Table 3: Comparison of predictability of fitted models using concordance measure: A multi-variable analysis

Table 4: Assessment of consistency of frailty model selection using subpopulation of size

Discussion:

In the present study, it was observed that log-normal frailty model was the best fit to model unobserved random heterogeneity in colo-rectal cancer survival. The choice of frailty distribution is one of the critical phases in frailty modeling. The selection of different models as frailty for survival data have been discussed extensively by Hougaard (2000). Zhou et al. (2015) used covariate adjusted frailty model for clustered time to event data and showed the superiority  of frailty over CPH model. Saeedi et al. (2017) considered cirrhosis patient mortality data and used gamma frailty to assess the risk. Frailty models and their merits over CPH model has been discussed by Yazdani (2019), Faradmal et al. (2012) and Talebi et al (2020) using cancer

survival data. Perperoglou et al. (2007) fitted frailty, CPH, time-depended Cox and cure rate models for breast cancer data with long-term survival and identified frailty as the significant model. Gurmu (2018) used parametric frailty model to assess the survival time among cervical cancer patients. The frailty models were looked into by several scholars (Hougaard, 1995, 2000; Therneau and Grambsch 2000; Noh et al., 2006; Liu et al., 2011; Zhou et al. 2015). Monaco et al (2018) introduced a general semi-parametric shared frailty model with gamma, log-normal, inverse Gaussian and power variance function as the frailty distributions, and provides consistent estimators of the standard errors of the parameters‟ estimators. This study indicated the applicability of frailty model in survival studies and their advantages over Cox model. Another on risk assessment for breast cancer using gamma frailty model by Krishna et al. (2021) also reported that frailty models as a better predictive model compared CPH model. All these studies suggested using frailty models to CPHM, which substantiates our findings of superiority of frailty model. Callegaro et al. (2012) introduce a new class of frailty model called log-skew-normal frailty model leading to an extension of the log-normal model. They compared Cox with log normal frailty model and log-skew-normal frailty model and illustrate it with a case study of multiple myeloma patients with autologous stem cells transplantation. They found that log-skew- normal frailty has a significant role in predicting the risk for survival. Nath et al (2020) studied a risk assessment in liver transplantation patients using frailty models. They compared the risk of survival by different frailty models using parametric approach namely, gamma, inverse Gaussian, positive stable and lognormal. Weibull and exponential distribution were considered as the baseline hazard. In this study they also used AIC and BIC criterion to identify better model. Legesse et al. (2022) used shared log-normal frailty model to know whether the recurrent event (time to recovery) is an associated factors of type 2 diabetes. This study also pointed that log-normal frailty model was better model compared to CPH. Similar to these studies, our study also identified log-normal frailty model as a better model compared to CPH model.

Among the covariates considered in the present study, metastases at presentation, stage at diagnosis and recurrence status have identified as significant factors on overall survival of colo- rectal cancer. The survival concordance was higher for log-normal frailty model in multivariable analysis also. In the multivariable analysis, covariates stage and recurrence jointly turned out to be significant prognostic factors and the estimated HR corresponding to log-normal frailty was higher than those obtained using CPHM. Though the frailty coefficient was insignificant, the concordance was always higher for log-normal frailty model. Hence it is suggested frailty models in cancer survival studies.

Jiang et al. (2020) used frailty models to identify the risk factors for hospital readmissions of colo-rectal cancer patients. They considered the covariates such as patient gender (male or female), tumour stage by Duke‟s classification (Stage A-B, Stage C, or Stage D), type of treatment (chemotherapy or radiotherapy), and Charlson comorbidity index (index = 0, 1 or 2, ≥3; time-dependent). They identified stage as one of the prognostic factor. From the simulation study, they concluded that the shared gamma frailty can provide reliable prediction on frailties even when the frailty distribution is miss-specified.

The present study used survival concordance to identify the best predictive model and was observed that log-normal frailty model as the best model based on the uni-variable and multi-variable analysis compared to CPH model. The predictability of log-normal frailty model using C-index ranges from 0.795 to 0.946 and that for CPHM ranges between 0.495 to 0.726, indicating lower predictability for CPHM. The measure of model discrimination analyses using concordance measures was not been discussed much in literature. The only article which highlighted the ability of concordance measure for model identification was by Krishna et al. (2021), which compared the predictability of the gamma frailty model with CPHM and found the gamma frailty as a better predictive model with high C-index.

Several studies have assessed the consistency of the derived model using simulated data. Kim (2019) used simulation methods to assess the consistency of posterior consistency in dispersion parameters and frailty coefficients. Austin (2012) used Monte Carlo simulation method to study the effect of time dependent covariates on CPH models. The present study assessed the consistency of the fitted frailty model  using computer generated random sub-population. Based on the randomly selected subpopulation of sample sizes 150 and 250, using AIC criterion, inverse Gaussian frailty was comparable with log-normal frailty model, however log-normal frailty was slightly better than all other frailty models, which establishes the consistency of the log-normal frailty model.

In conclusion, this study observed that frailty modeling is the best approach to incorporate the hidden random heterogeneity in cancer survival. Randomly selected sub-sample analysis would help to assess the consistency of the best fitted model.

Acknowledgement: The authors are grateful to Department of Health Research (ICMR), Government of India (Grant id: R.11012/03/2021-GIA/HR) for the funding support to carry out the study.

Conflict of interest: There exists no conflict among the authors.