Targeting: Logistic Regression, Special Cases and Extensions

Abstract

Logistic regression is a classical linear model for logit-transformed conditional probabilities of a binary target variable. It recovers the true conditional probabilities if the joint distribution of predictors and the target is of log-linear form. Weights-of-evidence is an ordinary logistic regression with parameters equal to the differences of the weights of evidence if all predictor variables are discrete and conditionally independent given the target variable. The hypothesis of conditional independence can be tested in terms of log-linear models. If the assumption of conditional independence is violated, the application of weights-of-evidence does not only corrupt the predicted conditional probabilities, but also their rank transform. Logistic regression models, including the interaction terms, can account for the lack of conditional independence, appropriate interaction terms compensate exactly for violations of conditional independence. Multilayer artificial neural nets may be seen as nested regression-like models, with some sigmoidal activation function. Most often, the logistic function is used as the activation function. If the net topology, i.e., its control, is sufficiently versatile to mimic interaction terms, artificial neural nets are able to account for violations of conditional independence and yield very similar results. Weights-of-evidence cannot reasonably include interaction terms; subsequent modifications of the weights, as often suggested, cannot emulate the effect of interaction terms.

Optimal Thresholding of Predictors in Mineral Prospectivity Analysis

“…In prospectivity analysis, the techniques of logistic regression and WofE are both applications of the method of maximum likelihood (Baddeley et al 2010;Schaeben 2014). Furthermore, the hypothesis tests reviewed in the section on SIGNIFICANCE TESTS are all obtained from the likelihood.…”

“…For a single binary predictor, WofE is equivalent to logistic regression (Agterberg 1974;Schaeben 2014;Schaeben and Semmler 2016), and is also equivalent to maximum Poisson likelihood estimation and to maximum entropy modelling (Baddeley et al 2010;Baddeley 2018;Baddeley et al 2015, Chapter 9). Equivalence means that these techniques use different formulations but yield the same predictions.…”

Comment on “Modified Weights-of-Evidence Modeling with Example of Missing Geochemical Data”

“…. , m, which are identical with the parameters of the corresponding logistic regression model [11,12] as we assumed joint conditional independence of all B ℓ given T. Otherwise, a linear relationship between weights-of-evidence and logistic regression parameters does not exist [11,12]. It should be noted that the total number of events B ℓ � 1 or B ℓ � 0 is n, which is the total number of pixels of the digital map image.…”