estimation of local disability schedules: an evaluation of relational models

POPULATION, SPACE AND PLACEPopul. Space Place (2012)Published online in Wiley Online Library

Estimation of Local Disability Schedules: anEvaluation of Relational ModelsAlan Marshall*

Cathie Marsh Centre for Census and Survey Research, Humanities Bridgeford Street building, University ofManchester, Manchester, UK

(wileyonlinelibrary.com) DOI: 10.1002/psp.1731

ABSTRACT

Local information on the age-specific prevalenceof disability, distinguishing disability type, isimportant for planning purposes to inform theprovision of specialist services appropriate to theneeds of those at different ages. Projections ofdisability, which require estimates of age-specificrates of disability as their base, are valuable asthey enable planners to prepare for future servicedemands. However, in the UK, estimates thatinclude detail of disability type, age, and sex arenot reliably available for sub-national areas. Thispaper evaluates relational models, a techniqueoriginally developed for the estimation of age-specific mortality rates, as a potential solution tothis information gap. Relational models areshown to be as successful in capturing sub-national variability in levels of disability asindividual-level synthetic regression models.The relational approach has importantadvantages over more commonly usedindividual-level synthetic regression models interms of parsimony of parameters andassumptions. As such, relational models offer avaluable new approach for health researchersand policymakers interested in the estimation oflocal age-specific rates of disability and moregenerally other health-related characteristicswith rates that followmortality-like age patterns.Copyright © 2012 John Wiley & Sons, Ltd.

Accepted 18 July 2012

Keywords: relational models; disability; syntheticestimates; schedules; small area estimation

*Correspondence to: Alan Marshall, Cathie Marsh Centre forCensus and Survey Research, Humanities Bridgeford Streetbuilding, University of Manchester, Manchester, UK.E-mail: [email protected]

INTRODUCTION

T his paper evaluates whether relationalmodels, a technique developed in the dem-ography literature for the estimation of

age-specific rates of survival, are suitable for theestimation of sub-national curves (or schedules)of age-specific disability rates (an age-specific rateof disability is defined here as the proportion ofthe population at a particular agewith a disability).This contribution is important for two reasons.First, it addresses an information gap, the lackof local survey data distinguishing disabilitytype and severity. Second, from a methodo-logical perspective, it offers an alternative tomore commonly used synthetic regression tech-niques for the local estimation of disabilityschedules, with advantages of parsimony ofparameters and assumptions.

Local estimates and projections of disability,distinguishing disability type and severity, areimportant for planning purposes to inform theprovision of specialist services, equipment, andsupport (Siegel, 2002). Estimates of schedules ofdisability rates are useful partly because thenature of disability service provision varies andis structured by age (Marshall, 2009) but alsobecause many disability types follow the samegeneral age pattern, with low rates across theyounger ages that rise with age reaching thehighest levels at the oldest ages. This age patternof disability rates holds across places (Figure 1)and over time. Projections of disability, whichrequire local age-specific rates of disability as theirbase, quantify the impact of trends in populationsize, age structure, and levels of disability, on thesize of future populations with various disabilities.Projections of disability are increasingly importantgiven the concerns surrounding the impact ofpopulation ageing on the size of the disabledpopulation and the associated costs of care.

Copyright © 2012 John Wiley & Sons, Ltd.

0.2

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Sight disability (England: females –2000/01)

Hearing disability (England: females -2000/01)

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LLTI, locomotor and personal care disability schedules (England: males –2000/2001)

LLTI in the districts of South Bucks, Bury and Easington (2001 – males)

Figure 1. Limiting long-term illness (LLTI) and selected disability schedules for England.

A. Marshall

An important information gap for those inter-ested in quantitative research on disability in theUK is the lack of a single source of informationthat contains both detailed geography and abreakdown of the nature and severity of disabil-ity. Sub-national survey estimates of disabilityare either very unreliable because of small samplesizes once disaggregated by age or are unavail-able for reasons of disclosure protection (Purdamet al., 2008). Although the UK census providesreliable data for very fine geographical areas,the 1991 and 2001 censuses are very limited interms of disability information that is collected,recording only the number of people who arelimited in work or everyday activities because ofan illness, disability, or health problem withoutany information on the nature of the limitingcondition. Once released, data from the 2011census will improve this situation slightly, sub-dividing limiting long-term illnesses (LLTI) anddisabilities into moderate and severe categoriesin England with some information on disabilitytype available in Scotland and Northern Ireland.


Although the Office for National Statisticsproduce estimates and projections of population,households, and labour force, they currentlydo not produce local estimates and projectionsof disability.

The data issues described previously are notunique to the UK. Countries such as Australia,Canada, and the USA also collect the most detaileddata on disability through specialist disabilitysurveys that tend to lack geography with moregeneral disability information available for finergeographies collected through the census. Thecensus questions on disability in Australia (2006),Canada (2006), and the USA (2000) are slightlymore detailed than in the UK, containing someinformation on the severity and/or nature ofdisability. However, both Canada and the USAhave moved away from the collection of all butthe most basic information in their most recentcensuses, and there are signs that the UK may fol-low suit for censuses beyond 2011 (Martin, 2006).

One response to the information gap describedearlier is to produce estimates (and projections) of

Popul. Space Place (2012)DOI: 10.1002/psp

Estimation of Local Disability Schedules

local populations with particular disability typesthrough multiplication of national age-specificrates of disability with local population struc-tures. For example, this approach is used by theInstitute of Public Care to create the ProjectingOlder People Information System (POPPI) (Instituteof Public Care, 2009a) and the Projecting AdultNeeds and Service Information System (PANSI)(Institute of Public Care, 2009b) that provide a setof estimates and projections of various disabilitytypes for all 354 local authority districts in England.The POPPI and PANSI websites are aimed atlocal authority planners and commissioners ofsocial care provision and are intended to helpthem to explore the possible impact that demog-raphy and certain conditions may have on popu-lations aged 65 and over.

The main weakness of using national sche-dules of age-specific rates and local populationage structures is that it fails to account forlocal deviation from national levels of disabilityfor which strong evidence exists (Bajekal andPrescott, 2003). The traditional statistical solutionto this weakness is provided by synthetic regres-sion models that combine survey and censusdata overcoming the weaknesses of each. Thisapproach involves dividing the population intogroups and modelling the effect that membershipof a particular group has upon the probabilityof having a certain type of disability. Thesepopulation group effects are then applied tothe aggregate data on proportions in each groupat small area level to derive local disabilityprevalence estimates. There is a wide body ofresearch on synthetic estimation, and a numberof models have been developed that differ interms of their complexity, data requirements,and the levels (area/individual or both) at whichthat they operate (Skinner, 1993; Rao, 2003;Bajekal et al., 2004).

This paper evaluates whether relational mod-els offer a viable alternative approach to syntheticregression models for the estimation of sub-national disability schedules. Relational modelswere originally developed for the estimation ofmortality schedules but have since been extendedto the estimation of fertility and migration. Therelational approach involves a ‘standard’ reliableschedule, and a relational rule that adjusts thisstandard curve of age-specific rates to representa schedule in a population where estimates areeither unavailable, unreliable, or lacking sufficient


age detail. The simplest form of relational modeldeveloped by Brass (1971) exploits the tendencyfor the logit transformation of the proportionssurviving to age x in different populations todisplay a remarkably linear relationship. Therelationship between two logit schedules ofsurvivorship probabilities can thus be expressedby two parameters (intercept and slope) (Newall,1988). More complex relational models involvingadditional parameters have been developed toimprove the model fit particularly at the oldestand youngest ages (Zaba, 1979; Ewbank et al., 1983).

The relational models of disability that areproposed here are based on those developed formortality by Brass (1971) and Ewbank et al. (1983).The census (2001) schedule of LLTI for England isused as the standard schedule and is adjusted usingtwo or three parameters to represent model sche-dules of various disability types for England[Health Survey for England (HSE)]. The parametersare stored and are then applied to local LLTIschedules to generate estimates of sub-nationaldisability schedules. A worked example using thesimplest Brass relational model is given in Figure 2.Marshall et al. (2012) validate the use of rela-tional models at national level developing specificrelational models for particular disability types.This paper evaluates the use of these relationalmodels for local areas testing the underlyingassumption that the relationship between LLTIand disability schedules remains constant be-tween areas. Research suggests this assumptionmay well be plausible; a range of different typesactivity limitation (use of stairs, bathing self,shopping, use of stairs, chair/bed transfers) areknown to be associated with self-assessed health(Valderrama et al., 2000).

The main advantage of relational models overthe synthetic regression models is that theyrequire fewer parameters with 25 variablesrequired to model all the disability types in thispaper compared with 62 variables for the syn-thetic regression model. This simplicity stemsfrom the assumption that the LLTI schedule givesa reliable template from which schedules ofother disability types can be accurately obtainedthrough adjustments to the level and shape ofthe LLTI curve. A parsimonious model is benefi-cial because it is less likely to be affected by errorin the estimation of parameters. Additionally, ifwe intend to use our models to project changesin disability over time, we can capture such


1. Schedules of rates of LLTI (census) and hearing disability (HSE) (England – Males)

0.2

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0 20 40 60 80Age

This graph shows age specific rates of LLTI (census) and hearing disability (HSE 2000/01) which display a similar shape curve but at different levels. The curve of rates for hearing disability curve is ragged due to fluctuations that stem from sampling variability

2. Age specific logit LLTI rates versus age specific logit hearing disability rates (England – Males)

-3-2

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-1-.

5

Logi

t of h

earin

g ra

te

-1.5 -1 -.5 0 .5

Logit of LLTI rate

If we take the logit transformation of the age specific rates of LLTI and hearing disability and plot these a linear relationship emerges. We can estimate this linear relationship (see line of best fit) using linear regression which gives an intercept of -0.87 and a slope of 0.97.

Note, by convention this graph shows the logit transformation of rates*0.5. The 0.5 multiplier was proposed by Brass in his original relational model, however, the inclusion of this multiplier makes no difference to final predicted values.

3. Logit LLTI and hearing disability schedules (observed and modelled) (England – Males)

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The slope (0.97) and intercept (-0.87) terms can be used to adjust the logit LLTI schedule (downwards) to represent the logit hearing disability schedule.

The negative intercept term moves the logit LLTI schedule downwards. As the slope term is close to 1 this means little change to the steepness of the logit LLTI schedule is required to relate it to the logit hearing disability schedule.

The modelled logit hearing schedule derived from this adjustment gives a good fit to the observed logit hearing disability schedule.

Figure 2. Worked example of the use of a Brass relational model to estimate hearing disability in Bury and SouthBucks districts. *Figure 2 continues overleaf

A. Marshall

trends more easily in a model with fewer para-meters. The main question that this paper investi-gates is whether the simplicity of relationalmodels over synthetic regression models is atthe expense of model fit.

The evaluation of sub-national relationalmodel estimates involves two tests in which esti-mates from a relational model are compared withdirect survey estimates and to model estimatesfrom a synthetic regression model. In the firsttest, relational and synthetic regression modelsare used to generate estimates of overall disabilityand four disability types (mobility, personal care,hearing, and sight disability) for the nine regions


in England using data from the HSE and thecensus. These estimates are compared with directsurvey estimates from the HSE. If relational mod-els capture the regional variability in levels ofdisability as well as synthetic regression models,this gives confidence for their use at lower geo-graphical levels where the release of HSE data issuppressed for reasons of disclosure protection.

The second test examines the robustness ofrelational models at district level (434 districts inthe UK, average population=120,000). Althoughthe HSE lacks data for finer geographies thanGovernment Office Regions, the Scottish House-hold Survey (SHS) does release data on disability


4. LLTI and hearing disability schedules (observed and modelled) (England – Males)

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We can convert the logit rates in the previous graph back to rates by reversing the logit transformation (see equations 12 and 13)

5. LLTI schedules and modelled hearing disability schedules– (Bury and South Bucks - Males)

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If we take the slope and intercept parameters that were estimated for England as a whole and apply them to logit LLTI rates locally (census) we can generate hearing disability schedules for areas that lack direct or reliable survey estimates.

In this graph these calculations are applied to LLTI data in Bury and South Bucks. Bury has higher levels of LLTI than South Bucks and this is reflected in the estimated rates of hearing disability.

Figure 2. (Continued)


for districts in Scotland. The SHS has a more lim-ited set of disability questions and different defini-tions of disability compared with the HSE, but itdoes measure two disability types (personal careand locomotor disability) in a similar way to theHSE. Although we would not expect the modelestimates based on HSE data to equal those fromthe SHS given the slightly different definitions ofdisability, we would expect there to be strongcorrelation if model estimates are reliable as theunderlying characteristics being measured aresimilar. Again, relational and synthetic model esti-mates are compared with direct survey estimatesof disability in this case for districts in Scotlandtaken from the SHS.

The meaning of ‘disability’ is not clear in thesame way as other demographic concepts suchas births or deaths. Inevitably, estimates of dis-ability prevalence vary between surveys accord-ing to the definitions of disability employed.The HSE is used here to develop model estimatesfor districts across the UK rather than selectingdifferent disability data sources for the variousparts of the UK so that a set of consistent set ofdisability estimates is generated. These estimatesare now incorporated within the POPGROUPpopulation projection software (http://www.ccsr.ac.uk/popgroup/) that allows users to


develop projections of population, households,labour force, and now of disability.

After this Introduction, the paper is organisedinto four sections. First, the data sources aredescribed. Second, the relational and syntheticmodels are specified. Third, results from thecomparison of synthetic and relational modelsare presented. Finally, conclusions are drawn.

DATA

Three data sources are used in this paper. Thecensus (2001) and the HSE (2000/2001) (Natcenand UCL, 2010; Natcen and UCL, 2011) are usedto develop sub-national estimates of variousdisability types. The HSE provides informationon disability with limited geographical detail,whereas the census supplies information oncorrelates of disability (age, sex, and LLTI) thatinform model estimates of disability for sub-national areas in England, Scotland, and Wales.Regional disability data from the HSE are usedto evaluate model estimates. The SHS (2001/2)(NFO Social Research et al., 2005) provides districtestimates of two disability types that are comparedwith model estimates (developed using the HSEand the census).


http://www.ccsr.ac.uk/popgroup/

http://www.ccsr.ac.uk/popgroup/

A. Marshall

The Census

The 2001 census question on LLTI (Table 1) isused to provide national and sub-national ageand sex-specific rates of activity limitation dueto illness or disability. A large body of worksupports the validity of self-assessed health(Mitchell, 2005) with LLTI found to be moststrongly associated with general health percep-tions, more serious health conditions (Manoret al., 2001), and physical limitations rather thanwith psychological health (Cohen et al., 1995).There are strong relationships between LLTIand other health outcomes including all causeand cause-specific mortality (Charlton et al.,1994; Bentham et al., 1995; Idler & Benyamini,1997) as well as sickness benefits claims from dif-ferent health conditions (Bambra and Norman2006; Norman and Bambra 2007).

It has been argued that perception of, andpropensity to report, ill-health may vary acrossthe constituent countries and regions of the UK(Mitchell, 2005; O’Reilly et al., 2005). For example,there is a lower level of illness in Scotland giventhe mortality levels in Scotland than in Englandand Wales (Mitchell, 2005). Any systematicdeviation in the relationship between disabilityand self reported LLTI would lead to bias in themodels fitted because they assume the national(age-specific) relationship between LLTI, anddisability in England holds for sub-national areasacross the UK. We evaluate model fit in theResults section.

The census data on LLTI are downloaded fromthe Table ST16 that records the population with(and without) LLTI with age and sex detail forthe household population. Census tabulations ofLLTI are released with quinary age detail and togenerate single year estimates (matching theavailability in the HSE); these five year rates aresmoothed using an Excel-based tool developedby Popgroup users specifically for this purpose.

Table 1. Limiting long-term illness question – census2001.

Do you have any long term illness, health problem ordisability which limits your daily activities or the workthat you can do? Include problems which are due toold age. (Yes/No)

Source: 2001 Census household questionnaire. Available at http://www.statistics.gov.uk/census2001/pdfs/H1.pdf


The Excel smoothing tool and more information onthe smoothing approach are available at http://www.ccsr.ac.uk/popgroup/about/manuals.html

Health Survey for England

The HSE was set up in 1991 to monitor the healthof the private household population in England.A module measuring disability is included in1995, 2000, 2001, and 2005. In this paper, the dataon disability in 2000 and 2001 are combined toincrease sample sizes and to overlap the datacollection date of the 2001 census, a feature thatis particularly useful for the models (relationaland synthetic) that combine HSE and censusdata. The total HSE sample included in thisanalysis includes 25,958 individuals. The depos-ited HSE data includes information on theGovernment Office Region (there are nine regionsin England) in which a respondent resides,but more detailed geographic information is sup-pressed to ensure that individuals cannotbe identified.

The 2000 HSE focused on disability amongstthe elderly with a boosted sample of elderlypeople including the elderly living in residentialand care homes along with a reduced sample ofthe general population (Bajekal and Prescott,2003). The older people in retirement homes arenot included in the models as the regions inwhich care homes are located are not given. Theabsence of information on care home locationprevents comparison of observed and modelrates for regions were this group to be included.

Disability is measured according to fivedomains: locomotion (mobility), personal care,sight, hearing, and communication. In this paper,model rates are produced for overall disabilityand each disability type with the exception ofcommunication disability that is omitted becauseit does not display the strong age pattern neces-sary for the relational models developed here,and it is not clear that regional differences aresignificant for this disability type. Althoughseverity of disability is recorded in the HSE, it isnot included in the models. The HSE allowsrespondents to take into account the use of aidsfor hearing and sight disabilities. However,for the other disability types, the use of aids to per-form tasks is not permitted. Data are collectedusing face-to-face computer-assisted personalinterviewing. The disability questions are asked


http://www.ccsr.ac.uk/popgroup/about/manuals.html

http://www.ccsr.ac.uk/popgroup/about/manuals.html

http://www.statistics.gov.uk/census2001/pdfs/H1.pdf

http://www.statistics.gov.uk/census2001/pdfs/H1.pdf


of all interviewees aged 10 or over. Parents answerfor children under the age of 13, but proxy answersare not permitted for adults. Bajekal and Prescott(2003) provide detailed information on the disabil-ity measures in the HSE.

Scottish Household Survey

The SHS was commissioned by the ScottishOffice Development Department in 1998 and isdesigned to provide accurate, up-to-date infor-mation about the characteristics, attitudes, andbehaviour of Scottish households and indivi-duals. The survey was first conducted in 1999,and over the first 4 years, it achieved a sampleof approximately 62,000 households collectedcontinuously. Statistically reliable results areavailable for larger local authority districts onan annual basis and for all districts, regardlessof size, every 2 years (1999–2000, 2001–2002,2003–2004, and 2005–2006).

The SHS contains amore limited set of disabilityquestions than the HSE but has an importantadvantage of containing disability data for dis-tricts. Although these estimates become veryunstable once disaggregated by age and sex, theydo provide some information to evaluate the HSErelational model estimates in Scotland. Althoughthe definitions of disability differ between the HSE

Table 2. Locomotor and personal care disability questions –H

Disability Scottish Household Survey

Locomotor* Which of the following would youhave difficulty managing on your own?

Climbing the stairsWalking for at least ten minutes

Personal care* Which of the following would youhave difficulty managing on your own?

Doing the houseworkDressingPreparing a main mealUsing a bus, train or car

Source: Bajekal M, Prescott A. Health Survey for England 2001: Disability. The1 (available at http://www.esds.ac.uk/findingData/snDescription.asp?sn=46*In both the HSE and the SHS, if an individual has difficulty with one or mocare disability.


and the SHS, the questions on personal care andmobility disability are sufficiently similar to allowcomparison (Table 2).

MODELS

Relational Models

Marshall et al. (2012) develop four relational models(Brass, Reduced Ewbank (l), Reduced Ewbank (k),and Piecewise Brass) to capture the relationshipbetween national schedules of LLTI and disabilitywith the choice of model varying according todisability type. The simplest model is based on thatdeveloped by Brass (1971) and includes two para-meters (a and b).

Ewbank et al. (1983) develop a more complexrelational rule with four parameters that allowsmore twisting of the reference schedule at theoldest and youngest ages. This four-parametersystem is an extension of Brass’ two parameterrelational model. The two additional parameters(l and k) only influence estimates at the oldestages and youngest ages, respectively. As l and kapproach zero, the Ewbank model reduces to aBrass relational model. Marshall et al. (2012)investigate the use of a Ewbank relational modelbut find parameter estimates to be unstablebecause of overparameterisation. However, two

ealth Survey for England and Scottish Household Survey.

Health Survey for England

What is the furthest you can walk on your ownwithout stopping and without discomfort?(200m is the threshold for disability)

Can you walk up and down a flight of 12 stairswithout resting?Can you, when standing, bend down and pickup a shoe from the floor?

Can you get in and out of bed on your own?

Can you get in and out of a chair on your own?Can you dress and undress yourself on your own?Can you wash your face and hands on your own?Can you feed yourself, including cutting up food?Can you get to and use the toilet on your own?

Stationery Office, London, 2003. Scottish Household Survey user guide42).re of the activities, they are classed as having a locomotor or personal


http://www.esds.ac.uk/findingData/snDescription.asp?sn=4642

A. Marshall

versions of the relational model developed byEwbank et al. (1983) provide stable model esti-mates and additional flexibility to capture theLLTI-disability age-specific relationship requiredfor some disability types. These models areknown as ‘Reduced Ewbank’ models becausethey have three parameters of the four para-meters in the original Ewbank model, droppingeither l or k. The decision as to which parameterto drop is based on a comparison of the residualsum of squares from each Reduced Ewbankmodel (Table 3).

Models are fitted for males and females separ-ately and use rates by single year of age up to theage of 84 with an age of 88 to represent all thoseaged over 84. The use of 88 as the upper age limitis based upon the average age of the populationaged over 84 as calculated using the HSE (2001).

The relational models are defined in thesucceeding text. For simplicity of notation, sex isdropped from the model notation.

Let

pxrd= prevalence of disability d (1 . . . 5) at age x(10, 11 . . . 84, 88) in sub-national area r (HSE00/01).

pxrl= prevalence of LLTI (l) at age x in sub-national area r (census01)

px+ d= prevalence of disability d at age x inEngland (HSE01/00). The + subscript indicatesthat the rate applies across all sub-national areaswithin England.

px+ l= prevalence of LLTI (l) at age x in England(census01)

Brass modelThe predicted prevalence of disability d for regionr and age x is derived from

12loge

pxrd1� pxrd

� �¼ a þ b

12loge

pxrl1� pxrl

� �� (1)

Table 3. Relational model selected for each disability type

Disability type Males

Overall disability Reduced EwbaLocomotor Reduced EwbaPersonal care Reduced EwbaHearing BrassSight Reduced Ewba


where a and b are estimated from Equation 2(England level data):

12loge

pxþd

1� pxþd

� �¼ aþ b

12loge

pxþl

1� pxþl

� �� þ ex

(2)

Reduced Ewbank model (l)The specification of the Reduced Ewbank modelinvolving the l parameter allows for greaterflexibility of LLTI adjustment at the older ages.

First, define the function T(pxrl; l) that appliesat the oldest ages, where pxrl≥ 0.5 (note: pxrl≥ 0.5when x≥ 76):

T pxrl; lð Þ ¼pxrl

1�pxrl

� �l� 1

2lifpxrl≥0:5 (3)

or= 1 if pxrl ≥ 0.5 and 0 otherwisewr = 1 if pxrl< 0.5 and 0 otherwise

Then, the predicted prevalence of disability dfor sub-national area r and age x is derived from

12

logepxrd

1� pxrd

� �� ¼ or a þ bT pxrl : l

� ��

þwr a þ b12loge

pxrl1� pxrl

� �� (4)

where a, b, and l are estimated from Equation 5using England level data:

12

logepxþd

1� pxþd

� �� ¼ ϖþ aþ bT pxþl : lð Þð Þ

þwþ aþ b12loge

pxþl

1� pxþl

� �� þ ex (5)

Reduced Ewbank model (k)The specification of the Reduced Ewbank modelinvolving the k parameter allows for greaterflexibility of LLTI adjustment particularly at theyounger ages.

.

Females

nk (l) Reduced Ewbank (l)nk (k) Reduced Ewbank (k)nk (k) Reduced Ewbank (k)

Brassnk (k) Reduced Ewbank (k)



Define the function T(pxrl; k) that applies at theyounger ages, where pxrl< 0.5 (note: pxrl< 0.5when x< 76)

T pxrl; kð Þ ¼1� 1�pxrl

pxrl

� �k2k

ifpxrl < 0:5 (6)

or= 1 if pxrl ≥ 0.5 and 0 otherwise

wr= 1 if pxrl< 0.5 and 0 otherwise

Then, the predicted prevalence of disability dfor sub-national area r and age x is derived from

12

logepxrd

1� pxrd

� �� ¼ or a þ b

12loge

pxrl1� pxrl

� ��

þwr a þ bT pxrl : kð Þ� �

þ ex (7)

where a, b, and k are estimated from Equation 8using England level data:

12

logepxþd

1� pxþd

� �� ¼ oþ aþ b

12loge

pxþl

1� pxþl

� ��

þwþ aþ bT pxþl : kð Þð Þ þ ex (8)

The relational model (see Table 3) selected foreach disability type is based on the extra sum ofsquares F test (Marshall et al., 2012).

Piecewise Brass relational model (sight disability)The sight disability schedule differs most fromthe shape of the LLTI schedule, and so thisdisability type may not be well predicted by aBrass or Reduced Ewbank relational model.Examination of the sight schedules (Figure 1)shows that age pattern is flat and very low up tothe age of 60 with a steady increase in prevalenceoccurring thereafter. Epidemiological researchconfirms the rarity of sight disabilities at theyounger and working ages and reveals that thecauses are often congenital in nature (Munieret al., 1998; Rahi and Dezateux, 1998). Sightdisabilities occur with increasing frequency atthe oldest ages with causes linked to the ageingprocess. Macular degeneration, glaucoma, andcataracts account for three quarters of sightproblems for those aged over 80 (Munier et al.,1998). The shape of the sight schedule may bebetter modelled with a piecewise approach usingan average prevalence up to the age of 60 and arelational model above the age of 60 where anage pattern emerges.


A piecewise Brass relational model is definednext.

Let

pxr= prevalence of sight disability at age x insub-national area r (HSE01)

pxrl= prevalence of LLTI at age x in sub-nationalarea r (census01)

gr= 1 if xr> 59 and 0 otherwise

υr= 1 if xr≤ 59 and 0 otherwise

Then, the predicted prevalence of sight dis-ability for sub-national area r and age x isderived from

12loge

pxr1� pxr

� �¼ υrd þ gr a þ b

12loge

pxrl1� pxrl

� �� (9)

where a and b are estimated from Equation 10using England level data:

12loge

pxþd

1� pxþd

� �

¼ υþdþ gþ aþ b12loge

pxþl

1� pxþl

� �� þ ex

�(10)

In the aforementioned model specification, theparameter d is constrained as

d ¼ 12log

p10�60þ1� p10�60þ

� �(11)

The a and b parameters have the same inter-pretation as in previous relational modelsadjusting the level and shape of the LLTI sched-ule to estimate the sight schedule; however, inthis model, they only have an effect over theage of 60.

All the aforementioned relational modelspredict the logit disability rate (Oxrd), for disabil-ity d, at age x (for males and females) within asub-national area r. These logit disability rates(Oxrd) can easily be converted to rates of disabil-ity (pxrd).Let


A. Marshall

Oxrd ¼ 12ln

pxrd1� pxrd

� �(12)

Then,

pxrd ¼ exp 2 �Oxrdð Þ1þ exp 2 �Oxrdð Þ (13)

The key focus of this paper is on the performanceof relationalmodels for sub-national areas, somodeland parameter statistics are given in the Appendix(Tables A1 to A3) rather than here. Marshallet al. (2012) discuss the model and parameterstatistics from national (England) relationalmodels in detail.

Synthetic Regression Model

A logistic regression model using data from theHSE (2000/2001) is used to predict the probabilityof a person having a particular disability based ontheir age, sex, and whether or not they have anLLTI. The non-linear nature of the logit transform-ation of the dependent variable limits the numberof explanatory variables in the model because allpopulation groups must exist as a cross-tabulationin a census table (Bajekal et al., 2004). Variables ofage, LLTI, and sex were chosen on this basis. It ispossible to develop more detailed census cross-tabulations, allowing inclusion of more explanatoryvariables, by combining data from the census andSample of Anonymised Records (Charlton, 1998;Simpson and Tranmer, 2005) or through the useof the Controlled Access Microdata Samples.Similarly, a more sophisticated model mightemploy the type of multilevel structure developedby Twigg et al. (2000) in their small area esti-mates of smoking and drinking. These optionswere not pursued here. The focus of this paperis to compare the results from synthetic regres-sion and relational models that use similardata and assumptions to consider the utility ofrelational models to produce robust estimatesof disability schedules for various disabilitytypes.

The synthetic regression model used in thispaper involves three stages. First, a logistic regres-sion model is fitted to predict the log odds ofan individual i, having a particular disability d(d=1 . . . 5) on the basis of their age x, sex (0=male,1= female), and whether or not they have an LLTI


(j= 1 indicates an individual has an LLTI; j= 0indicates an individual does not have an LLTI).Models are fitted separately for each disabilityand for males and females. For simplicity ofnotation, subscript for disability type and sexare dropped here.

Parameters quantifying the effect of age (b1),age squared (b2), and age cubed (b3) wereincluded in the model because of the strongrelationship between disability and age and thepossibility of non-linear growth in the probabil-ity of disability with age. υi is a dummy variablethat equals 1 if a male is aged over 64 (or if afemale is aged over 59). This dummy variableis combined with additional age parameters(j1 j2) to model the possibility of a slowing (oreven levelling off) of the increase in the probabil-ity of disability (and LLTI) around retirementage before the increase continues again through-out the very oldest ages (Figure 1). Finally, zij is adummy variable that equals 1 if a person has anLLTI and 0 otherwise. This dummy variable isused to capture the impact of having an LLTIon the log odds of disability (d0) and how thisimpact changes over the age range (d1). Waldtests were used to assess whether groups ofvariables (e.g. age, age squared, and age cubed)made a significant contribution to the model.Variables were included in the final model ifthe Wald test was significant at the 95% level.Most disability types included all the expla-natory variables except the two variables asso-ciated with the retirement kink. For personalcare disability (males) and hearing disability(females), all explanatory variables were foundto be significant and were included in the model.Model and parameter statistics are given in theAppendix (Tables A4–A6).

logpi

1� pi

� �¼ b0 þ

Xt¼3

t¼1

btxti

� �þ f1υi

þf2υixi þ d0zij þ d1zijxi

(14)

Second, the estimatedparameters fromEquation 14are used to generate model probabilities ofhaving a disability (pxþj) associated with popula-tion groups based on single year of age (x= 10,11. . . 84, 88), and LLTI status [j= 0 (‘no LLTI’)and j= 1 (‘has LLTI’)] (Equation 15). In total, 152model probabilities are calculated representing


pxþj ¼exp b0þ

X3t¼1

btxt

� �þ f1υx þ f2υxxþ b1zj þ d1zjx

!

1þ exp b0 þX3t¼1

btxt

� �þ f1υx þ f2υxxþ d0zj þ d1zjx

! (15)


the 76 single years of age for each of the two LLTIgroups (no LLTI and has an LLTI). The + in thesubscript of the pxþj term indicates that theseprobabilities relate to England as a whole. InEquation 15, υx is a dummy variable that equals1 if age (x) is greater than 64 for males and 60for females. zj is a dummy variable that equals1 if j= 1 (j= 1 indicates the population groupwith an LLTI) and 0 otherwise. Equation 15includes a transformation so that probabilitiesof disability, rather than log odds of disability,are calculated.

Third, Equation 16 uses census counts of popula-tion and the model probabilities from Equation 15to calculate (76) age-specific rates of disability( pxrþ ) for each single year of age x (x = 10, 11,12 . . . 84, 88) in a particular sub-national area r.

pxrþ ¼

X1j¼0

pxþjNxrj

Nxrþ(16)

A key assumption of the aforementionedsynthetic regression model is that the HSE andthe census measure explanatory variables in thesame way. Comparison of LLTI rates in the HSEand the census reveal inconsistencies; it isthought that the focus on health in the HSE leadsto higher estimates of LLTI prevalence comparedwith the census (Bajekal et al., 2003). At the oldestages, the census reminder to include problems

Table 4. LLTI differentials required to adjust census LLTI sc

Age group

LLTI prevalence(HSE)

LLTI prevalen(Census)

Males Females Males Fem

Pre-retirement ages 0.19 0.19 0.13 0.1Post-retirement ages 0.44 0.44 0.48 0.4

LLTI, limiting long-term illnesses; HSE, Health Survey for England.Source: Health Survey for England (2000/2001) and Census (Table ST16).Source: Author’s own calculations using the Health Survey for England (200


related to old age leads to a crossover and higherLLTI prevalence rates in the census comparedwith the HSE. To ensure consistency in the mea-sures of LLTI in the census and HSE, census LLTIschedules were adjusted using differentials atpre-retirement and post-retirement ages. Table 4shows these differentials that serve to movecensus rates upwards at the pre-retirement agesand downwards at the post-retirement ages.

To avoid a discontinuity in the adjusted LLTIschedule at retirement age, the pre-retirementdifferential was progressively reduced in the10years leading up to retirement age (65 for malesand 60 for females). Equations 17 to 19 illustrate thecalculation of adjusted census LLTI rates for males.

pxrl adjusted� � ¼ pxrl Censusð Þ p10�64þl HSEð Þ

p10�64þl Censusð Þ ifx≤55

(17)

pxrl adjusted� � ¼ pxrl Censusð Þ p10�64þl HSEð Þ

p10�64þl Censusð Þ65� x10

if 56≤x≤64 (18)

pxrl adjusted� �¼ pxrl Censusð Þ p65þl HSEð Þ

p65þl Censusð Þ if x>64

(19)

Model Fitting

Stata is used to fit all the models reported here.Relational models are fitted using least squares

hedules: HSE LLTI/Census LLTI.

ceAdjustment to Census rates (HSE rate/Census rate

ales Males Females

1 1.44 1.696 0.91 0.95

0/2001) and census (2001).


A. Marshall

regression. The regress (linear regression) com-mand is used to fit the Brass and piecewiserelational models. The nl (non-linear regression)command is used to fit the Reduced Ewbankmodels with starting values of 1 given to allparameters. Experimentation with other startingvalues did not alter the final parameter estimates.The logit command is used to fit the logisticregression models using maximum likelihoodestimation. A practical guide to developing sub-national estimates of disability in Stata using(Brass) relational and synthetic regression modelsis given by Marshall (2010).

RESULTS

Differences Between Relational and SyntheticModel Schedules

Relational and synthetic models provide esti-mates of the ‘true’ level of disability (by age andsex) within sub-national areas within the UK. Itis useful to compare the estimates of disabilityfrom each model to determine where we shouldfocus attention when evaluating the performance

Overall disability – Males – North East (high LLTI)

Overall disability – Male(low LLTI)

0.2

.4.6

.8

0 20 40 60 80age

0.2

.4.6

.8

0 20 40ag

Overall disability – females – South East (low LLTI)

Personal care – Males – LLTI)

0.2

.4.6

.8

0 20 40 60 80age

0.1

.2.3

0 20 40age

Figure 3. Comparison of relational and syntheti


of models against observed data. Figure 3 com-pares model schedules for a selection of disabilitytypes in the North East (high LLTI) and the SouthEast (low LLTI). The main differences occur at theolder ages (60+). Relational models tend to givehigher rates at the oldest ages for regions withhigh levels of LLTI with the opposite occurringin districts with low levels of LLTI. For regionswhere levels of LLTI close to the average, modeldisability schedules are similar across the ages.This finding holds across (almost) all disabilitytypes, although the differences in relationaland synthetic schedules are less pronounced forfemales. The inclusion of additional retirementage terms in the synthetic regression equationsfor hearing and sight disability (Equation 14)leads to synthetic schedules that are similar torelational model schedules for all regions.

As noted earlier, the sight disability schedulesdiffer most from the age pattern of age-specificLLTI rates and thus present the biggest challengefor relational models. The Reduced Ewbankrelational models are flexible enough to adjustsub-national LLTI curves so that model sight dis-ability schedules are similar to synthetic models

s – South East Overall disability – females – North East (high LLTI)

60 80e

0.2

.4.6

.8

0 20 40 60 80age

North East (High Sight – Males – North East

60 80

0.1

.2.3

.4

0 20 40 60 80age

c mode schedules in a selection of regions.



in many regions. However, in regions with highLLTI, such as the North East, there is evidencethat the relational approach may not be suffi-ciently flexible producing rates that are twicethose from synthetic models at the oldest age(85+). Figure 3 shows that a piecewise relationalmodel goes some way to address this issue pro-ducing a schedule that is closer to the syntheticschedule at the oldest ages and that may betteraccount for the tendency of sight disabilities todevelop throughout the oldest age rather thanthroughout the middle and older working ages.

This exploratory analysis suggests that the olderages is where we need to focus the comparison ofeach set of model estimates and that particularattention needs to be paid to sight disability whererelational models (Reduced Ewbank or piecewise)may give inaccurate results.

Regional Comparison

Table 5 shows the number of regions (out of nine)in which the model estimate of disability preva-lence (for the ages 10–59 and 60+) fell outside95% confidence intervals as recorded in the HSE(2000/01). Each model gives a good fit to the datasuggesting regional variation in population agestructure, and LLTI provides a means of captur-ing the deviation of regional levels of disabilityfrom national prevalence rates in most cases.

There is little difference between the perform-ance of the synthetic and relational models atthe age group 10–59 as we might expect from

Table 5. Number of regions where synthetic and relational esand sex).

Age Disability type Relationa

10 to 59 Disability 1Locomotor 0Personal care 0Hearing 0Sight 2Sight (piecewise model) 1

60+ Disability 0Locomotor 2Personal care 2Hearing 1Sight 3Sight (piecewise model) 2

Source: Author’s own calculations using data from Health Survey for Englan


Figure 3. At the older age group (60+), there isalso very little to choose between the two model-ling strategies with the exception of sight andoverall disability. For sight disability, there appearsto be a better fit under the synthetic model com-pared with the Reduced Ewbank relational model.This is likely to reflect the issue of the insufficientflexibility of relational models to replicate sightdisability schedules at sub-national levels. Thepiecewise relational model goes some way toaddress this issue giving a comparable fit to thesynthetic model. Relational models give a better fitfor overall disability amongst males at the oldestages compared with the synthetic regression model.

District Comparison

Figure 4 compares the model rates of mobilityand personal care disability with observed ratesfor each district in Scotland taken from the SHS.The positive correlation is encouraging suggest-ing that as the observed rates increase so domodel rates. The range of rates is smaller for bothsynthetic and relational estimates, suggestingthat the more extreme observed rates (smalland large) have been moved closer to the meanreflecting the likely involvement of the samplingprocesses in generating the most extreme surveyvalues. The correlations for the relationshipsbetween observed and model rates (Table 6) arealmost identical for relational and syntheticmodels suggesting that, as for the results at

timates fall outside regional confidence intervals (by age

Males Females

l Synthetic Relational Synthetic

1 2 20 1 10 0 01 0 02 1 2

n/a 1 n/a2 0 02 0 12 2 21 0 11 3 2

n/a 2 n/a

d (2000/2001) and the census (2001).


Table 6. Correlation between relational/synthetic model estimates and observed rates of disability at the 10–59 and60+ age groups for Scottish districts.

10 to 59 60+

Relational Synthetic Relational Synthetic

Personal care 0.8 0.8 0.76 0.76Locomotor 0.84 0.82 0.82 0.8

Source: Author’s own calculations using data from Health Survey for England (2000/2001), Census (2001), and Scottish Household Survey (2001/2).

Personal care (10-59) – Synthetic and observed Personal care (10-59) – relational and observed

.03

.04

.05

.06

.02 .04 .06 .08Survey rate

.03

.04

.05

.06

Mod

el r

ate

.02 .04 .06 .08Survey rate

Personal care (60+) – Synthetic and observed Personal care (60+) – relational and observed

.12

.14

.16

.18

.05 .1 .15 .2 .25Survey rate

.12

.14

.16

.18

.2M

odel

rat

e

Mod

el r

ate

Mod

el r

ate

.05 .1 .15 .2 .25Survey rate

Figure 4. Relationship between survey rates of personal care disability and estimates from synthetic/relational modelsat the ages of 10–59 and 60+.

A. Marshall

regional level, there is little to choose betweenthe modelling approaches based on the availablesurvey data.

Figures 5 and 6 display the relational modeldisability schedules in a selection of districts(Figure 7) with differing levels and shapes of LLTIcurves. The districts of South Bucks, Brighton andHove, Bury, Swansea, Glasgow, andMerthyr Tydfilare chosen because these districts capture thevariation in the shape and level of LLTI curvesacross the UK, from districts with smooth lowLLTI schedules (South Bucks) to areas withhigher LLTI curves with prominent retirementkinks (Merthyr Tydfil). If relational modelsproduce reasonable disability curves for a rangeof different district LLTI curves, this gives


confidence that the model is reliable across allUK districts. The relational approach producesreasonable schedules of the various disabilitytypes in each of the districts selected. The cross-over, where rates of overall disability exceedthose of LLTI, may initially appear concerning.However, this effect is noted in the literaturewhere it is suggested that older people tend tounder report LLTI as they consider activity limi-tation to be a normal part of the ageing process(Bajekal, 2003). Interestingly, the model estimateshere suggest this underreporting to be greater inthe least healthy areas, a result that is worthy offurther investigation.

For the districts that display a kink in theLLTI schedule around retirement age (Swansea,


South Bucks Brighton and Hove Bury

0.2

.4.6

.8

0 20 40 60 80age

LLTI DisabilityLocomotor Personal careHearing Sight

0.2

.4.6

.8

0 20 40 60 80age


0.2

.4.6

.8

0 20 40 60 80age


Swansea Glasgow City Merthyr Tydfil

0.2

.4.6

.81

0 20 40 60 80age


0.2

.4.6

.8

0 20 40 60 80age


0.2

.4.6

.81

0 20 40 60 80age


Figure 6. Disability schedules in a selection of districts (females).

South Bucks Brighton and Hove Bury0

.2.4

.6

0 20 40 60 80age

0.2

.4.6

0 20 40 60 80age

0.2

.4.6

.8

0 20 40 60 80age

Swansea Glasgow City Merthyr Tydfil

0.2

.4.6

.8

0 20 40 60 80age

0.2

.4.6

.8

0 20 40 60 80age

0.2

.4.6

.81

0 20 40 60 80age

Figure 5. Disability schedules in a selection of districts (males).


Glasgow, Merthyr Tydfil), this feature is pre-served in the model disability type schedules.The LLTI retirement kink is a feature noted


by other researchers. For example, Westerlundet al. (2009) report a retirement related improve-ment in self-reported health, particularly for those


Overall disability Locomotor

0.2

.4P

reva

lenc

e ra

te

Pre

vale

nce

rate

Pre

vale

nce

rate

Pre

vale

nce

rate

.6.8

0 20 40 60 80Age

0.2

.4.6

0 20 40 60 80Age

Hearing Sight

0.1

.2.3

0 20 40 60 80Age

0.0

5.1

.15

.2.2

5

0 20 40 60 80Age

Figure 8. Observed andmodel disability schedules - overall, locomotor, hearing, and sight disability (England - males).

Figure 7. Map showing the location of the six case study districts. Source: UKBorders (http://edina.ac.uk/ukborders/)

A. Marshall

Copyright © 2012 John Wiley & Sons, Ltd. Popul. Space Place (2012)DOI: 10.1002/psp


in poor work environments, in a longitudinalstudy of employees of the French national gasand electric company. Examination of disabilityschedules and relational model schedules forEngland suggests that the transfer of this kinkis not unreasonable (Figure 8).

A concern to note regarding the model sche-dules for sight is the extent of the jump inprevalence at age 60 for districts with the mostprominent LLTI kinks (e.g. Merthyr Tydfil). Thisjump is caused by the piecewise relational modelthat uses the average sight disability rate (forEngland) up to the age of 60 and then a Brass rela-tional model at the oldest ages. In districts such asMerthyr Tydfil that have high levels of LLTI, theextent of the jump in sight disability is larger thanwould be expected, and this is noted as aweakness.

CONCLUSIONS

This paper demonstrates that relational modelsgive as good as fit to sub-national disability esti-mates as individual level synthetic models thatare restricted to census data. Relational modelsproduce a plausible set of disability schedulesacross a selection of districts with varying levelsof LLTI with the possible exception of sight dis-ability. Relational models are valuable for thelocal estimation of disability schedules becausethey require fewer parameters and assumptionscompared with the synthetic regression modelreducing the risk of error.

The district disability schedules developed herehave now been incorporated into the POPGROUPpopulation projection software providing a valu-able resource for health researchers and policymakers. More generally, the relational method-ology has considerable applicability to the estima-tion of other disability types, combinations ofdisabilities, or other health-related characteristicsthat display a strong mortality-like age pattern.

ACKNOWLEDGEMENTS

I am very grateful to Paul Norman, Ian Plewis,Phil Rees, Ludi Simpson, Pia Wohland, and tothe two anonymous referees for their commentsand suggestions that have improved this paper.


REFERENCES

Bajekal M, Harries T, Breman R, Woodfield K. 2003.Review of disability estimates and definitions. HMSO:London.

Bajekal M, Prescott A. 2003. Disability. Health Survey forEngland 2001. The Stationery Office: London.

Bajekal M, Scholes S, Pickering K, Purdon S. 2004.Synthetic estimation of healthy lifestyle indicators: Stage1 report. NatCen: London.

Bambra C, Norman P. 2006. What is the associationbetween sickness absence, morbidity and mortality?Health & Place 12: 728–733.

Bentham G, Eimermann J, Haynes R, Lovett A,Brainard J. 1995. Limiting long-term illness and itsassociations with mortality and indicators of socialdeprivation. Journal of Epidemiology and CommunityHealth 49: S57-S64.

Brass W. 1971. Biological aspects of demography. Onthe scale of mortality. Brass W. (ed.) Taylor Francis:London; 69–110.

Charlton J, Wallace M, White I. 1994. Long-term illness:results from the 1991 census. Population Trends 75: 18–25.

Charlton J. 1998. Use of the Census Samples of Anon-ymised Records (SARs) and survey data in combin-ation to obtain estimates at local authority level.Environment and Planning A 30(5): 775–784.

Cohen G, Forbes J, Garraway M. 1995. Interpretingself-reported limiting long-term illness. British MedicalJournal 311: 722–24.

Ewbank DC, Gomez de Leon J, Stoto M. 1983. A redu-cible four-parameter system of model life tables.Population Studies 37(1): 105–127.

Idler E, Benyamini Y. 1997. Self-rated health and mor-tality: a review of twenty-seven community studies.Journal of Health and Social Behavior 38: 21–37.

Institute of Public Care. 2009a. Projecting Older PeoplePopulation Information System. Available at http://www.poppi.org.uk/

Institute of Public Care. 2009b. "Projecting AdultNeeds and Service Information System." Availableat http://www.pansi.org.uk/

Manor O, Matthews S, Power C. 2001. Self-rated healthand limiting longstanding illness: inter-relationshipswith morbidity in early adulthood. InternationalJournal of Epidemiology 30: 600–607.

Marshall A. 2009. Developing a methodology forthe estimation and projection of limiting long termillness and disability, PhD Thesis, School of SocialSciences, University of Manchester.

Marshall A. 2010. Small area estimation using ESDSgovernment surveys: An introductory guide.Online: http://www.esds.ac.uk/government/docs/smallareaestimation.pdf

Marshall A, Norman P, Plewis I. 2012. Developmentof a relational model of disability. CCSR Working


http://www.pansi.org.uk/

http://www.esds.ac.uk/government/docs/smallareaestimation.pdf

http://www.esds.ac.uk/government/docs/smallareaestimation.pdf

A. Marshall

Paper Series. Available at: http://www.ccsr.ac.uk/publications/working/

Martin D. 2006. Last of the Censuses? The future ofsmall area population data. Transactions of the Insti-tute of British Geographers. 31: 6-18. DOI: 10.1111/j.1475-5661.2006.00189.x.

Mitchell R. 2005. The decline of death – how do wemeasure and interpret changes in self-reportedhealth across cultures and time? International Journalof Epidemiology 34: 306–308.

Munier, A, Gunning T, KennyD, KeefeM. 1998. Causes ofblindness in the adult population of the Republic ofIreland. British Journal of Ophthalmology 82(6): 630–633.

National Centre for Social Research and UniversityCollege London. Department of Epidemiology andPublic Health. 2010. Health Survey for England, 20013rd ed. [computer file]. UK Data Archive [distribu-tor]: Colchester, Essex. SN: 4628. Available at:http://dx.doi.org/10.5255/UKDA-SN-4628-1

National Centre for Social Research and UniversityCollege London. Department of Epidemiology andPublic Health. 2011. Health Survey for England, 2000,4th ed. [computer file]. UK Data Archive [distributor]:Colchester, Essex. SN: 4487, http://dx.doi.org/10.5255/UKDA-SN-4487-1

Newall C. 1988.Methods and Models in Demography. TheGuildford Press: New York.

NFO Social Research, MORI Scotland and ScottishGovernment. 2005. Scottish Household Survey, 2001-2002 [computer file]. 4th Edition. Colchester, Essex:UK Data Archive [distributor]. SN: 4642, http://dx.doi.org/10.5255/UKDA-SN-4642-1

Norman P, Bambra C. 2007. Unemployment or incap-acity? The utility ofmedically certified sickness absencedata as an updatable indicator of population health.Population, Space & Place 13(5): 333–352.

O’Reilly D, Rosato M, Patterson C. 2005. Self reportedhealth and mortality: ecological analysis based on

APPENDIX

Table A1: R2 statistics for relational models.

Males

Disability Locomotor

Number of observations 76R2 0.94Females

Disability LocomotoNumber of observations 76R2 0.95

Source: Author’s own calculations using data from the Health Survey for En


electoral wards across the United Kingdom. BritishMedical Journal 331(7522): 938–939.

Purdam K, Afkhami R, Olsen W, Thornton P. 2008.Disability in the UK: measuring equality. Disability& Society 23(1): 53–65.

Rahi JS, Dezateux C. 1998. Epidemiology of visualimpairment in Britain. Archives of Disease in Childhood78(4): 381–386.

Rao J. 2003. Small area estimation. John Wiley & SonsInc.: New Jersey.

Siegel J. 2002. Applied Demography. Academic Press:London.

Simpson L, Tranmer M. 2005. Combining sample andcensus data in small area estimates: iterative propor-tional fitting with standard software. The ProfessionalGeographer 57(2): 222–234.

Skinner C. 1993. The Use of Synthetic EstimationTechniques to Produce Small Area Estimates. NewMethodology Series NM18. OPCS: London. Availableat: http://www.worldcat.org/title/use-of-synthetic-estimation-techniques-to-produce-small-area-esti-mates/oclc/30649044

Twigg L, Moon G, Jones K. 2000. Predicting small-areahealth-related behaviour: a comparison of smokingand drinking indicators. Social Science & Medicine50(7–8): 1109–1120.

Valderrama E, Javier G, Perez Del Molino J, RomeroLopez M, Lopez Perez M, Gavira Inglesias FJ. 2000.Association of individual activities of daily livingwith self-rated health in older people. Age and Ageing29: 267–270.

WesterlundH, KiyimakiM, Singh-ManouxA,MelciorM,Ferrie J, Pentti J, Jokela J, Leineweber C, Goldberg M,Zins M, Vahtera J. 2009. Self-rated health beforeand after retirement in France (GAZEL): a cohortstudy. The Lancet 374(9705): 1889–1896.

Zaba B. 1979. The Four-Parameter Logit Life TableSystem. Population Studies 33(1): 79–100.

Personal care Hearing Sight

75 69 74 640.94 0.86 0.85 0.68

r Personal care Hearing Sight75 0.88 71 71

0.91 0.91 0.81 0.71

gland (2000/2001) and the Census (2001).


http://www.ccsr.ac.uk/publications/working/

http://www.ccsr.ac.uk/publications/working/

http://dx.doi.org/10.5255/UKDA-SN-4628-1





http://www.worldcat.org/title/use-of-synthetic-estimation-techniques-to-produce-small-area-estimates/oclc/30649044



Table A2: Parameter estimates from relational models for overall disability and each disability type(males and females).

Males

Parameter Estimate Standard error t P> t 95% confidence intervalOverall disability a �0.17 0.03 �4.96 <0.0000 �0.24 �0.10

b 1.01 0.04 28.23 <0.0000 0.94 1.08l 1.05 0.11 9.85 <0.0000 0.83 1.26

Locomotor a �0.48 0.03 �15.19 <0.0000 �0.55 �0.42b 1.53 0.30 5.17 <0.0000 0.94 2.12k 0.57 0.04 14.76 <0.0000 0.50 0.65

Personal care a �0.93 0.04 �25.57 <0.0000 �1.00 �0.85b 1.19 0.34 3.47 0.001 0.50 1.87k 0.56 0.06 9.63 <0.0000 0.45 0.68

Hearing b 0.97 0.05 20.36 <0.0000 0.88 1.07a �0.87 0.04 �19.95 <0.0000 �0.95 �0.78

Sight a �1.65 0.10 �16.27 <0.0000 �1.85 �1.45b 3.02 0.76 3.96 <0.0000 1.49 4.54k �0.60 0.16 �3.73 <0.0000 �0.93 �0.28

FemalesParameter Estimate Standard error t P> t 95% confidence interval

Overall disability a �0.16 0.03 �5.28 <0.0000 �0.22 �0.10b 0.98 0.03 31.95 <0.0000 0.92 1.04l 0.96 0.06 15.24 <0.0000 0.84 1.09

Locomotor a �0.52 0.04 �13.66 <0.0000 �0.59 �0.44b 2.16 0.28 7.79 <0.0000 1.60 2.71l 0.47 0.02 19.52 <0.0000 0.43 0.52

Personal care a �0.93 0.03 �32.63 <0.0000 �0.99 �0.87b 1.07 0.20 5.45 <0.0000 0.68 1.46k 0.57 0.04 15.61 <0.0000 0.50 0.65

Hearing b 0.91 0.05 16.98 <0.0000 0.80 1.02a �1.09 0.05 �22.09 <0.0000 �1.19 �1.00

Sight a �1.36 0.12 �11.65 <0.0000 �1.59 �1.12b 1.74 0.48 3.6 0.001 0.78 2.71k �1.06 0.33 �3.18 0.002 �1.72 �0.39

Source: Author’s own calculations using data from the Health Survey for England (2000/2001) and the Census (2001).

Table A3: Parameter estimates from a piecewise relational models for sight disability (males and females).

Males

Parameter Estimate Standard error t P> t 95% confidence intervalSight d �2.33 – – – – –

a �1.48 0.06 �23.44 <0.0000 �1.61 �1.36b 2.05 0.32 6.35 <0.0000 1.41 2.70

FemalesParameter Estimate Standard error t P> t 95% confidence interval

Sight d �2.13 – – – – –a �1.32 0.06 �21.88 <0.0000 �1.44 �1.20b 1.38 0.23 6.13 <0.0000 0.93 1.83

Source: Author’s own calculations using data from the Health Survey for England (2000/2001) and the Census (2001).

The d parameter is constrained and so its standard error and associated statistics are not given.



Table A4: Synthetic regression model statistics.

Males

Disability Locomotor Personal care Hearing Sight

Number of observations 11,777 11,778 11,776 11,777 11,777Pseudo R2 0.34 0.42 0.32 0.15 0.17Females

Disability Locomotor Personal care Hearing SightNumber of observations 14,164 14,168 14,163 14,163 14,163Pseudo R2 0.35 0.41 0.30 0.17 0.15

Source: Author’s own calculations using data from Health Survey for England (2000/2001).

Table A5: Parameter estimates for overall disability and each disability type from synthetic regressionmodel (males).

Variables Coefficient Standard error t P> t 95% confidence interval

Overall disability LLTI 3.39 0.21 15.80 <0.0000 2.97 3.81Age 0.02 0.03 0.53 0.60 �0.04 0.07LLTI*age �0.02 0.00 �3.90 <0.0000 �0.02 �0.01Age squared 0.0002 0.001 0.26 0.80 �0.001 0.001Age cubed <0.0000 <0.0000 0.53 0.59 <0.0000 <0.0000Constant �4.39 0.45 �9.80 <0.0000 �5.27 �3.51

Locomotor LLTI 5.75 0.49 11.79 <0.0000 4.79 6.70Age 0.13 0.04 3.04 <0.0000 0.05 0.21LLTI*age �0.04 0.01 �5.33 <0.0000 �0.05 �0.03Age squared �0.001 0.001 �1.22 0.22 �0.003 0.001Age cubed <0.0000 <0.0000 1.23 0.22 <0.0000 <0.0000Constant �9.00 0.79 �11.36 <0.0000 �10.56 �7.45

Personal care LLTI 6.05 0.63 9.64 <0.0000 4.82 7.28Age 0.13 0.06 2.18 0.03 0.01 0.24LLTI*age �0.05 0.01 �4.70 <0.0000 �0.07 �0.03Age squared �0.002 0.001 �1.60 0.11 �0.004 0.0004Age cubed <0.0000 <0.0000 2.27 0.02 <0.0000 <0.0000Retirement kink 6.83 2.13 3.20 <0.0000 2.64 11.01Retirement kink*age �0.11 0.03 �3.24 <0.0000 �0.18 �0.04Constant �9.02 1.10 �8.18 <0.0000 �11.18 �6.85

Hearing LLTI 2.51 0.31 8.11 <0.0000 1.91 3.12Age �0.01 0.04 �0.24 0.81 �0.09 0.07LLTI*age �0.02 0.01 �4.61 <0.0000 �0.03 �0.01Age squared 0.001 0.001 1.33 0.18 �0.001 0.003Age cubed <0.0000 <0.0000 �1.07 0.29 <0.0000 <0.0000Constant �5.11 0.62 �8.26 <0.0000 �6.33 �3.90

Sight LLTI 1.96 0.48 4.04 <0.0000 1.01 2.91Age 0.11 0.05 2.05 0.04 0.005 0.22LLTI*age �0.005 0.01 �0.53 0.60 �0.02 0.01Age squared �0.003 0.001 �2.34 0.02 �0.01 �0.0004Age cubed <0.0000 <0.0000 3.00 <0.0000 <0.0000 <0.0000Constant �6.80 0.85 �8.04 <0.0000 �8.46 �5.14

LLTI, limiting long-term illnesses.

Source: Author’s own calculations using data from the Health Survey for England (2000/01).

A. Marshall


Table A6: Parameter estimates for overall disability and each disability type from synthetic regressionmodel (females).

Variables Coefficient Standard error t P> t 95% confidence interval

Overall disability LLTI 2.94 0.18 16.27 <0.0000 2.59 3.30Age 0.06 0.03 2.20 0.03 0.01 0.11LLTI*age �0.01 0.00 �2.44 0.02 �0.01 0.00Age squared �0.001 0.001 �1.52 0.13 �0.002 0.0002Age cubed <0.0000 <0.0000 2.47 0.01 <0.0000 <0.0000Constant �4.91 0.41 �12.00 <0.0000 �5.71 �4.11

Locomotor LLTI 4.45 0.30 14.62 <0.0000 3.86 5.05Age 0.15 0.04 4.12 <0.0000 0.08 0.23LLTI*age �0.02 0.005 �4.98 <0.0000 �0.03 �0.01Age squared �0.002 0.001 �3.35 0.001 �0.004 �0.001

Age cubed <0.0000 <0.0000 4.14 <0.0000 <0.0000 <0.0000Constant �8.01 0.68 �11.83 <0.0000 �9.34 �6.68

Personal care LLTI 4.59 0.39 11.70 <0.0000 3.82 5.36Age 0.20 0.06 3.39 0.001 0.08 0.31LLTI*age �0.03 0.01 �4.15 <0.0000 �0.04 �0.01Age squared �0.003 0.001 �2.38 0.02 �0.005 �0.0005Age cubed <0.0000 <0.0000 2.25 0.02 <0.0000 <0.0000Constant �9.76 1.02 �9.54 <0.0000 �11.77 �7.75

Hearing LLTI 2.01 0.35 5.80 <0.0000 1.33 2.68Age �0.10 0.06 �1.83 0.07 �0.21 0.01LLTI*age �0.02 0.01 �2.92 0.004 �0.03 �0.01Age squared 0.002 0.001 2.15 0.03 0.0002 0.005Age cubed <0.0000 <0.0000 �1.05 0.29 <0.0000 <0.0000Retirement kink 3.58 1.67 2.14 0.03 0.29 6.86Retirement kink*age �0.07 0.03 �2.31 0.02 �0.13 �0.01Constant �4.10 0.77 �5.31 <0.0000 �5.62 �2.58

Sight LLTI 1.16 0.36 3.21 0.001 0.45 1.88Age 0.04 0.04 0.91 0.36 �0.05 0.13LLTI*age 0.002 0.006 0.32 0.75 �0.01 0.01Age squared �0.0008 0.001 �0.86 0.39 �0.003 0.001Age cubed <0.0000 <0.0000 1.46 0.15 <0.0000 <0.0000Constant �5.52 0.63 �8.79 <0.0000 �6.75 �4.29

LLTI, limiting long-term illnesses.

Source: Author’s own calculations using data from the Health Survey for England (2000/01).



estimation of local disability schedules: an evaluation of relational models

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