Over thirty-nine million individuals globally are living with HIV, and has claimed millions of lives globally over the past decades. The study's aims to determine variables associated HIV/AIDS progression in individuals receiving antiretroviral therapy. The research employed a cohort analysis of surveillance data from the HIV Wellness Clinic in the province of Limpopo, South Africa, for 318 HIV-positive patients on antiretroviral therapy (ART). This study uses Aalen's additive regression (AAR) and Cox regression (CR) models in comparison. HIV/AIDS progression for patients receiving treatment therapy has a significant association with CD4 baseline, gender, age, and development of tuberculosis during treatment, according to results from AAR and CR models. With a very modest p value < 0.01, both models demonstrate that the CD4 baseline is highly significant when compared to the other factors in the models. Compared to female patients, male patients experience higher rates of immune system degradation. Compared to individuals with baseline CD4 counts over 350 cells/mm3, patients with baseline CD4 counts below 350 cells/mm3 have higher rates of immunological recovery. This study has reinforced the need for early detection of TB in people living with HIV so that these two diseases are concurrently managed for the benefit of the infected individual.
Research Article
Application of Aalen’s additive regression model and the Cox proportional hazards to model HIV/AIDS progression
https://doi.org/10.21203/rs.3.rs-5254940/v1
This work is licensed under a CC BY 4.0 License
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Aalen's additive regression model
Cox’s hazard regression model
antiretroviral therapy
treatment adherence
immune deterioration
cohort analysis
The HIV/AIDS disease progression is characterised by continuous destruction of the CD4 + cells leading to immuno-suppression, neoplasm, wasting, or low CD4 + T-cell count that defines AIDS1,2. Some clinical markers such as the CD4 + cell count and the RNA viral load help in providing information on HIV/AIDS disease progression. A normal CD4 + cell count varies from individual to individual, and it is usually between 500 and 1 400 cells per mm3. CD4 + cell count values below 500 are usually an indication of immune suppression and vulnerability to opportunistic infections3.
The Joint United Nations Programme on HIV/AIDS (UNAIDS) estimates show that approximately 39 million people globally are living with HIV and 40.4 million have died of AIDS-related illnesses since the beginning of the epidemic by the end of 2022. Although sub-Saharan Africa constitutes a small fraction of the world population, approximately 67% of the HIV/AIDS cases have been reported in sub-Saharan Africa4 with approximately 23.4 million HIV positive cases by 20225. South Africa was leading with 7.8 million followed by Nigeria with 3 million HIV-positive individuals. Between 2001 and 2011, South Africa, the country with the largest number of HIV infections, reduced new HIV infections by 41%. South Africa has one of the largest HIV treatment programmes in the world6. Therefore, it is important to identify factors associated with survival from HIV/AIDS for patients who are on antiretroviral therapy.
In 2016, Zhang et al.7 analysed the survival of patients receiving antiretroviral therapy. In their analysis. They concluded prolonged survival times for patients with HIV/AIDS and hence improved survival probability. They emphasised the need for constant follow-up, regular monitoring of CD4 + cell count, as well as timely and early antiretroviral intake to achieve better survival chances.
In 2019, Mangal et al.8 carried out a study on people living with HIV/AIDS using piecewise constant exponential models. The study shows that early treatment contributes to improved survival probabilities with the greatest benefit obtained in women and younger age groups.
In 2016, Kowaliska et al.9 used the Cox proportional hazard models to identify factors related to the first combination antiretroviral therapy (cART) modification in three age groups. Their results reveal significantly longer time on cART for the older age groups than their younger counterparts.
However, relatively few studies on HIV/AIDS progression have been carried out using both the Cox proportional hazards model and the Aalen additive hazards model. While the Cox proportional hazards model is frequently used to analyze survival in censored data, Basar10 contended that when the proportional assumption is not met, bias may result. Then, the additive model turns into an alternative option.
Thus, in this study, both the Cox proportional hazards model and Aalen's additive model are used. For this cohort, an investigation as to whether CD4 baseline (CD4BL), age, gender, TB before enrolment (TBB4), developed TB during treatment (DTB), adverse reaction to treatment (Reaction), WHO stage baseline (WSBL), Body Mass Index progression (BMI) and viral load baseline (VLBL) are associated with HIV/AIDS progression for patients under treatment therapy is done.
Ethical considerations
The data collection procedures used in this study were approved by the Research Ethics Committee of the University of Venda, SA (ref. no. SMNS/13/ MBY/01/0625), following the 1964 Declaration of Helsinki and its subsequent amendments. Additionally, permission to access health facilities was obtained from the Limpopo Provincial Department of Health, SA, and collaborating health facilities. Informed consent was obtained from study participants before their involvement, and data obtained were stripped of personal identifiers to ensure the anonymity and confidentiality of the participants.
Study population
The study used a cohort analysis of the surveillance data for 318 HIV-infected patients under anti-retro-viral therapy (ART) from the HIV Wellness Clinic in the Limpopo Province of South Africa. From these individuals, there were 227 females and 91 males. The survey was conducted between 2005 and 2009 and follow-up was done every 6 months. The age range of the patients at enrolment was from 2 to 77 years with a mean age of 36.47 years. The age group 31–40 years had the highest percentage of individuals with approximately 31.4%. Before the commencement of treatment, the body mass index (BMI) was calculated, and the mean BMI was 19.17 at enrolment. World Health Organisation (WHO) stage baseline and CD4 baseline were measured. The CD4 cell count at enrolment ranged from 1 cell/mm3 to 518 cells/mm3. 70% of the individuals had a CD4 cell count below 200, and 30% had a CD4 count between 200 and 750. There were no individuals with CD4 count above 750. Some of the patients were enrolled at the clinic with TB as the initial marker of HIV. During treatment, CD4 count was noted and any form of adverse reaction to treatment was noted as well as any development of a TB co-infection.
Variable coding
For this analysis, the variables were coded as follows: Gender: 1 - Male, 0 - Female; TB before enrolment (TBB4): 1 - yes, 0 - no; Developed TB on ART (DTB): 1 - yes, 0 - no; Reacted to ART: 1 - yes, 0 - no; The body mass index (BMI) was measured in kg/m2 and was calculated using the formula; \(\:BMI\:=\:Weight/(Height{)}^{2}\)where weight is measured in kilograms and height is measured in metres11. The CD4BL are as follows: 1 - \(\:CD4\:\ge\:\:750\); 2 - \(\:500\:\le\:CD4\:<\:750\); 3 - \(\:350\:\le\:CD4\:<\:500\); 4 - \(\:200\:\le\:CD4\:<350\); 5 - \(\:0\:\le\:CD4\:<\:200\). There were no individuals with a CD4 baseline in category 1.
Analysis was done using the R statistical software since it can handle Aalen's additive regression analysis.
Cox proportional hazards model versus Aalen's regression model in modelling HIV progression
The proportional hazards regression model, known as the Cox regression model, has been the popular choice for right-censored survival data12,13. Its popularity is based on the fact that it is simple to fit and that the results are easy to explain. However, Cox's regression model has its limitations. Firstly, the validity of the analysis relies heavily on the proportional hazards assumptions. Secondly, it cannot include time-varying covariate effects since the regression coefficients are assumed to be constants14. The alternative is Aalen's regression model. Aalen's model postulates a different relationship for the hazard and covariates than does the Cox model12,15. The coefficients in Aalen's model are functions that vary over time without any specific shape or reliance on the parameter functions16. Events that happen to one person are therefore presumed to be independent of those of other people. Because of this, it is non-parametric17 as opposed to Cox's proportional hazards model, which is semi-parametric. Thus, Aalen's model is ideally suited to investigate potential temporal variations in the variables' impacts.
It is argued18 that, it is not appropriate to use Aalen's additive hazards model for all data sets because the calculation of cumulative regression functions \(\:\beta\:\left(t\right)\) are restricted to time interval, and Z, the matrix of covariates, is full rank. This means that \(\:{Z}^{{\prime\:}}Z\) should be invertible, which is not always the case. However, with the Cox regression model, this is not a problem.
Cox proportional hazards regression model
Cox proportional hazards model is specified as \(\:h\left(t|\stackrel{-}{Z}\right)={h}_{0}\left(t\right)\text{e}\text{x}\text{p}\left({\beta\:}_{r}\stackrel{-}{Z}\right)\) where \(\:{h}_{0}\left(t\right)\) is a baseline hazard, which may vary arbitrarily over time, and \(\:\stackrel{-}{Z}=({z}_{1},\:{z}_{2},\dots\:,{z}_{r})\) is the covariate vector\(\:\varvec{\beta\:}={(\beta\:}_{1},\:{\beta\:}_{2},\:\cdots\:,{\beta\:}_{r})\) is a vector of covariate coefficients and is assumed constant. The baseline hazard is treated to be non-parametric. The null hypothesis to test for the significance of a given covariate \(\:{z}_{j}\)in the Cox model is given by:
$$\:{H}_{j}:{\beta\:}_{j}\left(t\right)=0$$
.
Where \(\:{\beta\:}_{j}\left(t\right)\) is the regression coefficient corresponding to the jth covariate 19.
If two individuals with covariate values, Za and Zb being compared, the ratio of their hazard rates at any time point simplifies to:
$$\:\frac{h\left(t|Z\right)}{h\left(t\right|{Z}^{*}}=\frac{{h}_{0}\left(t\right)\text{e}\text{x}\text{p}[{\sum\:}_{k=1}^{r}{\beta\:}_{k}{Z}_{ak}]}{{h}_{0}\left(t\right)\text{e}\text{x}\text{p}[{\sum\:}_{k=1}^{r}{\beta\:}_{k}{Z}_{bk}]}$$
$$\:=\frac{\text{e}\text{x}\text{p}[{\sum\:}_{k=1}^{r}{\beta\:}_{k}{Z}_{ak}]}{\text{e}\text{x}\text{p}[{\sum\:}_{k=1}^{r}{\beta\:}_{k}{Z}_{bk}]}$$
$$\:=\text{e}\text{x}\text{p}\left[{\sum\:}_{k=1}^{r}{\beta\:}_{k}({Z}_{ak}-{Z}_{bk})\right]$$
1
Where \(\:{h}_{0}\left(t\right)\) is the baseline hazard rate, \(\:{\beta\:}_{k}\) is the regression coefficient for the kth covariate. This ratio is considered constant throughout the study period.
A sample of size n yields data with F distinct failure times denoted by t1 < t2< … <tF and n-F censored times. t(i) is an ordered event while Tj is the follow-up time for the jth individual. The set of indices of subjects at risk at time ti is denoted by \(\:{R}_{i}=R\left({t}_{i}\right)\). The covariate for the individual who had an event at time ti is given by Z(i) to be distinguished from covariate Zj for individual j according to the previous study15. This allows the proportion for all the individuals at risk at time ti such that \(\:\left\{j\in\:{R}_{i}\right\}=\left\{j|{T}_{j}\ge\:{t}_{i}\right\}\). The partial likelihood for estimating the parameter β is given by:
$$\:L\left(\beta\:\right)=\prod\:_{i=1}^{F}\frac{\text{e}\text{x}\text{p}\left(\beta\:{\prime\:}{Z}_{\left(i\right)}\right)}{\sum\:_{jϵ{R}_{i}}\text{e}\text{x}\text{p}\left(\beta\:{\prime\:}{Z}_{j}\right)}$$
2
Where i denotes the subscript of the subject who dies at any time \(\:{t}_{\left(i\right)}\)20.
Aalen's additive hazard model
Aalen's additive regression model is constructed as follows: Suppose a possibility of a censored lifetime of a number of HIV patients is observed. Let \(\:{\lambda\:}_{i}\) denote the hazard rate of patient \(\:i\), \(\:n\) be the number of HIV-infected patients, and \(\:r\) the number of predictors in the analysis. The additive model, is constructed by taking into account the column vector \(\:\lambda\:\left(t\right)\) of hazard rates \(\:{\lambda\:}_{i}\left(t\right),i=\text{1,2},\dots\:,n\), is given by \(\:\lambda\:\left(t\right)=Y\left(t\right)\alpha\:\left(t\right)\), where the \(\:n\times\:(r\:+1)\) matrix \(\:Y\left(t\right)\) is built as follows: If the patient is a member of the risk set at time \(\:t\), them the \(\:{i}^{th}\) row of the \(\:Y\left(t\right)\) is the vector \(\:{\stackrel{-}{z}}^{i}\left(t\right)=(1,{z}_{1}^{i}\left(t\right),{z}_{2}^{i}\left(t\right),\dots\:,{z}_{r}^{i}\left(t\right)){\prime\:}\) where \(\:{z}_{j}^{i}\left(t\right),j=\text{1,2},\dots\:,r\), are time-dependent covariates values. The corresponding row of \(\:Y\:\left(t\right)\) contains only zeros if at time \(\:t\) the individual \(\:i\) is at risk. The vector \(\:\alpha\:\left(t\right)=({\alpha\:}_{0}\left(t\right),{\alpha\:}_{1}\left(t\right),\dots\:,{\alpha\:}_{r}\left(t\right)){\prime\:}\) contains the regression information where \(\:{\alpha\:}_{0}\) is a baseline function, while the remaining components are regression functions which quantify the impact of the corresponding covariates. Over time, their functions are free to change.
One primary question is "Do specific covariates have any influence on the distribution of lifetimes". This corresponds with the hypothesis; \(\:{H}_{j}:{\alpha\:}_{j}\left(t\right)=0,\) where \(\:j\) corresponds to the \(\:{j}^{th}\) covariate in the analysis. This corresponds to the null hypothesis in the Cox model. However, the alternative hypothesis for the Cox model \(\:{H}_{j}:{\beta\:}_{j}\left(t\right)\ne\:0\) is valid for all \(\:t\) whereas the alternative hypothesis in Aalen's model \(\:{H}_{j}:{\alpha\:}_{j}\left(t\right)\ne\:0\) is valid for some\(\:\:t\).
For Aalen’s additive model, cumulative regression functions or coefficients are estimated as:
$$\:{A}_{j}\left(t\right)={\int\:}_{0}^{t}{\alpha\:}_{j}\left(u\right)du$$
3
\(\:{\alpha\:}_{j}\left(t\right)\) is estimated from the slope of \(\:{A}_{j}\left(t\right)\), which is often done using kernel smoothing techniques.
Model Formulation
The HIV data used in this study has n = 318 patients and j = 9 variables. The covariates are: CD4 baseline (CD4BL), age, gender, TB before enrolment (TBB4), developed TB during treatment (DTB), adverse reaction to treatment (Reaction), WHO stage baseline (WSBL), BMI progression (BMI) and viral load baseline (VLBL). The effects of these covariates on the progression of HIV were analysed using the package "timereg" in R and the results are discussed in the next paragraphs.
The process of identifying the model that fits the data best is done using backward elimination. This involved starting by first considering a model in which the full set of covariates are included and non-significant covariates removed step by step. The effects of the removal of these variables are noted by checking the change in p-values of the global test statistics and also checking the changes in R2 for the fitted models. The analysis is done for both Aalen's and Cox's regression and the comparison of these two models is based on the p-values of each of the covariates as well as the p-values for the model and other statistics. The identification of the better model is done in four steps.
Results from the first step for both Aalen's and Cox's models with all the variables included is fitted. The Cox regression model fitted is:
$$\:h\left(t|\stackrel{-}{Z}\right)={h}_{0}\left(t\right)\text{e}\text{x}\text{p}({\beta\:}_{1}\times\:Gender+{\beta\:}_{2}\times\:WHOSBL+{\beta\:}_{3}\times\:Age+{\beta\:}_{4}\times\:CD4BL+{\beta\:}_{5}\times\:BMI+{\beta\:}_{6}\times\:Reaction+{\beta\:}_{7}\times\:DTB+{\beta\:}_{8}\times\:TBB4Enrl)$$
4
and the Aalen’s additive model is given by:
However, continuous elimination of non-significant is done to ensure that we remain with variables that explain the progression of HIV/AIDS better. The final theoretical models being estimated in the final step are:
$$\:h\left(t|\stackrel{-}{Z}\right)={h}_{0}\left(t\right)\text{e}\text{x}\text{p}({\beta\:}_{1}\times\:Gender+{\beta\:}_{2}\times\:Age+{\beta\:}_{3}\times\:CD4BL+{\beta\:}_{4}\times\:DTB)$$
6
and;
$$\:{\lambda\:}_{i}\left(t\right)={\alpha\:}_{0}\left(t\right)+{\alpha\:}_{1}\left(t\right)\times\:{Gender}_{i}+{\alpha\:}_{2}\left(t\right)\times\:{Age}_{i}+{\alpha\:}_{3}\left(t\right)\times\:{CD4BL}_{i}+$$
$$\:{\alpha\:}_{4}\left(t\right)\times\:{DTB}_{i}$$
7
where \(\:{\alpha\:}_{r}\left(t\right)\) for \(\:r=\text{0,1},2,\dots\:,4\) is the regression information, and \(\:{\beta\:}_{r}\left(t\right)\) for \(\:r=\text{1,2},\dots\:,4\) vector of covariate coefficient for Aalen’s additive model and Cox regression model respectively.
The Cox regression model and Aalen’s additive model are applied to the data. Estimates of the coefficients \(\:{\alpha\:}_{r}\left(t\right)\)for\(\:\:r=\text{0,1},2,\dots\:,8\), and \(\:{\beta\:}_{r}\left(t\right)\) for \(\:r=\text{1,2},\dots\:,8\) for Aalen’s additive model and Cox regression model respectively are shown in Table 1. The table also contains standard errors and p-values for the time-invariant effect for Aalen’s additive model and p-values for the constant covariate effect for the Cox regression model.
| covariates | Aalen’s additive model | Cox regression model | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Coef\(\:{\alpha\:}_{r}\left(t\right)\) | RR | SE | z | p-value | Coef \(\:{\beta\:}_{r}\left(t\right)\) | RR | SE | z | p-value | |
| factor(Gender)1 | 0.059 | 1.061 | 0.059 | 1.380 | 0.169 | 1.091 | 2.977 | 0.457 | 2.386 | 0.017 |
| WHOSBL | -0.062 | 0.940 | 0.046 | -2.220 | 0.027 | -1.435 | 0.238 | 0.283 | -1.538 | 0.124 |
| Age | -0.000 | 1 | 0.004 | -0.079 | 0.937 | -0.031 | 0.969 | 0.027 | -1.113 | 0.266 |
| factor(CD4BL)3 | -0.046 | 0.955 | 0.111 | -0.679 | 0.497 | -0.242 | 0.785 | 0.625 | -0.388 | 0.698 |
| factor(CD4BL)4 | -0.083 | 0.920 | 0.108 | -1.300 | 0.195 | -0.992 | 0.371 | 0.611 | -1.623 | 0.105 |
| factor(CD4BL)5 | -0.132 | 0.876 | 0.104 | -2.170 | 0.030 | -1.548 | 0.213 | 0.616 | -2.511 | 0.012 |
| BMI | 0.002 | 1.002 | 0.006 | 0.328 | 0.743 | -0.015 | 0.985 | 0.040 | -0.362 | 0.717 |
| factor(Reaction)1 | 0.005 | 1.005 | 0.061 | 0.082 | 0.935 | 0.027 | 1.027 | 0.419 | 0.064 | 0.949 |
| factor(DTB)1 | -0.074 | 0.929 | 0.049 | -1.430 | 0.151 | -0.951 | 0.386 | 0.415 | -2.294 | 0.022 |
| factor(TBB4Enrl)1 | 0.036 | 1.037 | 0.032 | 1.120 | 0.261 | 0.363 | 1.438 | 0.230 | 1.580 | 0.114 |
| Key: Coef = coefficient; SE = standard error; z = z score or z statistic or critical value; RR = relative risk | ||||||||||
The exponent coefficients or the relative risks in Table 1 are interpreted as additive effects or multiplicative effects for the Aalen model and the Cox model respectively.
The results show that for Aalen's additive model, two covariates turned to be significant at the 5% level, that is, WHO stage baseline (WHOSBL) and CD4BL5 (below 200 cells/mm3). The coefficient associated with WHOSBL is -0.062 with a standard error of 0.046 and hence a z-statistic of -2.220, giving a significant p value of 0.027. The associated relative risk (RR) is 0.940, implying that a unit increase in WHOSBL reduces the rate of immune deterioration by 0.940. The CD4BL5, has a coefficient is -0.132 and a standard error of 0.104, giving a z-statistic of -2.170 and hence a significant p-value of 0.030. The RR is 0.876, implying that a patient with \(\:0\:\le\:CD4\:<\:200\) has a slightly lower rate of immune deterioration than a patient with\(\:\:500\:\le\:CD4\:<\:750\). The Cox regression model is also fitted to the same data and three covariates turned out to be significant at 5% level, that is, Gender (male=1; female=0), CD4BL5 (individuals with CD4 baseline below 200 cells/mm3) and DTB1 (developed TB during treatment=1; did not develop TB during treatment=0). The coefficient of Gender is 1.091 with a standard error of 0.457, giving a z-statistic of 2.386 and hence a significant p-value of 0.017. The RR is 2.977, implying males increase the rate of immune deterioration by 2.977 when compared to females. The coefficient of CD4BL5 is -1.548 with a standard error of 0.616, giving a z-statistic of -2.511 and hence a significant p-value of 0.012. The RR is 0.213, implying that a patient with \(\:0\:\le\:CD4\:<\:200\) has a much lower rate of immune deterioration than a patient with\(\:\:500\:\le\:CD4\:<\:750\). The coefficient of (DTB)1 is -0.951 with a standard error of 0.415, giving a z-statistic of -2.294 and hence a significant p-value of 0.022. The RR is 0.386, implying that a patient who developed TB on ART (DTB) has a lower rate of immune deterioration than a patient who did not develop TB.
The results show that the relative risk of HIV progression, indicated by immune deterioration, for males, is 1.061 times or 2.977 times greater than that of their female counterparts if the Aalen additive model or Cox models, respectively, is used. Patients with a CD4 baseline below 200 cells/mm3 (CD4BL5) have a 0.876 times smaller risk of immune deterioration than patients who start treatment with a CD4 baseline between 500 and 750 cells/mm3 (CD4BL2). This is the lowest rate compared to all the other categories of higher CD4 baseline. Patients who react to treatment React1 have a 1.005 times greater risk of immune deterioration than patients who did not react to treatment.
The test for the significance of the models fitted in Table 1 is also performed at 5% level. The overall fit of Aalen’s additive model is investigated by the supremum-test of significance and the test for time-invariant effect is investigated using the Kolmogorov-Smirnov test and the results are shown in Table 2 below.
| Test for non-significant effects: | Test for time-invariant effects: | ||||
|---|---|---|---|---|---|
| Supremum-test of sig | p-value H0: \(\:{\beta\:}_{j}\left(t\right)=0\) | Kolmogorov-Smirnov test | p-value H0: constant effect | Cramer von Mises test | p-value H0: constant effect |
| \(\:{\beta\:}_{0}\left(t\right)\)=6.34 | 0.000 | \(\:{\beta\:}_{0}\left(t\right)\)=3.07 | 0.000 | \(\:{\beta\:}_{0}\left(t\right)\)=55.2 | 0.000 |
The output shown above contains a test for non-significance and time-invariance effects of the baseline rate for the fitted Aalen's additive model. The results from Table 2 above show that the tests are statistically significant as shown by small p-values (all equal to 0.000).
A goodness-of-fit test is also performed for the fitted Cox regression model and is investigated using three asymptotically equivalent test statistics; the likelihood ratio test, the Wald test, and the Score (logrank) test. Under the null hypothesis; \(\:{H}_{j}:{\beta\:}_{j}\left(t\right)=\beta\:\), these three statistics are asymptotically chi-squared distributed. The results are presented in Table 3 below.
| Statistic | Test statistic | Degrees of freedom | p-value |
|---|---|---|---|
| Likelihood ratio test | 26.52 | 10 | p = 0.003104 |
| Wald test | 27.66 | 10 | p = 0.002047 |
| Score (logrank) test | 30.06 | 10 | p = 0.0008389 |
| Concordance = 0.712 (se = 0.044 ); R-square(R2) = 0.244 (max possible = 0.999 ) | |||
The overall fit of the Cox model is investigated by the likelihood ratio, Wald and Score tests. The test is statistically significant with p-values of 0.003104, 0.002047, and 0.0008, respectively.
The analysis is done using a backward stepwise selection which starts with a model with all covariates and the removal of some covariates is based on the p-values. The first step, shown in Table 1, shows that for both models the variables Reaction1 (individuals who developed some adverse reaction to treatment) and body mass index (BMI) are non-significant (p-values above 0.9 and p-values above 0.7 respectively) to both the time-varying effect and the constant covariate effect of HIV/aids progression for individuals on drug therapy. Therefore, in step 2 the variable Reaction is removed from the analysis since it has the highest p-value.
The results from analysis for step 2 are not shown but they show the significance of the goodness of fit for both models. Results from step 2 still maintain non-significance for the variable Body Mass index (BMI) during treatment for both Aalen's regression model and Cox regression model. For this reason, BMI is removed from the models. The removal of BMI resulted in most of the variables having significant effects except for CD4BL3 (corresponding with CD4 baseline between 350 and 500 cells/mm3), TBB4Enrl1 (individuals who had TB before initiation of antiretroviral drugs), and WHO stage baseline. This results in the variable TB before enrolment being removed from the models. In the third step, the variable WHO stage baseline is removed from both models.
Results from Aalen's additive model and Cox regression model that best describe the HIV progression of individuals on drug therapy are summarised in Table 4 below.
| Covariates | Aalen’s additive model | Cox regression model | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Coef | RR | SE | z | p-value | Coef | RR | SE | z | p-value | |
| factor(Gender)1 | 0.086 | 1.090 | 0.050 | 2.56 | 0.011 | 0.959 | 2.609 | 0.351 | 2.74 | 0.006 |
| Age | -0.003 | 0.997 | 0.002 | -1.46 | 0.145 | -0.029 | 0.971 | 0.015 | -1.87 | 0.062 |
| factor(CD4BL)3 | -0.117 | 0.890 | 0.098 | -1.86 | 0.063 | -0.705 | 0.494 | 0.449 | -1.57 | 0.117 |
| factor(CD4BL)4 | -0.147 | 0.863 | 0.091 | -2.80 | 0.005 | -1.349 | 0.259 | 0.400 | -3.37 | 0.000 |
| factor(CD4BL)5 | -0.231 | 0.794 | 0.089 | -4.60 | 0.000 | -1.941 | 0.144 | 0.392 | -4.95 | 0.006 |
| factor(DTB)1 | -0.097 | 0.908 | 0.040 | -2.35 | 0.019 | -0.909 | 0.403 | 0.317 | -2.86 | 0.004 |
| Key: Coef = coefficient; SE = standard error; z = z score or z statistic or critical value, RR = relative risk | ||||||||||
Removal of the WHO stage baseline results in most of the variables being significant except for Age and CD4BL3 (corresponding to individuals who initiated treatment with a CD4 baseline between 350 and 500 cells/mm3). The p-values of the coefficients are 0.011, 0.145, 0.063, 0.005, 0.000, and 0.019 for gender, age, CD4 baseline 3, CD4 baseline 4 Cd4 baseline 5, and development TB respectively for Aalen’s additive regression model. This shows that there is an indication of some possible effect of some early commencing of ART (CD4BL3 = between 350 and 500 cells/mm3) and developing TB to HIV progression at 10% level. There is a significant effect for starting ART with CD4 baseline 4 and 5 (CD4 cell counts below 350 cells/mm3). The effect of gender is also significant to HIV progression at a 5% level.
For the Cox proportional hazards model, the p-values for the coefficients are 0.006, 0.062, 0.117, 0.000, 0.006, 0.004 for gender, age, CD4 baseline 3, CD4 baseline 4, CD4 baseline 5, and development TB (DTB1) respectively. This shows that there is some age effect on HIV progression at the 10% level of significance. There are also significant effects for gender, CD4 baseline 4, CD4 baseline 5, and DTB at a 5% level of significance.
For Aalen’s additive regression model, the coefficient associated with gender is -0.086 with a standard error of 0.050 and hence a z-statistic of 2.56, giving a significant p-value of 0.011. The associated relative risk (RR) is 1.090, implying that males have a 2.56 times greater risk of immune deterioration than their female counterparts. The coefficient associated with CD4BL3 is -0.117 with a standard error of 0.098 and hence a z-statistic of -1.86, giving a significant p-value of 0.063. The associated relative risk (RR) is 0.890, implying that patients starting ART with a CD4 baseline between 350 and 500 cells/mm3 have 0.890 times less risk of immune deterioration than patients who start ART with a CD4 baseline between 500 and 750 cells/mm3.
The coefficient associated with CD4BL4 is -0.147 with a standard error of 0.091 and hence a z-statistic of -2.80, giving a significant p-value of 0.005. The associated relative risk (RR) is 0.863, implying that patients starting ART with a CD4 baseline between 200 and 350 cells/mm3 have 0.863 times less risk of immune deterioration than patients who start ART with a CD4 baseline between 500 and 750 cells/mm3. The coefficient associated with CD4BL5 is -0.231 with a standard error of 0.089 and hence a z-statistic of -4.60, giving a significant p value of 0.000. The associated relative risk (RR) is 0.794, implying that patients starting ART with a CD4 baseline below 200 cells/mm3 have 0.794 times less risk of immune deterioration than patients who start ART with a CD4 baseline between 500 and 750 cells/mm3. The coefficient associated with DTB1 is -0.097 with a standard error of 0.040 and hence a z-statistic of -2.35, giving a significant p-value of 0.019. The associated relative risk (RR) is 0.908, implying that patients who develop TB and get treated have 0.908 times less risk of immune deterioration than patients who did not develop TB.
The results for the Cox proportional hazards model are almost similar with the results from Aalen’s additive model with some differences in the magnitudes. The other difference is that whereas CD4BL3 contributes significantly when Aalen’s model is used, it does not have any significant effect when the Cox proportional model is used.
Assessment of the fitted model
Goodness-of-fit tests were performed for the final models for both the Cox regression model and Aalen's additive model. The results are presented in the next sections.
Test for significance of Cox regression model
Test for the significance of the final Cox regression model is performed using the test statistics; likelihood ratio test, Wald test, and Score (logrank) test. The results are shown in Table 5 below. A further test is also performed using the Schoenfeld residual plots for the variables in the final model.
| Statistic | Test statistic | Degrees of freedom | p-value |
|---|---|---|---|
| Likelihood ratio test | 30.48 | 6 | 0.00003185 |
| Wald test | 32.79 | 6 | 0.00001152 |
| Score (logrank) test | 36.55 | 6 | 0.000002159 |
| Concordance = 0.686 (standard error = 0.038 ); R-square = 0.216 (max possible = 1 ) | |||
The overall fit for the final Cox regression model is statistically significant with p-values 0.00003184, 0.00001152, and 0.000002159 for the Likelihood ratio test, Wald test, and the score test, respectively. The likelihood ratio test for the final model is 30.48 (p-value = 0.00003185) which is higher than the likelihood ratio test = 26.52 (p-value = 0.003104) for the first model with a full set of variables. This indicates that the last model is preferable compared to the other models.
Tests for the proportional hazards assumption are done for each of the covariates together with a global test for the whole model. This test is based on the scaled Schoenfeld residuals. The scaled Schoenfeld residuals is the difference between the covariate at the failure time and the expected value of the covariate at this time. The results are shown in Table 6 below:
| covariates | rho | chisq | p-value |
|---|---|---|---|
| factor(Age) | 0.15211 | 3.93477 | 0.0473 |
| factor(Gender)1 | -0.00624 | 0.00623 | 0.9371 |
| factor(CD4BL)3 | -0.08836 | 1.12351 | 0.2892 |
| factor(CD4BL)4 | 0.08034 | 0.85007 | 0.3565 |
| factor(CD4BL)5 | 0.06781 | 0.59293 | 0.4413 |
| factor(DvpTB)1 | 0.10341 | 1.86523 | 0.1720 |
| Global | NA | 8.84077 | 0.1827 |
| Key: rho = correlation coefficient, chisq = chi-square statistic | |||
The results in Table 6 give strong evidence that the only variable displaying a significant deviation from the proportional hazards assumption is the age variable which has a p-value = 0.0473.
Figure 1a-f below shows the plots of the Schoenfeld residuals for the variables in the final Cox regression model. The plot of the Schoenfeld residuals is a useful diagnostic tool. A non-zero slope (\(\:\beta\:\left(t\right)\) will be a horizontal line) is an indication of the violation of the proportional hazards assumption.
The plots in Fig. 1b, 1d, 1e and 1f are relatively horizontal. This indicates that the proportional hazard assumption is satisfied for gender (Fig. 1b), CD4 baseline category 4 (starting ART when CD4cell count is between 200 and 350 (Fig. 1d)) and 5 (CD4cell count is below 200 (Fig. 1e)), and for patients who developed TB during the course of treatment (Fig. 1f). Figure 1a and 1c show a slightly increasing and decreasing trend, respectively. Thus, it seems there is not exactly satisfaction of the proportional hazard assumption for age (Fig. 1a) and patient in the CD4 cell count baseline category 3 (Cd4 baseline between 350 and 500 (Fig. 1c)).
Test for significance of Aalen’s additive model
Test for non-significance in Table 7 and the time-invariant effects were also performed for the final Aalen’s additive model and the results are presented below:
| Test for non-significant effects: | Test for time-invariant effects: | ||||
|---|---|---|---|---|---|
| Supremum-test of sig | p-value H0: \(\:{\beta\:}_{j}\left(t\right)=0\) | Kolmogorov-Smirnov test | p-value H0: constant effect | Cramer von Mises test | p-value H0: constant effect |
| \(\:{\beta\:}_{0}\left(t\right)\)=8.27 | 0.000 | \(\:{\beta\:}_{0}\left(t\right)\)=3.2 | 0.000 | \(\:{\beta\:}_{0}\left(t\right)\)=58.2 | 0.000 |
The results still maintain that the baseline rate is significant and time-varying as was shown by the first output containing all the variables.
Figure 2a-f below shows the cumulative regression functions for the final Aalen's additive model with variables, Age, WHO stage baseline, Gender, CD4 baseline, and development of TB during treatment as well as for the intercept.
confidence intervals based on Aalen’s additive model
Figure 2 indicates that the estimates of cumulative regression function for patient age (Fig. 2b), WHOSBL (Fig. 2f) are constant at a level of zero, hence the removal of these covariates except for age which was left because of its epidemiological importance. The estimated cumulative regression function plot for gender (Fig. 2c) increased rapidly after t = 4. The estimated cumulative regression function plot for DTB (Fig. 2e) and CD4BL (Fig. 2d) decreased slowly after t = 4. This could be an indication that the effectiveness of anti-retroviral therapy is notable after 4 six month periods (2 years) of treatment.
This paper presented a comparison of the Cox model and Aalen's additive model in analysing HIV progression in individuals upon initiation of treatment. The Cox model assumes constant proportional hazards. Aalen's additive model is a non-parametric model that allows time-varying covariate effects in which covariates are modelled as additive risks to a baseline hazard and are allowed to vary over time.
A comparison of the Cox model and Aalen's additive model is based on the p-values for the selected covariates. The results show that p-values in Aalen's additive model are fairly lower than the p-values in the Cox model. This could be because non-parametric models are generally less powerful in detecting significant effects compared to parametric models16. The overall significance of Aalen's model is higher than the overall significance of the Cox model. However, it is not possible to base our conclusion on this because Aalen's model uses a \(\:{\chi\:}^{2}\) test and the Cox model uses a likelihood ratio test. However, besides these differences, the Cox model and Aalen's additive model give almost similar results as far as the selection of covariates to remain in the final model is concerned. The signs of the coefficients of covariates for both models are also the same. Hence Aalen's additive model can be regarded as complementary to the Cox model. These findings agree with the results from studies on survival from breast cancer14,16.
The results from both Aalen's additive hazard model and Cox regression model show that the covariates; TB before enrolment (TBB4), adverse reaction to treatment (reaction), WHO stage baseline (WSBL), BMI progression, and viral load baseline (VLBL) are not significantly associated with HIV/AIDS progression for patients under treatment therapy. The other variables; CD4 baseline (CD4BL), gender, and developed TB during treatment contribute significantly to HIV/AIDS progression. These findings are in corroboration with the findings by Shoko and Chikobvu21. Although the variable age did not have any significant effect to the event of interest it was not removed from the model because of its epidemiological importance.
Results from this study show that males have poor adherence to treatment compared to their female counterparts. A study carried out in 2014 shows that males have higher patient attrition and mortality compared to females and this may be attributed in part to late presentation for HIV treatment and care22. Patients who develop TB whilst on treatment, if diagnosed and treated, are likely to have a better HIV treatment outcome than their counterparts. Another study also confirms that early recognition and appropriate management of these consequences can reinforce the successfully integrated therapy in HIV-infected patients with TB23. Commencing HIV/AIDS treatment when the CD4 cell count is below 350 cells/mm3 results in better treatment adherence than starting treatment when CD4 cell counts are above 350 cells/mm3. This finding is supported by the work from Sabin et al. which also recommended commencement of ART when the CD4 cell count is below 350 cells/mm3 and to improve adherence to treatment24;25.
However, for future studies on HIV/AIDS, the use of Markov multistate models is recommended because they allow monitoring of disease progression through multiple mutually exclusive states.
In this study, both the Cox proportional hazards model and Aalen’s additive model were used on HIV data. The Cox model and Aalen’s model both yield similar results with regards to the selection of covariates that contribute significantly to HIV progression. These covariates are gender, CD4 baseline, and development of TB. Males have an increased risk of immune deterioration than females. Patients who start ART with lower CD4 cell counts have smaller risks of immune deterioration. Developing TB on ART and having it monitored increases the rates of immune recovery.
SIGNIFICANCE STATEMENT
This study presents a comprehensive analysis of the determinants of HIV progression on patients receiving antiretroviral therapy using Aalen’s additive model and the Cox proportional hazard model. The models yield similar results. The study finding reinforce the need for early initiation of treatment for improved treatment outcomes and constant TB diagnostics to reduce mortality rates among HIV-co-infected individuals. This study suggests that in order to relay accurate information to the intended audience, at least two models can be used so that these models complement each other. Future prospects of this study is to model the progression of HIV among a TB co-infected cohort.
Conflict of interest:
The authors do not have any conflicting interests concerning the publication of this work.
Acknowledgment
We are grateful for the cooperation of the study participants in data collection. POB’s research was supported by the South African Medical Research Council (RCDI)through funding received from the South African National Treasury; the South African National Research Foundation (GUN109312, GUN86037), and the University of Venda. The views expressed are solely the responsibility of the authors and do not necessarily represent the official views of the South African Medical Research Council, the National Research Foundation, or the University of Venda.
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The authors declare no competing interests.