A MODIFIED GENERALISED-GAMMA MIXTURE CURE MODEL FOR SURVIVAL DATA ANALYSIS

Abstract

Cure models are special survival models developed to estimate cure rate in cancer research. Several cure models such as Lognormal-Mixture Cure-Models (LNMCM), Loglogistics-Mixture-Cure-Models (LLMCM), Weibull-Mixture-Cure-Models (WMCM) and Generalised-Gamma-Mixture-Cure-Models (GGMCM) have been used to study cure rates in epidemiology. The GGMCM has been established to have out-performed other parametric models in terms of Akaike Information Criterion (AIC) but could not handle acute- asymmetry in survival data. Therefore, this study was designed to develop a Modified Generalised-Gamma Mixture Cure Model (MGGMCM) that can handle acute-asymmetry in survival data. The GGMCM: whereα, β are the shape parameters and θ is the scale parameter was modified using a gamma generator: , where ωisthe shape parameter, and are cumulative density function (cdf) and probabilty density function (pdf), respectively. Life ovarian cancer data was obtained from Department of Obstetrics and Gynaecology, University College Hospital, Ibadan, Nigeria covering the period 2000-2015. The diagnosis time (in months) was until death. The simulation study utilised data based on the continuous uniform distribution with and using samples of sizes of 10, 20 and 50 each in 50, 100 and 500 replicates, respectively. The hazard-function of the MGGMCM was derived by of MGGMCM; the cure model was given as S(t) = c + (1 − c)Su(t) where S(t) is the survival function of the entire population, Su(t) is the survival function of the uncured patients and c is cure-rate.Parameters , (median time-to-cure) and of MGGMCM were determined using Maximum Likelihood Estimation. The MGGMCM was exhaustively investigated in terms of its parametric essence and against extant models of similar intent using relevant assessment criteria like scope, general estimability, AIC and relative efficiency of estimates of cure rate, median time-to-cure, variances, bias and mean square error where applicable.

The hazard-function of the developed model was obtained as:

The AIC, median to cure, c and variance(c) of ovarian cancer data were: WMCM (216.89, 60.07, 0.29, 0.097); LNMCM (205.98, 57.98, 0.37, 0.004); LLMCM (203.27, 56.87, 5.95, 0.052); GGMCM (206.20, 20.90, 0.11, 0.005) and MGGMCM (199.24, 11.82, 0.82, 0.001), respectively. For simulated data, Mean Square Error (MSE) and |bias|were obtained as follows: n=10,r=50: WMCM (772.11, 26.10); LNMCM (707.23, 26.70); LLMCM (791.30, 27.89); GGMCM (691.03, 25.33); MGGMCM (701.10, 25.81), respectively. At n=20,r=50: WMCM (611.59, 23.59); LNMCM (655.71, 25.31); LLMCM (695.0, 26.00); GGMCM (601.33, 23.57); MGGMCM (609.31, 23.89), respectively. At n=50,r=50: WMCM (700.18, 25.77); LNMCM (699.52, 25.83); LLMCM (719.52, 25.50); GGMCM (689.15, 25.59); MGGMCM (601.59, 23.19), respectively. At n=50,r=500: WMCM (623.90, 23.95); LNMCM (619.61, 24.01); LLMCM (644.59, 24.89); GGMCM (602.10, 24.01); MGGMCM (501.37, 22.11), respectively. The better model corresponds to the smallest MSE and |bias| values as sample size and replicate increases. The Modified Generalised-Gamma Mixture Cure Model was the better on the AIC criterion; the MGGMCM adequately handled the problem of acute-asymmetry associated with survival data and its robustness.