dc.description.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 1 exp
x
1 exp
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)]a1[1F(x)](b1)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 =
(yi )
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 (exp(zi)) (1 exp(
exp(zi))) [F(z)]a1[1F(z)]b1 where
F(z) =
+1
n
(1 exp(
exp(zi))) 1
+1 (1 exp(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 9112754:000 and 9112751:000 while the AIC
for the six generalised WW regression models were 474248:600 for log-beta Weibull,
3076234:000 for log-beta normal, 487430:400 for log-Weibull, 1541:182 for log-normal,
1102:662 for log-logistic and 252807:000 for log-weighted, respectively. Also the corresponding
BIC were 474249:500, 3076205:000, 487428:500, 1518:564, 1080:044 and
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. |
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