Research Article
Accelerated Failure Time Model with Application to Data on Tuberculosis/Hiv CoInfected Patients in Nigeria
Ogungbola OO* ^{}, Akomolafe AA and Musa AZ
Ogungbola OO^{1*}, Akomolafe AA^{1} and Musa AZ^{2}
^{1}Department of Statistics, Federal University of Technology Akure, P.M.B. 704, Akure, Ondo State Nigeria
^{2}National Institute of Medical Research, Yaba, Lagos State
*Address for Correspondence: Ogungbola Opeyemi Oyekola, Federal University of Technology, Akure,P. M. B. 704, Akure, Ondo State, Nigeria. Tel.: +2348060464240, ORCID: orcid.org/0000000307038047; Researcher ID: researcherid.com/rid/Q17542018; Email: ooogungbola@futa.edu.ng
Dates: 04 July 2018; Approved: 30 August 2018; Published: 03 September 2018
Citation this article: Ogungbola OO, Akomolafe AA, Musa AZ. Accelerated Failure Time Model with Application to Data on Tuberculosis/Hiv CoInfected Patients in Nigeria. American J Epidemiol Public Health. 2018;2(1): 021026.
Copyright: © 2018 Ogungbola OO, et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Keywords: Accelerated failure test model; Survival ratio; TB/HIV infection; Absolute lymphocyte count; Body mass Index
Abstract
In this research, we considered the Accelerated Failure Time (AFT) model for analyzing survival data on Tuberculosis/HIV coinfected patients in Nigeria We apply the methods to a cohort of these patients managed in tertiary Directly Observed Treatment Short Course (DOTS) Centre, Nigerian Institute of Medical Research (NIMR) for the period of 68 months. The effect of the accelerated failure time model was examined in determining the time to sputum smear and culture conversion following initiation of DOTS treatment and study the factors that influence it on TB patients who are coinfected with HIV. The research established that the model provides a better description than other commonly used models of the dataset because it allows prediction of Hazard function, survival functions as well as time ratio. AFT model was able to prepare some insights into the form of the baseline hazard. The result revealed that the weibull AFT model provided a better fit to the studied data. Hence, it is better for researchers of TB/HIV coinfection to consider AFT model even if the proportionality assumption is satisfied.
Introduction
Survival analysis is a statistical method for data analysis where the length of time, tc corresponds to the time period from a welldefined start time until the occurrence of some particular event or end point tc, i.e. t = tc  tc [1]. It is a common outcome measure in medical studies for relating treatment effects to the survival time of the patients. In these cases, the typical start time is when the patient first received the treatment, and the end point is when the patient died or was lost to followup.
Accelerated Failure Time model (AFT model) is a parametric model that provides an alternative to the commonly used proportional hazards models. Whereas a proportional hazards model assumes that the effect of a covariate is to multiply the hazard by some constant, an AFT model assumes that the effect of a covariate is to accelerate or decelerate the life course of a disease by some constant [2]. In clinical studies, the failure being investigated is often death. Survival analysis describes the methodologies used in Biostatistics to quantify and describe survival time and to examine the magnitude of differences in survival time [3]. Survival analysis is also appropriate for many other kinds of events such as criminal recidivism, divorce, childbearing, unemployment, and graduation from school [4]. Many studies in statistics deal with deaths or failure of components: the numbers of deaths, the timing of death, and the risks of death to which different classes of individuals are exposed. Then, in Uganda [5] used data to determine whether Tuberculosis (TB) preventive therapies affect the rate of Acquired Immune Deficiency Syndrome (AIDS) progression and survival in Human Immunodeficiency Virus coinfected adult by using Cox proportional hazards (PH) and Accelerated Failure Time (AFT) models and concluded that TB preventive therapies appeared to have no effect on AIDS progression, death and combined event of AIDS progression and death.
The AFT model is another alternative method for the analysis of survival data. This will be studied by means of real dataset which is from a cohort of TB/HIV coinfected patients managed in tertiary Direct Observed Treatment Short (DOTS) Course center for a period of six months among the Nigerian adults.
The time to sputum conversion is the smearpositive pulmonary TB (PTB +) cases registered in a specified period that converted to smear negative status after the standard two months of the intensive phase of treatment.
The main aim of this research is to fit the Accelerated Failure Test models in analyzing TB/HIV coinfected patients. The specific objectives are:
• To determine the effect of the three regimens of TB/HIV preventive therapy on TB/HIV coinfected adults.
• To determine the time to sputum conversion of TB/HIV patients on therapy.
• To determine the effect of the AFT model on time to sputum conversion in TB/HIV patients on therapy.
PH model and parametric AFT models was critically compared by [5]. The major aims of his work was to support the argument for consideration of AFT model as an alternative to the PH model in the analysis of survival data by means of real life data from TB and HIV in Uganda. There are two advantages of Cox proportional regression models, which are ability to incorporate time varying covariate effects and time–varying covariates [6]. The application of survival analysis has extended the importance of statistical methods for time to event data that incorporate time dependent covariates has used by [7]. The estimation of the longterm survivors’ proportion was extensively discussed by [8], also termed as “immune” or “cured” proportion, as well as the survival distribution, via parametric or nonparametric approach. In his work and other related works, ordinary maximum likelihood approach was successfully applied via mixture models to obtain estimators of the parameter P (the “susceptible proportion”) and the parameters associated with the proper survival function for the susceptible population, and their largesample properties are obtained via classic approaches. Some development that dealt with time varying effect of covariates was presented by [9]. He also emphasized the use of semiparametric models where some effects are timevarying and some are timeconstant, thus giving the extended flexibility only for effects where a simple description is not possible. Timevarying effects may be modelled completely nonparametrically by a general intensity model, ${\lambda}_{i}\text{(t)=}\lambda ({\text{t,X}}_{i}\text{(t))}$ . Smoothing techniques have been suggested for estimation of λ (.); see, e.g., [10] and the references therein. Such a model may be useful when the number of covariates is small compared to the amount of data, but the generality of the model makes it difficult to get a clear, if any, conclusion about covariate effects. Sy Han C proposed fast and accurate inferences for Accelerated failure time models with applications under various sampling schemes was proposed by [11]. In this research, we applied the model into the sputum conversion of the TB/ HIV which are coinfected patients managed in tertiary DOTS centre for a period of 6 months among the Nigeria adults. We also make use of the knowledge of percentage of censoring, variation in sample sizes. All these contribute to the existing knowledge.
Methodology
The study was carried out and obtained at the DOTS Clinic of the Nigerian Institute of Medical Research (NIMR). A parastatal under the Federal Ministry of Health that has treated over 5000 TB patients in the last 6 years (20112016). The Institute has a DOTS centre where it attends to patients infected with TB coinfected HIV.
All patients that were enrolled between 2011 and 2016 were included in the study; this will enable completion of the 6months treatment cycle for those enrolled in 2016.
We adopt AFT model in order to allow acceleration or deceleration of covariates effects which differentiates from PH models and during our analysis we found out that the data reveals just more AFT models and characteristics than other survival models.
Log rank test
This was used to compare the death rate between two distinct groups, conditional on the number at risk in the groups. The log rank test hypothesis that:
H0: All survival curves are the same
H1: Not all survival curves are the same.
Log rank test approximates a chisquare test which compares the observed number of failures to the expected number of failure under the hypothesis.
${X}^{2}=\text{}{{\displaystyle \sum {}_{i}^{k}}}_{=}{}_{1}({o}_{i}{E}_{i})/{E}_{i}\text{(3)}$
where, k −1 is the degrees of freedom. A large chisquared value implies a rejection of the null hypothesis for the alternative hypothesis.
Accelerated Failure Time Model
Exponential and Weibull AFT model
The exponential distribution was studied 1st in connection with kinetic theory of gasses [4]. The survival function of ${T}_{i}$ can be expressed by the survival function of . If the, has an extreme value distribution then ${T}_{i}$ follows the exponential distribution. The survival function of Gumbel distribution is given by
${S}_{\epsilon}{}_{{}_{i}}(\epsilon )\text{=exp(exp(}\epsilon \text{))}$
The Survival function of Weibull AFT model is given by
${S}_{i}(t)\text{=exp[`exp(logt}\mu \text{}{\beta}_{\text{i}}{\text{x}}_{\text{i}}\text{}\cdot \cdot {\cdot}_{\cdot \cdot \cdot}\text{}{\beta}_{\text{p}}{\text{x}}_{\text{p}}\text{]/}\sigma \text{(4)}$
And the cumulative hazard function of Weibull AFT is
${H}_{i}(t){\text{=logs}}_{\text{i}}\text{(t)=exp[`(logt}\mu \text{}{\beta}_{\text{i}}{\text{x}}_{\text{i}}\text{}\cdot \cdot {\cdot}_{\cdot \cdot \cdot}\text{}{\beta}_{\text{p}}{\text{x}}_{\text{p}}\text{]/}\sigma $
Lognormal AFT model
If the, has standard normal distribution then follows the lognormal distribution. The survival function of lognormal AFT model is given by
${s}_{i}(t)\text{=1}\varphi \text{[(logt}\mu \text{}{\beta}_{\text{i}}{\text{x}}_{\text{i}}\text{}\cdot \cdot {\cdot}_{\cdot \cdot \cdot}\text{}{\beta}_{\text{p}}{\text{x}}_{\text{p}}\text{]/}\sigma \text{(5)}$
The cumulative hazard function of Lognormal AFT model is
${H}_{i}(t){\text{=logs}}_{\text{i}}\text{(t)=log(1}\varphi \text{[`(logt}\mu \text{}{\beta}_{\text{i}}{\text{x}}_{\text{i}}\text{}\cdot \cdot {\cdot}_{\cdot \cdot \cdot}\text{}{\beta}_{\text{p}}{\text{x}}_{\text{p}}\text{])/}\sigma $
Loglogistic AFT model
If the 𝜀𝑖, has logistic distribution then 𝑇𝑖 follows the loglogistic distribution .The survival function of logistic distribution is given by
${s}_{{\epsilon}_{i}}\left(\epsilon \right)=\frac{1}{1+{e}^{\epsilon}}$
The survival function of lognormal AFT model is given by
${s}_{i}\left(t\right)=\left\{\frac{1}{1+{e}^{\left(\frac{(logt\mu {\beta}_{i}{x}_{i}\dots \dots {\beta}_{p}{x}_{p}}{\sigma}\right)}}\right\}\text{(6)}$
The cumulative hazard function of loglogistic AFT is given by
${H}_{i}\left(t\right)=log{s}_{i}\left(t\right)=\mathrm{log}\left(1exp\frac{(logt\mu {\beta}_{i}{x}_{i}\dots \dots {\beta}_{p}{x}_{p}}{\sigma}\right)$
Gamma AFT model
In survival literature, two different gamma models are discussed. The Standard gamma model with 2 parameters and the generalized gamma model with 3 parameters. In this study the standard gamma or gamma model is used.
The probability density function of gamma model
$f\left(t\right)=\frac{\alpha {\lambda}^{\alpha \gamma}}{\Gamma \left(\gamma \right)}{t}^{\alpha \gamma 1}exp\left[{\left(\lambda t\right)}^{\alpha}\right]t0,\gamma 0,\Gamma 0,\alpha 0$
Where γ is the shape parameter of the distribution. The exponential, Weibull and lognormal models are all special cases of the generalized gamma model. The generalized gamma distribution becomes the exponential distribution if $\alpha \text{=}\gamma \text{=1,}$ the Weibull distribution if 𝛾=1, and the lognormal distribution if $\gamma \to \infty $ .
Various Foodness of Fit Test
AIC
To compare various semiparametric and parametric models Akaike Information Criterion (AIC) is used. It is a measure of goodness of fit of an estimated statistical model. In this study, AIC is computed as follows
$AIC=2\left(log\text{likelihhod}\right)+2\left(P+K\right)\text{(7)}$
Where P is the number of parameters and K is the number of coefficients (excluding constant) in the model. For P = 1, for the exponential, P = 2, for Weibull, Loglogistic, Lognormal etc. The model which as smallest AIC value is considered as best fitted model.
For each distribution of 𝜀𝑖, there is a corresponding distribution for T. The members of the AFT model class include the exponential AFT model, Weibull AFT model, loglogistic AFT model, lognormal AFT model, and gamma AFT model. The AFT models are named for the distribution of T rather than the distribution of 𝜀𝑖 or $\mathrm{log}T$ as it is (Table 1).
Table 1: Summary of parametric AFT models. 

Distribution of ε  Distribution of T 
Extreme value(1 parameters)  Exponential 
Extreme value(2 parameters)  Weibull 
Logistic  Loglogistic 
Normal  Lognormal 
LogGamma  Gamma 
Model Diagnostics
The AFT model is well fitted. Log minus log also revealed the proportionality and the nature be it parallel or not.
Results
First, descriptive statistics are used to give us information about the distributions of the variables. We get the baseline characteristics in 452 participants using the descriptive statistics (Table 2).
Table 2: Baseline characteristics in 452 participants.  
Variable  T  Age  BMI  LYMPHABS  Creat  Haemo 
Total  3396  16627.32  158.11  97225.20  48092.80  23759.02 
Mean  7.5142  36.7861  0.3498  215.1000  106.4000  52.5642 
Std dev  1.2338  10.7473  0.1054  194.6253  64.3768  26.8265 
C.V  16.4196  29.2157  30.1315  90.4813  60.5045  51.0357 
Some continuous variables are grouped into categories according to clinical meaning. The KM curves show the shape of the survival function for each treatment arm. We can see from (Figure 1) that the cumulative survival proportion appears to be much higher in the AntiTB/HIV therapy (INHRIFEMB, INHRIFPZAEMB and INHPZAEMB) compared to the groups in which INHPZAEMB was used. In INHPZAEMB group, few participants resume this therapy. It would appear that INHRIFPZAEMB and INHPZAEMB of TB/HIV Therapy significantly prolong the time until participants resume event compared to the other interventions. The median survival time is at 40years of age for INHPZARIF combination TB/HIV therapy while 45years is expected for INPZAEMB therapy group. Many are censored in INHRIFEMB before reaching the age of 60years with regarding sputum conversion of TB patients on therapy in (Figure 1).
Note: EMB, INH, PZA, RIF and RPT represent Enthambutol, Isoniazid, Pyrazinamide, Rifampin and Rifapentine respectively.
Log Rank Test
Ho: The effect of the three regimens does not have significant factor to TB preventive therapy for TB/HIV coinfected adults.
H1: Not Ho: In the table 3, since Pvalue (.0192) < (α = 0.05), the effect of the three regimens does have significant to TB preventive therapy for TB/HIV coinfected adults. Then survival distributions are different in the population which make the result more statistically significance.
Table 3: Test for equality of Survival Distribution for Different level of TB/HIV Therapy.  
ChiSquare  df  Sig.  
Log Rank (MantelCox)  9.930  3  0.019 
Breslow (Generalized Wilcoxon)  8.570  3  0.036 
TaroneWare  9.055  3  0.029 
By the logrank test, in the preventive therapy, there is significant difference among three regimens of TB preventive therapy for TB/HIV coinfected adults, since the pvalue is 0.0192 against 5% level of significance. The KaplanMeier (KM) curves for time to educate length and time to combined event of the preventive therapy is presented (Figure 1), the age is just the median of the coinfected patients’.
In (Figure 2), the log minus log plot was not parallel which revealed to us approximately the suitability of AFT. For this reason, the investigation of AFT Model comes into play.
In (Figure 3), the Loglogistic, Weibull and Lognormal Density functions are approximately normal to the curve while others are not similar.
Accelerated failure time models were compared using statistical criteria (likelihood ratio test and AIC). The nested AFT models can be compared using the Likelihood Ratio (LR) test. The models in (Table 4 and 5) reveal that (covariates) age and sex are statistically significant while HAEMO GLUC, BMI and LYMPHABS are not significant with their pvalue greater than 0.05, whereas the (covariates) age, sex and haemo are statistically significant with their pvalues. In the AFT model, loglogistic AFT model and the Weibull AFT model are nested within the lognormal AFT model (Table 6). According to the Loglikelihood Ratio (LR) test, the weibull model fits better. However, the LR test is not valid for comparing models that are not nested. In this case, we use AIC to compare the models (Table 7) (The smaller AIC is the better). The Weibull AFT model appears to be an appropriate AFT model according to AIC compared with other models, although it is only slightly better than Loglogistic or Lognormal model. We also note that the Lognormal model is poorer fit according to LR test and AIC. At last, we conclude that the Weibull model is the best fitting the AFT model based on AIC criteria.
Table 4: Loglogistic AFT Model.  
Covariates  TR  Sig.  95.0% CI for TR)  
Lower  Upper  
AGE  0.022  1.571  0.035  1.210  1.932 
SEX  0.308  0.735  0.014  0.286  1.756 
LYMPHAB  0.013  0.989  0.689  0.922  1.056 
HAEMO  0.133  1.146  0.457  0.797  1.495 
CREAT  0.000  0.999  0.984  0.987  1.011 
BMI  0.561  1.753  0.410  0.528  2.978 
WEIGHT  0.061  0.928  0.510  0.738  1.118 
GLUC  0.022  0.978  0.168  0.947  1.009 
Loglikeliho  225.156 
Table 5: Weibull AFT model.  
Covariates  TR  Sig.  95.0% CI for TR)  
Lower  Upper  
AGE  0.018  1.018  0.023  0.618  1.418 
SEX  0.258  0.773  0.042  0.142  1.404 
LYMPHAB  0.011  0.919  0.500  0.852  0.986 
HAEMO  0.136  1.146  0.438  0.801  1.491 
CREAT  0.000  0.999  0.984  0.981  1.009 
BMI  0.336  1.396  0.371  0.659  2.133 
WEIGHT  0.075  0.908  0.440  0.718  1.098 
GLUC  0.022  0.978  0.145  0.949  1.007 
Loglikelihood  218.079 
Table 6: Lognormal AFT model.  
Covariates  TR  Sig.  95.0% CI for TR)  
Lower  Upper  
AGE  0.026  1.026  0.041  0.391  1.661 
SEX  0.158  0.854  0.036  0.437  1.271 
LYMPHAB  0.014  0.989  0.659  0.928  1.050 
HAEMO  0.146  1.158  0.009  0.842  1.474 
CREAT  0.000  0.999  0.079  0.987  1.011 
BMI  0.627  1.858  0.349  0.903  2.813 
WEIGHT  0.061  0.928  0.465  0.763  1.093 
GLUC  0.023  0.977  0.852  0.945  1.008 
Loglikelihood  235.009 
Table 7: The loglikelihoods and Likelihood Ratio (LR) tests, for comparing alternative AFT models.  
No of parameter  Loglikelihood  Testing against the Lognormal distribution  
Distibution  m  L  LR  df 
Loglogistic  2  100.532  326.46  1 
Weibull  3  263.762  440.452  2 
Lognormal  2  43.536 
Table 8: Akaike Information Criterion (AIC) in the AFT models.  
Distibution  Loglikelihood  k  c  AIC 
Loglogistic  100.532  6  2  225. 156 
Weibull  263.762  6  1  218. 079 
Lognormal  43.536  6  2  235. 019 
Discussion and Conclusion
In this paper, attempt has been made to find the best fitted model for studying the survival time TB/HIV patients. To meet the objectives various AFT models following distributions Weibull, Lognormal, Loglogisitic are fitted to survival data of TB/HIV patients collected from NIMC, Yaba, Lagos. Different statistical measures such as AIC, Cox Snell AFT residuals plots and Log minus are used to find the best fitted model. From the results, the Weibull AFT model is found to be the best fitted model.
In this study, the result shows that Weibull AFT model is better than the other models in case of explaining the survival esophagus cancer data. The factor, TB/HIV directed treatments, has a significant role in case of survival of TB/HIV patients. The patients who undergo the TB/HIV directed treatment other than surgery has lower risk of dying than the patients who has underwent the treatment of surgery and its combinations according to the laid down principles.
This study is based on a large number of participants from Lagos residents in Nigeria, where the prevalence of TB infection and HIV are very high. In this study, the Cox model and the Accelerated Failure Time model have been compared using TB/HIV coinfected data. Association of the TB/HIV preventive therapies with the sputum conversion is examined through the linkage of the signs and symptoms to replication of the virus.
The AFT model provides an estimate of the survival function time ratios. In this research, we have analyzed the TB/HIV dataset the methods. This study provides an example of a situation where the AFT model is appropriate and the description of the data reveal that logminuslog plot is not parallel.
Acknowledgement
We like to appreciate the management of National Institute Medical Research (NIMR) for their ethical approval to make use of their health survival data. God bless them all.
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