Robust diagnostic and estimation for binary logistic regression model in the presence of multicollinearity and high leverage points

The binary logistic regression model popularly used in medical data analysis. In spite of its popularity, there are only a few available robust methods for this model to encounter the effects of high leverage points and multicollinearity. Failure to address model adequacy when a combination of high leverage points and multicollinearity exist in data, lead to misleading and incorrect inferences. This study is aimed to develop new robust diagnostic and estimation for logistic regression (overlap cases) and hidden logistic regression (non-overlap cases). A new robust diagnostic called Logistic Influential Outlier Nominator (LION) is developed to identify influential outliers and the LION successfully detect the outliers in both x and y directions. Then, second robust diagnostic, namely Diagnostic Influential Observations (DIO) is developed, specifically to identify high leverage influential observations (HLIO). The DIO introduces two important stages whereby the initial stage employs the LION procedure and the confirmation stage comprises combine measures of Generalized Distance from the Mean and Generalized Standardized Pearson Residual to flag the HLIO. Adjusted Weighted Bianco and Yohai (AWEBY) is an improvisation on the Weighted Bianco and Yohai (WBY) robust estimator. The AWEBY is proposed to increase the efficiency of WBY estimator by constructing a "smooth rejection" to replace the "hard rejection" weight function. In the AWEBY, new robust weights are formulated based on the DIO and found to properly reduce the effect of HLIO whilst protecting the good leverage points. In combined problems of HLIO and multicollinearity for overlap cases, the AWEBY estimator is integrated for computing robust ridge parameter and formed Robust Ridge Logistic (RRL) iterative update scheme. By using the updated robust weights, the impact of the HLIO and multicollinearity will be toned down immensely. Adjusted Weighted Maximum Estimated Likelihood (AWEMEL) in hidden logistic regression is proposed to rectify the HLIO in separation problem. New robust weights in the AWEMEL is designed based on DIO which particularly down weighs the HLIO but not the good leverage points. Finally, Robust Ridge Hidden Logistic (RRHL) is proposed to remedy both HLIO and multicollinearity for separation problem. In RRHL's iteration, the AWEMEL estimator is employed to compute robust ridge parameter which is resistance towards the bad impacts of HLIO.

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