AN ALTERNATIVE GENERALISED WEIGHTED WEIBULL REGRESSION MODEL

Abstract

Classical Regression Model (CRM) such as Weibull regression is commonly used for estimating relationship among variables. The problem with CRM is its dependence on the assumptions of normality and homoscedasticity of the residual terms. However, the assumption of normality is not valid for several real life events especially time-to-event phenomenon where the data exhibit a high level of skewness. Previous research on CRM has generally excluded non-normality of the residual terms. Therefore, this study was aimed at developing an Alternative Generalised Weighted Weibull Regression Model (AGWWRM) for improved inference when the residual terms are not normal. The Weighted Weibull Distribution (WWD) f(x) = ( +1)

x ð€€€1 exp

ð€€€x

1 ð€€€ exp

ð€€€

x

where ; and are: weighted, scale and shape parameters. The WWD was rede ned by the introduction of two shape parameters, a and b to accommodate skewness in the data; based on the beta link function: g(x) = 1 B(a;b) [F(x)]að€€€1[1ð€€€F(x)](bð€€€1)f(x) where, is the beta function and F(x) is the distribution function of the WWD. To obtain a location-scale regression model that would link the response variable yi(= XT i + zi and zi =

(yið€€€ )

is the error term, where is the regression model, and are the location and dispersion parameters for i = 1; 2; ; n; to a vector X of p explanatory variables. The transformations Y = log(T); = 1 and = log( ) were used. T is a random variable having beta Weighted Weibull (BWW) density function and Y is a log-beta WW variable. The statistical properties namely: moments, moment generating functions, skewness and kurtosis were determined for the Alternative Generalised Weighted Weibull (AGWW) distribution. The performance of the AGWWRM was determined using secondary data on time-to-completion of a Ph.D. programme using a sample of 187 Ph.D. graduates from the University of Ibadan. The explanatory variables usedwere supervisor (x1), employment (x2), marital status (x3), age (x4) and faculty (x5),whileybeing dependent variable was time-to-completion. The AGWWRM was compared with six existing generi alised WW regression models: log-beta Weibull, log-beta normal, log-Weibull, log-normal, log-logistic and log-weighted. The Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) were used as the assessment criteria for AGWWRM. The derived AGWW distribution was g(z; a; b; ; ; ) = ( +1)

B(a;b) exp (ð€€€exp(zi)) (1 ð€€€ exp(ð€€€ exp(zi))) [F(z)]að€€€1[1ð€€€F(z)]bð€€€1 where F(z) = +1

n (1 ð€€€ exp(ð€€€ exp(zi))) ð€€€ 1

+1 (1 ð€€€ exp(ð€€€exp(1 + )(zi))) o . The developed AGWWRM was y = 0+ x1+ 2x2+ 3x3+ 4x4+ 5x5+ zi where 0 < ;1. The parameters of the regression model ( 0 ; 1 ; 2 3 ; 4 and 5 ) = (2:550; 3:250; 1:250; 4:150; 1:310; 5:150). The AIC and BIC for the AGWWRM were ð€€€9112754:000 and ð€€€9112751:000 while the AIC for the six generalised WW regression models were ð€€€474248:600 for log-beta Weibull, ð€€€3076234:000 for log-beta normal, ð€€€487430:400 for log-Weibull, ð€€€1541:182 for log-normal, ð€€€1102:662 for log-logistic and ð€€€252807:000 for log-weighted, respectively. Also the corresponding BIC were ð€€€474249:500, ð€€€3076205:000, ð€€€487428:500, ð€€€1518:564, ð€€€1080:044 and ð€€€252804:800, respectively. The assessment criteria for the AGWWRM were consistently lower than those from the existing generalised WW regression models indicating improved inference. The developed Alternative Generalised Weighted Weibull Regression model exhibited improved inference when the residual terms are not normal. Keywords: Beta link function, Log-beta distribution, Location-scale regression model, Log-normal distribution.