This paper identifies several serious problems with the widespread use of ANOVAs for the analysis of categorical outcome variables such as forced-choice variables, question-answer accuracy, choice in production (e.g. in syntactic priming research), et cetera. I show that even after applying the arcsine-square-root transformation to proportional data, ANOVA can yield spurious results. I discuss conceptual issues underlying these problems and alternatives provided by modern statistics. Specifically, I introduce ordinary logit models (i.e. logistic regression), which are well-suited to analyze categorical data and offer many advantages over ANOVA. Unfortunately, ordinary logit models do not include random effect modeling. To address this issue, I describe mixed logit models (Generalized Linear Mixed Models for binomially distributed outcomes, Breslow & Clayton, 1993), which combine the advantages of ordinary logit models with the ability to account for random subject and item effects in one step of analysis. Throughout the paper, I use a psycholinguistic data set to compare the different statistical methods.

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LetPi be the probability of a disease in one population and P2 be the probability of a disease in a second population. The ratio of these quantities, R = Pl/P2, is termed the relative risk. We consider first the analyses of the relative risk from retrospective studies. The relation between the relative risk and the odds ratio (or cross-product ratio) is developed. The odds ratio can be considered a parameter of an exponential model possessing sufficient statistics. This permits the development of exact significance tests and confidence intervals in the conditional space. Unconditional tests and intervals are also considered briefly. The consequences of misclassification errors and ignoring matching or stratifying are also considered. The various methods are extended to combination of results over the strata. Examples of case-control studies testing the association between HL-A frequencies and cancer illustrate the techniques. The parallel analyses of prospective studies are given. If Pi and P2 are small with large sample sizes the appropriate model is a Poisson distribution. This yields a exponential model with sufficient statistics. Exact conditional tests and confidence intervals can then be developed. Here we consider the case where two populations are compared adjusting for sex differences as well as for the strata (or covariate) differences such as age. The methods are applied to two examples: (1) testing in the two sexes the ratio of relative risks of skin cancer in people living in different latitudes, and (2) testing over time the ratio of the relative risks of