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Poisson regression and the regular multiple regression are almost similar. The major difference is that in Poisson regression, the dependent variable is the measured count that considers a Poisson distribution. As a result, our dependent variable can only have positive integers as possible values. Both Poisson regression and logistic regression have a discrete response value. However, the response in Poisson regression is not constrained to distinct values like in logistic regression.
Negative Binomial Regression
Negative binomial regression is a multiple regression with a variable that is an observed count that takes into account negative binomial distribution. It is a form of general Poisson regression with a loose limit to the assumption made by a Poisson model that variance and mean are equal. The negative binomial regression model is founded on the Poisson-gamma mixture distribution. This method is widely used because it supports using a gamma distribution to model Poisson heterogeneity.
Exact Logistic Regression
Exact logistic regression is suitable for binary outcome variables modeling. In this procedure, the outcome's log-odds are modeled as a linear integration of predictor variables. Exact logistic regression is used when the size of the sample is small and cannot be accommodated by the normal logistic regression. It can also be used when the categorical predictor variables create cells with no observations. Asymptotic results do not determine the predictions provided by exact logistic regression.
Robust regression is a substitute technique for least squares. It is used for data that has influential observations and outliers. Robust regression is also suitable for measuring and finding influential observations. An observation can be deemed to be influential if deleting it will substantially change the coefficients of the estimated regression. Robust regression is a popular strategy because it allows researchers to compromise between removing high leverage data points and treating these points equally in ordinary least squares.