R help - Dec 2008 - Prediction intervals for zero inflated Poisson regression

Dear all,

I'm using zeroinfl() from the pscl-package for zero inflated Poisson
regression. I would like to calculate (aproximate) prediction intervals
for the fitted values. The package itself does not provide them. Can
this be calculated analyticaly? Or do I have to use bootstrap?

What I tried until now is to use bootstrap to estimate these intervals.
Any comments on the code are welcome. The data and the model are based
on the examples in zeroinfl().

#aproximate prediction intervals with Poisson regression
fm_pois <- glm(art ~ fem, data = bioChemists, family = poisson)
newdata <- na.omit(unique(bioChemists[, "fem", drop = FALSE]))
prediction <- predict(fm_pois, newdata = newdata, se.fit = TRUE)
ci <- data.frame(exp(prediction$fit + matrix(prediction$se.fit, ncol 1) %*%
c(-1.96, 1.96)))
newdata$fit <- exp(prediction$fit)
newdata <- cbind(newdata, ci)
newdata$model <- "Poisson"

library(pscl)
#aproximate prediction intervals with zero inflated poisson regression
fm_zip <- zeroinfl(art ~ fem | 1, data = bioChemists)
fit <- predict(fm_zip)
Pearson <- resid(fm_zip, type = "pearson")
VarComp <- resid(fm_zip, type = "response") / Pearson
fem <- bioChemists$fem
bootstrap <- replicate(999, {
    yStar <- pmax(round(fit + sample(Pearson) * VarComp, 0), 0)
    predict(zeroinfl(yStar ~ fem | 1), newdata = newdata)
})
newdata0 <- newdata
newdata0$fit <- predict(fm_zip, newdata = newdata, type =
"response")
newdata0[, 3:4] <- t(apply(bootstrap, 1, quantile, c(0.025, 0.975)))
newdata0$model <- "Zero inflated"

#compare the intervals in a nice plot.
newdata <- rbind(newdata, newdata0)
library(ggplot2)
ggplot(newdata, aes(x = fem, y = fit, min = X1, max = X2, colour model)) +
geom_point(position = position_dodge(width = 0.4)) +
geom_errorbar(position = position_dodge(width = 0.4))


Best regards,

Thierry

------------------------------------------------------------------------
----
ir. Thierry Onkelinx
Instituut voor natuur- en bosonderzoek / Research Institute for Nature
and Forest
Cel biometrie, methodologie en kwaliteitszorg / Section biometrics,
methodology and quality assurance
Gaverstraat 4
9500 Geraardsbergen
Belgium 
tel. + 32 54/436 185
Thierry.Onkelinx op inbo.be 
www.inbo.be 

To call in the statistician after the experiment is done may be no more
than asking him to perform a post-mortem examination: he may be able to
say what the experiment died of.
~ Sir Ronald Aylmer Fisher

The plural of anecdote is not data.
~ Roger Brinner

The combination of some data and an aching desire for an answer does not
ensure that a reasonable answer can be extracted from a given body of
data.
~ John Tukey

Dit bericht en eventuele bijlagen geven enkel de visie van de schrijver weer 
en binden het INBO onder geen enkel beding, zolang dit bericht niet bevestigd is
door een geldig ondertekend document. The views expressed in  this message 
and any annex are purely those of the writer and may not be regarded as stating 
an official position of INBO, as long as the message is not confirmed by a duly 
signed document.

Thierry,

Simon had written some code for this but we never got round to fully 
integrate it into the "pscl" package. A file pb.R is attached, but as
a
disclaimer: I haven't looked at this code for a while. It still seems to 
work (an example is included at the end) but please check.

hth,
Z

On Tue, 16 Dec 2008, ONKELINX, Thierry wrote:
> Dear all,
>
> I'm using zeroinfl() from the pscl-package for zero inflated Poisson
> regression. I would like to calculate (aproximate) prediction intervals
> for the fitted values. The package itself does not provide them. Can
> this be calculated analyticaly? Or do I have to use bootstrap?
>
> What I tried until now is to use bootstrap to estimate these intervals.
> Any comments on the code are welcome. The data and the model are based
> on the examples in zeroinfl().
>
> #aproximate prediction intervals with Poisson regression
> fm_pois <- glm(art ~ fem, data = bioChemists, family = poisson)
> newdata <- na.omit(unique(bioChemists[, "fem", drop = FALSE]))
> prediction <- predict(fm_pois, newdata = newdata, se.fit = TRUE)
> ci <- data.frame(exp(prediction$fit + matrix(prediction$se.fit, ncol
> 1) %*% c(-1.96, 1.96)))
> newdata$fit <- exp(prediction$fit)
> newdata <- cbind(newdata, ci)
> newdata$model <- "Poisson"
>
> library(pscl)
> #aproximate prediction intervals with zero inflated poisson regression
> fm_zip <- zeroinfl(art ~ fem | 1, data = bioChemists)
> fit <- predict(fm_zip)
> Pearson <- resid(fm_zip, type = "pearson")
> VarComp <- resid(fm_zip, type = "response") / Pearson
> fem <- bioChemists$fem
> bootstrap <- replicate(999, {
>    yStar <- pmax(round(fit + sample(Pearson) * VarComp, 0), 0)
>    predict(zeroinfl(yStar ~ fem | 1), newdata = newdata)
> })
> newdata0 <- newdata
> newdata0$fit <- predict(fm_zip, newdata = newdata, type =
"response")
> newdata0[, 3:4] <- t(apply(bootstrap, 1, quantile, c(0.025, 0.975)))
> newdata0$model <- "Zero inflated"
>
> #compare the intervals in a nice plot.
> newdata <- rbind(newdata, newdata0)
> library(ggplot2)
> ggplot(newdata, aes(x = fem, y = fit, min = X1, max = X2, colour >
model)) + geom_point(position = position_dodge(width = 0.4)) +
> geom_errorbar(position = position_dodge(width = 0.4))
>
>
> Best regards,
>
> Thierry
>
> ------------------------------------------------------------------------
> ----
> ir. Thierry Onkelinx
> Instituut voor natuur- en bosonderzoek / Research Institute for Nature
> and Forest
> Cel biometrie, methodologie en kwaliteitszorg / Section biometrics,
> methodology and quality assurance
> Gaverstraat 4
> 9500 Geraardsbergen
> Belgium
> tel. + 32 54/436 185
> Thierry.Onkelinx at inbo.be
> www.inbo.be
>
> To call in the statistician after the experiment is done may be no more
> than asking him to perform a post-mortem examination: he may be able to
> say what the experiment died of.
> ~ Sir Ronald Aylmer Fisher
>
> The plural of anecdote is not data.
> ~ Roger Brinner
>
> The combination of some data and an aching desire for an answer does not
> ensure that a reasonable answer can be extracted from a given body of
> data.
> ~ John Tukey
>
> Dit bericht en eventuele bijlagen geven enkel de visie van de schrijver
weer
> en binden het INBO onder geen enkel beding, zolang dit bericht niet
bevestigd is
> door een geldig ondertekend document. The views expressed in  this message
> and any annex are purely those of the writer and may not be regarded as
stating
> an official position of INBO, as long as the message is not confirmed by a
duly
> signed document.
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>
>
-------------- next part --------------
## parametric bootstrap



predict.zeroinfl <- function(object, newdata, type = c("response",
"prob"),

                             se=FALSE,MC=1000,level=.95,

                             na.action = na.pass, ...)

{

    type <- match.arg(type)



    ## if no new data supplied

    if(missing(newdata)){

      rval <- object$fitted.values

        if(!is.null(object$x)) {

	  X <- object$x$count

	  Z <- object$x$zero

	}

        else if(!is.null(object$model)) {

          X <- model.matrix(object$terms$count, object$model, contrasts =
object$contrasts$count)

          Z <- model.matrix(object$terms$zero,  object$model, contrasts =
object$contrasts$zero)

	}

        else {

	  stop("no X and/or Z matrices can be extracted from fitted model")

	}

      if(type == "prob") {

        mu <- exp(X %*% object$coefficients$count)[,1]

        phi <- object$linkinv(Z %*% object$coefficients$zero)[,1]

      }

    }

    else {

      mf <- model.frame(delete.response(object$terms$full), newdata,
na.action = na.action, xlev = object$levels)

      X <- model.matrix(delete.response(object$terms$count), mf, contrasts =
object$contrasts$count)

      Z <- model.matrix(delete.response(object$terms$zero),  mf, contrasts =
object$contrasts$zero)



      mu <- exp(X %*% object$coefficients$count)[,1]

      phi <- object$linkinv(Z %*% object$coefficients$zero)[,1]

      rval <- (1-phi) * mu

    }   



    if(se & !is.null(X) & !is.null(Z)){

      require(mvtnorm)

      vc <- -solve(object$optim$hessian)

      kx <- length(object$coefficients$count)

      kz <- length(object$coefficients$zero)

      parms <- object$optim$par

      if(type!="prob"){

        yhat.sim <- matrix(NA,MC,dim(X)[1])

        for(i in 1:MC){

          cat(paste("MC iterate",i,"of",MC,"\n"))

          parms.sim <- rmvnorm(n=1,mean=parms,sigma=vc)

          beta <- parms.sim[1:kx]

          gamma <- parms.sim[(kx+1):(kx+kz)]

          mu.sim <- exp(X%*%beta)[,1]

          phi.sim <- object$linkinv(Z%*%gamma)[,1]

          yhat.sim[i,] <- (1-phi.sim)*mu.sim

        }

      }

      out <- list()

      out$lower <- apply(yhat.sim,2,quantile,(1-level)/2)

      out$upper <- apply(yhat.sim,2,quantile,1-((1-level)/2))

      out$se <- apply(yhat.sim,2,sd)

    }

    

    ## predicted probabilities

    if(type == "prob") {

      if(!is.null(object$y)) y <- object$y

        else if(!is.null(object$model)) y <- model.response(object$model)

	else stop("predicted probabilities cannot be computed for fits with y =
FALSE and model = FALSE")



      yUnique <- min(y):max(y)

      nUnique <- length(yUnique)

      rval <- matrix(NA, nrow = length(rval), ncol = nUnique)

      dimnames(rval) <- list(rownames(X), yUnique)

      

      switch(object$dist,

             "poisson" = {

    	       rval[, 1] <- phi + (1-phi) * exp(-mu)

               for(i in 2:nUnique) rval[,i] <- (1-phi) * dpois(yUnique[i],
lambda = mu)

             },

	     "negbin" = {

               theta <- object$theta

               rval[, 1] <- phi + (1-phi) * dnbinom(0, mu = mu, size = theta)

               for(i in 2:nUnique) rval[,i] <- (1-phi) * dnbinom(yUnique[i],
mu = mu, size = theta)

             },

	     "geometric" = {

               rval[, 1] <- phi + (1-phi) * dnbinom(0, mu = mu, size = 1)

               for(i in 2:nUnique) rval[,i] <- (1-phi) * dnbinom(yUnique[i],
mu = mu, size = 1)

	     })



    }

   

    if(se)

      rval <- list(rval,out)



    rval

}



################################################################

## test this code

require(pscl)

data(bioChemists)

obj <- zeroinfl(art ~ . | .,

                data=bioChemists,

                dist="negbin",

                EM=TRUE)



foo <- predict(obj,se=TRUE,type="response")



indx <- order(foo[[1]])

n <- length(indx)

plot(1:n,

     foo[[1]][indx],

     type="n",

     ylim=range(cbind(foo[[2]]$lower,foo[[2]]$upper)),

     xlab="Order Statistic",

     ylab="Predicted Value")

segments(x0=1:n,x1=1:n,

         y0=foo[[2]]$lower[indx],

         y1=foo[[2]]$upper[indx],

         col=gray(.45),

         lwd=.1)

points(1:n,foo[[1]][indx],cex=.5)