An algorithm for simulation in mixed models with crossed factors considering the sample sizes as random

Applying analysis of variance (ANOVA) when the sample dimensions are not known in advance is not such a rare situation. These situations frequently occur when a fixed time period is established for collecting the observations. In consequence, the number of observations that are obtained for each treatment is not fixed, depending on the arrival rate. So, since we cannot anticipate the exact sample dimensions, we should consider them as realizations of independent random variables. Therefore, the aim of this research is to extend the theory of the orthogonal mixed models to those situations, assuming the sample sizes as Poisson distributed. The model formulation is done considering stable statistics, which means that the test statistics, under the null hypothesis, have the same distribution whether referring to the fixed or to the random effects part of the model. The approach is extended to a general model with crossing of several factors. The applicability of the proposed approach is illustrated through an application based on real data, considering the incidence of unemployed persons in the European Union. The obtained results suggest that false rejections may be avoided when applying our approach, which is confirmed by carrying out two simulation studies.