Description

Application of Structural Equation
Models to Quality of Life
Sik-Yum Lee and Xin-Yuan Song
The Chinese University of Hong Kong, Hong Kong
Suzanne Skevington
University of Bath, Great Britain
Yuan-Tao Hao
Zhongshang University, China
Quality of life (QOL) has become an important concept for health care. As QOL is a
multidimensional concept that is best evaluated by a number of latent constructs, it is
well recognized that latent variable models, such as exploratory factor analysis

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Application of Structural EquationModels to Quality of Life
Sik-Yum Lee and Xin-Yuan Song
The Chinese University of Hong Kong, Hong Kong
Suzanne Skevington
University of Bath, Great Britain
Yuan-Tao Hao
Zhongshang University, China
Quality of life (QOL) has become an important concept for health care. As QOL is amultidimensionalconceptthatisbestevaluatedbyanumberoflatentconstructs,itiswell recognized that latent variable models, such as exploratory factor analysis(EFA) and confirmatory factor analysis (CFA) are useful tools for analyzing QOLdata. Recently, QOL researchers have realized the potential of structural equationmodeling (SEM), which is a generalization of EFA and CFA in formulating a regres-sion type equation in the model for studying the effects of the latent constructs to theQOL or health-related QOL. However, as the items in a QOL questionnaire are usu-ally measured on an ordinal categorical scale, standard methods in SEM that arebased on the normal distribution may produce misleading results. In this article, weproposeanapproachthatusesathresholdspecificationtohandletheordinalcategor-ical variables. Then, on the basis of observed ordinal categorical data, a maximumlikelihood (ML) approach for analyzing CFA and SEM is introduced. This approachproduces the ML estimates of the parameters, estimates of the scores of latent con-structs,andtheBayesianinformationcriterionformodelcomparison.Themethodol-ogies are illustrated with a dataset that was obtained from the WHOQOL group.
There is increasing recognition that measures of quality of life (QOL) or health-related QOL have great value for clinical work and the planning and evaluation of health care as well as for medical research. It has generally been accepted that
STRUCTURAL EQUATION MODELING,
12
(3), 435–453Copyright © 2005, Lawrence Erlbaum Associates, Inc.RequestsforreprintsshouldbesenttoSik-YumLee,DepartmentofStatistics,TheChineseUniver-sity of Hong Kong, Shatin, N.T., Hong Kong. E-mail: sylee@sparc2.sta.cuhk.edu.hk
QOL is a multidimensional concept (Staquet, Hayes, & Fayers, 1998) that is bestevaluated by a number of different latent constructs such as physical function,health status, mental status, and social relationships. As these latent constructs of -tencannotbemeasuredobjectivelyanddirectly,theyaretreatedaslatentvariablesin QOL analyses. The most popular method used to assess a latent construct is asurvey that incorporates a number of related items that are intended to reflect theunderlying latent construct of interest. Hence, QOL questionnaires often contain anumber of items that are treated as observed (or manifest) variables. For example,the WHOQOL-BREF assessment (The WHOQOL Group, 1998a) contains 24items for measuring four latent constructs of QOL.Exploratoryfactoranalysis(EFA)isastatisticalmethodthatisusedtogroupto-gether the items (manifest variables) that are related to a particular latent construct(factor) but relatively uncorrelated with other latent constructs. It has been used asa method for exploring the structure of a new QOL instrument (Fayers & Machin,1998;TheWHOQOLGroup,1998a).Confirmatoryfactoranalysis(CFA)isanat-ural extension of EFA that allows the identified latent constructs to be correlatedand any parameter to be fixed at a preassigned value. This model has been used toconfirm the factor structure of the instrument. Hence, hypothesis testing or modelcomparison among nested and nonnested models is a basic issue in CFA. It is im-portant to realize that latent constructs in a CFA model are never regressed on theother latent constructs. The basic goal of generalizing CFA to structural equationmodeling(SEM;Bentler,1992;Bollen,1989;Jöreskog&Sörbom,1996)istoadda component for regressing the endogenous latent construct to exogenous latentconstructs. Hence, the causal effects among the latent constructs can be analyzed.Pointing out some weaknesses of EFA, Fayers and Hand (1997) argued that SEMmay have great potential in QOL research. In fact, SEM based on the normal the-oryhaverecentlybeenappliedtoQOLanalyses(Meuleners,Lee,Binns,&Lower,2003; Power, Bullingen, & Hazper, 1999). However, as the factor score estimatesthat are obtained by the standard regression method (Bentler, 1992; Bollen, 1989)have several deficiencies (see, e.g., Fayers & Machin, 1998), they are rarely usedas a method of deriving outcome scores.Items in a QOL instrument are usually measured on an ordinal categoricalscale, typically with 3 to 5 points. It is well known in the psychometric and statis-tics literature that ignoring the discrete ordinal nature and treating the data as con-tinuousleadstoerroneousresults(Lee,Poon,&Bentler,1990;Olsson,1979;Poon& Lee, 1987). Consequently, a number of statistical methods, such as polychoricand polyserial correlations, logit and probit models, and ordinal regression, havebeen developed for achieving correct analyses. The discrete ordinal nature of theitemsalsodrawsmuchattentioninQOLanalyses(Fayers&Hand,1997;Fayers&Machin, 1998). It has been pointed out that nonrigorous treatment of the ordinalitems as continuous can be subjected to criticism (Glonek & McCullagh, 1995),and models such as the item response model and ordinal regression that take into
436
LEE, SONG, SKEVINGTON, HAO
accounttheordinalnaturearemoreappropriate(Lall,Campbell,Walters,Morgan,& MRC CFAS Co-operative, 2002; Olschewski & Schumacker, 1990).The aim of this article is to introduce recently developed methods in SEM withordinalcategoricalvariablesforanalyzingcommonQOLinstrumentswithordinalcategoricalitemsthatcanbemissingatrandom(MAR;Little&Rubin,1987).Theobjectivesare:(a)todeterminethemeasurementpropertiesofthelatentconstructsunderlying the QOL instrument, and to illustrate the method by using part of thedata that are obtained from the ordinal categorical items of the WHOQOL-BREFassessment (The WHOQOL Group, 1998a); (b) to assess intercorrelations amongthe established latent constructs; (c) to provide a more sensible estimate of latentconstruct scores (factor scores); (d) to establish a structural equation model with aregression type structural equation for assessing the casual effects of the latentconstructs to the overall QOL; and (e) to introduce a model comparison methodthat can be applied to compare nested or nonnested models.
BACKGROUND
Most QOL items are not only measured using the discrete ordinal scale, but arealso highly skewed; see Fayers and Machin (1998) for a good example in relationtotheHospitalAnxietyandDepressionScale.Hence,distributionsoftheitemsarenonnormal, and the results produced by the normal theory maximum likelihood(ML)approachmaybemisleading.Thebasicideaofthemultivariateprobitmodelapproach or the polychoric correlation approach is relating the ordinal categoricalitem to an underlying continuous normal distribution through a threshold specifi-cation. For example, let
z
be an ordinal categorical variable that corresponds to anordinal item that measures, say pain, with a 4-point scale (1–4). Suppose that for agivendataset,theproportionsof1,2,3,and4are.05,.15,.30,and.50,respectively(see the histogram in Figure 1a that is highly skewed to the right). Clearly, the ob-servationsof
z
cannotberegardedascomingfromanormaldistribution.However,they can be treated as manifestions of an underlying normal variable
y
that is di-rectly related to pain but exact continuous measurements of which are not avail-able due to the design of the discrete ordinal categorical scale in the questionnaire.The relation of
z
and
y
is defined as follows: for
k
= 1, 2, 3, 4
z
=
k
if
α
k
– 1
<
y
≤ α
k
(1)where –
∞
=
α
0
<
α
1
<
α
2
<
α
3
<
α
4
=
∞
, and
α
1
,
α
2
, and
α
3
are unknown thresholdparameters.Notethat
α
2
–
α
1
canbedifferentfrom
α
3
–
α
2
,henceunequal-intervalscales are allowed. For example, the ordinal categorical observations that give thehistogram in Figure 1a can be captured by a standard normal distribution,
N
(0, 1),with appropriate thresholds (see Figure 1b). For a random vector
z
= (
z
1
, ···,
z
p
) of ordinal categorical items, the distribution of the underlying continuous random
APPLICATION OF STRUCTURAL EQUATION MODELS TO QOL
437
vector
y
= (
y
1
, ···,
y
p
) is multivariate normal with a correlated structure. Dependingonthenatureoftherealproblem,thecorrelationstructurecanbeaparticularCFA,or a structural equation model. In this approach, a model is proposed for the latentrandom vector
y
, and a rigorous analysis is conducted on the basis of the observeddata set
z
1
, ···,
z
n
of ordinal categorical responses that are not assumed to have acontinuous distribution.As the probability density function of
z
involves a complicated integral of highdimension, the statistical analysis is nontrivial. Multistage estimation methods forstructural equation models that are based on the polychoric correlations have beenproposed(Lee,Poon,&Bentler,1995).Recently,theoptimalMLapproach(Shi&
438
LEE, SONG, SKEVINGTON, HAO
FIGURE1
(a) Histogram of a hypothetical ordinal categorical dataset; (b)The underlyingnormal distribution with a threshold specification.

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