Statistical testing in a Linear Probability Space

TL;DR

This paper proposes performing statistical tests in a linear probability space using log-odds, simplifying calculations like Bayes theorem and effect size, and replacing p-values with certainty measures for more accurate analysis.

Contribution

It introduces a linear probability space based on log-odds for statistical testing, offering a more accurate and intuitive framework than traditional probability spaces.

Findings

Statistical testing is more accurate in log-odds space.

Effect size is represented as impact (I) in this space.

Bayes theorem simplifies to addition of weights.

Abstract

Imagine that you could calculate of posttest probabilities, i.e. Bayes theorem with simple addition. This is possible if we stop thinking of probabilities as ranging from 0 to 1.0. There is a naturally occurring linear probability space when data are transformed into the logarithm of the odds ratio (log10 odds). In this space, probabilities are replaced by W (Weight) where W=log10(probability/(1-probability)). I would like to argue the multiple benefits of performing statistical testing in a linear probability space: 1) Statistical testing is accurate in linear probability space but not in other spaces. 2) Effect size is called Impact (I) and is the difference in means between two treatments (I=Wmean2-Wmean1). 3) Bayes theorem is simply Wposttest=Wpretest+Itest. 4) Significance (p value) is replaced by Certainty (C) which is the W of the p value. Methods to transform data into and out of linear probability space are described.