Structural Equation Modeling to Detect Predictors of CD4 Cell Count Change due to Long Term Antiretroviral Therapy Administered to HIV-Positive Adults at Felege Hiwot Teaching and Specialized Hospital, Bahir Dar, Ethiopia

Longitudinal data is a core for researching exploration of changes of the various outcomes across a wide range of diesoline and different techniques are existed for analyzing such data ¹. One of the approaches to study raw change scores which is computed as the deviation of outcomes/ values at time 1 and time 2 and these changes analyzed as a function of individual or group characteristics. The raw changes can be analyzed using t-test, ANOVA or multiple regression ². An alternative to this , residual change scores can be computed as the residual between observed at time 2 and expected at time 2 results/values predicted by the time 1 values/ outcomes ³. Such changes are typically analyzed using multiple regression or analysis of covariance (ANCOVA) ⁴. Both raw and residualized change scores are useful for changes between two discrete time points analyzed in longitudinal prospective research designs ⁵. Both raw and residualized change scores are useful for changes between two discrete time points analyzed in longitudinal prospective research designs. Frequently, it is interesting to modeling developmental trajectories or patterns of changes in multiple time points and common approache to study such trajectories is to use standard growth analyses like repeated measures MANOVA or Structural Equation Modelling (SEM). Standard growth analyses estimate a single trajectory obtained by averaging from the individuals trajectories of participants ⁶. Structural equation modelling (SEM) consists of a system of simultaneous linear equations that describes relations among different variables ⁷. It also consists of non-linear equations those describe pattern of variance and covariance among variables. It includes Confirmatory Factor Analysis (CFA), Path Analysis (PA), Partial Least Square and Latent Growth Modelling (PLSLGM)⁸. These models are applied to evaluate unobservable (latent variables) and call upon an evaluation model that refers to latent variable applying one or more observable variables ⁹. Structural equation modeling has been developed mainly in the social and behavioral sciences. Structural equation modeling represents the complex relationships of variables as a sequence of linear equations. A prominent feature that distinguishes SEM from the more mainstream linear statistical modeling is the inclusion of latent factors in SEM. Previous researchers analyzed the longitudinal data that referred to as latent (growth) curve modeling (GCM) ^{10, 11}. SEM is a power full mechanism that helps to combine complex path models with factors ¹². It is a technique which assists to specify confirmatory factor analysis, regression models as well as complex path analysis. ML estimation for the constraints of particular structural equation with inadequate source of information can be applicable in SEM ¹³. Two stage least square estimator with its asymptotic distribution is inversely incorporated in SEM for the purpose of parameter estimation ^{14, 15}. The link among the constructs of SEM may be approximate with self-determining regression equations or using advanced techniques. SEM has the ability to indicate associations among unobservable variables with manifest variables^{7, 16}. SEM revealed mathematical parameter estimations (arrows) in the model with the extent of the associations. Further, to evaluate the theoretical constructs, it gives a chance to discover observable predictors that considerably expresses factors / latent variables. Researchers frequently use longitudinal data analysis to study the trends of health related issues ¹⁷. They might study how an elderly person’s CD4 cell count changes over time or how a therapeutic intervention affects a certain behavior over a period of time. Earlier contributions are found in the sociological and psychological literature ^{18, 19}rather than modeling count or continuous response. The previous researches in SEM longitudinal data analysis mainly focused on main effects and not explored the interaction effects of covariates^{17, 19}.

This paper introduces the structural equation modeling (SEM) approach to analyzing longitudinal data applying SEM. Basic interest in structural equation modelling is the conceptual constructs denoted by unobservable variables.The main advantage of the SEM is the ability to estimate the direct and indirect pathways of the effect of the primary independent variable on the outcome, given sufficient sample sizes. In longitudinal studies, data are collected from subjects at several time points. The SEM framework is a general modeling framework and allows the modeling of potentially complex relationships among observed and latent variables and can be applied in the longitudinal data setting. The main purpose of longitudinal analysis is to study the trends or trajectories of the variables of interest. For example, after a medical intervention, health measures might be taken every few months to monitor the health status of patients. Will their health improve, decline, or stay the same in the subsequent months or years? Do all the patients show the same health trajectory? The primary objective of the analysis is to evaluate the overall effect (main and interaction effect) of predictor variables on CD4 cell count.

The baseline characteristics of study variables are indicated in Table 1.

As shown in Table 1, out of the sample of 792 patients, 40.9 % were rural residents, 50.6 % were females, 44.8 % were living with their partner, 72.6 % disclosed their disease to family members, and 50.5 % were owners of cell phones. Lastly, 68.2% of the patients had good adherence and the rest were non-adherent (Fair +Poor) patients. Few of patients (11.5%) were high income level and 20% of them had no education.

Model fitting for CD4 cell count data using structural equation modelling

One means of assessing the determinants of the change of CD4 cell count is Structural Equation Modelling (SEM). Considering the commonly significant covariates on the variable of interest on the previous chapters, let us apply structural equation modelling to see whether or not the significant variables found above are also significant in this case. In our case, all dependent and independent variables are manifest/observable variables. Let rectangles in Figure 1 represent the manifest variable and circles for errors that can be created during estimation of parameters²⁶.

Considering the current change of CD4 cell count as endogenous variable, the predictor variables (HAART adherence, weight, age, baseline CD4 cell count, visiting time and Cell phone ownership) found as significant variables from the previous chapters can be considered as exogenous variables. Since CD4 cell count results from the two previous results (prior one unit from the current and prior two units from the current) in transition model were significant for the current change, they had been included as predictor variables. Consider the first two lag variables (lag-2 and lag-1) as exogenous variable as shown in Figure 1²⁷.

Figure 1.Single factor measurement model for CD4 cell count change for adult HIV patients

In Figure 1, the change in current CD4 cell count at lag-2, CD4 cell count change at lag-1 and current CD4 cell count change were considered as endogenous variables whereas ownership of cell phone, baseline CD4 cell count, adherence, weight, age and visiting times were considered as exogenous variables. 0₁, 0₂, 0₃ are observed linkages with adherence to their endogenous variables (CD4 cell count change for lag-2, lag-1 and current CD4 cell count change) and 0₄, 0₅ , and 0₆ are linkages with weight to its endogenous variables. Similarly, let 0₇, 0₈ and 0₉ be linkages from age, 0₁₀, 0₁₁ and 0₁₂ be linkage with visiting times, 0₁₃ 0₁₃, 0₁₄ and 0₁₅ are linkages with initial CD4 cell count and 0₁₆, 0₁₇ and 0₁₈ are linkages from ownership of cell phone. (Table 2)

The goodness-of-fit statistic is indicated in Table 1. The chi-square test statistics is not significant at 0.05 which indicates that the model is good and accepted²⁸. The root mean square error approximation (RMSEA) was 0.00334 that is less than 0.05. The goodness-of-fit index (GFI) and adjusted goodness-of-fit index were 0.9973 and 0.9855 respectively. Such results indicate that the model was good to fit the data at 95% CI and gave, X²₁₅= 9.96, P-value = 0.09843.

From Table 2 (RMSEA model), the p-value for default model is 0.7820 which indicates that there is no evidence for rejection of the null hypothesis that states the model is good. So, we have to choose the saturated model rather than the null model. (Table 3 and Table 4)

(Table 5) indicated that current CD4 cell count change increased as adherence increased. Similarly, CD4 cell count change at these stages increased for patients having cell phone. In addition to predictors to CD4 cell count change, the previous two responses (CD4 cell count change at lag2 and lag1) had significant effect on the current CD4 cell count results. The expected log of CD4 cell count change increased as level of adherence increased. Likewise, for one unit increase of the expected log of CD4 cell count change at lag-2, the expected log of the current CD4 cell count change increased by 0.24 cells per mm3, keeping the others constant.

The correlation structure between E1<--->E4, E2<--->E3, E3<--->E5, E2<--->E8, E6<--->E8, E4<--->E8, E7<--->E8 and E3<--->E8 had significant effect on the relationship between endogenous and exogenous variables. In order to assess estimated value of linkage for each covariate on CD4 cell count change, standardized regression weights and the structural equation model are needed.

In current investigation, the structural equation models for analysis of longitudinal data on univariate models of observable variables (CD4 cell count change) that are conditional to the other variables (time-varying or time invariant) were reviewed. Hence, in the paper keeping the statistical theory to be a practical guide for analysing longitudinal data, SEM applied for analysis of longitudinal data(CD4 cell count change and its predictors). However, considering the readers’ concept and prior knowledge of SEM, the investigators largely avoid dwelling on the basis of SEM. Although common software packages such as SAS and R have the capability to run SEMs, software designed specifically for SEMs ²² may be more intuitive and user-friendly in model specification, particularly in the development of highly complex models.

The current study examines one specific setting of mediated longitudinal data. Other situations with different data structures where mediation is present could also be explored, e.g. situations where the mediator and the primary independent variable as well as the outcome are repeatedly measured, categorical outcomes, and settings with more complex pathways between variables. In addition, we specifically explored the question of whether the LMM performs sufficiently in a setting favorable to the SEM. Future studies examining broader settings where the data arise from non-SEMs would provide further insight into the use of the LMM and SEM in mediated longitudinal settings. First, we found the factor loadings and intercepts of HD (health distress) and EF (energy and fatigue) not to be invariant across measurement occasions and, second, we found direct effects of CD4-cell count on EF and RF (role-functioning)²⁹. The first two findings of measurement bias are considered as response shift by definition, as the measurement invariance is violated by the time of the measurement occasion. However, upon inspecting the HD and EF parameter estimates (Table 2) there did not appear to be an obvious substantive explanation for the changes in the factor loadings of HD. The other two findings of measurement bias are considered as response shift only if they vary with time. The bias in EF with respect to CD4-cell count is consistent over time and therefore not considered as response shift ³⁰. The bias in RF with respect to CD4-cell count did vary with time, but again, it was difficult to provide a substantive explanation for this so-called response shift ³¹. Perhaps some of our results are chance findings, despite our best attempts to guard against such findings.

The Bonferroni adjustment of the level of significance guards against inflation of the family-wise error rate, but the chi-square difference test can still be affected by model complexity and sample size ³². In a simulation study, Cheung and Rensvold (2002) considered various alternatives to the chi-square difference test for testing across group constraints in multi-group factor analysis, and recommended inspection of differences in Bentler’s (1990) comparative fit index (among others). In our longitudinal factor analysis, we complemented the chi-square ifferences with ECVI differences, really only in order to provide additional information about the necessity of further modifications that cannot be substantively justified. In the present analyses, the ECVI differences generally agreed with the chi-square difference tests at Bonferroni adjusted levels of significance. One notable exception was that according to the 90% confidence interval of the ECVI difference, the fit of Models 2.2 and 2F was essentially equivalent, suggesting that constraints on EF factor loadings and intercepts could have been retained.

It should be noted that most response shift researchers in substantive areas of psychology contend that response shifts are the result of some catalyst event, such as an intervention in educational research (Howard et al. 1979), or a health state change in medical research (Sprangers and Schwartz 1999). In the HRQL study of HIV/AIDS patients, there is not a well defined event that all respondents have in common, other than having been diagnosed with HIV or AIDS some time ago. However, the time since diagnosis and the time on HAART vary greatly across patients and cannot be considered true catalysts. The one thing all patients have in common is that they participate in the HRQL study, and that they complete HRQL tests every half year. The test taking itself can have an effect on their response behaviour, which may change with time. The patients may become more accustomed to both their disease and taking the test, which perhaps induces a response shift. It should also be noted that most work on response shift in substantive psychological research was not aimed at investigating measurement invariance, but rather at explaining paradoxical intervention effects. Seeing that research into response shift was hampered by researchers having different conceptions of response shift, Oort (2005b) proposed to formally define response shift as a special case of measurement bias, although some researchers may still have another perspective on response shift (Oort et al. 2009).

As is illustrated by the empirical example, Step 2 and Step 3 of the detection procedure are laborious and time consuming. Especially if the numbers of observed variables and exogenous variables are large, these two steps involve the fitting of numerous models, in order to evaluate the chi-difference tests. An advantage of using modification indices is that, within each iteration, the researcher only has to fit a single model. Therefore, although perhaps less sound (Kaplan 1990), we explored the use of the modification index as an alternative to the global tests with multiple degrees of freedom.

When we evaluated the modification indices with the Bonferroni adjusted levels of significance, none of the findings were significant because of the large number of tests under consideration (e.g. 120 in Step 2). When testing at less conservative levels of significance, for example by considering tests of intercept constraints first and factor loading constraints second, or by simply raising the family-wise level of significance, there was a number of modification indices that reached significance.

However, as multiple modification indices were about equally large, the choice of which constraint to remove first seemed arbitrary, yet highly consequential for the removal of constraints in subsequent iterations, leading to very different conclusions. In addition, we also had to be careful not to run into constraint interactions. Still, the most important problem with relying on modification indices and less conservative testing was that many of the modifications were difficult to interpret and that the number of iterations grew very large. Saris et al. (2009) suggest only modifying models if the modification indices are associated either with moderate (instead of high) statistical power or with substantial expected parameter changes. When statistical power is high, one can only rely on substantive arguments for modification (ibidem), which we did, as in the present analyses the power to find medium sized differences was consistently above 99%.

In such situations, the decision making becomes increasingly subjective, as researchers will have to base their decisions between modifications and when to stop modifications on the interpretability of the different modifications. It is therefore highly likely that different researchers, with different substantive knowledge and different interpretation skills, will end up with different conclusions when analysing the same data. As can be seen from the procedure using modification indices, subjectivity in measurement bias detection influences whether and where bias is found. Notall researchers may want to test every possible combination of tenable equality constraints.

When this is the case, a priori hypotheses driven by theory should be stated before analysis and only these tests should be conducted. Under these circumstances, chance findings may further be reduced and more generalisable results found.

The problems associated with devising an objective procedure for measurement bias detection is common to specification searches in general. Bollen (2000): “Modelling strategies are subject to debate for virtually all statistical procedures. Witness the sharp disagreements over stepwise regression, the interpretation of clusters in cluster analysis, or the identification of outliers and influential points. The largely objective basis of statistical algorithms does not remove the need for human judgment in their implementation.” Similarly, when investigating measurement invariance, it is impossible to completely remove the element of human judgement. This is certainly true for the substantive interpretation of apparent measurement bias. However, we think that the procedure presented in this paper, with its safeguards against chance findings, at least helps to more objectively decide which measurements are biased and which are not.

Not applicable

Amhara Region Health research & laboratory Center at Felege-Hiwot Referral Hospital, Ethiopia is gratefully acknowledged for the data supplied in our health research.

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