# Use of the F statistic in logistic regression

This paper uses a generalised linear mixed model assuming a binomial distribution for the errors. In the results section, the F statistic and associated P-value is used for the model (page 2150, paragraph beginning 'Males and females also differed')

I thought the F statistic could only be used in ANOVA and linear regression. Could anyone tell me how and why the F statistic is being used in the logistic regression in linked paper?

I'd say their use of $F$ statistics ranges from OK to slightly dodgy, but probably not actively harmful.

The authors say:

We used generalized linear mixed models (GLIMMIX) assuming a Poisson distribution for the call rates and latency data, and a binomial distribution for the likelihood of response in the test section. The tests assuming Poisson distributions were corrected for over-distribution to avoid an increase in type 3 errors (Littell et al. 2006).

(I think they mean "overdispersion", not "over-distribution", and "type I" rather than "type 3" errors).

In the case of overdispersed Poisson results, it sounds like they are essentially using quasi-likelihood, so that they are effectively estimating a scale parameter $\varphi$; Venables and Ripley (2002, p. 187) say

if $\varphi$ is not known, by analogy with the Gaussian case it is customary to use the approximate result $\frac{(D_{M_0} - D_M)}{\hat \varphi(p-q)} \sim F_{p-q,n-p}$ although this must be used with caution in non-Gaussian cases.

[they actually use an "approximately distributed as" symbol instead of $\sim$]

So I would say that an $F$ test makes sense for cases where the scale parameter is estimated, but not otherwise. However, note that the authors are actually using GLMMs (generalized linear mixed models) rather than GLMs. In this context Stroup (2014, "Rethinking the Analysis of Non-Normal Data in Plant and Soil Science" doi:10.2134/agronj2013.0342) says:

[Non-integer denominator df], and the $F$ and $p$ values, reflect the procedure developed by Kenward and Roger (2009) to account for the effect of the covariance structure on degrees of freedom and standard errors. Although the Kenward–Roger adjustment was derived for the LMM with normally distributed data and is an ad hoc procedure for GLMMs with non-normal data, informal simulation studies consistently have suggested that the adjustment is accurate. The Kenward–Roger adjustment requires that the SAS GLIMMIX default computing algorithm, pseudo-likelihood, be used rather than the Laplace algorithm used to obtain AICC statistics. Stroup (2013b) found that for binomial and Poisson GLMMs, pseudo-likelihood with the Kenward–Roger adjustment yields better Type I error control than Laplace while preserving the GLMM’s advantage with respect to power and accuracy in estimating treatment means.

• So I think of the F statistic as a ratio of variation explained by predicted values to variation explained by residuals. Is using a scale parameter equivalent to estimating variation explained by residuals? – luciano Apr 1 '14 at 17:23