# Paper No. 09-36

**I Kosmidis**

**On iterative adjustment of responses for the reduction of bias in binary regression models**

**Abstract: **The adjustment of the binomial data by small constants is a common practice in statistical modelling, for avoiding sparseness issues and, historically, for improving the asymptotic properties of the estimators. However, there are two main disadvantages with such practice: i) there is not a universal constant adjustment that results estimators with optimal asymptotic properties for all possible modelling settings, and ii) the resultant estimators are not invariant to the representation of the binomial data. In the current work, we present a parameter-dependent adjustment scheme which is applicable to binomial-response generalized linear models with arbitrary link functions. The adjustment scheme results by the expressions for the bias-reducing adjusted score functions in Kosmidis & Firth (2008, Biometrika) and thus its use guarantees estimators with second-order bias. Based on an appropriate expression of the adjusted data, a procedure for obtaining the bias-reduced estimates is developed which relies on the iterative adjustment of the binomial responses and totals using existing maximum likelihood implementations. Furthermore, it is shown that the bias-reduced estimator, like the maximum likelihood

estimator, is invariant to the representation of the binomial data. A complete enumeration study is used to demonstrate the superior statistical properties of the bias-reduced estimator to the maximum likelihood estimator.

**Keywords:** bias reduction, adjusted responses, adjusted score functions