Health-related quality of life in HIV-1-infected patients on HAART: a five-years longitudinal analysis accounting for dropout in the APROCO-COPILOTE cohort (ANRS CO-8)

Abstract

Background

The long-term efficacy of Highly Active Antiretroviral Therapies (HAART) has enlightened the crucial role of health-related quality of life (HRQL) among HIV-infected patients. However, any analysis of such extensive longitudinal data necessitates a suitable handling of dropout which may correlate with patients–health status.

Methods

We analysed the HRQL evolution over 5 years for 1,000 patients initiating a protease inhibitor (PI)-containing therapy, using MOS SF-36 physical (PCS) and mental (MCS) scores. In parallel with a classical separate random effects model, we used a joint parameter-dependent selection model to account for non-ignorable dropout.

Results

HRQL evolved according to a two-phase pattern, characterized by an initial improvement during the year following HAART initiation and a relative stabilization thereafter. Immunodepression and self-reported side effects were found to be negative predictors of both PCS and MCS scores. Hepatitis C virus coinfection and AIDS clinical stage were found to affect physical HRQL. Results were not significantly altered when accounting for dropout.

Conclusion

Such results, obtained on a large sample of HIV-infected patients with extensive follow-up, underline the need for a regular monitoring of patients–immunological status and for a better management of their experience with hepatitis C and HAART.

Appendix

The linear random effects model

Let y _ij be the transformed score (MCS or PCS) measured for patient i at time s _j , i = 1,...,N,j = 1,...,7. We used a piecewise linear random effects model with a change in the slope at M12, written as:

Here x _1i (t) and x _2i (t) are vectors including fixed and time-varying explanatory variables, and (t _j -12)⁺ = t _j - 12 if t _j ≥ 12, and 0 otherwise. U _0i is the subject-specific intercept, U _1i and U _2i are the random slopes, modelling the true individual level trajectories after they have been adjusted for the overall mean trajectory and the other fixed effects, such that (U _0i , U _1i , U _2i )′ ˜ N(0, G and ε _ij ˜ N(0, σ₂) are mutually independent measurement errors. The first term α₀ + x _1i ′(t _j )′ ₀ + U _0i includes principal effects of the factors, the second term α₁ + x _2i ′(t _j )′ ₁ + U _1i represents the changes in the short term evolution (M0–M12), and the third term α₂ + x _2i ′(t _j )′ ₂ + U _2i represents the changes in the long-term evolution (M12–M60).

We used the random effects covariance matrix G to perform a likelihood ratio test for choosing the number of random effects in the model. We tested whether we needed a model with q = 3 random effects (i.e. two random slopes: U _1i , U _2i ) or whether a simpler model, with only two random effects, would be more adequate. We fitted both models with the same set of fixed effects and recorded the deviance (− Log Likelihood) for each of them. The distribution of the difference in deviances is a mixture of a χ² with q degrees of freedom and a χ² with q− degrees of freedom [45].

For the MCS (resp. PCS) transformed score we found an increase in the deviance of 16.2 (resp. 23.6) for q = 3 degrees of freedom, so the test rejected the specification with two random effects in both cases. Nevertheless, the estimated G matrix for the model with three random effects was almost singular, which suggested that the variability in the last slope was very small after considering heterogeneity which was accounted for by the fixed factors. Moreover, the estimated parameters and their P-values were very similar for the two models, so we finally decided to include only two random effects in the longitudinal model, which allowed a faster estimation of the joint model.

The time-to-dropout model

Patients had scheduled follow-up visits at fixed times (M0, M12, M28, M36, M44, M52, M60). The proposed model allowed the investigation of the relationship between times-to-dropout and a set of possibly time-dependent explanatory variables. When dropout time is discrete with tied events without underlying ordering, this model is equivalent to a logistic model (“discrete-time logit model– for a data set with patient-time as the unit of analysis, which is the same data set used for the mixed longitudinal analysis [46]. For each of these observations the response variable w is dichotomous, corresponding to 1 = “dropout–and 0 = “still in the study–in that time interval. Dropouts due to death or withdrawal from the study, as well as all observations corresponding to patients followed until the end of the study (M60) were censored. A censored observation can be modelled by having a record of all zeros until the end of the follow-up.

The time scale of the study follow-up is partitioned into six disjoint intervals, say [t _h , t _h+1), H = 1,...,6. Let T _i be the discrete time-to-dropout variable of the patient i. Then the hazard of dropout in the interval [t _h , t _h+1) for the patient i given covariates z _i is the conditional probability that its dropout is at time t _h+1, given that it was still in the study at the beginning of the interval:

Using elementary properties of conditional probabilities, it can be shown that Suppose that [) is the observed interval of dropout of the ith subject, and δ _i = 1 if and 0 otherwise. Let h(i) be such that . The likelihood for the discrete time data is given by where w _ij = 1 if j = h(i) + 1 and 0 otherwise. Prentice and Gloeckler [16] have shown that where λ_0h is the logarithm of the integral of the baseline hazard on the relevant interval [t _h , t _h+1). This can be written equivalently, Given this complementary log–log transformation, the parameter γ is interpreted as the effect of covariates in z _i on the hazard rate of dropout, assuming the hazard rate to be constant on each interval of the study.

As noted by Vonesh et al. [4], one advantage when using such a model is that it allows the inclusion of time-dependent covariates, defined at the start of each interval [t _h , t _h+1. In this way, the inclusion of time-dependent covariates is not affected by the fact that an actual measurement may not be available for them at the time-of-dropout. Moreover, the specification of the time-to-dropout model gives a closed-form expression for the marginal and joint likelihood, allowing us to use existing standard software like the SAS procedure NLMIXED to estimate both the separate and joint model (see also Guo and Carlin [8]).

The log-likelihood function for the sample of individuals used in this study can then be given by This takes the form of a “sequential binary response–model with data in the “vertical–form with the ith subject contributing h(i) observations. Allison [47

47] and Jenkins [48–50] have shown that, for such suitably organized data, the log-likelihood function is the same as the log-likelihood for a generalized linear model of the binomial family with complementary log–log link function.

The joint model

In the joint approach, the linear random effects model and the time-to-dropout model are estimated simultaneously (see Guo and Carlin [8]). The discrete-time proportional hazard model is now written as follows: where U ₀ and U ₁ are the random effects that appear in the linear random effects model. In this way, the HRQL model and the time-to-dropout model are connected through the stochastic dependence on unobserved factors represented by these random variables. The parameters γ₀ and γ₁ in the time-to-dropout model measure the association between the two submodels induced through these two common latent variables. In the absence of association between the two models, the parameters γ₀ and γ₁ are not statistically significant and there is no gain from the joint analysis.

As in Guo and Carlin [8], the binary explanatory variables were coded with 1 for the active modality and − for the reference modality.

The SAS codes used to estimate the separate and joint models for the MCS transformed score are given below.

Determination of factors–contribution to the untransformed HRQL scores changes between M0 and M12

In order to have an idea of the contribution of each factor on the HRQL scores changes between M0 and M12 on their original scale (varying between 0 and 100), we adopted the following approach. For each factor, the corresponding univariate two–phase random effects model on transformed HRQL scores was translated back onto the original scale using the inverse transformation of -ln (100-score). Then, the factor’s contribution to the HRQL scores evolution during the first year of HAART was calculated as the difference between the score values estimated by the back translated model at M12 and M0. The same approach was used to determine the intercepts–contribution, from the two-phase random effects model without covariates.

Random slopes and variance components parametrization for the mixed model:

• proc mixed data = mydata covtest method=ml; class patient; model lnmcs = obstime1 obstime2 sexe_r naif_tr naif_tr_1 naif_tr_2 part_pr confort confort_1 confort_2 homo toxico cd4t_200 nbs30 nbs30_1 nbs30_2 nbtm_1 nbtm_2 / influence(iter = 5 effect = patient est) s cl; random obstime1 intercept / subject = patient type = fa0( 2 ); ods output covparms = cp; ods output solutionf = solf; ods output Influence = inf; run;

Here obstime1 and obstime2 are the two time variables defined in the “Statistical methods–section (i.e. (t _j , 12) and (t _j − 12)₊ respectively). The legend for the rest of the covariates is given at the end of this Appendix. Multivariate influence analysis was performed using the INFLUENCE option of the MODEL statement, in order to test the stability of the linear mixed model to perturbations of the data. According to the likelihood distance and Cook’s D measure [51], two patients identified as influential observations were dropped from the basis for each HRQL score.

As in Guo and Carlin [8], we used the FA0(q) structure (No Diagonal Factor Analytic) for the G matrix in the RANDOM statement of SAS PROC MIXED, where q is the number of random effects. This parametrization is equivalent to specifying a Cholesky root for the G matrix, to constrain it to be non-negative definite. The lower triangular Choleski root of G is a lower triangular matrix C, with non-negative diagonal elements that is a ‘square root’ of G in the sense that CC–G.

The time-to-dropout model

• proc logistic data = mydata; model w(event=––= time1–time6 homo toxico ue_bis cd4t_200 naif_tr confort sexe_r ageinc logem alc / noint link = cloglog technique = newton; run;

Here w is the dropout indicator variable, and time1 to time6 are the indicator variables for the six intervals between the visits. The legend for the rest of the covariates is given at the end of this Appendix.

The NOINT option was specified to prevent the redundancy in the estimation of the coefficients of time1–time6 indicator variables. The Newton–Raphson algorithm with Gauss–Hermite quadrature was used for the maximum-likelihood estimation of the parameters.

The joint model

The following statements were used to estimate the joint model with two random effects (u0 and u1, corresponding to the intercept and the first slope), and the same list of predictors used in separate analyses. This is a modified version of the model XI in Guo and Carlin [8] (see also their SAS code supplied at http://www.biostat.umn.edu/~brad/software.html).

The estimate of the longitudinal and dropout separate models were combined in a table named SeparateEstimates. This data set was used to provide starting values for the joint model, cf. PARAMETERS statement.

Unlike Guo and Carlin’s [8] program, we used a discrete-time model, rather than a continuous one. This way, the computation of the log likelihood contribution of the dropout data is not made when the last observation of a subject is reached, but instead for each observation in the vertical dataset of follow-up visits.

Variance and covariance of the random effects were obtained by the delta method, using ESTIMATE statement. Approximate 95% prediction intervals could then be obtained by assuming asymptotic normality.