Goodness of fit for the logistic regression model using relative belief

Luai Al-Labadi,Zeynep Baskurt,Michael Evans

doi:10.1186/s40488-017-0070-7

Luai Al-Labadi, Zeynep Baskurt + Show 1 more

Open Access

https://doi.org/10.1186/s40488-017-0070-7

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Abstract

A logistic regression model is a specialized model for product-binomial data. When a proper, noninformative prior is placed on the unrestricted model for the product-binomial model, the hypothesis H0 of a logistic regression model holding can then be assessed by comparing the concentration of the posterior distribution about H0 with the concentration of the prior about H0. This comparison is effected via a relative belief ratio, a measure of the evidence that H0 is true, together with a measure of the strength of the evidence that H0 is either true or false. This gives an effective goodness of fit test for logistic regression.

Highlights

IntroductionAl-Labadi et al Journal of Statistical Distributions and Applications (2017) 4:17
Suppose there is a response Y ∈ {0, 1} related to k predictors (X1, . . . , Xk) via the logistic regression model p(x β) = P(Y = 1 | X1 = x1, . . . , Xk = xk) where p(x β) = exp x β / 1 + exp x β . (1)and x = (x1, . . . , xk), β = (β1, . . . , βk) ∈ Rk
Evidence can be obtained in favor of the logistic regression model, as opposed to only evidence against as with p-values, and there is no appeal to asymptotics

Summary

Introduction

Al-Labadi et al Journal of Statistical Distributions and Applications (2017) 4:17 In this case the logistic regression is just a reparameterization of the product-binomial model. Evidence can be obtained in favor of the logistic regression model, as opposed to only evidence against as with p-values, and there is no appeal to asymptotics. To see this suppose that we have a statistical model {fθ : θ ∈ } for the data x, and a prior π. Note that is easy to generate from both the prior and posterior of θ(X)

Hypothesis assessment via relative belief and concentration

Conclusions