3.1. Patients’ Characteristics
In this study, the time to death of TB patients was collected from a total of 604 patients identified retrospectively who had undergone anti-TB treatment. Complete follow-up information was available for 558
TB patients with mean (SD) age at baseline
years and median age 43 years. It was observed that 46
TB patients died during the follow-up period and the rest of the TB patients were censored at the end of the study. Regarding the sex composition of patients, 300
were males, among them
died, while 27
deaths occurred among the female patients (
Table 1). It was also observed that 282
patients were HIV-positive, of whom 27
later died. In addition, of 232
pulmonary TB patients, 15
died during the study period, while 25
deaths occurred among the negative pulmonary TB patients. Furthermore, 339
TB patients had no previous treatment history at baseline, of whom 28
later died.
Table 2 presents descriptive summaries for the continuous covariates. The median follow-up time was four months for patients that were censored and about
of the patients had three months of follow-up (lower quartile). The median time of death was five months. This indicates that most of the events or deaths occurred in the earlier months of the anti-TB treatment. At the start of the study, the baseline average (SD) age and weight of TB patients was
and
, respectively. Fifty percent
of the sampled patient ages fell between a lower quartile of
and an upper quartile of
, whereas the patients’ weight fell between a lower quartile of
and an upper quartile of
.
3.2. Comparison of Survival Curves
The survival distributions of time to death of the TB patients were estimated for each group using the Kaplan–Meier (KM) method to compare the survival curves of two or more groups. The Kaplan–Meier estimated survival curves in
Figure 1 highlight the overall estimated survival function using different groups of covariates. The overall estimated survival curve for the time to death of TB patients is shown in
Figure 1a. In addition, the survival curves of TB patients were different for HIV status, smear result, category of TB, and type or status of TB patient (see
Figure 1b–d), whereas among gender, previous treatment history, and place of residence there were not clear differences (figure not shown).
The observed differences in survival experiences in different patient groups was also assessed using the log-rank and Breslow tests [
23].
Table 3 shows that there was a significant survival time difference among type of TB, patient category, and HIV status for TB patients at the 5% significance level. Since the null hypothesis was rejected for the covariates, type of TB and category of TB, post hoc analysis was performed using pairwise comparisons among the categories. The survival times or time to death of TB patients was not significantly different between pulmonary negative and pulmonary positive TB patients (
p-value
), whereas extrapulmonary patients experienced significantly different survival times when compared to pulmonary negative and pulmonary positive patients (
p-value
). The interactive effects of gender, smear result, previous treatment history, and place of residence with survival time of TB patients were statistically non-significant (result not shown).
3.3. Model Selection and Inference
The Weibull, log-logistic, and log-normal distributions for baseline hazard function were also assessed. The likelihood ratio test (LRT) result for testing the regression parameters equal to zero was rejected for all baseline hazard distributions. The LRT, parameter estimates, standard error, and
p-value of the models are displayed in
Table 4. The parameter estimates obtained were consistent in sign for all the baseline distributions. However, the standard error was relatively smaller for the Weibull baseline distribution when compared to the other baseline distributions. The age, weight, HIV status (positive), smear result (positive), extrapulmonary TB type, retreated, and return-after-default were important covariates related to the survival of TB patients. The loglikelihood value for the loglogistic baseline was
with an LRT chi-square value of
and a
p-value
. For the Weibull and lognormal baseline, the loglikelihood values were
and
, respectively. It was observed that the Weibull baseline had the highest loglikelihood value of
, indicating that the Weibull baseline was better fitted to the data than the others. The survival model fitted with the Weibull baseline was significant with
and
p-value
.
The Gamma and inverse-Gaussian distributions are the most common and widely used distributions cited in the literature for modeling the frailty effect [
25,
26]. Thus, considering the Gamma and inverse-Gaussian distributions for frailty random variates (hospitals), parametric shared frailty models were fitted. The parameter estimates, standard error (se) and
p-value (
p) of Gamma and inverse-Gaussian shared frailty models are displayed in
Table 5. The loglikelihood value for the Gamma frailty model with Weibull distribution for baseline hazard function was
, which was the largest among the models considered. This indicated that the Weibull baseline hazard distribution for both frailty variates fitted better for the time to death outcomes of TB patients. The Weibull–Gamma shared frailty model had a minimum
value of
compared to the other models, suggesting that it is the preferred model for describing the TB dataset (see,
Table 5).
The heterogeneity test of the frailty terms for the selected model was performed using the one-sided likelihood ratio test (LRT) proposed by Claeskens et al. [
31]. The LRT for testing
for a shared Gamma frailty model with a Weibull baseline hazard function has an asymptotic
mixture distribution. The loglikelihood value for the null model (Weibull baseline model) was
, and for the full model (Gamma frailty model) was
, with an observed value for the LRT statistic of
with a
p-value of
. Thus, the variances of the random effect of the Gamma shared frailty model were significant at the
level of significance.
This result indicates the existence of unobserved heterogeneity between the hospitals and that the frailty component in the model was important. The heterogeneity estimate was
and the dependence within the clusters (treatment centres) was measured by Kendall’s tau as
(see,
Table 5). Therefore, the time to death of TB patients was modeled by a Gamma shared frailty model with hospitals as a clustering effect and a parametric Weibull distribution for the baseline hazard function.
The parameter estimate, standard error, and
p-value for the Weibull–Gamma shared frailty model displayed in
Table 5 show that age in years, initial weight, category of TB patient, extrapulmonary type of TB, and HIV status were important factors affecting the time to death of TB patients. However, the variables, gender, smear result, previous treatment history, and place of residence had no significant effect on the time to death of TB patients.
Conditional on the hospital frailty, the estimated acceleration factor or risk of death for the age of TB patients was , with a confidence interval of . After adjusting for other covariates, a one year increase in the age of a TB patient increased the risk of death by 1.022 times, whereas, an increase in age of 10 years resulted in an increased risk of death by times. These results imply that the risk of death was higher for older TB patients compared to younger patients.
The estimated acceleration factor and its for initial weight were and (0.9400, 0.9805), respectively. For a one kg increase in the weight of a TB patient, the risk of death decreased by 0.9627 times, holding the effects of other covariates constant and accounting for frailty effects. Thus, patients with low weight measurements at baseline had a higher risk of death.
The hazard rate or risk of death for extrapulmonary TB patients was significantly different from that for pulmonary negative TB patients (p-value ), while the risk of death was not significantly different for pulmonary positive patients. The estimated acceleration factor for extrapulmonary TB patients at any time t was , with 95% Wald CI (1.534, 11.993). Thus, conditional on the frailty effects, the risk of death for extrapulmonary TB patients was increased by times compared to that of pulmonary negative patients, holding the effects of other covariates constant.
The estimated acceleration factor and its 95% CI for HIV-negative TB patients were and , respectively. The point estimate indicates that HIV-negative TB patients were about times less likely to die than HIV-positive TB patients. This indicates that HIV-negative patients had a higher survival probability compared to HIV-positive patients.
According to the result of the Weibull–Gamma shared frailty model, the estimated acceleration factor for re-treated TB patients was with a CI of , while, for returned TB patient’s after treatment, the default was with a CI of . Thus, conditional on the covariate and frailty effect, the hazard or risk of death for retreated TB patients at any time t was times that of newly diagnosed TB patients, while TB patients who returned after treatment default were times more likely to die compared to new TB patients.