Application SEM to QOL. Lee n Song 2005.pdf

Application of Structural EquationModels to Quality of Life

Sik-Yum Lee and Xin-Yuan SongThe Chinese University of Hong Kong, Hong Kong

Suzanne SkevingtonUniversity of Bath, Great Britain

Yuan-Tao HaoZhongshang University, China

Quality of life (QOL) has become an important concept for health care. As QOL is amultidimensional concept that is best evaluated by a number of latent constructs, it iswell 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, wepropose an approach that uses a threshold specification to handle the ordinal categor-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, and the Bayesian information criterion for model comparison. The methodol-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 ofhealth 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.

Requests for reprints should be sent to Sik-Yum Lee, Department of Statistics, The Chinese Univer-sity of Hong Kong, Shatin, N.T., Hong Kong. E-mail: [email protected]

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-ten cannot be measured objectively and directly, they are treated as latent variablesin 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.

Exploratory factor analysis (EFA) is a statistical method that is used to group to-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; The WHOQOL Group, 1998a). Confirmatory factor analysis (CFA) is a nat-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) is to adda 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-ory have recently been applied to QOL analyses (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-tinuous leads to erroneous results (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 theitems also draws much attention in QOL analyses (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

account the ordinal nature are more appropriate (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 withordinal categorical variables for analyzing common QOL instruments with ordinalcategorical items that can be missing at random (MAR; Little & Rubin, 1987). Theobjectives are: (a) to determine the measurement properties of the latent constructsunderlying 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 relationto the Hospital Anxiety and Depression Scale. Hence, distributions of the items arenonnormal, and the results produced by the normal theory maximum likelihood(ML) approach may be misleading. The basic idea of the multivariate probit modelapproach 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 agiven dataset, the proportions of 1, 2, 3, and 4 are .05, .15, .30, and .50, respectively(see the histogram in Figure 1a that is highly skewed to the right). Clearly, the ob-servations of z cannot be regarded as coming from a normal distribution. 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 αk – 1 < y ≤ αk (1)

where –∞ = α0 < α1 < α2 < α3 < α4 = ∞, and α1, α2, and α3 are unknown thresholdparameters. Note that α2 – α1 can be different from α3 – α2, hence unequal-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 = (z1, ···, zp) ofordinal categorical items, the distribution of the underlying continuous random

APPLICATION OF STRUCTURAL EQUATION MODELS TO QOL 437

vector y = (y1, ···, yp) is multivariate normal with a correlated structure. Dependingon the nature of the real problem, the correlation structure can be a particular CFA,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 z1, ···, zn 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, the optimal ML approach (Shi &


FIGURE 1 (a) Histogram of a hypothetical ordinal categorical dataset; (b)The underlyingnormal distribution with a threshold specification.

Lee, 2000) and the Bayesian approach (Lee & Zhu, 2000) have been developed us-ing powerful tools in statistical computing such as the Monte Carlo EM algorithm(Wei & Tanner, 1990) and the Gibbs sampler (Geman & Geman, 1984). It has beenshown (Lee, Song, & Lee, 2003; Song & Lee, 2002) that these computing tools areefficient and able to handle data that are MAR. The analysis presented here isbased on a structural equation model with the ML approach.

METHODS

Instrument and Data

The WHOQOL–100 (The WHOQOL Group, 1995, 1998b) assessment was devel-oped by the WHOQOL Group in 15 international field centers. As this assessmentmay be too lengthy for some uses, the WHOQOL-BREF (The WHOQOL Group,1998a) instrument was derived from it by selecting 24 ordinal categorical items outof the 100 items. It has been shown that the WHOQOL-BREF is highly correlatedwith the WHOQOL–100 and is useful to health professionals (The WHOQOLGroup, 1998a). This instrument was established to evaluate four latent domains. Thefirst 7 items are intended to address physical health, the next 6 items are intended toaddress psychological QOL, the 3 items that follow are for social relationships, andthe last8 itemsare intended toaddress theenvironment. Inaddition to the24 itemsat-tributed to domains, the instrument also includes 2 items for the overall QOL andgeneral health, giving a total of 26 items. All of the items are measured with a 5-pointscale; for example, 1 (not at all/very dissatisfied), 2 (a little/dissatisfied), 3 (moder-ate/neither), 4 (very much/satisfied), and 5 (extremely/very satisfied).

In this article, only data that were obtained from the United Kingdom with asample size n = 475 are used. This dataset contains a number of incomplete obser-vations, which were assumed to be MAR. The frequencies of all the items and thenumber of incomplete observations are presented in Table 1. As can be seen fromthe table, several items tended to take maximum values for most patients, andhence skew to the right. Examples of these items are Q1, Q4, Q6, Q8, Q16, Q18,Q24, Q25, and Q26. Treating these data as coming from normal is not recom-mended. There were a number of missing observations, especially for Q17, whichis related to the question about sexual activity. For simplicity, missing data wereassumed to be MAR. The number of fully observed observations is 416. In contrastto the listwise deletion approach, incomplete observations with missing entrieswere included in statistical analysis.

Statistical Analysis

Usually, QOL analyses start with an EFA model to identify the appropriate numberof latent constructs that are likely to be present in an instrument. The conclusion


that is drawn from the EFA analysis is then confirmed via CFA, by testing thegoodness-of-fit of the conclusion to the data. For instruments that are specificallydesigned to have certain items relating to a particular latent domain, such as theWHOQOL-BREF instrument (The WHOQOL Group, 1998a), there is sufficientprior knowledge to suggest that the instrument can be represented by some latentdomains or factors. Hence, statistical analyses can be started with a CFA modelwith a given number of factors or a structural equation model with a specific pathdiagram (see the examples in Bollen, 1989; Fayers & Hand, 1997; Fayers &Machin, 1998). Based on a posited model, the first basic statistical analysis is toobtain good estimates of the unknown parameters, and then on the basis of the esti-mates and a goodness-of-fit statistic, evaluate the fit of the posited model to thesample data. Finally, it is desirable to have a method with which to choose a bettermodel from a number of competing models.


TABLE 1Frequencies of the Ordinal Scores of the Items

Ordinal Scores Number ofIncomplete

ObservationsWHOQOL Items 1 2 3 4 5

Q1 Overall QOL 3 41 90 233 107 1Q2 Overall health 32 127 104 154 58 0Q3 Pain and discomfort 21 65 105 156 127 1Q4 Medical treatment dependence 21 57 73 83 239 2Q5 Energy and fatigue 15 57 166 111 118 8Q6 Mobility 16 36 58 120 243 2Q7 Sleep and rest 28 87 95 182 83 0Q8 Daily activities 7 73 70 224 100 1Q9 Working capacity 19 83 88 191 91 3Q10 Positive feelings 2 30 141 241 59 2Q11 Spirituality/personal beliefs 13 45 149 203 61 4Q12 Memory and concentration 4 40 222 184 21 4Q13 Bodily image and appearance 9 46 175 137 106 2Q14 Self-esteem 13 72 130 210 50 0Q15 Negative feelings 4 54 137 239 39 2Q16 Personal relationships 8 46 68 218 134 1Q17 Sex 25 55 137 149 76 33Q18 Social support 2 23 84 228 136 2Q19 Physical safety and security 2 25 193 191 62 2Q20 Physical environment 4 29 187 206 43 6Q21 Financial resources 27 56 231 105 54 2Q22 Information and skills 5 27 176 194 70 3Q23 Recreation and leisure 10 99 156 163 47 0Q24 Home environment 9 27 53 235 151 0Q25 Health accessibility and quality 0 17 75 321 61 1Q26 Transport 8 38 61 253 113 2

Estimatesof theunknownparametersareobtainedonthebasisof theordinalcate-gorical data that are correctly treated by a underlying multivariate normal distribu-tion with a threshold specification. As the approach that is based on the polychoriccorrelation cannot be applied to large dimensional problems (e.g., EQS [Bentler,1992] only allows a maximum of 20 ordinal categorical variables) approaches thatuse advanced tools in statistical computing have been recommended (Lee & Zhu,2000; Shi & Lee, 2000). We analyzed the ordinal categorical data with the expecta-tion-maximization (EM) algorithm (Dempster, Laird, & Rubin, 1977). The EM al-gorithm is perhaps the best known tool for analyzing missing data and latent con-structs in statistical computing, and it has been extensively used for estimating manyimportant models in biostatistics (see Bock & Gibbons, 1996; Laird & Ware, 1982,among others), psychometrics (see Bock & Aitkin, 1981; Rubin, 1991, among oth-ers), and statistics (see Meng & van Dyk, 1997; Shi & Copas, 2002, among others).This algorithm is particularly useful for analyzing CFA or structural equation mod-els by treating the latent constructs as hypothetical missing data in the model (Jedidi,Jagpal, & DeSarbo, 1997; Lee & Tsang, 1999; Rubin & Thayer, 1982; Song, Lee, &Zhu, 2001). In this article, ML estimates of the unknown parameters in the CFA orstructural equation model with the incomplete ordinal categorical data are obtainedon the basis of the EM procedure that is given in Lee, Song, and Lee (2003), Jedidi etal. (1997), and Song et al. (2001). This procedure has the following advantages overthe multistage approaches (Lee et al., 1995) that are based on the polychoric correla-tions: (a) It gives the ML estimate, which is not only more optimal but also useful inthe computation of the Bayesian information criterion (BIC) for model comparison;(b) it is more efficient and can handle problems with a large number of ordinal cate-gorical items (variables); (c) it produces estimates of the latent construct scores (fac-tor scores) as by-products; and (d) it can handle missing data much more efficiently.A brief description of this procedure is given in the appendix.

Assessing the goodness-of-fit of a posited model is an important issue in apply-ing SEM to QOL analysis. This is usually done by testing a null hypothesis thatspecifies the proposed model fits the sample data. The traditional approach is touse a significant test on the basis of p values that are determined by the asymptoticχ2 distribution of the likelihood ratio test (Bentler, 1992; Jöreskog & Sörbom,1996). As pointed out in the statistics literature (Berger & Delampady, 1987; Kass& Raftery, 1995), there are some difficulties with this approach. For instance, the pvalue is only a measure of evidence against the null hypothesis; hence, not reject-ing the null hypothesis does not mean the null hypothesis is better than the alterna-tive hypothesis. Moreover, it cannot be used to test nonnested hypotheses. Hence,methods that do not have these difficulties are proposed in statistics as well as inSEM. One common approach is to consider the comparison of two nonnested com-peting models, say M1 and M2 (when applying to hypothesis testing, M1 and M2

represent two hypotheses, say H1 and H2, respectively). In particular, the followingBIC has been widely applied in the field of SEM (Lee & Song, 2001, 2003a;


Raftery, 1993). Given an observed dataset Z that may or may not involve incom-plete data,

where ��k is the ML estimate of the unknown parameter vector under Mk, dk is thenumber of the parameters in ��k, n is the sample size, and L0 (Z ; ��k, Mk) is thelog-likelihood function of the observed dataset Z evaluated at ��k under Mk. Accord-ing to Kass and Raftery (Raftery, 1993), the BIC12 is commonly applied to modelcomparison as follows: It gives evidence of support to M1 (hence accepting H1) ifBIC12 < 0; and gives evidence of support or strong evidence of support to M2 (ac-cepting H2) if BIC12 > 2 and BIC12 > 6, respectively; and when M1 is nested in M2,it cannot give a definite conclusion if 0 < BIC12 < 2. For structural equation modelswith fully observed data that come from the normal distribution, the log-likelihoodfunction values in BIC12 can be obtained directly using LISREL (Jöreskog &Sörbom, 1996) or EQS (Bentler, 1992); hence, the computation is easy. For struc-tural equation models with incomplete ordinal categorical data, the log-likelihoodcannot be directly obtained. In this article, it was computed via path sampling(Gelman & Meng, 1998), a well-known technique in statistical computing that hasbeen successful applied to structural equation models (Lee & Song, 2003a,2003b).

RESULTS

CFA

To study the measurement properties of the QOL instrument, we only used the datathat corresponded to the 24 ordinal items that reflected the latent constructs. Weproposed a CFA model with four correlated factors for accessing the domains:physical health, psychological QOL, social relationships, and environment. Ac-cording to the meaning of the questions, the structure of the proposed CFA is dis-played by the path diagram in Figure 2 (see Fayers & Hand, 1997 for a good de-scription of a path diagram). This CFA model defined a specific covariancestructure for fitting the data. To assess its goodness-of-fit to the sample data, we re-garded it as M2 and compared it with the general model M1, which has a generalnonstructured covariance matrix. In covariance structure analysis terminology, M1

is known as a saturated model. ML estimates of the parameters in the competingmodels were obtained by the EM algorithm, and the BIC12 was computed viaEquation 2. The value of BIC12 is 687.0. According to the criterion given inRaftery (1993), the proposed CFA model is definitely selected. Hence, the ob-served data give strong evidence of support to the CFA model with four correlated


� � � � � �12 0 1 1 0 2 2 1 2ˆ ˆBIC 2 log ; , log ; , log (2)L M L M d d n� ��

Z Z� �

factors. This conclusion is more meaningful and stronger than the conclusion ob-tained from the common asymptotic chi-square test.

To give a further illustration of the BIC application, we compared the selectedCFA model M2 with M3, which is a similar CFA model with four factors that areuncorrelated. We found that BIC32 = 378.25, and hence M2 should be selected.That means that a CFA model with correlated factors can fit the data much better.

The completely standardized (see Jöreskog & Sörbom, 1996) ML estimates ofthe factor loadings and factor correlations in the selected CFA model with four cor-related factors are given in Figure 2. To save space, threshold parameter estimatesare not presented. On the basis of the meaning of the questions, the latent factors


FIGURE 2 Path diagram and completely standardized ML estimates of the parameters in theCFA model. In all diagrams, double arrows are the factor correlations.

were interpreted as physical health, ξ1; psychological QOL, ξ2; social relationship,ξ3; and environment, ξ4. This interpretation agrees with that given by theWHOQOL Group (1998a). All of the factor loading estimates, except for that cor-responding to sexual activity and social relationship, which is 0.126, were high.This indicates a strong association between each of the latent constructs (factors)and its respective items.

Each zi (yi) in the factor analysis model was associated with a vector of factorscores,�i =(ξ1i,ξ2i,ξ3i,ξ4i).Theproposedestimationproceduregave the factor scoreestimates, � , , , ,� i i m�1� as by-products. Histograms that were obtained from{� , , , },ξki i n�1� k = 1, 2, 3, 4, are presented in Figure 3. It is evident that the empiricaldistributions of the factor scores were close to normal with zero mean values. Thesample correlations {φ12, φ13, φ14, φ23, φ24, φ34} that were obtained from this sample


FIGURE 3 Histogram of the factor score estimates.

of factor scores estimates were {0.74, 0.55, 0.92, 0.70, 0.87, 0.84}. Although all ofthese correlation estimates were generally larger than the ML estimates of the corre-lations given in Figure 2, the differences were not substantial. Hence, these factorscore estimates can be used for further statistical inference.

Structural Equation Models

To address the causations of the four identified latent constructs to the health-relatedQOL,weappliedSEMto thedataset thatadditionallycontained theoverallQOLandgeneralhealth items.Wefirst studied therelationofQOLandthe latentconstructsξ1,ξ2,ξ3, andξ4 bymeansofSEM.Thepathdiagramof theproposedstructuralequationmodel for analyzing the overall QOL with the 24 ordinal categorical items is dis-played in Figure 4. This structural equation model, which is called M4, defines acovariance structure for the 25 ordinal categorical items. To assess its good-ness-of-fit to the sample data, it was compared with M5, the general saturated modelwith 25 observed variables that had a general nonstructured covariance matrix. Wefound that BIC54 = 492.58. This result clearly indicated that M4 is much better thanthe saturated model. ML estimates of the parameters in M4 are also presented in Fig-ure 4. As the structural equation model was different from the previous CFA model,the estimates of the factor correlations among the latent constructs ξ1, ξ2, ξ3, and ξ4

were not exactly the same as those obtained on the basis of the previous CFA model.However, asexpected, theirdifferenceswereveryminor (in the thirddecimalplaces;hence, some estimates in Figure 3 appeared to be the same as those in Figure 2). Thisis also true for the factor loading estimates, although the differences were larger. Themost important additional result that can be obtained using a structural equationmodel is the followingestimatedstructural equation in relation to theendogenous la-tent construct QOL(η) and the exogenous latent constructs ξ1, ξ2, ξ3, and ξ4:

η = 0.46ξ1 + 0.24ξ2 + 0.21ξ3 + 0.13ξ4 + ζ (3)

with an estimated residual variance 0.19. This estimated structural equation indi-cates that physical health (ξ1) had the most substantial causal effect on QOL(η),which was larger than the causal effects of psychological QOL (ξ2) and social rela-tionship (ξ3), which were in turn larger than the causal effect of environment (ξ4).It is also interesting to observe that causal effects of psychological QOL and socialrelationship were basically the same. Moreover, it is also nice to see that the esti-mated residual variance was not large.

Furthermore, we investigated the relation of the health-related QOL (HRQOL,η) with the latent constructs. The latent construct HRQOL was formed by the over-all QOL item and the general health item. The path diagram of the proposed struc-tural equation model, which is called M6, is displayed in Figure 5. To assess thegoodness-of-fit of M6, we again compared it with the saturated model M7 with 26observed variables. We found that BIC76 = 401.35, which clearly indicated that theproposed model M6 was much better than the saturated model. The ML estimates


of the parameters in M6 are presented in Figure 5. We also observed that the behav-ior of the factor loading and factor correlation estimates was similar to that in theanalysis of M4. Now, the estimated structural equation that addresses the causal ef-fects of the latent constructs ξ1, ξ2, ξ3, and ξ4 to HRQOL(η) is:

η = 0.68ξ1 + 0.28ξ2 + 0.08ξ3 + 0.05ξ4 + ζ (4)

with an estimated residual variance of 0.08. Hence, for the health-related QOL,which involves an additional component on the general health of the patients when


FIGURE 4 Path diagram and completely standardized ML estimates of parameters in thestructural equation model for the overall QOL.

comparing the single overall QOL, only the physical health (ξ1) and psychologicalQOL (ξ2) had substantial causal effects, whereas social relationship (ξ3) and envi-ronment (ξ4) were less important.

DISCUSSION

In QOL or HRQOL analysis, the fundamental aim of factor analysis and SEM is tofind an appropriate model for assessing the relations among the observed items in


FIGURE 5 Path diagram and completely standardized ML estimates of parameters in thestructural equation model for the HRQOL.

the questionnaire and the underlying endogenous and exogenous latent constructs.EFA and CFA models are used to explore and confirm the appropriate number oflatent domains for explaining the items. A main objective of this article is to applySEM to study the causal relations of the latent constructs to the QOL or theHRQOL.

A slightly different approach using a hierarchical measurement model, which isbasically a second-stage factor analysis model, may also be considered. In this ap-proach, a new latent variable η* for addressing QOL was constructed from the la-tent factors (physical health, mental health, social relationships, and environment).This η* can be compared with η in Figure 4 that is directly obtained from the over-all QOL (Q1). However, to avoid confusing the SEM approach with a second-orderfactor analysis approach, detailed analyses are not included here.

As the observed ordinal categorical observations are clearly nonnormal, morerigorous statistical methods that take into account the discrete nature of the ob-served data may be preferable to the standard methods in the commonly usedpackages. This article introduces the ML approach for analyzing structuralequation models with ordinal categorical data that can be MAR. We utilized re-cently developed tools in statistical computing to obtain the results. It is not toodifficult to implement computer programs to obtain these results. In fact, thesoftware BUGS (Gilks, Thomas, & Spiegelhalter, 1994, freely available atwww.mrc-bsu.com.ac.uk/bugs), can be utilized to complete the E step of theproposed EM algorithm. However, as far as we know, there is no user-friendlysoftware for obtaining the ML solution. Admittedly, this is a weakness of therigorous ML approach. In behavioral, educational, medical, and social-psycho-logical research, the use of recently developed tools in statistical computing suchas the EM algorithm and the Markov chain Monte Carlo (MCMC) methods arebecoming more and more popular, mainly because they are very helpful in solv-ing complex problems. Due to the existing demand, we expect that user-friendlysoftware will soon be developed. In the meantime, we are willing to share ourprogram with others. For the general problem of assessing QOL, an alternativeapproach is to provide interval response scales in the questionnaires (Skevington& Tucker, 1999).

In this article, we assume for brevity that missing data are MAR with anignorable mechanism. The observed-data log-likelihood is L0(Zobs;�). For themissing data associated with the sensitive question Q17, it may be better to analyzethem with a nonignorable missing mechanism. To consider this more general situ-ation, we need to model the matrix of missing indicators, R, via a distribution witha vector of unknown parameters, say �. The observed-data log-likelihood L0*(Zobs,R; �, �) is more complicated. The EM algorithm can be applied to L0(Zobs; �) orL0*({Zobs, R; �, �). However, the treatment of L0* is much more technical. Hencewe regard it as a topic for future research.


It is well known that QOL and HRQOL are complicated concepts, and the un-derlying observed data are very complex in general. Generalizations of the stan-dard structural equation model, for example, the nonlinear structural equationmodels (Lee & Song, 2003b; Lee, Song, & Poon, 2004), multilevel structuralequation models (Song & Lee, 2004a; Lee & Song, 2004), diagnostic analysis(Lee & Lu, 2004; Song & Lee, 2004b; Lee & Xu, 2004), and finite mixtures ofstructural equation models (Lee & Song, 2003a; these are generalizations of multi-ple-group structural equation models), appear to have the potential to provide rig-orous statistical analysis of these complicated concepts in a variety of situations.

ACKNOWLEDGMENTS

This research was supported by Hong Kong UGC Earmarked grant 4243/03 H. Weare thankful to Professor M. Power for helpful comments. We are solely responsi-ble for any mistakes.

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APPENDIX

Let � be the overall parameter vector of the CFA or structural equation model of in-terest, and let Zobs be the observed data of the ordinal categorical items. As thestructure of Zobs is rather complicated, obtaining the ML solution by directly maxi-mizing the observed-data log-likelihood L0(Zobs; �) is very difficult. Like many ap-plications in statistics, biostatistics, and psychometrics, the data augmentationstrategy is applied here to solve the difficulty. The key idea of data augmentation isto augment the observed data Zobs with the latent quantities, which include Y =(y1,···, yn), the continuous measurements underlying Z, and �, the latent con-


structs in the model. For example, for a CFA, � = (ξ1,···, ξn), the collection of alllatent factors corresponding to Y. In the complete dataset (Y, �, Zobs), Y and � aretreated as missing data. The well-known EM (Dempster, Laird, & Rubin, 1977) al-gorithm is applied to obtain the ML estimate of �. This algorithm is implementedas follows. At the rth iteration of the algorithm with a current value �(r):

1. E-step: Evaluate Q(�; �(r)) = E{Lc(Y, �, Zobs; �)|Zobs, �(r)}, in which theexpectation is taken with respect to the conditional distribution of (Y, �)given Zobs at �(r).

2. M-step: Determine �(r+1) by maximizing Q(�; �(r)).3. Check convergence. If the algorithm has not converged, then update r to r +

1 and return to the E-step of the (r + 1)th iteration.

For strucutral equation models with ordinal categorical data, a common anduseful method for evaluating the E-step is to approximate the conditional expecta-tion by a sufficiently large number of observations that are simulated from the un-derlying conditional distribution (Y, �) given Zobs. The basic concept is to approx-imate the mean by sufficiently large observations of the underlying distribution.The task of drawing observations from the conditional distribution of (Y, �) givenZobs at �(r) is conducted by the most popular MCMC method, namely the Gibbssampler (Geman & Geman, 1984). The algorithm produces a sample {(Y(j), �(j)): j= 1, ···, J} for every E-step at the rth EM iteration. This sample will be used to ap-proximate the E-step of the EM algorithm. More specifically,

The M-step updates the unknown parameters by solving the following system ofequations

In this article, following the recommendation of Gilks et al. (1994), a closed formsolution of the preceding equation is obtained by conditional maximization (Meng& Rubin, 1993). Convergence of this Monte Carlo type EM algorithm is monitoredby the approach given in Shi and Copas (2002).

At the E-step of the last iteration of the EM algorithm, where the current valueof the parameter vector is ��, the ML estimate of �, we have a sample {�(j); j = 1,···,


� � � �( ) ( ) ( )

1

1; , , ;

Jr j j

obsj

Q LcJ �� Y Z��

� �( );0

rQ��

�

� �

�

J} that is simulated by the Gibbs sampler from the conditional distribution of �given Zobs. Hence, estimates of the factor scores in � can be obtained from

These sampling estimates have better statistical properties than those obtainedfrom standard pa\ckages (Bentler, 1992; Jöreskog & Sörbom, 1996) via the regres-sion method (see Bock and Aitkin, 1981, and Rubin, 1991).


( )

1

1ˆJ

j

jJ �

� ��

Documents

Application SEM to QOL. Lee n Song 2005.pdf