Modeling time-to-cure from severe acute malnutrition: application of various parametric frailty models

Malnutrition is a common cause of morbidity and mortality in developing countries [1]. It is a risk factor for over 50% of the 11 million annual childhood deaths [2]. Severe acute malnutrition (SAM) is defined as a very low weight for height less than 70% NCHS median or less than 115 mm of MUAC [3]. In developing countries, about 3.5% of children aged 0–5 years are suffer from SAM [4], which is the most important nutritional disease because of its high prevalence and relationship with child mortality [5]. Currently, there are a number of studies on time of recovery from acute malnutrition in various parts of the world. Survival data is a term used for describing data that measure the time to a given event of interest. The term survival data is often used for describing data that measure the time to the occurrence of a given event of interest. The event of interest can be seen as a transition from one state to another. In this study, the event of interest was the time-to-cure from SAM from the day of diagnosis. One of the major aim of this analysis was to model the time-to-cure from SAM among less than five year inpatient children in Wolisso St. Luke Catholic hospital and to compare the efficiency of various parametric frailty models using the same dataset. The classical model for this kind of data is the proportional hazards model popularized by Cox [6]. However, correct inference based on Cox’s model needs identically and independently distributed samples. One of the reasons why this model is so popular is the ease with which technical difficulties such as censoring and truncation are handled. This is due to the appealing interpretation of the hazard as a risk that changes over time. Naturally, the concept allows for entering covariates to describe their influence and model different levels of risk for different subgroups. Nonetheless, subjects may be exposed to different risk levels, even after controlling for known risk factors; as some relevant covariates were often unavailable to the researcher or even unknown (univariate frailty case). The study population may also be divided into clusters so that subjects from the same cluster behave more cohesively than subjects from different clusters (multivariate frailty case). The frailty model, introduced in the statistical literature by Vaupel et al. [7], and discussed in details [8‐10], accounts for heterogeneity in baseline. It is an extension of the proportional hazards of Cox’s model in which the hazard function depends upon an unobservable random quantity, the so-called frailty that acts multiplicatively on it. Study subjects (children) in this study came from clustered community and hence clustered child survival data may be correlated at the Kebele level. In this research, shared frailty models were explored assuming that children within the same cluster (kebele) share similar risk factors, which will take care of the frailty term at kebele level. This model is a conditional independence model where the frailty is common to all individuals in a cluster, and therefore responsible for creating dependence between event times. This is because ignoring the full dependence among observations may lead to standard errors that are understated and parameter estimates that are both biased and inconsistent [11]. Estimation of the frailty model can be parametric or semi-parametric. In the first case, a parametric density is assumed for the event times, resulting in a parametric baseline hazard function. Estimation is then conducted by maximizing the marginal log-likelihood function. In the second case, the baseline hazard is left unspecified and more complex techniques are available to approach that situation [12]. Even though semi-parametric estimation offers more flexibility, the parametric estimation will be more powerful if the form of the baseline hazard is somehow known in advance [13]. In this study, parametric frailty models were used to investigate the relationship between different potential covariates (sex, age, type of malnutrition and co-infection) and time to cure from SAM for clustered survival data with random right censoring. The choice of distribution for the hazard is very important than the choice of frailty distribution [14]. Hence, in this research exponential, weibull and log-logistic hazard functions were used and compared for their efficiencies. Regarding the frailty distribution, we assumed gamma and inverse Gaussian distributions. For comparison of different distributions, the AIC criteria were used, but for comparing nested models, likelihood ratio test were used.

Of all 855 malnourished patients, 711(83.16%) were cured and the median cure time from SAM was 14 days, while the minimum and the maximum cure times observed were 7 and 63 days, respectively (Table 1).

Using all the multivariable frailty models, the covariate co-infection was significant, indicating that it was the most important prognostic factor for the time-to-cure from SAM. Age group was significant in the three models namely, weibull-gamma, weibull-inverse Gaussian and log-logistic-inverse Gaussian frailty models. Type of malnutrition was not a significant factor for time-to-cure from SAM using all the models. The variance of the random effect (θ) was significant for the weibull and log-logistic baseline frailty models but not significant for the exponential baseline frailty models. It was highest when we assume the inverse Gaussian distribution (θ = 0.21) followed by the gamma distribution (θ = 0.172) with the log-logistic baseline hazard function. This term was again higher when we assume the inverse Gaussian frailty distribution (θ = 0.169) than the gamma distribution (θ = 0.163) for the weibull baseline hazard function. The Kendall’s tau (τ) was higher for the higher θ values. Accordingly the dependence within the clusters for the log-logistic-inverse Gaussian frailty model (τ = 0.081) was the maximum followed by the log-logistic-gamma frailty model (τ = 0.079). The AIC value of the log-logistic-inverse Gaussian model 4941.63, was the minimum among all the other AIC values of the models indicating that it was the most efficient model to describe the SAM dataset using various parametric frailty models (Table 2).

Analysis based on log-logistic-inverse Gaussian frailty model showed that age group of the children and presence of co-infection were significant. That is the confidence interval of acceleration factors(ϕ) for both Age group and co-infection did not include 1 (Table 3).

Children with age group between 12 and 23 months had significantly different curing time than the reference groups (aged 0–5months) with acceleration factor (ϕ = 1.18) and confidence interval (1.014, 1.374), did not includes 1, at 5% level of significance. An acceleration factor of greater than 1 indicates prolonging the time-to-cure from SAM. Therefore, children aged between 12 and 23 months had prolonged cure time by a factor of 1.18 than the reference groups. But the other age groups were not significantly different from the baseline age group at 5% level of significance. The confidence interval of the acceleration factor of co-infection was (1.093, 1.268), did not include 1, indicating co-infection is also significant prognostic factor for time-to-cure from SAM. It prolonged curing time by a factor of (ϕ = 1.177) at 5% level of significance.

The value of the shape parameter in the log-logistic-inverse Gaussian frailty model was (ρ = 3.56). This value greater than unity indicates that the shape of hazard function is unimodal, i.e., it increases up to some time and then decreases. The variability (heterogeneity) in the population of clusters (kebeles) estimated by our working model was θ = 0.21, and the dependence within clusters was about τ = 8.1%.

The predicted frailty values increases with range of 0.731 to 1.272 with increasing the median time of the cluster on log-logistic- inverse Gaussian frailty model (Figure 1). That is, these values are lower for lower values of event times and higher for higher value of event times. The median value of the frailty distribution is around 1.

The hazard functions given the 25 ^th , 50 ^th and 75 ^th quartiles of frailty distribution (Z = 0.837,Z = 1andZ = 1.167, respectively) were plotted for the log-logistic-inverse Gaussian frailty model (Figure 2). The parameter estimates obtained by maximizing the likelihood of the log logistic-inverse Gaussian frailty model were λ = 1.306e-4, ρ = 3.56(S.E = 0.134) and θ = 0.21(S.E = 0.055).

The conditional hazard functions given the 75 ^th quartile frailty values was greater, followed by the conditional hazard functions given by the 50 ^th (median) and 25 ^th quartile frailty values of the clusters respectively. From this what we can observe is that the more frail groups (Z > 1) were more likely to get events earlier (cure earlier) and the less frail groups (Z < 1) had prolonged cure time. All the conditional hazard functions were almost equal at the beginning time (t = 0). But the gap widens through time, specifically at mid time. The pattern of all the hazard function is unimodal (increases up to some point and then decreases) as the shape parameter for the baseline hazard function is greater than unity (ρ = 3.56).

To check the adequacy of our baseline hazard, the exponential has been plotted by the cumulative hazard function with time-to-cure from SAM. Similarly, the weibull has been plotted by the logarithm cumulative hazard function with the logarithm of time-to-cure from SAM and log-logistic has been plotted by the logarithm of the failure odds with the logarithm of time-to-cure from SAM (Figure 3). The plot of log-logistic was more linear than the other plots, though only few observations were scattered at the beginning time. The patterns suggested that the log-logistic hazard function was appropriate in the model.

The Cox-Snell residuals together with their cumulative hazard function were obtained by fitting the exponential, weibull and log-logistic models to our dataset, via maximum likelihood estimation (Figure 4). The plots showed that the Cox-Snell residuals fitted to assess the log-logistic model for the dataset were nearest to the line through the origin as compared to the other models, again indicating that this model described the SAM dataset well.

A quantile-quantile or q-q plot was made to check if the accelerated failure time provided an adequate fit to the data using two different groups of population. We checked the adequacy of the accelerated failure-time model by comparing the significantly different age groups (children in the age group 0–5 months and 12–24 months); as well as the co-infected and non co-infected groups of patients (Figure 5). The figures appear to be approximately linear for both covariates age group and co-infection with slopes equivalent to the acceleration factors 1.180 and 1.177, respectively. Therefore the log-logistic baseline model used for time-to-cure from SAM was accelerated failure time model.

The main aim of the study was to model time-to-cure from SAM using appropriate survival model among various parametric frailty models. The comparison of distributions of the models was performed using the AIC criteria, where a model with minimum AIC is accepted to be the best [13]. Accordingly, the log-logistic-inverse Gaussian frailty model which has AIC value of 4941.63 was the most appropriate model to describe the SAM dataset.

This study also showed that there was a clustering (frailty) effect on modeling time-to-cure from SAM which might be due to the heterogeneity in kebele from which the child came, assuming children coming from the same kebele share similar risk factors related to SAM. Therefore, it was important considering the clustering effect in modeling the hazard function. Clusters with minimum median time have smaller frailties, so that these clusters are predicted to have a high hazard [9], more probable to cure in this case. Our data demonstrated that the nuisance (frailty) terms modified the hazard function, and therefore the the hazard function was evaluated conditionally on this effect. Kebeles that frail more were more likely to cure than the less frail kebeles (since the event is positive).

The inverse Gaussian frailty generates very strong dependence at mid time [16]. According to the conditional hazard function given the 25 ^th , 50 ^th and 75 ^th quantile frailty values, the hazard function depends on these values especially at the mid time (Figure 1). This frailty distribution introduced as an alternative to the gamma distribution [17] was better in this dataset compared to the gamma frailty distribution. The heterogeneity in the kebeles was estimated to be θ = 0.21, and the dependence within clusters is about τ = 8.1%. These values were the maximum among the variance of the random effects and the Kendall’s tau of all the models, which consolidates the idea that the better the model, the less unobserved heterogeneity will be [18].

Nonetheless, the most acknowledged parametric model is the weibull, which allows the proportional hazards and accelerated life time model [8]; the SAM data set was best described by the log-logistic baseline as compared to the exponential and weibull hazard functions. According to the diagnostic plots the log failure odds of log-logistic baseline with log time was more linear as compared to the plots of exponential (cumulative hazard versus time) and weibull (log cumulative hazard versus log time), showing the SAM dataset was best described by the log-logistic baseline. This result was also confirmed by the cumulative hazard plots for the Cox-Snell residuals of the exponential, weibull and the log-logistic models. The plot was more approached to the line in case of the log-logistic model, indicating that the log-logistic was best. A q-q plot was done to check if the accelerated failure time provided an adequate fit to the dataset and the log-logistic as baseline was accelerated failure time model. Hence, a survival model need not be chosen arbitrarily to fit event times, the baseline hazard function as well as the frailty distribution should be compared and the most appropriate model should be selected for appropriate inference.

The prognostic factors considered were the age group of the child, type of malnutrition and presence or absence of co-infection, which were significant covariates using univariable analysis. Analysis using the best model, log-logistic-inverse Gaussian frailty model showed that the age group of children and presence of co-infection(s) were the determinant factors for the time-to-cure from SAM. Children aged between 12–23 months had prolonged cure time as compared to older age groups. This age group is known to be the time when the prevalence of SAM is the highest [19‐21]. This may be related to the fact that they start sub-optimal complementary feeding and compromise breast feeding practice which is important in preventing malnutrition among children [22]. Literatures like [23‐25] identified infection as a prognostic indicators, likewise, co-infection(s) prolonged the time-to-cure from SAM in this study. However, our findings showed that the time of curing did not depend on type of malnutrition. Similarly, Efrem et al. [26] showed that there was no significant difference in average length of stay among patients with severe wasting and edematous malnutrition. Therefore, a special attention should be given to children with SAM between one to two years old and/or co-infected by other disease(s) in order to control excess numbers of children in the program at any one time which might increase the cost of the program. A program is well functioning and considered acceptable if the length of stay in a hospital is less than 4 weeks with acceptable cure proportion of greater than 75% [15]. This study revealed that the median time-to-cure from SAM for patients in Wolliso St. Luke Catholic hospital was 14 days with maximum cure time of 63 days of which (83.16%) were cured. This implies that the program was acceptable as per the above standard.

We classified SAM cases as marasmus, kwashiorkor and marasmic-kwashiorkor, a naming might imply that the cause is protein and calorie deficiency perse. However, the intent was to see if there are differential in recovery time between edematous and non-edematous cases of SAM. We acknowledge the limitation of our study for not being able to isolate SAM cases with micronutrient deficiency.