Flexible Parametric Survival Model for Analysing Censored Time-To-Event Data Among Tuberculosis Patients.

Conclusion

Using the flexible parametric survival model hazard can avoid splitting the survival data into intervals and avoid making an assumption of constant hazard rates within the time intervals in estimating the hazard rates and probabilities. The study highlights the importance of fitting models by modelling the baseline, which is easier to make different types of predictions and allows for non-proportional hazards since it is an interaction. The results from applying the three models revealed that sex, ART, and diabetes covariates were found to statistically significantly improve the survival rate of TB treatment prognostic. The flexible parametric survival model hazard ratios were compared with the additive hazard model and Cox proportional hazard model, and the effect estimates obtained from the hazard models were different. The study recommends hospital authorities pay attention to female TB patients who are HIV positive, on ART, and diabetic. The rate of mortality during TB treatment was high for these groups. TB patients should be advised to adopt healthier lifestyles during treatment, which can include better nutrition and increased physical activity. These lifestyle changes can reduce the risk of developing diabetes.

Acknowledgments

Not applicable.

Authors’ contributions

AA, OG, AT, and OA designed the study. AA, OA, and MR performed the data analysis and drafted the manuscript. OA, MP, AT and RM supported the analysis and provided revisions and improvements to the manuscript. All authors have read and approved the final manuscript.

Funding

No funding

Availability of data and materials

Data will be available on request. All data analysis codes used can be found on my GitHub page.

Ethics approval and consent to participate

The study was approved by the Ethic Committee of the University of Fort Hare (REC-270710-028-RA) with reference number: QIN11SAZE01 and also approved by the Eastern Cape Department of Health (EC_202008_007). Informed consent was not required as we had no contact with the study individuals nor did our analyses contain any personal identifiers. All methods were performed according to relevant guidelines and regulations.

Consent for publication

Not applicable.

Competing interests

We declare that they have no conflict of interest.

Author Details

1. Introduction

Tuberculosis (TB) is primarily caused by Mycobacterium tuberculosis, typically affecting the lungs [1]. Annually, 10 million people contract TB and 1.5 million people die from the disease, making it one of the deadliest infectious diseases globally. It is the primary cause of death for those with HIV and significantly contributes to antimicrobial resistance [1]. Poor outcomes from TB treatment are driven by the high rate of death and significant loss of follow-up [2], [3]. This issue of loss to follow-up is a major challenge experienced by the South Africa National TB Programme [4]. One of the key advantages of shorter treatment regimens for tuberculosis-resistant treatment is a reduction in the loss to follow-up rate [5].

The treatment success rate in South Africa for new smear-positive and smear-negative/ extrapulmonary TB patients has improved by 79% and 76%, respectively [6]. This was achieved as a result of higher cure rates and a decrease in the treatment default rate. However, the treatment success rate for retreatment cases remains low at 66.3% [6]. Of particular concern is the fact that up to 25% of sputum smear-positive TB cases are lost to follow-up before treatment initiation, which may contribute to ongoing transmission of the disease and an increased risk of death [7]. Furthermore, the mortality rate remains high even after completion of TB treatment, likely due to HIV disease [8]. To address this issue, there is a need to expand access to antiretroviral therapy (ART) for all HIV-infected TB patients to reduce HIV-related mortality among individuals with TB.

Understanding the effect of TB treatment on the time-to-death of TB patients by covariates such as gender, HIV status, age, and many more, may provide valuable insights for health programs in South Africa and globally. To achieve this target, flexible parametric hazards (PH) models and Additive hazard (AH) models were used in survival analysis to study the time-to-event data, which provide a flexible and versatile framework to capture complex hazard functions and assess the impact of covariates on survival probabilities in a more flexible way.

The concept of the flexible PH models is to use restricted cubic splines to approximate the baseline hazard function in the context of the Cox proportional hazards model [9] and the AH model using kernel smoothing techniques [10].

These models are a more adaptable approach to modelling survival data, which accommodates both non-linear and time-dependent effects. The integration of time-dependent covariates is used to examine the changes in risk factors over time. Furthermore, these models offer greater flexibility in capturing a wide range of survival patterns, from monotonically increasing or decreasing hazards to more intricate shapes, by altering the number and placement of spline knots [11]. Parametric models offer distinct advantages, including better suitability for prediction, extrapolation, quantification of risks, modelling time-dependent effects, enhancing understanding, and handling complex large datasets. However, the estimates from flexible parametric survival models are often similar to those obtained from the Cox model.

In Cox regression, the baseline hazard function is not estimated, which can limit its ability to capture the true underlying hazard function. In contrast, flexible parametric models offer an alternative approach by explicitly modelling the baseline hazard using splines, enabling more accurate representations of complex hazard patterns and facilitating better predictions, especially in scenarios with non-proportional hazards. The models provide a parametric estimate of the baseline hazard without usual shape restrictions, making it highly flexible [9]. It can be applied to both standard and relative survival models and is capable of fitting relative survival cure models [12]. It can also be estimated on the log-hazard scale [13].

For this work, we investigated the use of flexible parametric methods to analyse censored time-to-event data in patients with tuberculosis in a small population of South Africa. We described and compared flexible parametric hazards (PH) models and Additive hazard (AH) models using a real-life case study of individual-level censored data from the Tuberculosis Hospital and linked mortality data for the general population and people with HIV status and stratified by ART status.

2. Methods

The Cox Proportional Hazards (Cox PH) model

In survival data analysis, the Cox model is a widely used statistical method to assess the relationship between covariates and the hazard rate over time while making no assumptions about the shape of the hazard function. The model is written as:

where  is the hazard rate at time  for a given set of covariates ,  is the baseline hazard rate, and  is the coefficients of the covariates  on the hazard rate. The integrated form of the model is:

where  is the cumulative hazard function.

The model does not assume a specific distribution for survival times but estimates the relative risk of covariates about the shape of the baseline hazard function. But, the underlying shape of the hazard function is often ignored [14].

Additive hazards

The additive hazards models have a general form of . Recently, the default model specification for survival without specifying the smoothness has cumulative hazard function as:

where  is a natural spline design matrix with parameter , and  is the parameter for . The hazard function is given as

Let  be the hazard rate at time  for a specific event of interest, the additive hazards model can be expressed as:

where  is the hazard rate for the event,  is the vector of covariates,  is the vector of regression coefficients, and  is the function of time  that captures the smooth or time-varying effects. The additive hazards model assumes that the log-hazard rates for the event are linearly related to the covariates  and include time-varying components . The implementation of Additive hazard models offers flexibility by allowing the modelling of the baseline hazard using splines and accommodating both constant hazards and smooth time-varying effects.

3. Flexible Parametric Survival Models

Let the survival function  for a random variable  at time  with covariate  written as:

where  is the parameter,  is the coefficient of covariate indexed , and  is the parametric smooth function. Within this framework, a smooth function is used to model the baseline log cumulative hazard function and a linear predictor to model the covariates. However, the hazard function and cumulative hazard function are modelled as:

Now, considering two sets of covariates,  and , the hazard ratio can be expressed as:

If  and  for , then the hazard ratio is equal to  for all  and other covariate values. The model can be incorporated with time-dependent effects of covariates  on the log-hazard scale given as:

where  is the restricted cubic spline function,  denotes the spline function for the pth time-dependent effect, and  is the number of time-dependent effects.

The restricted cubic splines within the models are employed to model the log cumulative hazard or the log cumulative odds [9], [11]. These splines are piecewise cubic functions connected at specific positions referred to as knots. To ensure smoothness, the first and second derivatives of the overall function are enforced to be continuous at the knots, and the function is linearly constrained before the first knot and after the last knot. The complexity of these spline functions is dictated by user-defined degrees of freedom, which equate to the number of knots minus one. Knot positions can either be defined by the user or set to be evenly spaced percentiles of the observed event-time distribution [13].

However,  with knots  is expressed as:

where  is the basis function for the lth time defined as:

where  and  are the knots boundaries and .

Using restricted cubic splines in flexible parametric survival models would help to capture both simple and complex hazard functions in situations where standard parametric models may have challenges [15].

Study setting and design

This was a hospital-based retrospective individual-level censored data in TB patients reported for the treatment in East London Central Clinic-TB unit, Eastern Cape, South Africa. This clinic is a specialized facility funded by the provincial government, dedicated to the diagnosis, treatment, and prevention of TB, especially in patients with HIV co-infections. The clinic offers a range of services, including antiretroviral treatments (ARTs), TB services, and diagnostic tools with standard TB treatment protocols. Patient medical histories, from their initial consultation to discharge, are recorded in the TB treatment registry, which is the official record-keeping system for TB treatment under the Department of Health in South Africa.

Data collection

In this study, data were gathered from medical records of hospitalized TB patients with HIV coinfection. Information was extracted from TB record files and patients' medical files using a standardized TB card format recommended by South Africa's Department of Health. The information includes gender, age, location, TB category, HIV status, diabetes, weight, and antiretroviral therapy (ART).

Statistical analyses

We initiate our analysis by applying basic proportional hazard models to the TB dataset to identify the contributing covariate factors. Initially, we employed a Cox regression model was used to analyze each single covariate to determine whether the covariate is associated with improved survival of the TB patient. A flexible parametric survival model was fitted with an additional argument (df =4) to specify four different degrees of freedom for the baseline smoother. We compare the survival estimate of the flexible parametric survival model with predictions from non-parametric Kaplan-Meier and Additive hazard model curves. All analyses were done in R using rstpm2 and flexsurv packages.

Ethics

Ethical clearance for the study was obtained from the Ethics Committee, University of Fort Hare, and Department of Health, Eastern Cape chapter, South Africa.

Results

More than half of the patients were male (63.2%). The mean age of TB patients on treatment was 39.4 ± 17.3 years (range: 14-80 years). Of all the TB patients, 79 (45.4%) were HIV-positive TB patients, 78.2% had pulmonary TB, 77.6% had drug-resistance TB, 21.8% were placed on antiretroviral therapy and almost two-thirds (60.9%) of the patients were treated without diabetes (Table 1).

Table 1: The Demographic Characteristics of the TB patients

As we did not assume proportional hazards for TB treatment risk and aimed to evaluate the risk ratio instead of the attributable risk, we opted for a flexible parametric survival model. Unfortunately, the results from the additive hazards regression models did not provide satisfactory outcomes. We considered three different models for fitting the data, and based on the Akaike Information Criterion (AIC=1421.628) and the log-likelihood estimate (-2 log L=1389.628), we determined that the flexible proportional hazards (PH) model was the most suitable model. The results are summarized in Table 2, with the flexible PH model being the preferred choice.

Table 2: Selection criteria for best model fit

The analysis results revealed that certain variables, including sex, ART, and diabetes, were identified as treatment risk factors affecting the survival of TB patients, while the other clinic characteristics were not statistically significant (Table 3). Notably, the result of the Cox proportional hazards model was similar to the flexible parametric survival model in detecting the TB treatment risk factors, whereas the results from the additive hazards model were notably different. From the model output. the hazard ratios in the flexible model for estimating the TB treatment risk factors were lower and had more narrow confidence intervals compared to the Cox regression model. Specifically, the analysis showed that sex was significantly associated with improved treatment survival for TB patients (HR=0.49, 95% CI: 0.38, 0.62). Furthermore, ART was found to be statistically significantly associated with improved treatment survival for TB patients (HR=0.53, 95% CI: 0.34, 0.78), and diabetes exhibited a similar statistically significant association with improved treatment survival for TB patients (HR=0.58, 95% CI: 0.41, 0.78).

Table 3: Hazard estimates of TB treatment factors based on Cox model and Flexible model

Flexible parametric survival models are capable of estimating a wide range of parameters. However, the prediction estimates from the flexible model were compared with predictions from the Additive model and non-parametric Kaplan-Meier curves (Figure 1). The plot shows that the Flexible parametric survival model has a better-predicted survival probability at each time point compared to other models. The shape of the flexible model indicates a higher and improved survival rate among TB patients, followed by the additive model and KM model suggests a lower survival rate among TB patients. The overall pattern of the curves is steadily decreasing indicating consistent association with improved treatment survival for TB patients, which contributes to the survival estimates up to the last observed time.

In addition to the mortality rates, we obtained smooth predicted survival curves to facilitate comparisons among different covariate groups within various sub-groups and time intervals. Figure 1 illustrates the predicted survival probabilities on the time scale for gender and the use of antiretroviral therapy (ART), which were found to significantly contribute to improved survival outcomes among TB patients. The left panel shows the smooth predicted survival curves for the three models over the time since the initiation of TB treatment, categorized by patient gender. The right Panel shows the corresponding smooth predicted survival curves for patients on ART.

Figure 1: Predicted survival rate for covariate sex and ART among patients on TB treatment

Figure 2 displays the predicted survival probabilities on the time scale for diabetes and HIV status, both of which were identified as significant contributors to enhanced survival rates among TB patients. The left Panel exhibits the smooth predicted survival curves for the three models over the time since the TB treatment initiation for diabetic patients. Meanwhile, the right Panel displays the equivalent curves for patients with HIV status. It can be seen from both figures that the survival proportions are higher for TB patients in the flexible model compared to other models.

Figure 2: Predicted survival rate for covariate diabetes and HIV status among patients on TB treatment

The spline coefficients are not interpretable on their own but they are used to predict the shape of the hazard surface at different covariate values. Figure 3 displays four panels for viewing the estimated mortality rates among TB patients from the flexible parametric model. The upper left panel shows the estimated mortality rates of female patients on TB treatment and male patients on TB treatment. The upper right panel shows the estimated mortality rates of patients without ART on TB treatment and patients with ART on TB treatment. The lower left panel shows the estimated mortality rates of patients without diabetes on TB treatment and diabetic patients on TB treatment. The lower right panel shows the estimated mortality rates of HIV-negative patients on TB treatment and HIV-positive patients on TB treatment.

Figure 3: Predicted estimate of Hazard ratios with 95% C.I for sex, ART, diabetes, and HIV status groups among TB patients

The covariate sex panel shows that the mortality rates for both females and males are decreasing with male patients having significantly lower mortality rates (improved survival rates) compared to female TB patients. There was a complete overlap between the sex group's estimated mortality rates and survival proportions. Patients treated with ART therapy also have lower mortality rates than patients without ART therapy. There was a small overlap between the ART group's estimated mortality rates. Diabetic patients were observed to have a lower mortality rate compared to patients without diabetes. The high mortality among patients without diabetes can be due to a more severe disease for this group of patients. The overlap between the diabetes group's estimated mortality rates is very small. There was little or significant overlap between the HIV group's estimated mortality rates and survival proportions (Figure 4). Moreover, the peak of the surface, marked with green and red bands, reflects group hazard ratios of 95% CI. As time progresses on the time scale, the mortality rate surface widens, showing predicted clinical treatment observations [16].

Figure 4: Overlap Predicted estimate of Hazard ratios with 95% C.I for sex, ART, diabetes, and HIV status groups among TB patients

We assessed the time-varying covariate effects on TB treatment by estimating survival differences and hazard differences to compare hazard and mortality rate ratios along with their 95% confidence intervals over time for each group (Figure 5). We initially defined the survival differences based on the covariate pattern and subsequently transformed them into an 'exposed' covariate pattern using the exposed function. From Figure 5, we observed that the relative risk effect (hazard differences) of sex, ART, and diabetes slightly increases and the mortality effect (survival differences) decreases rapidly with time after 100 days of TB treatment initiation. Meanwhile, the relative risk effect (hazard differences) and mortality effect (survival differences) of HIV status among TB patients are the same ( no increase or decrease) after 100 days of TB treatment initiation.

Figure 5: The time-varying covariate effects on TB treatment using survival and hazard differences

Discussion

The flexible parametric survival model has been widely used in various fields, including applications in relative survival and clinical decision-making [9], [11], [17]–[21]. However, its utilization in the context of TB treatment has been relatively limited. In this article, we have employed the flexible parametric survival model to estimate survival probabilities for patients undergoing TB treatment and juxtapose the results with additive hazard models and the Cox proportional hazards model.

The flexible parametric model is an alternative approach for estimating survival probabilities. Unlike conventional methods that rely on a priori transition probabilities, this approach uses individual patient data directly to model survival. The integration of individual patient data with additional covariate information could be worthwhile and lead to improved accuracy in predicting survival probabilities [22]. In our analysis, the flexible parametric survival model provided reliable and smooth estimates of the baseline cumulative hazards [18].

According to the analyses of the flexible parametric survival models, the results obtained indicate that sex is significantly associated with improved survival rates of patients on TB treatment. The result indicates that male is significantly associated with a 51% lower risk of mortality rate [0.490 (95% CI: 0.382, 0.617), p-value=0.002] compared to female. This result is consistent with other earlier studies [23]–[25] in which standard survival analyses were applied. Hence, it can be concluded that male patients had a higher improved survival rate of TB treatment. There may be biological differences between males and females that can impact how they respond to TB treatment, such as hormonal differences, which play a role in immune response and may explain this pattern, which has also been observed in TB research studied in other areas [26], [27].

Our findings indicated that individuals who began the ART regimen experienced a 47.4% improved survival rate following the initiation of TB treatment [0.526 (95% CI: 0.336, 0.777), p-value=0.0019]. This result supports the recommendations to commence ART at earlier stages for all symptomatic individuals, irrespective of their CD4 cell counts. Previous studies have consistently demonstrated that the initiation of ART serves as a significant predictor of mortality among TB patients concurrently receiving ART [28]–[33].

The findings from our study also revealed that diabetes was significantly associated with improved survival rates among TB patients. The result showed that TB patients have a 42% lower risk of mortality rate [0.579 (95% CI: 0.414, 0.784), p-value = 0.0015]. This is consistent with some studies with lower risk [34]–[36] and dissimilar to other studies with higher diabetes risk among TB patients in different countries [37]–[39]. The observation of improved survival rate among the patients may be attributed to several factors. TB and diabetes are known to have complex biological interactions. TB can lead to a temporary state of insulin resistance, which can affect glucose metabolism. This can result in lower blood sugar levels, reducing the likelihood of a diabetes diagnosis during TB infection. TB and diabetes share some common symptoms. These overlapping symptoms might make it more challenging to diagnose diabetes in TB patients, potentially leading to underdiagnosis. TB can be a severe disease, and individuals with TB may not survive long enough to develop diabetes. This selective survival effect could contribute to the observed improved survival rate among TB patients.