  Examining residuals such as Pearson and deviance residuals, is a standard
tool for assessing normal regression. However, for discrete response, these
residuals cluster on lines corresponding to distinct response values. Their
distributions are far from normality; graphical and quantitative inspection of
these residuals provides little information for model diagnosis. Marshall and
Spiegelhalter (2003) defined a cross-validatory predictive p-value for
identifying outliers. Predictive p-values are uniformly distributed for
continuous response but not for discrete response. We propose to use randomized
predictive p-values (RPP) for diagnosing models with discrete responses. RPPs
can be transformed to "residuals" with normal distribution, called NRPPs by us.
NRPPs can be used to diagnose all regression models with scalar response using
the same way for diagnosing normal regression. The NRPPs are nearly the same as
the randomized quantile residuals (RQR), which are previously proposed by Dunn
and Smyth (1996) but remain little known by statisticians. This paper provides
an exposition of RQR using the RPP perspective. The contributions of this
exposition include: (1) we give a rigorous proof of uniformity of RPP and
illustrative examples to explain the uniformity under the true model; (2) we
conduct extensive simulation studies to demonstrate the normality of NRPPs
under the true model; (3) our simulation studies also show that the NRPP method
is a versatile diagnostic tool for detecting many kinds of model inadequacies
due to lack of complexity. The effectiveness of NRPP is further demonstrated
with a health utilization dataset.
