ABSTRACT
We also introduce an alternative test specifically designed for overdispersion due to the existence of a latent process in a Poisson observation process. Since this test uses higher moment properties of Poisson observations it was considered as a possibly more powerful test statistic than Sw Also it might be more appropriate for the lagged regression models introduced in Sec. 3 since for these the distributional theory of Sa is not clearly appropriate. Under the null hypothesis that there is no latent process (i.e. E1 = I) the Pearson residuals
have approximately zero mean and unit variance. Hence the statistic
where
Using 1000 replications of a time series of length n = 168 obtained from simulating independent Poisson variates (i.e., with no latent process present) with mean f1. 1 = {l1 , the GLM fit to the polio data considered in Zeger ( 1988), gave simulated type I errors for Q which are severely lower than the nominal values. The observed mean and standard deviation of Q are -0.23 and 0.788, respectively, explaining the low coverage type I errors rates observed. To adjust for the negative bias alternate estimates of residuals which adjust for the effect of fitted values {l 1 could be used. For example one could use the divisor n - p instead of n in the numerator of Q. The resulting statistic had appreciably better performance than Q but was still on the conservative side. A second reasonably simple approach would be to use standardized Pearson residuals. Standardized Pearson residuals are
With this definition we get based on the same 1000 replications as used for Q above, mean and standard deviation of Q as 0.0 II and 0.826, respectively, and clearly improved, although still low, type I errors as
Empirical P(Q > =t-n) 0.073 0.037 0.022 0.004
with corresponding significance points
(.\' 0.100 0.050 0.025 0.010
and Sw First the size properties are as follows for a simulation of 1000 replicates assuming no latent process is present:
it 0.100 0.050 0.025 0.010
Empirical P(Q >=I-n) 0.089 0.048 0.026 0.003 Empirical P( Sa > = 1-o) 0.085 0.045 0.027 0.01 I
a 0.100 0.050 0.025 0.010
Empirical P(Q > z1_0 ) 0.077 0.038 0.021 0.013 Empirical P(Sa > z1_0 ) 0.099 0.056 0.025 0.009
In addition the power of the test to detect departures from the null hypothesis were investigated using I 000 replicates with the same regression models. The latent process was generated using a log-normal distribution. The latent process had variance a; = 0.05 chosen to give a small deviation from the null hypothesis. The autocovariance was simulated using an autoregressive process with ¢ = 0 and ¢ = 0.9. The results, again based on 1000 replications are for a size 0.05 test using empirical significance points obtained under the null hypothesis.
