These infinite estimates can be viewed as arising from separation of the outcomes by the covariates ( 7, 8). In these cases, ML estimators are not even approximately unbiased, and ML estimates of finite odds ratios may be infinite. Nonetheless, ML estimation can break down with small or sparse data sets, an exposure or outcome that is uncommon in the data, or large underlying effects, especially with combinations of these problems ( 1– 6). Logistic models are almost always fitted with maximum likelihood (ML) software, which provides valid statistical inferences if the model is approximately correct and the sample is large enough (e.g., at least 4–5 subjects per parameter at each level of the outcome). Logistic regression is a standard method for estimating adjusted odds ratios. Logistic regression, maximum likelihood, penalized likelihood, separation, small samples, sparse data We provide an illustration of ideas and methods using data from a case-control study of contraceptive practices and urinary tract infection. We discuss likelihood penalties, including some that can be implemented easily with any software package, and their relative advantages and disadvantages. These methods improve accuracy, avoid software problems, and allow interpretation as Bayesian analyses with weakly informative priors. We then describe methods that remove separation, focusing on the same penalized-likelihood techniques used to address more general sparse-data problems. We discuss causes of separation in logistic regression and describe how common software packages deal with it. In practice, however, separation may be unnoticed or mishandled because of software limits in recognizing and handling the problem and in notifying the user. In theory, separation will produce infinite estimates for some coefficients. It is most frequent under the same conditions that lead to small-sample and sparse-data bias, such as presence of a rare outcome, rare exposures, highly correlated covariates, or covariates with strong effects. Separation is encountered in regression models with a discrete outcome (such as logistic regression) where the covariates perfectly predict the outcome.
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