برآوردگر تابع اامتیاز صلاح شده برای رگرسیون لجستیک چندگانه در نمونه های کوچک

Logistic regression modelling of mixed binary and continuous covariates is common in practice, but conventional estimation methods may not be feasible or appropriate for small samples. It is well known that the usual maximum likelihood estimates (MLEs) of the log-odds-ratio parameters are biased in finite samples, and there is a non-zero probability that an MLE is infinite, i.e., does not exist. In this paper, we extend the approach proposed by Firth (Biometrika 80 (1993) 27) for bias reduction of MLEs in exponential family models to the multinomial logistic regression model, and consider general regression covariate types. The method is based on a suitable modification of the score function that removes first order bias. We apply the method in the analysis of two datasets: one is a study of disease prognosis and the other is a disease prevention trial. In a series of simulation studies in small samples, the modified-score estimates for binomial and trinomial logistic regressions had mean bias closer to zero and smaller mean squared error than other approaches. The modified-score estimates have properties that make them attractive for routine application in logistic regressions of binary and continuous covariates, including the advantage that they can be obtained in samples in which the MLEs are infinite.

Methods for logistic regression modelling of nominal categorical responses based on the multinomial logistic likelihood are now generally available in standard statistical packages, and have been applied in the analysis of case-control studies with multiple case or multiple control groups, and in randomized trials and cross-sectional surveys with categorical responses. One of the concerns of investigators is the valid estimation of model parameters in the finite sample sizes encountered in practice. In finite samples, the usual maximum likelihood estimates (MLEs) of the log odds ratios are biased, and the bias increases as the ratio of the number of observations to the number of parameters (n-to-p ratio) decreases ( Cordeiro and McCullagh, 1991; Bull et al., 1997). This is of particular concern when there are several response categories and multiple covariates because the number of parameters can become large. We consider an alternative estimation method for small samples based on a modification of the score function that removes first order bias and is equivalent to penalizing the likelihood by Jeffreys’ prior (Firth, 1993). We extend the modified score function method to the multinomial logistic regression model with nominal response categories, and compare the modified estimates to the usual MLEs and to the MLEs corrected by an estimate of the asymptotic bias. As the sample size increases, the modified-score estimates become equivalent to the usual MLEs. As systematic small sample comparisons of this approach have not been reported previously for binomial or multinomial logistic regression models, we also present a Monte Carlo simulation study in which we compare the mean bias and mean squared error (MSE) of the modified estimates to the MLEs, and to the MLEs corrected by the estimated asymptotic bias. Considering the same series of logistic regression models studied previously (Bull et al., 1997), we find that the modified-score estimates are competitive and often superior to the other approaches.