| Title: | Parametric Regression for Discrete Response |
|---|---|
| Description: | Regression for a discrete response, where the conditional distribution is modelled via a discrete Weibull distribution. |
| Authors: | Veronica Vinciotti <[email protected]> |
| Maintainer: | Veronica Vinciotti<[email protected]> |
| License: | GPL (>= 2) |
| Version: | 2.0 |
| Built: | 2024-12-14 06:49:17 UTC |
| Source: | CRAN |
Density, distribution function, quantile function and random generation for the discrete Weibull distribution with parameters q and beta.
ddw(x,q=exp(-1),beta=1) pdw(x,q=exp(-1),beta=1) qdw(p,q=exp(-1),beta=1) rdw(n,q=exp(-1),beta=1)ddw(x,q=exp(-1),beta=1) pdw(x,q=exp(-1),beta=1) qdw(p,q=exp(-1),beta=1) rdw(n,q=exp(-1),beta=1)
x |
quantile |
p |
probability |
n |
number of observations |
q, beta
|
Parameters of the distribution |
The discrete Weibulll distribution has density
for .
If q or beta are not specified they assume the default values of exp(-1) and 1, respectively. In this case, DW corresponds to a geometric distribution with p=1-q.
ddw gives the density, pdw gives the distribution function, qdw gives the quantile function, and rdw generates random samples from a DW distribution with parameters q and beta.
Veronica Vinciotti
Nagakawa T, Osaki S. The discrete Weibull distribution. IEEE transactions on reliability 1975; R-24(5).
x<-rdw(1000,q=0.9,beta=1.5) hist(x) plot(x,unlist(lapply(x,ddw,q=0.9,beta=1.5)),ylab="density") plot(x,unlist(lapply(x,pdw,q=0.9,beta=1.5)),ylab="cdf")x<-rdw(1000,q=0.9,beta=1.5) hist(x) plot(x,unlist(lapply(x,ddw,q=0.9,beta=1.5)),ylab="density") plot(x,unlist(lapply(x,pdw,q=0.9,beta=1.5)),ylab="cdf")
Mean and variance of a discrete Weibull distribution with parameters q and beta.
dw.meanvar(q,beta,M)dw.meanvar(q,beta,M)
q, beta
|
Parameters of the distribution |
M |
Maximum value of the summation. Default value is 1000. |
The mean and variance are computed using the following approximations:
The function returns the mean and variance of a DW distribution with parameters q and beta.
Veronica Vinciotti
Khan M, Khalique A, Abouammoth A. On estimating parameters in a discrete Weibull distribution. IEEE transactions on Reliability 1989; 38(3):348-350.
dw.meanvar(q=0.9,beta=1.5) #compare with sample mean/variance from a random sample x<-rdw(1000,q=0.9,beta=1.5) mean(x) var(x)dw.meanvar(q=0.9,beta=1.5) #compare with sample mean/variance from a random sample x<-rdw(1000,q=0.9,beta=1.5) mean(x) var(x)
Estimation of the parameters q and beta of a discrete Weibull distribution
dw.parest(data,method,method.opt)dw.parest(data,method,method.opt)
data |
Vector of observations |
method |
Either "likelihood" or "proportion" |
method.opt |
Optimization criterion used in maxLik (default is "NR") |
If method="likelihood", the parameters q and beta are estimated by maximum likelihood.
If method="proportion", the method of Araujio Santos and Fraga Alves (2013) is used, based on count frequencies.
The function returns the parameter estimates of q and beta.
Veronica Vinciotti
Araujo Santos P, Fraga Alves M. Improved shape parameter estimation in a discrete Weibull model. Recent Developments in Modeling and Applications in Statistics . Studies in Theoretical and Applied Statistics. Springer-Verlag, 2013; 71-80.
x<-rdw(1000,q=0.9,beta=1.5) dw.parest(x) #maximum likelihood estimates dw.parest(x,method="proportion") #proportion estimatesx<-rdw(1000,q=0.9,beta=1.5) dw.parest(x) #maximum likelihood estimates dw.parest(x,method="proportion") #proportion estimates
Parametric regression for discrete response data. The conditional distribution of the response given the predictors is assumed to be DW with parameters q and beta dependent on the predictors.
dw.reg(formula, data,tau=0.5,para.q1=NULL,para.q2=NULL,para.beta=NULL,...)dw.reg(formula, data,tau=0.5,para.q1=NULL,para.q2=NULL,para.beta=NULL,...)
formula |
An object of class "formula": a symbolic description of the model to be fitted. |
data |
An optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which dw.qr is called. |
tau |
Quantile value (default 0.5). This is used only to extract the conditional quantile from the fitted distribution. |
para.q1, para.q2
|
logical flag. If TRUE, the model includes a dependency of q on the predictors, as explained below. |
para.beta |
logical flag. If TRUE, the model includes a dependency of beta on the predictors, as explained below. |
... |
Additional arguments to the maxLik function |
The conditional distribution of Y (response) given x (predictors) is assumed a DW(q(x),beta(x)).
If para.q1=TRUE,
If para.q2=TRUE,
This is equivalent to a continuous Weibull regression model with interval-censored data.
If para.q1=NULL and para.q2=NULL, then q(x) is constant.
If para.beta=TRUE,
Otherwise beta(x) is constant.
A list of class dw.reg containing the following components:
call |
the matched call. |
data |
the input data as a list of response and covariates. |
coefficients |
the theta and gamma estimated coefficients. |
loglik |
the log-likelihood of the model. |
fitted.values |
fitted values (on the response scale) for the specified quantile tau. |
fitted.q |
fitted q values. |
fitted.beta |
fitted beta values. |
residuals |
randomised quantile residuals of the fitted model. |
tTable |
coefficients, standard errors, etc. |
tTable.survreg |
Only for the model para.q2=TRUE. Coefficients, standard errors, etc from the survreg parametrization. These estimates are linked to changes of log(Median+1). |
Veronica Vinciotti, Hadeel Kalktawi, Alina Peluso
Kalktawi, Vinciotti and Yu (2016) A simple and adaptive dispersion regression model for count data.
#simulated example (para.q1=TRUE, beta constant) theta0 <- 2 theta1 <- 0.5 beta<-0.5 n<-500 x <- runif(n=n, min=0, max=1.5) logq<-theta0 + theta1 * x - log(1+exp(theta0 + theta1 * x)) y<-unlist(lapply(logq,function(x,beta) rdw(1,q=exp(x),beta),beta=beta)) data.sim<-data.frame(x,y) #simulated data fit<-dw.reg(y~x,data=data.sim,para.q1=TRUE) fit$tTable #simulated example (para.q2=TRUE, beta constant) theta0 <- -2 theta1 <- -0.5 beta<-0.5 n<-500 x <- runif(n=n, min=0, max=1.5) logq<--exp(theta0 + theta1 * x) y<-unlist(lapply(logq,function(x,beta) rdw(1,q=exp(x),beta),beta=beta)) data.sim<-data.frame(x,y) #simulated data fit<-dw.reg(y~x,data=data.sim,para.q2=TRUE) fit$tTable fit$survreg #real example library(Ecdat) data(StrikeNb) fit<-dw.reg(strikes~output,data=StrikeNb,para.q2=TRUE) fit$tTable fit$survreg#simulated example (para.q1=TRUE, beta constant) theta0 <- 2 theta1 <- 0.5 beta<-0.5 n<-500 x <- runif(n=n, min=0, max=1.5) logq<-theta0 + theta1 * x - log(1+exp(theta0 + theta1 * x)) y<-unlist(lapply(logq,function(x,beta) rdw(1,q=exp(x),beta),beta=beta)) data.sim<-data.frame(x,y) #simulated data fit<-dw.reg(y~x,data=data.sim,para.q1=TRUE) fit$tTable #simulated example (para.q2=TRUE, beta constant) theta0 <- -2 theta1 <- -0.5 beta<-0.5 n<-500 x <- runif(n=n, min=0, max=1.5) logq<--exp(theta0 + theta1 * x) y<-unlist(lapply(logq,function(x,beta) rdw(1,q=exp(x),beta),beta=beta)) data.sim<-data.frame(x,y) #simulated data fit<-dw.reg(y~x,data=data.sim,para.q2=TRUE) fit$tTable fit$survreg #real example library(Ecdat) data(StrikeNb) fit<-dw.reg(strikes~output,data=StrikeNb,para.q2=TRUE) fit$tTable fit$survreg
Quantile-Quantile plot of the randomised quantile residuals of a DW regression fitted model with 95% simulated envelope.
res.dw(obj,k)res.dw(obj,k)
obj |
An object of class "dw.reg": the output of the dw.reg function. |
k |
The number of iterations for the simulated envelope. |
Diagnostic check for a DW regression model. The randomised quantile residuals should follow a standard normal distribution.
A q-q plot of the residuals with 95% simulated envelope
Veronica Vinciotti, Hadeel Kalktawi
Kalktawi, Vinciotti and Yu (2016) A simple and adaptive dispersion regression model for count data.
#simulated example (para.q2=TRUE, beta constant) theta0 <- -2 theta1 <- -0.5 beta<-0.5 n<-500 x <- runif(n=n, min=0, max=1.5) logq<--exp(theta0 + theta1 * x) y<-unlist(lapply(logq,function(x,beta) rdw(1,q=exp(x),beta),beta=beta)) data.sim<-data.frame(x,y) #simulated data fit<-dw.reg(y~x,data=data.sim,para.q2=TRUE) res.dw(fit,k=5) ks.test(fit$residuals,"pnorm") #real example library(Ecdat) data(StrikeNb) fit<-dw.reg(strikes~output,data=StrikeNb,para.q2=TRUE) res.dw(fit,k=5) ks.test(fit$residuals,"pnorm")#simulated example (para.q2=TRUE, beta constant) theta0 <- -2 theta1 <- -0.5 beta<-0.5 n<-500 x <- runif(n=n, min=0, max=1.5) logq<--exp(theta0 + theta1 * x) y<-unlist(lapply(logq,function(x,beta) rdw(1,q=exp(x),beta),beta=beta)) data.sim<-data.frame(x,y) #simulated data fit<-dw.reg(y~x,data=data.sim,para.q2=TRUE) res.dw(fit,k=5) ks.test(fit$residuals,"pnorm") #real example library(Ecdat) data(StrikeNb) fit<-dw.reg(strikes~output,data=StrikeNb,para.q2=TRUE) res.dw(fit,k=5) ks.test(fit$residuals,"pnorm")