# Interpretation of F-statistics in a linear mixed model

I was reading a paper yeaterday, and in their results they reported an F-score for each of their fixed effects in a linear mixed effect model.

Here, lux is a catagorical variable, but the rest are continuous. I haven't come across this before. Normally, I'm used to seeing Z or T scores, and these represent Wald tests --- examining the the regression slope for a given predictor variable is significantly different than 0.

Can someone explain to me:

• What is the F-statistic testing in this context?
• How are the numerator and denominator degrees of freedom calculated in this context (just in a GLM sense, we don't have to worry about the issues that come along with mixed models here if there are some).
• And finally, if its not implicitly answered in the first point... why would an author opt to do hypothesis testing with F scores, rather than Z or T scores.

My first intuition was that perhaps all the variables were categorical, so this is just an ANOVA... but I'm quite sure most of the variables are not categorical here.

### Reference

Riley, W. D., Davison, P. I., Maxwell, D. L., Newman, R. C. and Ives, M. J. (2015). A laboratory experiment to determine the dispersal response of Atlantic salmon (Salmo salar) fry to street light intensity. Freshwater Biol 60, 1016–1028.

There is a valid point in the comments about degrees of freedom in the mixed model. However, I suspect that this knowledge will lead you towards an answer, and it’s too long for a comment.

The F-test can test groups of variables, such as dog/cat/horse, which you would represent with $$(0,0)$$, $$(1,0)$$, and $$(0,1)$$. To be consistent with what they were doing with the factor variables with multiple levels (like dog/cat/horse), they did an F-test on the continuous variables.

The F-test of one continuous (or just not non-binary categorical) variable is equivalent to the t-test. The F-stat is the square of the t-stat, and both tests give the same p-value (assuming a two-sided t-test). Let's simulate this in R.

set.seed(2020)
x <- rnorm(100)
y <- x + rnorm(100)
L <- lm(y~x)
summary(L)


The result...

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  -0.1121     0.1144  -0.980     0.33
x             0.9675     0.1022   9.463 1.78e-15 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.139 on 98 degrees of freedom
Multiple R-squared:  0.4775,    Adjusted R-squared:  0.4721
F-statistic: 89.54 on 1 and 98 DF,  p-value: 1.775e-15


As you can see, except for some tiny rounding differences, the t-test on the x-coefficient is the same as the F-test. (This F-test compares the given model to the intercept-only model.)

• How about degrees of freedom in linear mixed-effects models?
– chl
Commented Oct 21, 2020 at 13:04
• Cool, so i now understand the interpretation, but I'm a little fuzzy on how to think about how the F test does it when applied to each variable of interest (as opposed to applied to the full model). Should I think about it as a analysis of variance between the intercept only model and a intercept plus that variable of interest model? Unfortunately, the wikipedia page is quite empty on this topic.
– user35780
Commented Oct 21, 2020 at 18:52
• It works by comparing the full model (with that variable or variables) to a reduced model (without that variable or variables).
– Dave
Commented Oct 21, 2020 at 21:58