) q ( . j ) O(ÛR~.) q(O^R.trl *T(ÔR~. fl((G.w 1
U.UO0.I 30% 7.2512.32 7.0tif '2.46 9.14f3.8 6 . 9 6 S. t i9 i4 . 19 9.43k5.3
Talrltl B.-!: i r i i o l t o . S I : .-Ill/Sotliiiig. t r u e \ ' E = 70%
1 l t i
Table B.5: Final esti~nates of stanciarcl deviatioris for various riieasures of I - E fi-urti
siitiiilatioris witti 11011-iiriifurrii riiisitig (a( ivr ) ~..i,(,,.., ). .\Iotlel: Leaky. t r w I - E = WX.
blodel: Leaky. n u e VE=90% Fraction
C'ac . Tirnr- Slatched (.'iirriiiIativt. Iriciderici~
20%/407r ('riide 1 . W f 0.05 1.15f 0.07 l .S6f 0.1 0.-I.Sf 0.0.5
Poolr(l 1 .SC3kU.05 O.%f 0.06 1 Xf 0.1 1 .:lS&O. L I O.:K!f 0.04 O..3.5f 0.07
:10%/GU% ('riide 1 .C2f 0.07 I .27fO. 1 1.59f 0.15 O.95kU. 12
Pooled l..ïrjfO.Oü 1.05f 0.09 1 . 4 1 1 1.731 0.17 O.Ï.L*O.l2 l.UTf O.l(j
40(7c/S0'7( ( 'rtidr 6 1 . 3 2.4Cf 1.38 3.6f 2-05 :!.48*4.O.5 Pooleil 2.2 1 . 1.9-lf 1.32 :I.O6f 2.14 :I .Wf l .ÏG 2 . 9 4 . i 1 3.4.5f -1.24
\.accine Distribution Moderately C'liistered
i'accine Dist ribiition Highly ( 'Iiistt.reti
20%/4O% C'rude 1 4 . 0 0.6.5~k0.0.5 0.ïXf 0.01 O.?* 0.0-
Pooled l.-l6f O. 1 0.47&0.0.5 0.73I0.08 0 . Ï l k O. l l O. l f 0.02 0.2-If 0.05
:JO%/GO% ('riide l . l2f 0.07 0.6 1f 0.05 0 . ï Ï f 0.07 U.32f 0.04
Poolecl 1.07f 0.08 O . : K k t 0.0.5 O.6:jf 0.07 0.73f O. 1 1 O. l Ï f 0.O-l 0.32+U.OÏ
4 0 5 /$O% ( 'rude 1.01if 0.70 0 . 7 5 f 0 3 1 1.12* 1.6 0.7 1 H . S 6
F'ooltd 6 1-03 O.R9f 0.89 0 . ï ï f 1-48 O . 1 U S O.5Sf O.Sl
Table B.6: Final estirnates of standard errors for \-arious Iiieasures of 1.-E fruni siiiiiila- tions witli non-uniforrii iiiisiiig IVE ) I.s ,( , . , ) ). .\Iodel: ;\II/Notliitig. t rut> I - E = !)05.
Pooleci 1.19kO.OS 0.Ï:If 0.1 O.Wf 0.13 1 . 0 1 O..jlf O. L I U.,S-!f0.1.5
-IO%/SO% ('rude 2 0 . 5 2 2.06f 0.54 2.Wf 0.S-l ?-;"li-+O.S.?
Pooled l.S-!f 0.41 1 . 2.4Yf 0.75 :I.O9f 0.1) Z.2lkO.h.5 ?.!J 1kl.l:i
C'accine Distribution llodrrately ('Iilstered
-O%/-IO% C'ri& 1. l S f 0.08 O.57f 0.05 0.63~k0.06 O.??* O.UZ
Pooled I.13f 0.08 0.4310.04 O..jgf 0.06 O.7lfO. 1 0.13jAl.02 U.32kO.Oti
3O%/h'O% C'rudt- O.%* 0.07 0.5-lf 0.06 O.6W 0.08 0.3Sf O.U.5
POOM 0.9f0.07 o.-l*~1.0.5 O..x*o.w O.W. IS o. 19*0.05 o . . i ~ . 1
O C'rude 0 1 O.tjÏ_tU. 16 O.99f 0.2 O.Il& O.22 Pooled 0.ZkkO. 1 0 .44~0.14 O.7lf 0.18 1.07f 0.28 O.- l : ! iO. 19 O.ÏZf 0.31
- - - - - - pp - -
C'acci~ie Distribut ion Higtili. ('liiçtered
Table B.:: Final estirriates of standard errors for varioils nieasures of I * E frurii sirri- iilatioiis witli non-iiiiiforrri riiisiiig ( u ( r e ) I ..;,(,,, ). c l : Leaky. triir I - E = 70%.
Model: Leaky. Trile VE=70% Fraction
i,-ac . Tinie- A1 atclietl ( 'iirriirlativt~ Iticitftwcr
\,'accine Evonly Distributeci
Lhccine Distribution S l o d c r a t ~ l y C'liistered
\,'accine Distribut ion II ighly C'liistered
O 0 C'rude 2.13fO.02 1 0 . 4 l.GÏfO.i:3 0.-16fO.U(i
Poolerl 2.OSfO.02 1 O . l 1.56~kO.14 1 . 1 . 1 7 U.37f0.07 0.5fO.OS
30C7c/tj0L7r C'ritclr 1 .S2f O.O:I 1.49f 0.1 1 l . X f 0. 15 0.6rifO.W
F'ooled l . 7 I f 0.02 1.2fjf 0.1 1 1.5k0.16 1..52kO.l9 O.(jf 0.15 U.tj2f O.09
40% /SOS 7r ('rude l .78f 0.023 I .64f 0.14 2.19=t0.2 1 1.2Y*O.ls
Pooleti 1 .4 Ï f0 .06 1.26fO.l.î 1.6YfO.Z:I 1.78fO.2.5 I.OSfO.3 1.01f0.19
Table B S : Final estiniates of standard errors for varioi.1~ nieasures of L'E frorti siriiirIa- t ions \vit h rion-uni forin niixiiig (a(r7e) i s,(,,) ). Slodel: r\ll/Sot liiiig. triitB I ' E = 70%.
ModeI: All/Nothilig. Triie VE=70% Fraction
L'ac . Time-~ Ia t chec l C'itriiiilative I n c i d r ~ t i c i ~
C'accine Evenly Distributed
Appendix C
Computational Details for V E
Estimators in Simulation Study
Fur tlir. piirposes of the siriiulation stiidy. the I * E aiici -4 R valiirs iii Tal>lt~s -1.14.12
aiid Figiires 4.1-4.20 werc coinputecl usirig t lie followirig for~iiiilar. Tlir strat.ititvl
vwsioris are tiot statecl esplicitly here hi i t werr c-alcirlated i t i açror(iaric.r \vit Il t I i v
i i i t ~ t liud uiit l i t i t ~ ( l iri Sert ioii 2.3 iisiiig reciprocal 1-ariarice as t lie ivriglit.
Rtv-d l tliiit at tiiiir 0. wtb starter1 w i t h va~-(-itlateci aiid i ) ( ~ 1111vacx.iiiattv1 i~ltli-
ri(liials i i i t lie pupialat ioii. Le t .\',.(t) (kiiote t lie total tiiiiiil)rr uf iiidii-idiials iv i t l i
i-a-ciriat ioii stat 11s 1. irlio riiterrd t tir inkctioiis statr 1,)- tiiiiv /. :\/W. let .r,.(l ) (Itwut ( %
the riurnber of iridivicliials ~ v i t l i vaccination statiis t* wlio enter the iiifectioris stntc i t i
t . l i r iiiterval ( t - 1.11. Tlirrefore
Also. detiiir . \*(t) = .Yo( t ) + .Yl ( t ) and s ( t ) = ~ ~ ( t ) + ~ ~ ( 1 ) .
So. the attack rate at r for vaccination group 1 . is giveii bj-
.Y,( r ) . - \Rv(7) = -. r* = 0.1.
I l ,
TIitmreforr. Imsetl on t liese qiiantit ies we can easily conipute
For cornpiit in% t be iiriiriat clietl ( iinpaired) odtls ratio untler (.'iirtiiilat i \-e- 1 1 1 ~ - i c l r r i t - t a
sariipliiig L - E ~ O R ~ - ) ( Ï ) was detinetl as
u-Iiwe C , ( r ) tlenotes the number of controls with t-acciiiation statiis I * wlio u.ertx
srlrcted frorii t tiose wlio were still siisceptible at tinir T . Siriiilarl~.. for tlie itiatc-lirtl
( pairecl) oclcls ratio. 1 - E ( O R.ir ) ( r ) was defineci as
i-asrs aiid iirivar(-iiiated corit rols. alid .Iloi ( r ) represriits t lie riiirtilwr of pairs wiisist iiig
of un\-acciiiatrd cases ancl vaccinated coritrols. ;\gain. coritrols w r t . selrctrtl fioiii
t liose wlio tverc still susceptible at tinie r .
I'iider Titiir-~iiatclietl sampliiig the almve forriiiilas va-r t l i t a sanie for t l i r uclcl::
susceptible at Ï only. Frorii t hese saniples. cadi case \vas paircrl i ~ p wit li a (-ont roi t u
c.It.teriiiiiie t lie qiiaiit ities r r l ) aritl rno1 ( 1 ) . The sartie forniiilar i v c w iistd as al)ovr
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