Class Activity, November 9

Diagnostics for ZIP models

In this class activity, we will create quantile residual plots to check the shape assumption for ZIP models

Questions

  1. Run the code below to generate ZIP data for which the shape assumptions are satisfied for both the logistic and the Poisson components.
x <- rnorm(1000)
alpha <- exp(x)/(1 + exp(x))
lambda <- exp(1 + x)
z <- rbinom(1000, 1, prob=alpha)
y <- 0*z + rpois(1000, lambda)*(1 - z)
  1. Now use the code below to fit a ZIP model, and create a quantile residual plot for your fitted model. Are the fitted regression coefficients close to the true parameters?
library(pscl)
library(tidyverse)

## randomized quantile residuals for a ZIP model
## zip_m = fitted ZIP model from pscl
qresid_zeroinfl <- function(zip_m){
  y <- zip_m$y
  pred_probs <- predict(zip_m, type="prob")
  resids <- c()
  for(i in 1:length(y)){
    cdf_b <- sum(pred_probs[i,1:(y[i]+1)])
    if(y[i] == 0){
      cdf_a <- 0
    } else {
      cdf_a <- sum(pred_probs[i,1:y[i]])
    }
    
    resids[i] <- qnorm(runif(1, cdf_a, cdf_b))
  }
  return(resids)
}


m <- zeroinfl(y ~ x | x)
summary(m)

data.frame(x = x, resids = qresid_zeroinfl(m)) %>%
  ggplot(aes(x = x, y = resids)) +
  geom_point() +
  geom_smooth() +
  theme_bw()

  1. Now generate data so that the Poisson or logistic shape assumption is violated. Re-fit the ZIP model, and make a quantile residual plot for the new model. Can you detect the shape violations in the residual plot?

  2. If you see a violation in the quantile residual plot, will you be able to tell whether the violation is in the Poisson component or in the logistic component?