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The self-controlled case series (SCCS) method is often used to examine the temporal association between vaccination and adverse events using only data from patients who experienced such events.Conditional Poisson regression models are used to estimate incidence rate ratios, and these models perform well with large or medium-sized case samples.Firth's idea has been applied in logistic regression (19, 20) to reduce the bias in cases of data separation and in Cox regression (21) to handle the problems of monotone likelihood, when at least 1 parameter estimate diverges to negative or positive infinity.In this study, we evaluated the performance of Firth's bias-prevention method and the CM bias-correction method for correcting MLE bias in studies using the SCCS design with rare adverse events.In this study, we used simulations to evaluate 2 bias correction approaches—the Firth penalized maximum likelihood method and Cordeiro and Mc Cullagh's bias reduction after maximum likelihood estimation—with small sample sizes in studies using the SCCS design.The simulations showed that the bias under the SCCS design with a small number of cases can be large and is also sensitive to a short risk period.However, in some vaccine safety studies, the adverse events studied are rare and the maximum likelihood estimates may be biased.
Often when a sample size is small, data separation occurs if no events are observed in one of the 2 groups defined by a dichotomous covariate (or no events are observed in either the risk period or the control period(s) in the SCCS design), and no MLE is produced.Therefore, in this study, we used the profile penalized likelihood ratio test statistic to obtain confidence intervals and values for the Firth method while using the Wald test statistic for the CM and ML methods.Cordeiro and Mc Cullagh (15) developed a bias correction method for generalized linear models with a canonical link function, such as linear logistic models for binomial data and log-linear models for Poisson data. This method also eliminates the first order of bias from ML estimation, and the bias calculation is straightforward. We conducted a simulation study to evaluate the performance of these two bias correction methods in SCCS designs with rare adverse events.Therefore, the conditional Poisson regression model can be implemented using a Cox proportional hazards framework, thus simplifying the analysis of SCCS data using standard statistical packages.In addition, Heinze and Schemper (19) have demonstrated that inference based on the profile penalized likelihood is preferable to the Wald test statistic for the Firth correction method.
Several bias correction methods have been studied in generalized linear models for matched case-control studies (11–13), but none of these correction methods have been evaluated in the SCCS study design.