A D -7 66 780

-LOGISTIC ANALYSIS OF BRUCETON DATA

L.; D- Harnpton, a6t a!

NevvalI_ Or dn ance Labor~toryWhit. Oak, Maryland

23 July :1973

DISTRIBUTED BY:

Naticual Technical Idfermetle sericU. S. DEPARTMENT OF COMMERCE5285 Port Royal Road, Springfield Va. 22151

NOLTR 73-91

I LOGISTIC ANALYSIS OF BRUCETON DATA

It "

BYL.D. HamptonG.D. Blum

J.N. Ayres

23 JULY 1973 SEP logo

NAVAL ORDNANCE LALORATORY, WHITE OAK, SILVER SPRING, MARILANDA O L, ,

[. -,,,NICA

APPROVED FOR PUBLIC RELEASE;0 DISTRIBUTION UNLIMIT EDz

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DOCUMENT CONTROL DATA• R & Dl"l -,,', l fla I0 U1 or ri A ,J.uin• ,n l,.n . I. erlerrd .hrn he ,I r f l u ea-, I II rt I I.d"

]. ' . .*• (crp. . atdhorl R COfT CSCCjIRTI C LA IýS TIC C

"Naval Ordnance Laboratory tJNCLASSIFIED ED'White Oak, Siiver Spring, Maryland 20910 ....

LOGISTIC ANALISIS OF BRUCETON DATA I

L. D. Hampton, G. D. Blum, and J. N. Ayres C 0

5 1P -- 0.TC78,TO IL 40 OF 'AQES 6., P40 or PK[F

23 July 19738

Task NOL-443/NWL i *b ..... c T No NOLTE 73-91

CRD-033 234/O92-1/FO0B•08-11 i. . O C C RK KCmOKT NO(9) (Any o• tnr,. •-.b@e•g y be rvfamo a

Approved for public release; distribution unlimited.

SN-Ik.E A- NO E . PONSOeN-0 -L...TARY ACTlVI TVNaval Weapons LaboratoryDahlgren, Virginia 22448

Naval Ordnance Systems Cornand1, ,,,1i1C T washblatan, D.C. 20!60

detailed study of the Bruceton test is presented when the logistic distributionrather than the Gaussian is assumed. Included are: recapitulation of underlyingstatistical concepts; discussion of the strategy and technique for performinglogit Bruceton tests; details of logit Bruceton data analysts; and Monte Carlotests which show good agreement with theory for very large samples, but J

A •, considerable differences when ordinary sample sizes are used.

hI

D D I°F"oi.,.1473 ( ) UNCLASSIFIEDS/N 0IG 807 6GI /ecurity 0I,,T7ti6o8

Zvcuuity Classification

' 4 UNCVASCIRI% LINK A LINK 0 LINK C

MOLE MIT NOLEt ? ROLE 9 !

Bruceton

Logistic distribution

Go/no-go data analycsa

Monte Carlo

Sensitivity

Explosives

fllFORM I47 6hACK4 UNCLASSIFIEDDD NOVJ 473

(PAGE- 2) security Classificationt

NOLTR 73-9i

LOGISTIC ANALYSIS OF BRUCETON DATA

ByL. D. Hemptons G. D. 3lum, J. N. Ayres I

ABSTRACTs A detailed study of the Bruceton test is presented whenthe logistic distribution rather than the Gaussian is assumed. Includedare: recapitulation of underlying statistical concepts; discussion ofthe stretegy and technique for performing logit Bruceton tests; detailsof logit Bruceton data anlysis, derivation of asymptotic equations forthe logit Bruceton analysis; and Monte Carlo tests which show goodage•ement with theory for very large samples, but considerable differ-ences when ordinary sample sizes are used,

EXPLOSION DYNAMICS DIVISIONEXPLOSIONS RESEARCH DEPARTMENT

2.AVAL 01DNANC,-, LABORATORYq!IT-" OAK. SILVL1V• SPRING, MARP/LAND

i &

NOLTR 73-91 2. ._iy "97ý

LOGISTIC ANALYSIS OF BRUCETON DATA

Previous work has shown that there are numu ous occasions wheresensitivity data should be interpreted with the logistic rather thanthe normal (Gaussian) an the assumed distribution function. One ofthe tools for collecting and assessing sensitivity data is theBruceton test and analysis method. The analysis, as it wasoriginally developed, assumes the normal distribution.

This report contains a theoretical development and study of aBruceton analytical method assuming the logistic distributionfunction. Also, Monte Carlo techniques wore used to study theaccuracies of the estimating parameters and how the accuracies areaffected by sample size. Although the work was begun over sevenyears ago under the HERO (Hazards of Electromagnetic Radiation toOrdnance) Project, Task NOL-443/NWL and completed somewhat lateriunder ORDTASK ORD-033 234/092-l/F008-08-1l Problem 001, Reliabilityand Gap Sensitivity of Explosives, it was not published because twoof the authors left the Laboratory. The methods described are ofcontinuing utility however and warrant publication.

Since the Bruceton test method is widely used for collectingsensitivity data for predictions of safety and reliability, thiswork should be of interest to systems analysts as well as to thoseworking directly with explosives in the fields of research, design,and mannfacturing quality control.

ROBERT WILLIAMSON II

C. J. ARONSONBy direction

NOLTR 73-91

CONTENTS

page

I. INTRODUCTION .... . ........... . 1

1I. STATISTICAL BACKGROLND .. . . 2Critical Level . . . . . . .'. , . . . . .. . . . . 2Distribution 2....... .........Dosage vs Stimulua .............. 3Probability Density Function . . . . . . . . . . . 4Cumulative Prob~bility Function . . . . . . . . 4The Oauusi-n and Logistic Distributions . . . . . . . 6Proba)%ility Space . . . . .. . . . . 6Slope and Variance ............ . . 8

IXX, BkUCETON METHODYS FOR DATA COLLRCTION AND ANALYSIS • 9Strategy . . . . . . . . . . . . . . . . . 9initial Conditions fow Bruceton Test . . . . . . . . 11Th* Bruceton Analysis Method . . . . . . . . . . . . 13Xllustrative Computations, First Example . . . . . . 15X11ustretive Computations, Second Example . . . . . 20

IV. THEORETICAL STUDIES FOR T79 LARGE SAMPLE LOGITBRUCETON ....... . ... 27

V. MONTE CARLO STUDIE3 . . . . . . . . ......... 37Bias in the Estimate of Gamma . . .. . . . . 39Effect of Sample Size on Estimates ot n H ... . 39Effect of Starting Level.. ........... 41Choice of Step Size. ................. 46

-A PWEDIX A ... . ........... ........ . A-4

APPENDIX B ..................... ......................... .. B-I-- APPENDIX C . . . . . . . . . . . . . . . . . . . . . . . . . .C-1

TABLESTable Title Page

1 COMPARISON OF THE GAU5SSAN AND LOGISTIC

VARIOUS Y-AXTS PAR.AMETERS FOR THE GATIS5IANAND LOGISTIC PROBABIL.. Y SPACES. . . . .......... 8

3 ALGORITHM FOR COMPUTATrON OF t WITH f DEGREES

OF FREEDOM FOR SELECTED VALUES OF P ... ....... ... 14

4 TYPICAL CALCULATIONS OF ASYMPTOTIC VALUESFOR M, G &H ...................... 33

. ." --.

NOLTR 73-91

CONTENTS (continued)

TAPLES

Table Title Page

5 BIAS IN ESTIMATE OF GAMMA . . . . . . . . . . . . . 40

6 BIAS IN ESTIMATE OF GAMMA FOR SAMPLESIZE OF TWENTY ............ . . ... . . 40

7 EFFECT OF SAMPLE SIZE ON VALUE Or G . . . . . . . 42

8 EFFECT OF SAMPLE SIZE ON VALUE OF H . . . ... 43

9 EFFECT Or POOR STARTING POINT ON DETERMINATIONOF THE MuA 6 44

10 EFFECT OF P0OO STARTING POINT ON THE

DETERMINATION OF GAMMA ........ . . . 45

11 EFFECT OF POOR STARTING POINT ON VALUE OF H .... 45

A-i TABULAR VALUES OF THE CORRECTION FACTOR 6 . . ... A-6

A-2 TABULAR VALUES OF G ....... .......... .A-8

A-3 TABULAR VALUES OF H ...... . . .............. . A-9

A-4 COMPILATION OF EQUATIONS NEEDED TO COMPUTETHE LOGIT R38PONSE LEVEL AND CONFIDr.NCR LIMITS A-10

9-1 ASYMPTOTIC CONVERGENCE OF THE LOGISTICDISTRIBUTION FUNCT IO14 .. .. .. .. .. .. . .. B-10

C-1-i SIZE 10,POPULATION MEANt 19.5 .. .. .. .. .. .. . .. C-1

C-2 MONIE CARLO RESULTS, SAMPLE SIZE 10,POPULATION MEAN 20.0 ...................... C-2

C-3 MONTL CARLO RESULTS, SAMPLE SIZE 10,POPULATION MEAN 20.5 ................. . . . . . . C-3

C-4 MONTE CAPLO RESULTS, SAMPLE SIZE 10,POPULATION MEAN 25 ... ....... . ......... C-4

C-5 IONTE CARLO RESULTS, SA•PLE SIZE 20,POPULATION MEAN 19.5 . . . . .............. C-5

C-6 MONTE CARLO RESULTS, SAMPLE SIZE 20,POPULATION MEAN 20 ........... .......... C-6

iv

NOLTAR 73-91

CONTENTS (Continued)

TABLES

Table Title Pageo

C-7 MONTE CARLO RESULTS, SAMPLE SIZE 20,POPULATION MEAN 20.5. . . ... ..... ............... C-7

C-8 MONTE CARLO RESULTS, SAMPLE SIZE 20,POPULATION MAN 25. . ........ . .. . . .C-8

C-9 MONTE CARLO RESULTS SAMPLE SIZE 50)POPULATION MEAN 19. . . . . . . . . . . . . . . . C-9

C-10 MONTE CARLO R.-SULTS0 SAMPLE SIZE 50,POPULATION MEAN 20 . . . . . . . . . . . . . . . . . C-10

C-1l MONTE CARLO RESULTS, SAMPLE SIZE 50,POPULATICN MEAN 20.5 . . . . . . . . . . . . . . . . C-Il

C-12 MONTE CARLO RESULTS, SAMPLE SIZE 50,POPULATION MEAN 25 . . . . . . . . . . . . . . . . . C-12

C-13 4ONTE CARLO RESULTS, SAMPLE SIZE 100,POPULATZON MEAN 19.5 .... ......... ... C-13

C-14 MONTE CARLO RESULTS, SAMPLE SIZE 100,POPULATION MEAN 20. . . ............ .. C-14

C-15 MONTE CARLO RESULTS, SAMPLE SIZE 100,POPULATION MEAN 20.5 . . . . . . . . . . . . .. . C-15

C-16 MONTE CARLO RESULTS, SAMPLE SIZE 100,POPULATION MEAN 25 . . . . . . . . . . . . . . . . . C-16

C-17 MONTE CARLO RESULTS, SAMPLE SIZE 500,POPULATION MEAN 19.5 .......... ................. C-17

C-18 MONTE CARLO RESULTS, SAMPLE SIZE 500,POPULATION MEAN 20. ......... ....... C-18

C-19 MONTE CAkLO RESULTS, SAMPLE SIZE 500,POPULATION MEAN 20.5. .*.................... . . C-19

C-20 MONTE CARLO RESULTS, SAMPLE SIZE 500,POPULATION MEAN 25 . . . . . . . . . . . . . . . . C-20

V

NOLTR 73-91

CONTENTS (Continued)

ILLUSTRNT IONS

Figure Title Page

1 A PROBABILITY DENSITY FUNCTION (FREQUENCY

2 RESPONSE TO A PARTICULAR STIMULUS . . . . . . . ... 4

3 SIGHOID CUMULATIVE PLOT OF A LOGISTICDISTRIBUTION ..... ......... . ........ . .5

4 STRAIGHT LINE CUMULATIVE PLOT OF ALOGISTIC DISTRIBUTIONs . ....... ....... 5

5 TALLY OF TYPICAL RESULTS OF A BRUCETON TEST . . . . 10

6 TALLY SHEET FOR SECOND BRUCETON EXAKPLZ . . . . . . 21

7 M FOR LOGISTIC BRUCETON ..... ........... .. 34

8 G FOR LOGISTIC BRUCETON . . . . . . . . . . . . . . 35

9 H FOR LOGISTIC BRUCETON .............. 36

A-I FLOW CHART FOR COMPUTATION OF THELOGIT-BRUCETON PARAMETERS xgSm, AND Sg . . . . . . . A-2

A-2 VALUES OF 6 AS A FUNCTION OF D AND M1FOR M <0.60 TO BE USED IN EQUATIONE - M 0 .. . . 0.. . . . . 4.. 0.. 0.. . . . . . .a A-3

A-3 VALUES OF G AS A FUNCTION OF D AND E . . .. . .. . A-4

A-4 VALUES OF H AS A FUNCTION OF D AND E ........ A-5B-I ASYMPTOTIC CONVERGENCE OP THE VARIANCEVp

OF THE LOGISTIC DISTRIBUTION FUNCTION . . . . . . . . B-11

I,.

vi

_OLTR 73-91

I. INTRODUCTION

1. The statistical treatment of experimental results in which theresponse of an item to a stimulus is of a quantal (yed or no) natureis becoming an increasingly important problem in current research.The Bruceton test was designed to handle Aust such exL)eriments whenthe stimulus is continuously controllable *. Its sequential, "up-and-.down," method of testing is ideally suited to destructive testing(any situation in which each item can only be tested once) becausl ittends to be economical in the number of items consumed in testingThis economy has led to its being used extensivoly in explosivestechnology arnd, to some degree, in the field of experimental bi~ology/medicine.

S.!

2. Another recent development in statistical data analysis is theincreased use of the logistic lietribution'. The expression for thecumulative logistic probability is

x I 1

r(x) -cosn ' ' (9)

ill

This distribution hao been shown to be preferable to the normnl** in,aertain applications '.

j. Earlier work on tho Bruceton test has based the cnlculations on

the asbumption that the populetion from which the items under test aredrawn is normally distributed '2. Part II is a recapitulatiol of thestattietical concepts that we use in the ensuing presentations. Thosewith an Ictive familiarity with the field of statistics of attributescan skim or skip thie• section. Part III i& a discussion of trestrategy and technique for carrying out a Bruceton test and analyzingthe results. The dl,stimilarities between thi Gaussian-Bruceton andthe logistic-Bruceton analyleh -- e pointed out. Two numerical examplesare used to demonstrate the of a Bruceton analysis. The graphsand tables used in carrying ou a logistic Bruceton analysis arecollected into one place (Appendix A) along with a summary of appro-priate equations to aid in the carrying out of the analyses ofexperimental data. Part IV of this paper is a redevelopment of thecalculational procedure of the Bruceton teit using, as a basicassumption, the hypothesis that the population is logistically distri-buted. Part V is a report of an extensive serLes of MoXte Carlo testsof the logistic-Bruceton analysis, showing the effects of finitesample size and related factors on the theoretical assumptions.

• eferences are on page 4..

"*The comparable expression for the Gaussian distribution is

p(x) -exp .. Lx-u d27 (2)

7- T1

NOLTp 73-91

II. STATISTICAL BACKGROUND

Critical Level

4. Impli.cit in the following methodolcgy in the concept of "criticallevel". An explosive system will respond if the appropriatetriqgsring signal is large enough. There ii some signal amplitu~ebelow which the system will not respond but at which, or above, itwill respond. Thia signal amplitude id the critical level. A groupf explosive systems, even when made as nearly identical to each

other as humanly possible, caninot be expected to have identicalcritical levels. In general the critical levels wi.11 be groupedaround a central value, some being further away (above and below)than others. 1

Distribution,

5. Experience has shown that this grouping, or distribution, can berepresented by a bell-shaped plot, as in Fig I. Here p(x) representsthe relative probability of encountering a device whose criticallevel is precisely equal to the stimulus, x. If the curve is

LL

FIG. A PROBA 'L'TY D•.SI1 jNC:C FP!.JE CV r)."'

2

NOLTR 73-91

symmetrical, the median, mode, and m2-an* all fnll on the samestimulus level (p in Fig 1). The Greek letter w in defined as themean of the populations the population being all possible devsicesthat have been made, that exist, or that will be made.

Dosage vs Stimtlus

6. We distinguish between dosage and stimulus. The term 'dosage'is ieserved to designaue the magnitude of the physical parameterwhich the experimenter adjusts to apply the desired signal to theexplosive system. The stimulus, which is some Conformal functionof the dosage, serves to transform the probability density functionto some assumed distribution, which in the 2resent cas is the logistic.Without this transformation the statistical method to be used would inot be val:,~d. 1'

7. As an example of this differences BED's can be fired from acharged capacitor. The dosage might be the potential to which thecapacitor is charged, or the stored energy. But from the propertiesof the logistic p.d.f. (probability density function), namely thatthe curve approaches zero probability as x approaches the limits ofminus and plus infinity, we are confronted by a logical inconsistency. IThe idea of firing an EED with negative energy has no meaning. Bytaking the logarithm of the energy as the stir.ulua we force thelower asymptotic limit to'be zero energy. While there are manyother transforms which could be used this is one of the simplest andhas been used quite widely.

*The mean is defined as (Ex)/n. That is, if we have a group ofvalues, observations, or measurements (xI,xg,xs,etc) the mean would bethe total of the magnit'2des of the observations divided by the numberof cbservation3. The mode is the particular x (or interval about x)for which there is the greatest number of observations. The median isthe particular x which divides the group into upper and lower halves.Unless we are dealing with an absolutely symmetrical distribution wecannot expect these three measures of central tendency to be the same(especially for small groups). For instancte, in this group of fifteennumberas

6,6,7,7,7,7,7,8,9,9,9,10,12,15,16we find that the mean is 9, the mode is 7, and the median is 8. There-fore, the level of stimulus in a Go/No-Go test at which 50% responseis observed (or expected) is the median. To refer to this level asthe mean is correct only if the distribution is symmetrical about themedian.

3., , .

NOLTR 73-91

Probability Density Function

8. The probability density function is not particularly useful whenworking with explosives since we can never know the critical levelof each individual explosive unit. For instance, let us take brepresentative sample from the population of Fig 1. To each memberof the e ýmpla we apply a particular stimulus, x1 . Those embers whosecritical levels are equal to or less than x, will respond. The nuaberof members responding is represented in Fig 2 by area a. Theremaining members of the sample will not respond (area b).

NON

SG. 2 RESPONSE TO A PARTICULAP ýTI W LUS

But once the members of area a have responded they no longer exist.We, therefore, cannot find, for any member, whether or not it wouldhave responded at a lower level. Nor can we assume (without someindependent source of information) that the members which did notrespond were not changed as a result of being subjected to stimulus x1 .We can only assume that they may have been changed ant thexefore areno longer members of the population from which the original sample wasdr3wn.

Cumulative Probability Function

9. To compute area a, Fig 2, we need to integrate the p.d.f. from thenegative limit up to xi. A plot of the integral of the p.d.f. for allvalues of x ranging from x - -- to x - +m gives a cumulative curve,such as in Fig 3. From this curve we can deduce the proportion ofthe population that would respond if the entire population were sub-jected to a particular stimulus level.P The population proportion thatwould respond can be considered as either the expecied Tractionlresponse of a finite sample to a given stimulus or the expected proba-bility of response of a particular ite;n to the same stimulus.*DerLving the cumulative form by integration of the p.d.f. appliesstrictly to the Gaussian distribution since, as a m~ttrr of history,ýqujO~itij di3tribution was conceived and is only used in the

a orm.

i~l- !- .11l~l! !1 1 11 1"1I.... w4

NCLTR 73-91

I--100

Z 75-

""P (X) - 50%" .1000

x

50-

'• 25- NOT =1o00-0110

0 -. -25 NOi ITEJ: I . 1OO0'

950 1000 1050

FIG, 3 SiGMOID CUMULATIVE PLOT OF A LOG'STIC DISTRIBUTION

4.0 - 98.2

2.0 P(X) 50% -88.1 z

C) L(X) =00

1.0 .0 73.1 0.

>- 0.0 50. 0- /S-.1 .0 - 26.9

0IL 0

"-2.0 11.9 c-NOTE I 1000

~10-3,0 4.7

-4.0 I.8950 Co0 1050

FIG. 4 STRAIGHT LINE CUMULATIVE PLOT OF A LOGIST:C DISTRIBUTION

MOLTR 73-91

The Gaussian and Logistic Distributions

10. 'n Table 1 we compare the mathematical expressions for thep.d.f.'s and their integrals for the time-honored Gaussian (or normal)distribution and for the logintLc function that we will employ inthis paper.

Table 1

Comparison of the Gaussian and Logistic Functions

Gaussian p.d.f. Logistic p.d.f.

1 (4)Px W (3) p(x) = - (4)

j Gaussian~ Cumulative Form Logistic Cumulative Poi'm

Sex " l-exd"pWx L dx (2) p W) (1)

The evaluation of the Gaussian integral, in tabular form, isavailable in many collections of mathematical tables. Because theintegral. cannot be evaluated in closed form the use of the Gaussianfunction in high-speed computer programs depends upon either tabularlook-up (which is apt to be inefficient in computer storage space andprocessing timt) or else approximation expressions (which have to bedevised or found). on the other hand, the logistic cumulative andits transformations involve only logarithmic and exponential functionswhich are already available in most machine languages.

Probability Space

11. Even the cumulative curve shown in Fig 3 is not in its mostaseful form. The vertical axis can be transformed so that the 'ess"-shaped curve changes to a straight line, as in Fig 4. we shall usethe symbol, L(x), to indicate the logit of x ... the value of thetransformation of p(x). The logit transform can be expressed in anumber of equivalent forms:

= •. m .i,=.== • -=='* - m,= ,L"= = r • i": I ' l ... .•"|i i i i~ llll I l u i • ,- , . . . . •... ::6

NOLTR 73-91

L x) - (5a)

- W (5b)

- • (5c)

where $a is the mean and y is the vaziability parameter. By simple ialgebraic manipulations the preceding equ3tions can be sclved fcr

ptXx),

p exp (6)

p(x) - 1 (7

which can be differentiated, as in shown in Appendix B, to yield theprobability density function, Eq (4) or Eq (B-13). Even thoughBerksona expressed the logit transform a6 a 2onventional linearequation, L(x) - Ax+B, we prefer and will use the form of Eq's (5).The nature of the straight-line transformation can be better under-stood by making substitutions in the equations. If we Lmt p(x) - 0.5then the logit of x will be the natural logarithm of 1, or zero.If we let x be larger than u by one gamma-unit (this is cowparableto "one sigma above the mean" in the normal distribution) we wouldfind that

L(x)- - 1

As a consequence, p(x)/q(x) -exp(1) -2.71828. Hence, p(x) wust be0.73106.

12. A similar y-Lxis transformation cart be made to produce a straightline for the Gaussian cumulative function. The units of the verticalcoordinate usually have been designated normits, with zero normitsbeing made to correspond to a probability of 0.50. Finney, in hisprobit analysis, uses units of probits which are numerically 5.0greater than the normit.

7

NOLTR 73-91

Slope and Variance

13. The slope of the normal cumulative curve (in the normal proba-bility space) is 1/0 where a2 is the variance. The slope of thelogistic cumulative curve (in the logistic probability space) is1/Y. The expression for the variance of the logistic distributionis given by the following equation

(The derivation of this expression, as well as other moments of thep.d.f., is given in Appendix B.) Table 2 has been included to showthe relationships between the logistic and two Gaussian straight linecoordinate systems and their corresponding probabilities.

Table 2

Various Y-Axis Parameters for the Gaussian andLogistic Probability Spaces

Gaussian Probability Loglstic ProbabilityN P Cumulative L Cumulative

Normit Probit Probability Logit Probability

4 9 0.999683 4 0.982014

3 8 0.998650 3 0.952574

2 7 0.977250 2 0.880797

1 6 0.84134 1 0.7310590 5 0.500000 0 0.500000

-1 4 0.15866 -1 0.268941

-2 3 0.022750 -2 0.119203

-3 2 0.001350 -3 0.047426

-4 1 0.000317 -4 0.017986

i 8

)

NOLTR 7391

III. BRUCETON METHODS FOR DATA COLLECTION AND ANALYSIS

14. The Bruceton plans for collecting and analyzing data &redescribed in the literature'12 . However, at least some of thesesources mpy not be readily available. We have decieed to includean exposition of the procedures so that this report can be self-sufficient. Some of the notation has been changed to exploit theadvantages of the logistic distribution.

StrategyL

15. The goal of the test method is to concentrate the trials at afew levels centered about i, the stimulus which will cause 50%repponse. The units are tested one at a time, the test level beingiiselected in a manner which is most apt to cause a response oppositeto the one just observed. The Bruceton data collection plan iscarried out in the following ways

a. Select a number of equally spaced stimulus levels x1, xg,x%) .x where the lowest and highest are aboutequidistant fpom the expected R (50% response level.) Let drepresent the step size (the difference in magnitude ofstimulus between any two adjacent levels).

b. Choose a level (closest to the expected 50% point) and testthe first item at this level.

c. Test all succeeding items, one at a time, by repeated 4applications of the following process:

(1) Note the level at which the last item was tested and

note the behavior of this item;

(2) If the item responded, test the next one at the nextlower level; or

(3) If the item did not respond, test the next one at thenext higher level.

(4) Record the result as a response or non-response at theappropriate level.

16. Recalling that stimulus is defined in such a way that we expecta greater probability of response with an increased (higher level of)stimulus we see that the above procedure concentrates the testing aroundthe 50% response point. If, for instance, we happen to be far abovex, the probability of response will be quite close to 1.0 which meansthat (by application of rule c.(2) the succeeding test level willprobably be closer to x. similarly, a trial far below x would beexpected to cause the succeeding level to be higher thon the pre-ceding test level. Use of the Bruceton method causes a hoveringaround i leading to a zig-zag pattern when the results are recordedsequentially on a tally sheet such as in Fig 5.

mC

NOLTR 73-91

1400 XTI7 7 'H• 1 3.80 _x - -

3.60 x x .1 .l X3.40 0. ,x x 0 ... -. . .

,, 3.20 1 10 0_ _ _ ... .• - . i i

TRIAL NO, 1 2 3j4 5 6 7 1 19 10 111 1211314115 16 17118119 20212jZ

- 4.00 - TS3.800 xS 3.60 00 10 1 x x x x xf 0X oX

P 3.4 - - 0 o_ j

c 3,20 x Q43.00 __0 3 73 94[124

TRIAL NO, 23 2425i2627 28 29 30131 32 33 34 35 3613714199 40t41 4243

NOTE: X SIGNIFIES A RESPONSE0 SIGNIFIESA NON-RESPONSE

FIG. 5 TALLY OF TYPICAL RESULTS OF A BRUCETON TEST

10

NOLTR 73-91

Initial Conditions for Bruceton Test

A

17. Three ... '-' must be made before a Bruceton test can beperfcrined t

a. the proper dosage-sti:nulus transform (paragraph 6);

b. the starting level; and

c. the step size.

18. In order to use the Bruceton method for analyzing the data, thetest levels must be set at equal spacinqs of increasinq intensityof stimulus. Under actual testing conditions we sometimes find(particularly when the step size is small compared to the mean)that there is no practical difference between liri.ar and locarithmicsteps. In such a case no transform is necebsary. In some caserlogarithmic steps would complicate the experiment, as for instance,adjusting the drop height on a stab primer test machine which has aratchet-detent mechanism with stops at quarter-inch intervals. Insuch cases, it sY,,uld be possible to choose a set of logarithmicallyspaced numerical values which will have a minimum deviation fromthe experimental levels. This will introduce an error which will haveto be lived with.

19. For best efficiency the starting level should be close to x. Dataobtained previous to the first alternation of responses and non-responses should not be used in the data analysis. For example, inFig 5 trials 4 and 5 are the first alternation (reversal), and, as aconsequence, trials 1, 2, and 3 must be di.1carded. The early encounter Iof the first reversal is made more likely by starting the test close

to the 50% point. Also, as has been brought out in a previous study*,a starting level some distance away from the mean tends to bias theestimate of the mean for small-sample Brucetons.I

20. The choice of d, the step size, depends upon what kind ofinformation is needed and often entails a compromise. If a closeestimate of the mean is desired without much accuracy in the estimateof the standard deviation, then d might be 1/4 to 1/2 of 1 (if aGaussian distribution is assumed) or 1/2 to 1 of Y (for a logisticdistribution). A better estimate of the standard deviation with acorresponding sacrifice of an accurate estimate of the mean wouldrequire the use of d about equal to 2C (or 4Y). When one desiresestimates of both parameters then the best compromise seems to beto set d about equal to a (or 2y). As a passing note we caution thata single run of six or more steps with obviously different 50% points,before and after the run, may indicate that the experiment is out ofcontrol.

21. If the step size is very large compared to the variabilityparameter, an alternating two-level pattern may be obtained. Thiswill happen when one level is far below x and the next higher levelis far above R. The probability of observing any responses at thelower level or any non-responses at the higher level is very small.

kNOLTR 73-91

On the other hand, a three-level pattern may evolve when the middlelevel is close enough to R that a mixed response can be expected, andthe other levels are so far removed from R that the probability ofobserving all responses on one and all non-responses on the other isvery large.

22. In either of these cases (where the step size is large) we candeduce that we have found the limits within which R i located. Wealso may ba able to guess at an upper limit for the size of thevariability parameter. But these estimates and guesses may' not bevery informative unless the size of the step is small enough,physically) compared to the desired accuracy.

23. Confidence limits can be assigned to a stimulus, Xp, associatedwith an expected response of p as follows. (A numerical example isgiven in steps 9 and 10 of the first illustrative example, paragraph26.) The quantities mm and s are calculated in the first part ofthe analysis and from these ws derive Sp using the variance* equations

s2 - a' + L(x) . (9)

Having obtained mv we can then write the lower and upper confidencelimits for sp as

L = xp - top (10a)

U - Xp + tsP (10b)

where the t is the Student's t with the assumed confidence and withthe proper number of degrees of freedom. For 95% two-sided confidencelimits we should enter a t table with P - 97.5% since this gives us2.5% outside each side of the stated limits. The authorsof ReferenceI assumed a very large sample, therefore they used t with an infinitenumber of degrees of freedom. For small samples which are actuallyused in experimental work it would be better to use the value of twith the number of degrees of freedom equal to the number of firesor fails observed (whichever was used in the computations). Forhigh-speed computer use we can approximate the values of t quite wellby the algorithm

t w R + (11

Sand , are the variabilitiefi of the estimate of Y, of the

mehn, and of aMy particular per cent response point respectively.The variance equation referred to can be found on page 6 of Ref. 6.

12]

NOLTR 73-91

where f is the number of degrees of freedom and the values of R, S,and V depend upon the desired value of p. *rhe following are examples:

P R S V

90 1.282 1.2228 1.1345

95 1.645 0.6602 0.627997.5 1.960 0.4210 0.4774

The resulting values agree in general with the tabulated values oft when f_? 6 for P - 97.51 for f2 7 for P a 951 and for f > 9 forP - 90. Table 3 gives t values for five values of P and compares theresults of using the algorithm with published tabular values. Notethat the constant R is actually t. We emphasize that the tabularvalues are the correct values and that this algorithm is an approxi-mation designed for computer use.

The Bruceton Analysis Method

24. The objective of any Bruceton experiment is to obtain R and g(the estimates of the population parameters w and Y) and statisticalparameters which can be used to give the probable error of theseestimates. Note that the Greek letters " and Y are reserved for thepopulation parameters whose values are, in real life, not availableto the experimenter. The computational procedure is quite simple,starting with a tabulation of the usable fires and fails observed* atthe various test levels. As an example, the data of Fig 5 would betabulated as follows:

Stimulus Number of Responses Number of Non-__X nxai Responses noi

4.00 1 03.80 2 13.60 9 2

3.40 7 10

3.20 1 7

3.00 0 1S-no -21

25. For the analysis a choice is made between responses or non-responses, whichever has the lesser total quantity. In the example theresponses therefore are chosen. Each test level is assigned an index*In this report the number of fires at a particular level (the i-thlevel) will be represented by a double subscripts nx ii and the numberof fails, similarly: noi' J

-M

NOLTR 73-91

TAiLE 3

ALGORITHM FOR COMPUTATION OF t WITH f DEGRE!•S OFFREEDOM FOR SELECTED VALUES OF P

Equations t-R+l/ (sf-V)

P:90t PU95 P097.5 P-99 P=99.5%S1.0703 .645 1890 20326 2.576

1 1.2228 0.6602 0.4210 0.26687 0.20276V 1.1345 0.6279 0.4774 0.37980 0.33618

f Calc Tab Calc Tab Calc Tab Calc Tab Calc Tab I5 1.4828 1.476 2.U190 2.015 2.5744 2.571 3.3736 3.365 4.0518 4.0326 1.4432 1.440 1.9450 1.943 2.4481 2.447 3.1447 3.143 3.7119 3.7077 1.4166 1.415 1.8954 1.895 2.364', 2.365 2.3979 2.998 3.4992 3.4998 1.3976 1.397 1.8598 1.860 2.3059 2.306 2.8958 2.896 3.3537 3.3559 1.3833 1.383 1.8331 1.833 2.2619 2.262 2.!!206 2.821 3.2477 3.250

10 1.3721 1.312 1.8123 1.812 2.2279 2.228 2.7629 2.764 3.1672 3.16915 1.3401 1.341 1.7528 1.753 2.1313 2.131 2.6020 2.602 2.9457 2.94720 1.3248 1.325 1.7245 1.725 2.0859 2.08b 2.5277 2.528 2.8449 2.84525 1.3159 1.316 1.7079 1.709 2.0595 2.060 2.4849 2.485 2.7873 2.78730 1.3101 1.310 1.6971 1.697 2.0422 2.042 2.4571 2.457 2.7500 2.75040 1.3029 1.303 1.6837 1.684 2.0211 2.021 2.4231 2.423 2.7046 2.70450 1.2986 1.298 1.(758 1.616 2.0086 2.009 2.4031 2.403 2.6780 2.67860 1.2958 1.296 1.0706 1.671 2.0003 2.000 2.3900 2.390,7.660• 2.66080 1.2923 1.292 1.6641 1.664 1.9901 1.990 2.3737 2.374 2.6390 2.63900 1.2902 1.290 1.6602 1.660 1.9840 1.984 2.3640 2.365 2.6262 2.62600 1.2861 1.286 1.6526 1.653 1.9719 1.972 2.3449 2.345 2.6009 2.60100 1.2836 1.283 1.6480 1.648 1.9647 1.965 2.3335 2.334 2.5859 2.586

1.2820 1.282 1.6450 1.645 1.9600 1.960 2.3260 2.326 2.5760 2.576

Note: This is a single-s'.ded t table.

I

NOLTR 73-91

number, i, which increases in integral steps per level with increasingintensity of stimulus. Customarily the zero-th level (i - 0) isassigned to the lowest level at which tests were carried out, but itcan be put wherever the experimenter wishes. By setting i - 0 at thelevel at (or near) which the largest number of observations was made,the sizes of the numbers are reduced, often making the computationspossible by inspection or simple mental arithmetic. The two methodswill be carried out side by side to facilitate comparison. Thederivations of the equations and detailed explanations of suchvariables as M, D, E, 6, G, and H will be given in Part IV of thisreport.

Illustrative Computations, First Example

26. Step .. Assignment of indices, xo and d.

d - xl-x 0 (12)

*.. d w 0.20

.x i nx Xi x i n C. i

4.00 5 1 4.00 2 1

3.80 4 2 3.80 1 2

3.60 3 9 xo" 3.60 0 9

3.40 2 7 3.40 -l 7

3.20 1 1 3.20 -2 1xO" 3.00 0 0 3.00 -3 0

Step 2. Computations of N, A, and B.

_ n__X_ i inxi 0. _ _i _i _. __ in i ion xi

5 1 5 25 2 1 2 44 2 8 32 1 2 2 23 9 27 81 0 9 0 0

2 7 14 28 -1 7 -7 7

1 1 1 1 -2 1 -2 4

0 0 0 0 N-20 A- -5 B-17-- 20 A-55 BS=167

NOLTR 73-91

(26. Illumtrative Computations, First" Example, Continued.)

Step 3. Computation of g, the stimulusi for 50% response.

If N E Enxi (for the fires) then x - x- . (13a)if N "r (for the fails) then R - xo+d(+ + . (13b)

using the firest

: :3.:0+0.2 - :1 3.6:+o.2 0 - [2S-3.45 ff=3.45

Note that x is the same for both setsof indices.

Step 4. Computation of 14.

2- - (A -, (14)

3 a167 - (55- 17 -

X - 8.35 - 7.5625 M - 0.85 - 0.0625

M - 0.7875 M - 0.7875

Note again that the difference in indices didnot che 7s the final answer. .?

-f

NOLTR 73-91

(26. Illustrative Computations, First Example, Continued.)

Step 5. Determination of D.

D - (15)

where x° is the test level closest tO X.

D = 13.40-3.451 0.250.20

The value of D will have the limits 0.0 . D 1 0.5 andwill not be needed if M is greater than 0.65, which isthe case in the present example.

Step 6. Determination of E.

E - M, for the limits of 0.65 < X < 10, (16a)

E - M + 6, when M S 0.65 . (16b)

The correction term 6 is a function of both X and Dand can be found either from Fig A-2 or in Table A-1.For M greater than 10 the value of E begins again to

depend upon both M and D. However, when M approaches10 it is known that a much too small step size isbeing used. Since the results of such a test areapt to be very misleading we have stopped the tablesat this point.

17

-. Ohl

NOLTR 73-91

(26. Illustrative Computations, First Example, Continued.)

Step 7. Computation of g, am' and a ,

g E.d , (17)

a (m.g)//•, and (18)

* (H.g•)/'¶ , (19)

where 0 and H are functions of E alone, when E > 0.65.When E < 0.65, a and H are functions of both E and L.The values of G and H are given in Tables A-2 and A-3and plotted in Figures A-3 and A-4 in Appendix A.

For the numerical exampleo

g n (0.7875) (0.2) - 0.1575

a a (1.630)(0.1575)/(4.472) - 0.05741

• - (1.626)(0.1575)/(4.472) - 0.05726

Step 8. Computation of stimulus, Xp, at a specifiedprobability, p.

Eq. (5), L(x) - n - can be rewritten in

terms of p and the estimates of the population parameters

and solved for Xp:

X -XIn -i (20)

p"xp + g. -I+g- x

For example, suppose we wisi' to estimate the 99% point:= ~99

x - 3.45 + 0.15751(n 99Np T1

- 3.45 + 0.1575(4.5951)

- 4.174

Note that L(xp) = 4.5951.p

NOLTR 73-91

(26. Illustrative Computations, First Example, Continued.)

Step 9. Computation of a

The variance associated with the estimated stimulus, xp.is given by applying Eq. (9)o

&a a + L(x) *•p m L p. g

The term, L(x ), has been evaluated in Stop 8. Thestandard deviftions of the moan and of g have been,determined in Step 7. (Note that when p - 0.50,L(Xp) - 0 and s,.go properly reduces to s ) Follod-ing along with the illustration we find tNat

p- /(0.05741), + (4.595l),(0.05726)- 0. 2 6 9 3 .

Step 10. Two-sided confidence interval about x p'

If, for instance, we wish to find the two-sided 95%coifidence interval about the 99% point we would use(LU), 5 - (Xp - tg*.5 sp, Xp + t,*.% Cp) (10)where xp is the 99% point and t is the 97.5% Studmnt'svalue, with f degrees of freedom where f-N-20. Thevalue of t (from Table 3) is 2.08591 and Xp, fromStep 8 above, is 4.174. Hence,

" s - 4.174 - (2.086) (0.2693) - 3.61

Us - 4.174 4 (2.086)(0.2693) - 4.74

step 11. Singlc-sided limit on x .

If, instead of an error band or tolerance intervalabout x , we wish to find only an upper limit (or onlya lower limit) for the estimate of a particul3r responselevel we would use either

bib _ - ' • -ii...

MNCLTR 73-91

(U).. p + ttasp, or (22a)

(L)x. p - tisp a (22b)

Thus for the example we would use a value of t(with 20 degrees of freedom since N - 20) of1.725 which would give

"') (U)oe - 4.174 + (1.725) (0.2693)

-"4.64

SNotico that t%12 sin~le-sided upper 95% limit -- 4.64 Is!l

closer to the value of x.- 4.174 - than the uppertw-•-ided 95% limit -4 This is as it should be.

Illustrative Computations, Second Example

27. A second example is included to show the procedure when M happensto come out les than 0.65. The data are shown in Fig 6.

Step 1. Assignment of indices, xO and d.

X i n.,!

3.80 2 0 Note: Use "fails" since there3.60 1 2 are fewer "fails" than "fires".

x0 3.40 0 13 Xo - 3.40t d - 0.20.3.20 -1 1

Step 2. Computation of N, A, and B.

2 0 0 01 2 2 20 13 0 0-1-. -1 +1

20

NOLTR 73-91

3.60 x x x _x 0x x IxI_ 3.40 0 0 0 o o x o

3.20 - - -o

TRIALNO i 8 9N,0 1 ,4 35 45 ,617

3.80 ! 1 V ! x_ 3,60 x x x I x o, x x xz 3.40 0 0 0 0 1 0 0*- 3.20

TRIAL NO, Is19 2021 2212324 25262728129130 I32[33

FIG. 6 TALLY SHEET FOR SECOND BRUCETON EXAMPLE

21

NOLTR 73-91

(27. Illustrative Coraputationa, Second Example, Continued.)

Step 3. Computation of R.

R - . + (A+ 1(13b)

- 3.40 + 0.2( 1 + 1>

- 3.5125.

Step 4. Computation of M.

a4 9 -1 WNA (14)

- (3)(16)-l

- 0.1836.

Step 5. Determination of D.

D X -" Id

1 3.60-3.512510.2

- 0.4275.

Step 6. Determination of E

From the graph, Fig A-2, or by double interpolation

from Table A-1, we find that 8-0.0409 and E-0.225.

22

VOLTR 73-91

(27. Illustrative Computations, Second Example, Continued.)

Tabular 6 Values

D0.1800 0.1850

0.4000 0.0399 0.03840.5000 0.0456 0.0440

we show the actual process of interpolation:

D0.1800 0.2836 0.1850

0.4000 0.0580 0.0388" 0.03840.4325 0.0409**0.5000 0.0456 0.0444* 0.0440

*First Interpolation"*Second Interpolation

Step 7. Computation of g, amp Ag.

g - Ed (17)

- (0.225) (0.2)

- 0.0450

From the graph, Pig A-3, or by double interpolationfrom Table A-2, we find that G - 2.4930.

NOLTR 73-91

(27. Illustrative Computations, Second Example, Continued.)

Tabular G Values

S0.2000 0.2500

0.4000 2.666 2.2650.5000 2.769 2.306

(2.4930) (0.0450; (18)

16

" 0.028

Similarly from Figure A-4 or Table A-3, H - 1.146:

Tabular H Values

0.2000 0.2500

0.4000 1.141 1.1740.5000 1.105 1.140

S(1.1458) (0.0449)

" 0.0129

24

SGzTq 73-91

(27. ZLlustrati,*,e Computations, Second EXample, Continued.)

step 8. Cowputmtion of xp.

Tske p, lot us say, as 0.95 (the 95% point)s

X p - + g.,•.• r4(21)

- 3.5125 + 0.0449 (In 19)

" 3.6447

step 9. Computation of ap.

- (oo)'+ Lj )], -(oo)'

1/0 2 + 2 4) (0.0129)9

" 0.0472

25

NOLTR 73-91

(27. Illustrative Computations, Second Example, Continued.)

Ste 10. Two-sided confidence interval about the 95% point.Take the confidence level to be 90%. The student's t valuewill be at a 95% level for f - N - 16

* •t - 1.645 + (0.6602) (161 - 0.629()

"- 1.7457

L - 3.5125 -(1.7457) (0.0472)30

- 3.43

U a 3.5125 + (1.7457) (0.0472)to

a 3.60

Step 11. Single-aided lower limit on the 95% point at j90% confilence

We find thats

t -1.282 +1 (1.2228) (16) - 1.1345

- 1.3363

The lower limit, then Lut

L - 3.5125 -(1,3363) (0.0472)00

"n 3.45

2

26

1

.------..---- '----

NOLTR 73-91

IV. THEORETICAL STUDIES FOR THE LARGE SAMPLE LOGIT BRUCETON

28. In Section IIZ procedures, tables, and graphs were given toestimate the statistical parameters *, g, am, and s from Brucetondata on the assumption of a logistically distributes population.The tabular (and graphed) values of 8, G, and H are asymptoticvaluesy that is, they were computed using large-sample statisticaltheory and taking into account the properties of the Brucetondata collection method. The approach is patterned after theoriginal work done by the Applied Mathematics Panel assuming theGaussian distribution (references I and 2).

29. The computations of the 50% point should be the same for eitherdistribution, since the assumption that the distributions aresymmetrical about the mean is valid for both. Consequently, theBruceton estimate of the mean isi

(M Einx± _)d +X 0 (23.)mB (-r•, i-

or

(T i n,,1 ±mB Eno + 1 d + xi (23b)

These are equivalent to equations 13a and b of the previoussection a

R XK0 + d A (for the fires), or (13a)2"

9 X0 +d(A 1 (for the fails) (13b)£~ ~~ I -X+d -- + - ,

30. The intermediate parameter, M, for estimatir.q variabilityporataeters was defined in the save way for tho lo.(stic Bruceton axfor the Gaussian:

M (24)

4

Where, in the asymptotic cast, either nax,i or nxo car, be usedfor n-.

27

NOLTR 73-91

This is equivalent to equation 14 in th. previous section.

S-B ----- (14)

The value of q (the estimate of V) would be found by app~.yingequation 16 followed by equation 17:

X = + 6 (16)

where 8 0 when 0.65 4M<10i and

g n Esd. (17)

31. The variance of R, (Vp), and g, (V ), can be estimatedusing the method of maximum likelihood'. The likelihood function is

N NP- nr Pil qj (25)1 1

&nd its logarithm N N

L *~in Pi + F, n qi (26)

where Nx is the number of responses at all levels, and N is the0

number of non-responses at all levels. Maximum likelihood theorygives the asymptotic variance of R and g as

- 1V an6 (27)

28

NOLTR 73-91

V g , 1 (28)

The symbol e is used to indicate the expected values of thederivatives. The expeuted values for these derivatives can be evaluatedto give

V - - and (29)rPliqi

V 7YVg " --- '-- , where yi is defined as (30)

g iYi xi-uYj - and

where the summation is over all trials at a level as well as overall levels of the Bruceton test. Since for the usual Brucetonnotation the summation over i moans a summation only over the levels,the above equations must be rewritten for consistency: doing thisand solving for the standard deviation gives

a * and (31)m 1" (n ,i + no,i)piqi

S - -. where (32)g g rnxi+ nl~~~

am ahd sR are used interchangeably.

As with the case in which the normal distribution is assumed, weshould be able to express the values of sm and s aSb

m n. and (33)

X (34) 1

Here N is the sinaller of the two values N and N , but in theasymptotic case N o so that either o~e can ge used. Solving

29

NOLTR 73-91

(31) and (33) for 0, and solving (32) and (34) for H, gives

3 N and (35)

H, * (36)~En i + p~pqy

The parameters 0 and H would then be used in equations 18 and 19 tocompute s and .9"

5m - (0.)/1 n i)and

Sg - (Sg)l./" . (19)g

32. We derived numerical values for the correction terms 6, G, and Hby generating expected very-large-sample Bruceton results for varioescombinations of step mine and spacings of the mean from the closesttest level. By expressing the step size in units of the populaticay, and the spacin of the mean in units of step size, we geaerateddimensionless variables which are independent of the actual distrtbu-tion 0 and •. From the expected Bruceton results for each combina-tion of stop size and mean spacings 6, 0, and H were computed usingequations 15, 17, 24, 35 and 36.

33. Arbitrarily we chcee ý. to be 1000 and N to be 10. we thenchose step sizes ranging from d - 0.I• to d w ION. For each of thesecenter levels we then set up an array of levels Xi (spaced d unitsapart) above and below the center leval and including the center level.For each of the levels in the ray we then computed the logit value,ofYL, associated with Xi by

Xi - 1000

The expected Pi and qi for each level was computed from y by equa- • Ition 5. We then chose arbitrarily a level several steps belowthe mean. At this bottom level, Xi, we stipulated that there would

30

NOLTR 73-91

be one trial which would be a fail.* (As will be seen, this levelmust be chosen far enough below the mean that the probability of fiisis quite small.) The expected number of fire* at the next higherlevel is equal to the number of fails at the previous level.t*That is, for all levels we can say that

nx,(i + 1) " no,i t37)

At any level above the lowest the expected number of fails willdepend on the probability of firing at the level and the expectednumber of fires as generated by equation (37). The expected numberof fails is

Anx,(i + 1) q(i + 1) (38)

no,( )+ +)

in our work no was rounded off tr. the nearest whole number. If theBruceton test size was less than 100,000,000 the process was repeatedstarting at lower levels. Also the process wad iterated similarlyuntil successive values of 6, a, and H showed no chanqes in thefirst four significant figures.

34. A specific computation is included to show the process:

We remember that 0 - 1000 and > - 10.We wish to generate the values for the case w0-n the step

size is 40, that is, when y/d - 0.41 and when the centrallevel (the level closest to the mean) in 0.4 of a stepaway from •. The central level is therefore, 1016.

We assign i - 0 to the central level and then set up anaxry of levels and indices below and above x. The levelsare 40 units apart: 896, 936, 976, 1016, 1056, 1096;the corresponding indices are -3, -2, -1, 0, +1, +2.

*We could have started with two or more fails (and no fires) at thebottom level without changing the results provided the proba-.bility was small enough.

"**There are certain properties of the Brucwton data collection planwhich can be seen by inspection. First we define a closed Brucetontest as onc irn which the test at the last level would lead to thesame level as the one at which the test began. For iny closedBruceton test the total number of fires is equal to the totalnumber of failas also, the number of fails at any level is equalto the number of fires at the next higher level. If the test isnot closed, the number of fails at one level will differ from thenumber of fires at the rnext higher level by no more than 1. However,with a large number of trials at all levels except the top andbottom an assumption of eq:uality will introduce negligible error.

7I

NOLTR 73-91

At each level, xi, the values of Yi' , and are computed.

Using the algorithm of equations 37 and 38 and starting ati - -3 the values of n xi and no0 1 are computed at eachlevel, yielding the results given in Table 4.

The various summations overall levels are computed3FYnx, i' I Lnx, i^I^Ei nx, i I E(nx, i '+ no, i) kn +npx,i + no'i)piqiYi

The values of M, G, and H are computed using equations24,35, and 36.

When the lowest levwl is set at 856 rather than 896 an eight-level pattern is generated. The values of the summations and M,0, and H for this latter case are also shown in Table 4. There&re differences between the six-level and sight-level results, thegreateat difference being less than 0.05%. Since the eight-level pattern results were the same as larger-level patterns theycan be used as the asymptotic values.

35. The process described abovs was carried out for values ofV/d ranging from 0.1 to 10.0. Even though the algorithm wouldwork outside of the above range we elected not to go furtherbecause we see no pra-tical use for the results. In fact, eitherwith such extremely large steps (y/d <0.1) or with extremely smallsteps (y/d >10.0) a Bruceton test would be of dubious value.For values of v/d from 0.65 to 10.0,M was equal to y/d to sixsignificant figures and independent of D (the spacing between thecentral level and A). For y/d running from 0.1 to 0.60, cal-culations were made for values of D of 0.0, 0.1. 0.2, 0.3,0.4, ainr' 0.5. V ,, H .v:,r similrly depenlent upcn D for v/, 'helcwapproximately 0.65 and otherwise independent of D. The asymptoticvalues of M, G, and H are plotted in Figures , , and g.

36. These graphs should not be used to find values for use inBruceton computations, because they have been made to show thecomplete range of values and to show the shapes of the functions.Detailed plots can be found in Appendix A. There are some otherdifferences between the two sets of graphs. The asymptoticgraphs (Figures 7, , iri' 9) are of the various pdrametersversus Y/d. But of course in an actual Bruceton test we cannotknow -'r we can compute x from equation 13, M from equation 14,and D from equation 15. The variable E has been defined anthe ratio of the expected value of g to the step size, d (equation 17).When M is less than 0.65 it will be a function of both E andD as can be seen from the following typical values:

]

3?i

NOZTR • 3- 91

TAfALE 4 TYPITAL CALCULATICNS OF ALY?,SP''OTIC VALULS FOR M, Gr & H

x Pi n xn 0 i

2 1096 9.6 0.999932 5 0

1 1056 5.6 0.996316 1339 5

0 1016 1.6 0.832018 6634 1339

-1 976 -2.4 0.0831727 602 6634

-2 936 -6.4 0.0016588 1 602

-3 896 -10.4 0.0000304316 0 1

six levels eight levels*

-, n 8,581 281,984,179x~i

FinX,i " 745 24,496,639

- 1,965 64,572,209

E(n + n'po 'q-•i . 1,672 05 54,946,457

,(n- +• n y 6,226.62 204,614,821

M - 0.221456 0.221455

G - 2.265092 2.265388

H - 1.173932 1.173439

*See .xplaination in latter part of paragraph 34.

33

It

NOLTR 73-91

0

0

m0

C 0

340

NOLTR 73-91

0

C4

0

z0

-- .-

35 I-

a5

NOLTR 73-91

00

0

- Y

CN4

36

NOLTR 73-919

D o0.20 0.250 0 266

.0-6 0.2806 0.2860 0.291

0.10 0.2724 0.2789 0.2856

0.20 0.2!17 0.2605 0.2694

0.30 0.2271 0.2386 0.250ý

0.40 0.2080 0.2214 0.234

0.50 0.2058 0,2150 0.228

However during an ordinary Bruceton data analysis it is necessaryto derive E from the value of M obtained from equation (14). Bya four-term Lagrangian interpolation the numbers can be A

reorganized into a table of E values:

0,200 0.220 0,240 0,260 0.280 0.30d00 - - - 0.1890 0.2588 0.272E

C,•1037 0.1352 0,1787 0,2189 0.2517 020

0,20 0.1810 0.2036 0.2265 0,2494 0, 8 0 2930,30 0.2170 0 2339 0.2512 0.2689 0.2871 0,50.40 0,2342 0.2489 0,2642 0.2799 0.2962 0. 30,50 0.2394 - 0.2536 0.2682 0.2834 0.2991 0.3 5

We found that a better graphical presentation could be made by plotting5 - f(M,D) rather than E - f(M,D) where E - M + 5. The values of6 which correspond to the E values above are:

0.200 1 0.220 0.240 0o.60 0.280 0.309

0,00 1 - - -0.0710 -0,0212 -0.0271

g.10 -0.0963 -0,0848 -0,0613 -0,0411 -0.0283 -0.020d

0.20 -0,0190 -0.0164-0135 -0,0106 -0.0082 -0.006A

0.30 +0.0170 +0,0139 +0.0112 +0.0089 +0,001 1 +0.0054

0.40 +0.0342 +0.0289 +0.0242 j0.0199 +0,0162 +0.013

0,50 +0.0294 +0.0336 +0.0282 +0.0234 +0.0191 +0.015i

The tabular and graphic values of G and H are given as functionsof E and D rather than of M and D because, once E is found, it isthe best available estimate of N/d and should be used as such.

V. MONTE CARLO STUDIES

37. In order to check the results of the theoretical work a MontqCarlo investigation was carried out. A program for the IBM 7090computer was prepared to generate random numbers with a logisticdistribution to represent the sensitivities or critical levels of

NQ)LTR 73-91

the test items. The distribution parameters, - and N, and thesample size, Ns, were adjustable. A run of one hundred Brucetontests was made for each combination of the quantities investigated.The sample mean, ms, and the variability, gs, were computed foreach test. The starting level was fixed at 20.0 and the atepsize at 1.0. For each Brocetn, tem, the estimates ms an& M asdefined in the previous seo-'on, aquations 23 and 24, were obtained.(Subscript B indicates an eotimate obtained from a Bruceton testand subscript a refers to , value for the sample.) For each runthe average and standard 6Aviation of each of the quantities mBand M was found.

38. We showed in the previous section that g/d can be taken as equalto M if M > 0.65. For this computer study (and for that matter inordinary testing), -/d can be taken as equal to M if M > 0.4. It wasconvenient to use this relationship for the Monte Carlo studies.Since the step size, d, was chosen as unity we can therefore tak-.-M, in this range, as equal to 9 B' the Bruceton estimate of ga.In this case, then, the average and standard deviation of M willbe the average and standard deviation of NB. The computation ofgs when N/d < 0.4 depends upon the relative po0ition of the meanand the nearest step level, For each run we know this for thepopulation mean. If the sample mean were the same as the populationmean we could use the curves of figure 7 to estimate YB from thevalue of M. Inspection of the curves of figure 7 shows that therelative position of the mean is not as critical when the meanis midway between step levels as when it is on a step level. Forthe runs in which the population mean was midway between steplevels the valuesof the mean and standard deviation of gB wereestimated from the mean and standard deviation of M by using t'hecurve of figure 7 for the case with this position of the mean.The values of the mean and standard deviation of g given in thetables of Appendix C for y/d less than 0.4 were obhained in thisway. Since d - 1, the value of -*/d is to be found in the secondcolumn, headed by -y. These values of gB are too large and theirstandard deviations are too small since the means of the individualtests do not fall midway between the step levels. (This can beseen by comparing the results which would be obtained from thecurve for m - h - 0.3 with the results obtained from the assumptionthat the mean is midway between levels.) We did not estimatethe mean and standard deviation of gB for tests with y/d lessthan 0.4 and with the population mean on a step level.

39. For each run of 100 trials the average and standard deviationof the obee'ved mB and gB were calculated. But the standarddeviation of mE is the o served su, and similarly the standarddeviation of g is a . Substituting am and a into equations 33and 34, using •he average gB for y and settin• N (the number offires or fails) equal to one half of Ns permits us to find valuesof G and H which are representative of the experimental conditions.This is because t-he experiments must use observed values sincewe cannot knov the population parameters.

NOLTR 73-91

40. Since theoretical development in the previous section assumeslarge samples the first interest in the Monte Carlo investigation_was the effect of sample size on these results. Runs weretherefore made with samples as small as ten and as large asfive hundred. It is not difficult to see that for small samples 1a few unexpected results in the early part of the test couldinfluence the estimates of the mean and the dispersion unduly.This would be especially the case if the step were small or, 11in other words, if the value of gamma were large in comparisonwith the step size. In our program the step size was always one.The effect of the relation of step size to the value of gamma wasrinvestigated by using population values of gamma from 0.1 to 100.(d/-y from 10 to 0.01) The effect of choice of starting level wasinvestigated by using population means of 19.5, 20.0, 20.5, and25.0. Since the starting level was always at 20.0 these gavetests in which the starting level was one half-step above, on,one half-step below, or five steps below the mean. The resultsobtained are summarized in the tables of Appendix C.

Bias in the Estimate of Gamma

41. Other Monte Carlo investigations have shown that the Brucetontest, based upon the assumption of a normal distribution, givesa biased estimate of the standard deviation (references 7, 8).It is evident from our results that a similar bias eyists in theestimate of gam=a for the logistic distribution. The amount ofthic bias is shown in the last column of tables Cl to C20 as theratý.n of g. to gB" Tables 5 and 6 are condensations of these* retts. We observe that: the bias decreases as the sample sizeis increased, the bias is greater as the ratio -'/d increases.that is, as the step becomes small with respect to the value ofgarnTa. Our results do not show a bias when Ns - 20 with startingl-"el at thc.) mean and the ratio •i'd equal to or greater than one(medium or s.,ali step size). This should not be taken as a recom-mendation for a test plan since uncertainty in the knowledgeof the value of the mean would make it impossible to plana test so that the mean would be on a test level. It is atemptation to use the bias ratio, gs/gB, as a fudge factor totry to improve the estimate of -. We counsel strongly againstsuch a measure because the individual g. values are scatterecso greitly about their average.

Effect of Sample Size on Estimates of G and H

42. The values of G and H as obtained from the Monte Carloinvestigations agree well with the values predicted by thetheoretical computations when the sample size was large (fivehundred items) and the step size was not extremely large orsmall. The computation of the values of G and H was based uponthe values of gamma obtained by the Bruceton analysis. Forsamples of less than five hundred the Monte Carlo values ofG and H were greater than the theoretical values. Part, butnot all, of this variation is due to the fart that the value ofgamma is underestimated for these smaller samples. The Monte

NOLTR 73 -91

Table 5 Bias in Estimate of Gamma

7-.O. -. 0 2.0___ 3.0__0.5 1.02.0 3.0 Upper figure

1.2015 1.3300 1.7119 1.987 - 19.5

0 1.1954 0,9992 0.9965 1.002E u - 20.0

1.1099 1.4242 1,8175 1.9508 L - 20.5

1.0828 1.1754 1.2577 1.379450 1.0881 1.1185 1.2921 1.425

1.0255 1.1252. 1.2274 1.378¢

1.0402 1.0533 1.1411 1.1933

100 1.0914 1.0634 1.1860 1.2209

1.0476 1,0667 1.1071 1,1979

0.9988 0.9871 1.0068 1.0330

500 0.9920 1.0252 1.0498 1.0367

0,9914 1.,0198 1.0302 1.040 -

Table 6 Bias in Estimate of Gammafor Sample Size of Twenty

-y7/d •--19.5 4-20.0 :1 U-20.5 -2 5. q0.5 1.215 1.1954 11099 1.13910.7 1.2915 1.2603 1.2267 1.128

0.85 1.3939 1.2124 1.1624 1.132

1.0 1.3300 0.9992 1.4242 1.0022

1.5 1.7225 0.9982 1.6289 1.011

2.0 1.7119 0.9965 1,5175 1.021

2.5 2.0594 0.9989 2.0226 1.125

3.0 1.9870 1.0026 1.9508 1.455

4,0 2.4999 1,0050 2.5306 1.352]

'C,.

NOLTR 73-91

Carlo investigations indicate that values of am and a will beseriously undcrestimated when the formulas bared on t~e assumptionof large samples are used with smaller samples.

43. As an examp2e of tids underestimation take the results for asample size of 50 with the population mean at 20.5, population gammaat 1, as given in Table C-li. In this case the average estimate ofgamma obtained from the one hundred tests was 0.89. From Tables A-2and A-3, we would then get 1.585 and 1.745 as values of G and H forthis value of gamma. Using equations (18) and (19) with N - 25 (halfof the sample), we would obtain am - 0.282 and ag w 0.311. Table C-liqives am n 0,274 and ag - 0.346 so that we have reasonable agreementfor sm, but the value of sg an obtained from use of Table A-3 istoo small.

44. Tables 7 ana 8 give the effect of sample size on the valuesof G and H as computed from the Monte C 1rlo investigations. Thetheoretical values are yiven under the infinite sample size entry.Since there is no reason to expect the values of G and H to be affectedby the position of the mean when the ratio Y/d is greater than 0.4,theobserved differences between values for the same sample size and Y/dratio can bo considered to be due to random error. The three valuesin the tables were therefore arrayed. It can be seen that within theestimate of random error noted above the values of o for largersamples agree with the theoretical values. As the sample size de-creases the value of G increases. The values of H also agree well forlarge samples. However, for values of V/d of 1.0 and 2.0, the valueof H decieases for small samples. This is due to the fact that in Ithese cases the step is so small that the test does not have time tocover the full range of levels before the sample is exhausted. Theresult is,therefore,that an artificially piecise,but quite inaccurate,determination is made of gamma.

"2ffect of Starting Level

45. The effect of a poor choice of starting level was investigatedbý Monte Carlo runs in which the mean of the population was at 25.The starting point of the test was kept at 20.0. As might be expected,the value of the mean as computed from the Bruceton tests tended to benear the starting point when a small number of items was tested andalso for a la:ge value of the ratio Y/d, that is, when the distribu-tion function was relatively flat. When a small number of items istested the estimate of the dispersion parameter is too small and showsmore variation from sample to sample than would be expected `,om thelarge sample theory. The results are given in Tables C-4, C-8, C-12,C-1S, and C-20. The comparative effect on the mean is shown for threestep sizes in Table 9 which gives the values of (u - ms)/Y and thecorresponding true percent response of the estimated fifty percentpoint. The effect on the value of gamma is shown in Tables 10 and 11.Table 10 gives the values of the bias factor gs/gB. Comparison with thevalues of this bias found in tests with better choice of starting pointshows little difference. Table 11 gives the values of H, which is ameasure of the variability of the estimate of gamma. It will beseen that for small samples these are larger than when a good choice

Al

NOLTR 7.3-e1l

Table 7 tffect of Sample Size on vaLie of G

SaI ple .•-19.5 .- 20. j .- 20.5 average

10 2.6980 2.5871 2.7994 2. 6 9 4 8

20 2.4133 1.9499 2.0331 2.1321

50 1,7944 1,7714 1.9956 1.8538

100 1.6856 1.5412 1.8502 1.6923

500 1.9019 1.5720 1,4675 1.6471

Inf1nite - - L_.

Y/d 1.0

Sample 71§. 5 ý.20.0 [ -20.5 average

10 3.4533 2.6696 3.2983 3. 4=4

20 2.1599 2.1014 2.0653 2,108950 1.8659 1.7569 1.5375 1.7201

100 1.7452 1.7947 1.7345 1.7581

500 1.3624 1.7222 1.4635 1.5160

Infinite - - -

•/d - 2.0

Sample •-19.5 k-20.0 .- 20.5 aveorage

10 3.1579 3.5380 3.6335 3.44=3

20 2.5096 2.7503 2.3119 2.5239

50 1.7646 1.7569 1.5577 1.6931

100 1.6388 1.6205 1.4989 1.5861

500 1.5508 1.4057 1.5202 1.4922

Infinite 1- 1.501

NOLTR 73-91

Table 8 Effect of Sample Size on value of H

- --/ d 0 . 7 a e,Sampe ý-9, i•20.0 i p-20. 5 _ vrgo

10 2.4122 1.7636 1.4802 1.8853

20 1.7724 1.3645 2.0666 1.7345

50 1.7218 1.6561 1,7294 1,7024

"100 1.4804 1,7171 1.4852 1.5609

500 1.5743 1.5356 1.6419 1.5839

InfinitO - - 1,577

z2iiie=, 20,0 k=20.510 1.9087 1.5850 1.7838 1,1591

20 1.7603 1.7916 1.5^60 1.6926

50 1.6482 1.8939 1.V407 1.0276

100 1,4554 1.7629 1.7095 1.6426

500 1.6723 1.9354 1.8860 1.8312

Infinite - I - - 1, 745

/d- 2.0

Sample-• 16-19.5 -6.h20.0 1.-2015 average10 2.3629 1.6225 1.6173 1.8676

20 2.0225 1.6898 1.7771 1.8298

50 2.0313 1.6939 2.0827 2.0026

100 2.0174 2.1640 2.1115 2.0976

500 2.1511 2.3735 2.1579 2.2275

Infinitl - - - 2.240

NOLTR 73-91

of starting level was ,',iade. Similar effects were observed in MonteCarlo tests with populations having a normal distribution.(Reference 8).

Table 9Effect of Poor Starting Point on Determination of the Mean

(Table gives values of (G - m;)/- below the lineand the corresponding true rcent point abovethe line for -/d values of 0.7, 1.0, and 2.0)

.. 7 1.0 2.0

Startin, 22erent point 0.10 0.67 7260

$ample size 10 .6 0*0-" 70.

5000 .006. 01 oZ.o. 2

50 0.07 2 007 1 0 -150 L0-•-U! 0o0 0Tj

.t

5 1

4NOLTR 73-91

Table 10

Effect of Poor Starting Point on the Determination of Gamma

(Valuea of thQ Bins Factor g*/g 8 )

Samp e 017 1.0 2.0

10 1.6500 1.3934 2.4678

20 1.1288 1.0028 1.0211

50 1.0649 0.9842 0.9453100 1.0200 1.0162 0.9655500 0,9881 1.0104 1,0158.

Table 11

Effect of Poor Starting Point on Value of H(Values of the Bias Factor g6/g 3 }

Samole ./ .07 1.0 2.0

10 3.2567 3.3538 2.9715

20 2.2232 2.4274 2.422850 1.7776 1.9905 2.0749

100 1.6393 1.8627 2.2386

500 1,7682 1,6963 _2.0778

NOLTR 73-91

Choice of Step Size

46. Some generalizations can be made. The choice of the step sizeto be used in the test depends upon what is already known aboutthe distribution and upon whether we are more interested in deter-mining the fifty percent point or the value of gamma. The followingpoints should be noted.

a. For large steps (larger than five times gamma) the valueof gamma becomes indeterminate if the fifty percent point falls ona test level (Figure 7). For this reason large steps should beavoided unless the position of the fifty percent point is at leastapproximately known.

b. A small step size would not be good with a test of asfew as twenty items unlearn we can be sure that the starting levelis near the population fifty per cent jýint. For example see theMonte Carlo determination of the fifty percent point for testsof twenty items with the population mean at 25 and a value for gammaof 2.5. Here the starting level is at the 12 percent point of thepopulation and the step size (d/N) is 0.4. The average of theBruceton means was 23.827 which Js the 38.5 per cent point of thepopulat ion.

47. The precision with which the values of the mean and ganrra canbe measured under ideal conditions for different step sizes can bodeduced from the shape of the G and H curves (Figures - " ^;.

a. The values of G show that the most precise measurementsof the mean are obtained for small steps. However these idealconditions are not ordinarily met. An we have just shown a smallstep qives a poor estimate of the medn when we have a short testwith a poor choice of starting level.

b. The minimum value of H occurs when ,/d is equal to 0.5 ifthe mean is on a test level or 0.2 when the mean is midway betweentest levels. This gives the most precise determination of gammawhen the step is two to five times the value of gamma dependingupon the position of the mean with respect to the test levels. Ifwe know the mean well enough so that we can be fairly certain thatit in nearly halfway between at*; levels and if we expect to testenough items to obtain a good determination of the response atthe two levels nearest the mean a step size as large as five timesthe expected value of gamma will be best for measuring the dispersionparameter. In this case we would be testing at the estimated 8 and92 percent points. This would mean that the test should include atleaat fifty items if we expect to get fairly good estimates ofthese points. If we cannot tese as many as this a smaller stepsize must be chosen.

I - • i ii i i i [ I

NOLTR 73-91

REPFRENCES

1. Statistical Research Group, Princeton University, "StatisticalAnalysis for a New Procedure in Sensitivity Experiments', AMPReport 1Ol.lR, July 1944.

2. Dixon, W. J. and Mood, A. M., "A Method for Obtaining andAnalyzing Sensitivity Data", J. Am. Stat. Assoc., 43, pp 109-126(1948).

3. Berkson, Joseph, "A Statistically Precise and Relatively SimplaMethod of Estimating the Bio-Assay with Quantal Response on theLogistic ?unction", Jour. Am. Stat. Assoc. , 48, pp 56s-599(1953).

4. Hammer, Carl, "Statistical Methods in Initiator Evaluation",The Franklin Institute Laboratories for Research and Development,Interim Report No. 1-1804-1, 1955.

5. Hampton, L. D. and Ayres, J. N., NAVWEPS Report 7347, "Characteriza-tion of Squib Mk 1 Mod 0, Determination of the Statistical Model"(1961).

6. Hampton, L. D. and Blum, G. D., NOLTR 64-238 (AD 622199),"Maximum Likelihood Logistic Analysis of Scattered Go/No-Go(Quantal) Data", (1965).

7. Martin, J. W. and Saunders, J., "Bruciton Testot Results of aComputer Study on Small Sample Accuracy", Int'l Confer,'nce onSensitivity and Hazards of Explosives, London, 1-3 Oct 1963.

8. Hampton, L. D., NOLTR 66-117, "Monte Carlo Investigations ofSmall Sample Bruceton Tests", (1967)

4

I

ii

YOUR 73-91

APPIMDIX A

Appendix A is a summry of the process for carrying out alogistleno testes analysis, Figure A-I is I flow chart giving thevarious operations and equations to obtain x, g, am, and ag.Fýquwres A-2j A-3, and A-4 present graphically the vprious relation-apf needed for the computations olled out in the flow chart.

Tables A-Is A-2, and A-3 from which tigures A-2j A-3, and A-4were constructed, can be used for greeter accuracy. Table A-4 is acompilation of the equations needed to compute any desired responsestimulus level and associated one- and two-sided confidence limits.The student t approximation equation, and constants for variouspercentage levels, are also included in this compilation.

A-i

NOLTR 73-91

F CliOOSE INDICEý-FSTABLISH x0od

(AND x', IF NEFDED)

N. N, t.,

Is A- '-';mo,,A, NO YY EE 55

mo 11 2p L i 8- ý.' 1 0, 19-xo*d(N- 1112) ;,Xo-d(N+ 1 /1)

r

BN-A2

N2

is YESMlý 0. 65 D Lxl -!-I

d

NO 4A -YMD)

TABLE A-115 TILT I Iis YES STiP SIZE IS FIGURE A-2

MUCH TOOSMALL. E-M-ASEEq26 g-E d

glEd STEP 6

c'. f 3 (E, D)GýF2(E) TARLE A-2TABLE A-2 rIGURE A-3FIGURE A-31

i(EH- F4 (E) TAiLE A-3

TABLE A-2 FIGURE A-4FIGURE A-3

G g

r

FIG. A- I FLOW CHART FOR COMPUTATION OF THE LOGIT-BRUCETON PARAMETERS xg, Sm AND S

A-2

NOLTR 73-91

0.08

-0.04

0.100.1 0. 0 00.0

0.0 ,__ __ _ __ _

-- 0.02

Z 0D5 0 0.6-0.08D 0.M

-0..12AUSOF1A AFNTONO ADM ORM 06TO1 0.E USED 0.N EQATO,30M

0. 02 r - I3

NOLTR 73-91

7.0 3.0

D- 0.5D=0.5

6.0 G2.0

D-0.4\5.01.

0 2.0 4.0 6.0E

G4.0

3.0G

2.0

1.01.10.10 0.15 0.20 0.25 0.30

EIFIG A -3 VALUES OF G AS A FUNCTION OF D AND E

A- 4

NOLTR 73-91

7,0 4,0

2.0SH ! H

6.0 -H

0

0 I I I I

0 2.0 4,0 6.0E

5,0

4.0.

310-

..

4,,2

3,0 4

0.10 0.15 0.20 0.25 0.30E

FIG. A-4 VALUES Oý H1 AS A FUNCTION OF D AND E

A-5

NoL:R 73-91.

ofný C-c

~; 000 1 00

C;d 0 000 ~0

0l rl C. --1 00 0 C) 0

C 0O I Io C; C;i~ +

000~ 0

00 0 0

01 0000ý (

~~~-.~CY 00 0 00 0 0 0 0 00 00

A-64

NOLTE 73-11

0~ 0oi

1' -o 0

VN P-44

1? 0 1

144 P4cc~o o

00 UA188IC3A-7

NOLTR 39

pi HNNlý 0Le4NNNClf lHaffNN 0]H,..m. N N n . a .- aw N CYc .C r - 41-

,.. ff 4 4 C4 * ci oH J4~J' 0 1 'JN 0! Ht4 4 -4 H 4 12jt:4

lN N; M~ 0 I6 1 ; Cý4 Fn0 -HN N.N~ CQI 8 1 .4,4NNN 0ý Hz 4 F-4 PA -1 4 A

N Lr*u4 O4CJC-AO ýoq ., 09.H--

cy. N ~ N-4r N N' N 1;- 0, A A 4 A 48 r

133

4kN 0Hy* 0N * OHHNNN 4 1; 4

000000ý 0000Ied l4 C C 40 00 000000N i 4 4 8 00

A-8

YoLTR 73-91

I*e i-- It4.4,44 4 0A .4 .4p.4H A 6aol Cd 0 C; ; 04

ann I \IV t-~4~r

dou*c4Ao eýp4*dIan

Ai~ :i o;4A. Y 4. 44A. 4P 4m 4Po,.A 4 A 8 4, 4rri~ A p.g4A 4 4

0\OIC%.FA-IF4 P4 " 1 4~~4p4OI ,- 4,-~-p4pI-.-4,-61 4 -l 4

0 a a 004 4 0o cic mA4 oAyA , 14444-444C ý 4 C1r~u 14 Ee5 k uf~

Uoo oo 00 00 kgr00 0 00 .8I- ii

.9 - 44,- C AP4 4 4 4 r r AA ý r N04 elm t""-a-F , , , 4 ýc

NOLTF7 3-91

TABLL A.4 com4rInmoQ 01, EqUA~'i0Ns m= To COHet~rE THELOGIT "M0NSE LEVEL AND C0NFI~iCE LIMITS

I.. To compupte response stimaus~ level, xlj Probabil.ity of response is p,

31 + G - in (-- I + a -LVX)1-p

L(zY) . in

2. To ooqpube the standa~rd deviation or zr,.

3. To approximate the value of Student's t.

t 0 R +4S f-v

sin&le-Sided TWo-Sidod UsableConfidence Level, Confidence for

90 1.2820 1.2229 1.1345 so 995 1.6450 o.6602 0.6279 go 797.5 1.9600 0.21.20 o.47714 95 699 2.3260 0.26687 0.3798c 98 759.5 2.5760 0276 0.33618 997

4. To coepute two-sided interval abmA t at a confidence of P%:

(L,U) - [(xp -tss), (xP + * )3

At the 50% Points2

(L,tJ) - [( t-am), +. +-m]

5. To compute the lover single-sided limit at a confidence of P%t

(L) - x - t-sy

6. Tlo compute the up~per single-sided limit at a confidence of P%:

(U) - X + ta

A-10

NOLTR 73-91

APPENDIX B

DERIVATION OF VARIOUS MMNTS OF THE LOGISTIC DISTRI:UTICN

1. Ordinarily the logistic distribution is expressed in the cumulative formt

L(2-px x)-i B1

where p is the cumulative probability at a stimulus level of x. The p.d,f.(probability density ftunotio) is Oefined am the derivative of the cumulativeprobability:

p ( ) x ) . _____x (B-P)dx dx

Equation (B..) can be molved for p to Live

X z) .iL. ...1L.. , where (3-3)

•_ Y

By letting u I 2 + ey, the equations can be made less avwkwrd to write

p(x) (B-4)U

The p.d.f., then, becomes

(x)d x

and the individual derivatives can be seen to be

n M 5) (B.6)

du , .Y, and (B-7)

d~p(x)] I (B-8

Thus,

dpx) p. di. f.~ *)(~ (B-9)

B .1Y

NOLTE 73-9;

Sm

24-2(- .Z

) (B-12)

Y

2. The integral of the p.d.f., between the liaits of - and 01, ioud beequal to 1 for a well-behaved distribution function:

4,M

A 2 I__ Q !I (B-14)

X-a

mince

dy \

-Y

U ~. (B-15)

d (B-1.6)2

let w eY, & .Yd.y

(B -17)

0 %3t5

L IilmitsL-y i+04- +

VOLTR 73-91

0

0 -o - (-I)-- * 1. .... .Q .3.D.

3. By definition, the man should be the first uovent about the origin:

M4 . mean j (p.d.f.)md . (B.18)

In our notation ve shotJd be able to shov that 14 equals P. Since x = 1 + Yyequation (B-18) bee•e,,

S(=~)y (B-19)

I

+ (B..20).. 1+000h y 2JXQe-

But the secood term in an odd function. The integral of an odd funaction tetweenUinits wbich are sytric about zero, is itself zero. The first tezr. of theabove equation is the aime as equation (3-15), multiplied by k. Hence

14~* w.1 ~Q.E.D.

4. By definition, the variance should be the second mment about the mean:

Y41 * rpd.f.)(z - mean)gdx (B-21)

+0

-(B-22)

Y iAgain, since x *k + yy the precedin~g equation becomes

_?dy and (-32 J l+couh Y (-3

(B-24)

B-3

loiLn' 73-91

but,

+co-h y " ) 7

MAn thus

o ( 0 +) 2

By breaking (B-26) into 9s and .L-L. and, integrating by parts the e~ressionboom*&

But the term, - .L.,is zero at ym o and becoisme ser as y e * thu~s

as. hy ..s

0 eY+1

If nov y is transformed by 7 m 1n u, tb-n dy azd

let •

u

u v0 dv

3B-

3•-4

NOLTR 73-91

0 1+v

Break•n• this into in v and and Integrating by pats, yields

ON 149[..inV) in ( .+V) +fIln(1+v) &VI.

The term (ln v)ln (1+v) is saro Aen v u 1 but beemes indeterminate at v * 0.By rewriting the teru in the form

2Z

in (,.v)1tu

0

Ncjv, a well. kjiow series exists:

The integral between the limits of zero %nd one gives

fan~l+v~dv . v' vO ArvVr2-

o 0

Therefore

B-5

,LTR 73-91

5. The preceeding derivation of the variance of the logistic distri-bution was performed by Dr. A. H. Van Tuyl of this Laboratory.Yndependent of his effort were two other pieces of work. One ofthese w3s a derivation of the variance (by Mr. Blum) following amarkedly different chain of logic. The other was an experimentalverification using a high-speed Monte Carlo simulation. A descriptionof the second derivation and of the Monte Carlo experiment follow,in that order.

b. Derivation of Standard Deviation of Logistic Distritution

N-0

= (x-")"p (x) dx P(x) - Y -

. (x-u)'e-'-VMgL dxr '%

iY. l+y(x-U)

as *Yare y~eydy

.-= , 1+e-y -•-V " '-a. - d

,-" (ae-e-Y)17 Y lim - --- dy

I Y2 li fo ý2 (_______ dyK- -rl+e-Y12/

S*a u" e-y (X-B)e-YdK -- i Y2 -i (i +L-Y)2

YOUTR 73-91

as ys Ai As' ux1 lot U - 0-,du - -- d

go Ys Ai 80 D'IC,2-K) B(m,n) xr M-d

B(m,n) a - lrnVA (rM+n)

as a sAi rx (2-Kj r1f(n) - (n-l)!K -0 [r(2) J

fl(x+l) - xr(x)

r(2c)r(l-2c) Tr

x 01

as- y" Aim r ~ (-~o¶KK K B7~

an' M ye Ai cooarYx uiniTK (fr coonli- (K-1) rm sinrr) -2 (K-~1) ?TO cons ¶1(1

K siin K J

F~eo (2'X~-l-cos' (M~) +r (K-1) &;in (2vrK)'a YeAim [2coo an) by L'Ro16pitaiusK* 3sin T1XcoaYK rule

B. .7

BOLT' 73-91

a a a ye.2 1- in2TI+r (1-K) co@2",•K I. m in2T'K

3 eroYO " 3.2898Y

Reference: Apotolo, Mathematical Analysis. [Theorem 14-24 edThe order of differentiation and improper integration may be exchangedunder the following conditions.

(1) ý exists for all a,b in 1--,-1•

;: eKdy(2) (1 +0 y)' converges pointwise for all K in [c,d] for some

cdj

(3) L_ & a * must be continuous on the strip

S (4) I •: [.ir YK. dy must converge uniformly on (c.d).(4,) . - 4 L.-Y)•,

"(1) and (2) follow by limit-comparison with spy for appropriate valuesof p [theorem 14-4]1 (3, follows from the continuity of the sumjproduct, and quotient of two continuous funictions [theorem 4-10], and(4) follows from a Cauchy condition [theorem 14-18a if we choose anyc, d satisfying O<C4dm2.

B-8

ANOLTR 73-91

7. Monte Carlo studies on asymptotiq convergence of the varianceof the logistic distribution function.

We wished to verify the algorithm for producing numbers distributedrandomly in the logistic distribution and at the same time demonstrate the natureof the effect of the smp4ae size on the sampling error of the variance.

We assume that we know the critical levels of the devices being tested.

Furthermore, we set the population parameters to convenient values... ýk w 0 and

Y 1 1. The logistic equation reduces to:

L(x) I.n x1-p Y

Using a high-speed computer it is possible to generate a sequence of numbersPi, pr,, p ],N ........ which have the property of falling between the limits ofzero and one and having a rectangula distribution. Ohat is, if any particular

value is chosen between zero and one that value will be the same, in the limit,as the decimnl fraction of the numbers that will be generated which will fallbetween zero and that particular value. For instance if a value of .85 is chosenand if many hundreds of the random numbers are generated and classified accordingto whether they are larger or sma.ler than .85 we will find that 85 out of 100 onthe average will fall in the lower classification.

The random rectangular distribution was converted to a random logisticdistribution of values of x (with a 4 of 0 and a Y of 1) merely by letting

xP

In the Monte Carlo experiment 100 trials were run on each of 9 sample sizes wherethe sample size, N, took values of 15, 20, 40, 63, 80, ioC, 200, 500 and 1000. Foreach trial of NA samples a variance was computed using the equation

When one hundred such variancen ha been obtained the avrage of these was comutedas well a nhe 95% confidence band about the average. We also noted the smallestand largest value of v for each of the 9 different sample sizes. The observedvalues given in -able B-1 have been plotted in Figure B-I. To facilitatecomparison with the theoretical value, IP/3, the observed value, normalized bydividing by L0/3,which is 3.28987, are also tabulated.

13-q

i -' I" ' .--, '

k4OLTR 73-91

T-Ai3LE L-I

A-SYi4PTOTIC CON RG-'N; OF THAW L4.GZSTIC L;ZSTRZ8U.'IQN FU.;C'TI0N

14 1wN 44 >'"4 0 -44

snize 0~

15 1.070 2.936 3.040 3.143 8.518I

20 1.162 3.161 3.286 3.411 6.736

40 1.766 3.156 3.260 3.363 7.732

60 1.893 3.240 3.318 3,396 6.392

80 2.139 3.288 3.351 3.414 5.099 Variance

100 1.962 3.362 3.428 3.404 5.353 Vle

200 2.362 3.263 3.300 3.338 4.161

500 2.748 3.268 3.293 3,119 3.906

1000 2.719 3.262 3.282 3.300 3.694

15 .325 .862 .924 .986 2.5a9

20 .353 .925 .999 1.073 2.048

40.537 .929 .991 1.052 2.350

60 .575 .962 1.008 1.055 1.943 values

80 .650 .981 1.019 1.056 1.550 gorm~alized

100 .596 1.003 1.042 1.081 1.627

200 .718 .981 1.003 1.025 1.265

500 .835 .986 1.001 1.0).6 1.187

1000 .845 .986 0.997 1.009 1.123

B-10)

NOLTR 73.91

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NOLTR 73-91

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C-3

NOLTR 73-91

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NOLTR 73-9 1

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m ~~~ 0~ NO M D m. m

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Nc)LrR 73-91

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NCLTR 73-91

it N %V 0 ON %D N NN~ 0 f, ~Wt' r- ON r- %Q0

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