Bartlett-type correction terms for tests in intensity regression models

For multi-factor intensity regression models both computational and theoretical difficulties arise when one tries to find an exact and useful correction term to adjust the likelihood ratio test for deviations from the asymptotic chi-square distribution. In cases with low intensities, short observation intervals, or small samples, the Bartlett adjustment cannot be recommended as a correction even for simple exponential distribution models, for the convergence of the expansion of the expected likelihood ratio is too slow. In this paper some suggestions for approximate correction terms are discussed. These corrections are constructed as sums of correction terms for simple censored exponential distribution models, with a structure similar to that of the test statistic in analysis of variance. A simulation study is described, and results concerning the likelihood ratio statistic for some simple models are presented. The study indicates that the Bartlett corrected test tends to be too conservative, and that the use of parameter estimates instead of the true parameter values in the correction formula makes the test even more conservative. Finally, we discuss the choice of degrees of freedom for likelihood ratio tests for data sets with practically empty cells. For these cases, we suggest a very careful use of a chi-square statistic with a decreased number of degrees of freedom.