The Importance of the Frailty Effect In Survival Models: For Multidrug-resistant Tuberculosis Data

Sizwe Vincent Mbona1, *, Henry Mwambi2, Shaun Ramroop2
1 Department of Statistics, Durban University of Technology, Durban, 4001, South Africa
2 School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, King Edward Avenue, Pietermaritzburg, 3209, South Africa

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© 2023 Mbona et al.

open-access license: This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0), a copy of which is available at: This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

* Address correspondence to this author at the Department of Statistics, Durban University of Technology, 41-43 ML Sultan Road, Mariam BEE Building, Durban, 4001, South Africa; Tel.: +27 31 373 5703;
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Frailty models have been proposed to analyse survival data, considering unobserved covariates (frailty effects). In a shared frailty model, frailties are common (or shared) amongst groups of individuals and are randomly distributed across groups.


In this paper, the authors compared the semi-parametric model to shared frailty models by studying the time-to-death of patients with multidrug-resistant tuberculosis (MDR-TB).


Secondary data from 1 542 multidrug-resistant tuberculosis patients were used in this study. STATA software was used to analyse frailty models via the streg command.


Of 1 542 patients diagnosed with MDR-TB, 245 (15.9%) died during the study period; 77 (5.0%) had treatment failure; 334 (21.7%) defaulted; 213 (13.8%) completed treatment; 651 (42.2%) were cured of MRD-TB; and 22 (1.4%) were transferred out. The results showed that 797 (51.7%) were females, and the majority were aged 18 – 30 and 31 – 40 years (35.5% and 35.7% respectively). Most of the patients (71.3%) were HIV-positive. The results also showed that most patients (95.7%) had no previous MDR-TB episodes, and 792 (51.4%) had no co-morbidities. The estimate of the variance for the frailty term in the Weibull gamma shared frailty model was 2.83, which is relatively large and therefore suggests the existence of heterogeneity.


The Laplace transform of the frailty distribution plays a central role in relating the hazards, conditional on the frailty, to the hazards and survival functions observed in a population.

Keywords: Frailty, Hazards, MDR-TB, Risk factors, Survival data.