The statistical model considered in this study for the data analysis belongs to the class of the varying-coefficients regression models as introduced by [12] after [13]. In this setting the regression coefficients are allowed to vary with respect to some covariates called effect modifiers or moderators. When dealing with the framework of the linear model, the dependence of the regression coefficients β j of a predictor X j on effects modifiers should be understood as the expression of an interaction between X and the effects modifiers. Let’s consider a response variable Y conjointly with covariates (X, Z) where X = ( X 1 , ⋯ , X p ) and Z = ( Z 1 , ⋯ , Z q ) . Let’s denote D ( X ) and D ( Z ) respectively the spaces of the potential values of X and Z and μ ( x , z ) = E ( Y | X = x , Z = z ) with x in D ( X ) and z in D ( Z ) ; the varying-coefficients model will take the following form: μ ( x , z ) = β 1 ( z ) x 1 + ⋯ + β p ( z ) x p . For the present study the functions β j ( z ) are modeled through a binary tree as follows:

β j ( z ) = β j 1 I n d ( z ∈ N 1 ) + ⋯ + β j K I n d ( z ∈ N K )

where ( N k , k = 1 , ⋯ , K ) is rectangular partition of D ( Z ) and I n d ( ⋅ ) is the indicator function. This choice offers is a flexible way to consider interactions between the covariates Z and the covariates X without a tight specification of such interactions at the model statement step. Moreover this framework allows the identification of subgroups of individuals with specific shape of CD4 variation over time. Finally the model that will be considered for the data analysis will be expressed as follows:

Y | X = x , Z = z , b ∼ N ( x t β ( z ) + b I , σ 2 I ) (1)

where

β ( z ) = ∑ k = 1 K     I n d ( z ∈ N k ) β k (2)

b ∼ N ( 0, σ b 2 ) (3)

and I n d ( ⋅ ) is the indicator function. This is a special case of varying-coefficient regression model [12].

We chose a generalized linear mixed model (GLMM) conditional inference tree algorithm for tree construction [14]. It helps to overcome some instance instability and bias problems that are common in recursive binary tree models fitting [1] [15]. In brief, at each iteration, the algorithm combines the fixed-effect estimation by a model-based partitioning algorithm (MOB) [2] with a linear mixed model random effect prediction. MOB is based on parameter instability tests [16]. Instability refers to a significant difference in coefficient estimates for two subsets of the dataset. The algorithm builds the tree after repetitions of the following steps:

GLMM tree algorithm stopping criterion is based on the linear mixed model log-likelihood denoted l ( b ) . One strategy to fit the model to data is outlined in the following algorithm [14]:

We used R package glmertree [14] for model fitting.

We have looked for an optimal joint value of a minimal terminal node size (minsize) and a numeric significance level α by using Bayesian Information Criterion (BIC). That model selection strategy avoids to choose arbitrary values for minsize and α . In our application, the linear mixed model includes two fixed-effects (Treatment and clinical visit index) and one random effect (patient ID). The covariates used to defined the subgroups are Age, Gender, Marital status, baseline CD4 count, WHO clinical stage, the mode of entry into the active list, the history of opportunistic infection at ART initiation, the HIV type and Body Mass Index.

Stability is an essential property of a fitted regression tree model. It ensures that the potential instability of the model is minimized and then the model use, as for prediction task. A model stability can be assessed by fitting the model on data obtained by bootstrap resampling of the training dataset. Variables and cutpoints that were not selected by the original tree may be selected by replicate samples. Metrics for stability assessment include the relative variable selection frequency, the mean frequency of the variable selections per tree and the frequency of each cutpoint over the trees [17]:

• The relative variable selection frequency for a partitioning covariate z m , m = 1 , ⋯ , q equals the total number of replicate trees that have selected z m at least once, divided by the total number of replicate trees.

• The mean frequency of the covariate selections per tree for z m is the total number of times z m is selected for partitioning by a replicate tree over the repetitions, divided by the total number of replicate trees.

• The relative frequency of a cutpoint c ( z m ) equals the total number of replicate trees that have selected c ( z m ) to split the variable z m , divided by the total number of replicate trees.

A covariate selection is stable if its average split count is close to its number of selections in the original tree and its frequency of selection is close to 100%. Graphical methods are used to analyze variable cutpoints’ variability. A histogram is used to illustrate the cutpoint variability when the partitioning variable is numerical. It is expected that the cutpoints selected in the original tree have the highest frequencies (one or more peaks in the histogram). For an ordered categorical variable, a barplot is used to show the frequency of all possible split points. For an unordered categorical variable, a specific plot is used to visualize the partitions’ variability over the replicates. The same color is used for categories that belong to the same node. The combination of categories that corresponds to a partition observed in the original tree is marked on the right side of the plot by a solid red line. In addition, two dashed lines enclose the area representing the partition. Number(s) on the right side of the area indicate the level(s) of the corresponding split(s) in the original tree. In conclusion, a split point is stable if it is selected by most replicate trees.

We have implemented a non parametric bootstrap method for generalized linear mixed models [18] [19]. Mainly, the method consists of three steps. First, for all i = 1 : n , we computed scalars b i * and ϵ i * by centering and scaling the predicted random variable b ^ i and the predicted random variable ϵ ^ i respectively. Consider b ¯ and e ¯ the empirical means of b i and e i . Denote

σ b * 2 = 1 n ∑ i = 1 n ( b ^ i − b ¯ ) 2     and     σ ϵ * 2 = 1 n ∑ i = 1 n ( e ^ i − e ¯ ) 2 (4)

the empirical variances of b ^ i and of e ^ i respectively. The marginal residuals e ^ i is also centered and scaled. we compute the predicted values y * by replacing b ^ i by σ b * b * and ϵ ^ i by σ ϵ * ϵ * . Note that the two random variables follow a standard normal distribution. Thirdly, to obtain each bootstrap dataset, the response y is replaced by a bootstrap of the predicted response y * .

To study the plausibility of the model assumptions, we proceed in two steps. First, we computed the least confounded residuals [20] and assessed the normality assumption using quantile-quantile plot. Secondly, we evaluated the homoscedasticity assumption by plotting predicted values against standardized conditional residuals. For this purpose, we wrote a R code for parametric bootstrap that account for fitted mixed model as add-on for the package stable learner [17] to realize the stability analysis of linear mixed model based recursive binary tree.

Our model highlights seven subgroups of patients with different temporal variations of CD4 counts (Figure 1). The set of patients at an advanced disease stage (CD4 < 225) is the most splitted. It consists of at least four subgroups of chronological increase of CD4 count. Baseline CD4 count is selected for the first split. It is the most correlated with the temporal variation of CD4 count. Baseline CD4 count is a predictor of sustained virologic response [21]. In the subgroup of patients with baseline CD4 count ≤ 73 cells/μl, Gender is the most correlated with the CD4 count variation. Female patients have a faster chronological increase of CD4 counts than male patients ( β ^ 2 = 1.02 and β ^ 2 = 0.82 respectively).

Uptake and adherence to HIV services may explain this difference. In the subgroup of patient with baseline CD4 counts between 73 cells/μl and 129 cells/μl as well as in the subgroup of patients with baseline CD4 counts between 129 cells/μl and 225 cells/μl, Age is correlated with CD4 count temporal variation. In these subgroups, patients with aged 34 years or below have a slightly higher chronological CD4 count increase compared with older patients. The variable Age selection might be explained by what is called discordant immunological response, likely to happen among older patients [22]. Conversely, the group of patients with baseline CD4 count ≥ 225 cells/μl is homogeneous according to partioning rules.

Table 2 shows a chronological increase of CD4 count on average in all subroups ( β 2 > 0 and p -value < 0.05 ). This finding corroborates the fact that ART treatment aims to reduce the viral load and progressively to restore the immune system. The increase is faster for patients with an initially weakened immune system. On average the β 2 coefficient is small when baseline CD4 count is large. In addition, the ratio σ ^ b 2 / ( σ ^ 2 + σ ^ b 2 ) = 57.07 % . This shows that individual random effect is important in the analysis of CD4 cell counts time variation.

Subgroups also differ by the difference in treatment response between patients treated with 2NRTI + 1NNRTI regimen and those treated with other ART regimens (2 NRTI + IDV, 2 NRTI + LPV/r, 2 NRTI + NFV). It is commonly admitted that there is no optimal treatment for all patients. Chronological CD4 count levels are higher for patients treated with 2NRTI + 1NNRTI than for those treated with other ART regimens group in the subgroup of patients with baseline CD4 count ≤ 129 and aged 34 and over as well as in the subgroups of patients with baseline CD4 counts between 129 cells/μl and 225 cells/μl (Figure 2). The difference in chronological CD4 count levels between ART regimens is highest

in patients aged 34 or younger and with baseline CD4 counts between 129 cells/μl and 225 cells/μl. In the remaining subgroups, we found no significant difference ( p > 0.05 ).

We performed the stability analysis with 500 bootstrap samples. Relative frequencies of selecting baseline CD4 count, Gender, Age are all equal to 100% (Table 3 and Figure 3). Their mean frequency of selections per tree are respectively equal to their selection frequency in the initial tree (Table 3). In addition, Figure 4 shows that all the bootstrap trees have splitted baseline CD4 count on the same levels as in the initial tree. Age is splitted at age 34 on level 3 as in the original tree. Similarly, Gender is splitted on level 3. Thus, baseline CD4 count, Gender and Age can be considered as stable. Most of the bootstrap trees (97.6%) and the original tree are identical (Figure 3). Finally, variable EntryMod need not be retained. It is only selected by 2.4% of bootstrap trees. To sum up, the fitted tree is stable. Finally, Figure 5 and Figure 6 show that there is no evidence against normality and homoscedasticity assumptions respectively.