Background: Human immunodeficiency virus/acquired immune deficiency syndrome (HIV/AIDS) have caused the world most shocking tragedy and risk. Mortality among patients on HAART is associated with high baseline levels of HIV RNA, WHO stage III or IV at the beginning of treatment, low body mass index, severe anemia, low CD4+ cell count, type of ART treatment, gender, resource-poor settings, and poor adherence to HAART.

Objective: The main objective of this study was to make use of appropriate modeling approach to CD4+ cell progression and identify the potential risk factors affecting the CD4+ cell progression of ART patients in Hossana District Queen Elleni Mohamad Memorial Hospital.

Methods: In this longitudinal retrospective based study secondary data was used from Hossana District Queen Elleni Mohamad Memorial Hospital. The study population consists of 222 HIV-1-positive patients, measured repeatedly at least one time on each patient who are 15 years old or older those treated with ART drugs from September 2011 to May 2014. The data was analyzed using SAS 9.2 version procedure NLMIXED. Poisson, Poisson-gamma, Poisson-normal, and Poisson-normal-gamma models were applied to study over-dispersion and correlation in the data.

Results: A total of 222 adult ART HIV-1-positive patients were included in this study. Out of these ART patients, 131(59%) were female patients and 91(41%) were male patients; 65(29.30%) were followed the drug combinations properly; the mean and standard deviation of baseline CD4+ cell counts were 355.9 and 321.4 cells per milliliter of blood, respectively; the mean and standard deviation of age of patients (p=0.0001) were 31.06 and 8.50 years, respectively; patients were followed for a mean of 24 months (p=0.0001). The analysis showed that the covariates significant for the progression of CD4+ cell counts were age of the patient, time since seroconversion, and sex at 5% level of significance.

Conclusion: On average CD4+ cell count increases after patients initiated to the HAART program (the disease rate declines). The progression of end outcome depends on patient’s baseline socio-demographic characteristics. For the presence of over-dispersion, and clustering, the Poisson-normal-gamma model results in improvement in model fit.

Keywords: CD4+ cell count; Poisson-normal-gamma model; Over dispersion; Correlation

Background

Human immunodeficiency virus/acquired immune deficiency syndrome (HIV/AIDS) have caused the world most distressing tragedy and danger. More than 25 million people worldwide have died of AIDS since 1981, as reported by Avert Org.¹ In 2005 Ethiopia launched free ART, over 71, 000 were initiated by the end of November 2006. 241 hospitals and health centers are now providing HIV care and treatment services in regions of the country.

According to Bayeh et al.,² enumeration of CD4+ T cell count has been useful to initiate and monitor therapy in HIV infected individuals taking potent ART.

Count data are collected repeatedly over time in many applications, such as biology, epidemiology, and public health. Such data are often characterized by the following features: correlation due to the repeated measures is usually accounted for using subject-specific random effects, which are assumed to be normally distributed. The sample variance may exceed the mean, over-dispersion. Hence, appropriate modeling approaches which can overcome these issues and which lighten data analysis are needed.

Statement of the problem

The CD4+ cell count still remains the major determinant or measure of the cell mediated immunity. Currently, there is no enough evidence showing that all the ART centers in Ethiopia have implemented research tools to monitor patient’s immune (CD4) response to HAART within a specified time frame and identification of factors that might be associated with the poor CD4-Lymphocyte response to HAART.

Hence, this study seeks to answer the following questions:

1. Does HAART have a positive effect on the HIV/AIDS patients immune system based on an indication of their gained CD4+cell counts at Hossana District Queen Elleni Mohamad Memorial Hospital, SNNPR, Ethiopia?
2. What are the appropriate longitudinal models to handle over- dispersed and correlated individual subjects in this data?
3. What are important potential determining factors in HIV/AIDS patient’s response to HAART at Hossana District Queen Elleni Mohamad Memorial Hospital, SNNPR, Ethiopia?

Objectives of the study

General objective: To make use of appropriate modeling approach to CD4+ cell counts progression and identify the potential risk factors affecting the CD4+ cell progression of ART patients in Hossana District Queen Elleni Mohamad Memorial Hospital.

Specific objectives

1. To explore how CD4+ cell counts of HIV-1-positive patients under ART in Hossana District Queen Elleni Mohamad Memorial Hospital change over time;
2. To fit an appropriate statistical model for the average evolution of CD4+ cell counts of HIV-1 positive patients;
3. To identify the potential factors affecting the evolution of CD4+ cell counts among HIV-1 positive patients under ART.

Significance of the study

This study will have the following benefits:

1. It helps to identify the potential risk factors influencing the absolute CD4 count measurements in HIV infected patients.
2. It helps the respective policy makers of the health sector monitoring frequency of CD4 Count, monitoring therapeutic response, and judge urgency of ART initiation using CD4+ cell Counts of the patients.
3. It can be used as a reference for those who want to apply the techniques of handling correlation and over-dispersion in counts longitudinally collected data analysis.

Source of data and sample size

The data was obtained from Hossana Queen Elleni Mohamad Memorial Hospital, SNNPR, Ethiopia from September, 2011 to May, 2014. The hospital is found in Hosanna town which is 235 km away from Addis Ababa. Persons living with HIV/AIDS, age greater than or equal to 15 years and started ART treatment in Hossana Queen Elleni Mohamad Memorial Hospital were included in the study. Individuals of 222 HIV patients in the HAART with a minimum of one and maximum of nineteen measures were selected. A retrospective longitudinal study design was conducted. Data was analyzed by SAS version 9.2 using NLMIXED procedure.

Study variables

Dependent/response variable

Enumeration of CD4+ cell counts (CD4+ T cells) per mm3 of blood of ART patients, which is count.

Independent/explanatory variables

Eleven covariates were used to meet the goal of the study. These are:
Baseline age, Time since a month of seroconversion, Sex of the patient, Level of Education, (WHO) Clinical Stage, Area/residence of the patient, Adherence to any of the drugs, Employment status, Religion, and Marital Status.

Method of data analysis

Explanatory data analysis:

1. Exploring the individual profile: Provide some information on within and between subject variability.
2. Exploring the Mean Structure: Its purpose is to choose the fixed effects for the model.

Statistical models

Generalized linear model/poisson model

Generalized linear models are usually in use for modeling univariate non-Gaussian data.

The Poisson distribution belongs to the exponential family and is commonly used for the analysis of count data.

The distribution of outcome is Y ~ Poisson  $\left(\theta \right)$ , thus:

$f({{\displaystyle y}}_i,{{\displaystyle \theta }}_i)=\frac{{{\displaystyle \theta }}_i^{y_i}{{\displaystyle e}}^{-\theta }}{y_i}$ (1)

The Poisson regression model, with $\beta$  a vector of p fixed, but unknown regression coefficients is given by

$$\log \left(\lambda \right)={{\displaystyle X}}_i^0\beta$$

Over-dispersion model

For count data, we consider the assumption that the variance proportional to the mean (in Poisson distribution). Specifically, $\mathrm{var}\left(y\right)=\phi E\left(y\right)=\phi \lambda$

Where, $\phi$  is over dispersion parameter if $\phi =1$ , then the variance equals the mean and we obtain the Poisson mean-variance relationship. When the mean and variance are not equal (over-dispersion), often the Poisson distribution replaced with Negative Binomial Distribution.

Start from a Poisson regression model and add a multiplicative random effect $\theta$  to represent unobserved heterogeneity.

$y/\theta \sim P\left(\theta \lambda \right)$  (2)

$\theta$ has a gamma distribution with parameters $\alpha$  and $\beta$  unconditional distribution of the outcome, which happens to be the negative binomial distribution.

Generalized linear mixed model

According to Engel & Keen,³ Molenberghs & Verbeke,⁴ and Pinheiro & Bates,⁵ when non-Gaussian data are repeatedly measured like CD4+ cell counts, the GLM is usually extended to GLMMs, with a subject-specific random effect, usually a Gaussian type, added in the linear predictor to capture the correlation.

GLMMs combine the properties of linear mixed models and generalized linear models.

$g\left(\lambda \right)=In\left(\lambda \right)=\gamma {{\displaystyle X}}_{ij } +{{\displaystyle b}}_i {{\displaystyle Z}}_{ij}$ (3)

< $\lambda \left(X_{11},\mathrm{...},X_{jk}\right)=\exp \left(\gamma {{\displaystyle X}}_{ij}+b_{i }{Z}_{ij}\right)$

Where, $\lambda$ : The mean of which is related to the covariates of X by link function

$X_{ij}$ : Covariates of the i^th patient for the j^th time

$\gamma$ : Regression coefficients of $X_{ij}$

$Z_{ij}$ : The covariates of the random effects of the i^th patient at j^th time

Poisson-normal-gamma model

Correlation and over-dispersion can crop up simultaneously, in practice. Combining ideas from the over dispersion and the GLMM the Poisson model with normal and gamma random effects can be specified as:

$Y_{ij}\sim Poi\left({\lambda }_{ij} ={\theta }_{ij} K_{ij}\right)$ (4)

with $K_{ij} = \exp \left(X_{ij} \gamma + Z_{ij} b_i\right)$  [6], where $Y_{ij}$  be the j^th outcome measured for subject $i = \mathrm{1...}N , j = 1,\mathrm{...}, n_i, b_i \sim N (0, D)$ and ${\theta }_{ij\sim \gamma }(\alpha , \beta ), {{\displaystyle X}}_{ij} and {{\displaystyle Z}}_{ij}$ p-dimensional and q dimensional vectors of known covariate values, and $\gamma$ a p-dimensional vector of unknown fixed regression coefficients.

Model comparison/selection and variable selection

NLMIXED procedure fits nonlinear mixed models by maximizing an approximation to the likelihood integrated over the random effects.

It uses adaptive Gaussian quadrature method (MLH) by default.

Although the AIC can be used in association with mixed models, it is not common to be used with the models discussed above to select either the optimal set of explanatory variables or other structures.
Hence, finally the four models are compared to select the best one using -2log-likelihood comparison technique.

In all models, to select significant variables, first the main effect and main effect by time interaction were incorporated to the initial candidate model and, then avoid non-significant and so on (Back ward selection technique).

Author declares that there are no conflicts of interest.

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