Maths behind phia functions

What is the meaning of the column "Value" in the tables returned by testInteractions?

In general, it is a combination of differences between expected values of the output of the model, for given contrasts of factor levels.

For instance, let's consider a linear model called mod containing the factor treatment with levels placebo and medication, and the factor gender with levels male and female. You can test the contrasts of treatment for both genders, through:

testInteractions(mod, pairwise=treatment, fixed=gender)

This will give a table with two rows, which may bear the labels medication-placebo : female and medication-placebo : male. The value reported in both rows will be the difference between the expected output of the model for cases of medication minus the expected output for placebo, for each gender.

However, the default behaviour of the function will be reporting the pairwise contrasts between both factors. I.e. if you call:

testInteractions(mod)

You will ge a table with only one row, labelled something like medication-placebo : female-male. The value reported in this case is the "difference of differences", and (for a linear model at least) will coincide with the difference between the two rows of the previous example. In models with more factors or more levels within each factor there may be many more rows, and more levels of nested differences, but the rationale is the same in all cases.

The test reported in each row of the table is related to the "relevance" of those values. In the simplest case of linear models, it means to what extent they are significantly different from zero.

Please read chapters from 3 to 5 of the vignette to find many more examples with actual data.

How does testInteractions test pairwise comparisons: by Scheffé's method, Tukey's...?

testInteractions is a wrapper of the function linearHypothesis from package car. As told in the manual of that package, it carries out a Wald test, which in this case compares the expected value of the interaction reported in the column called Value with the null hypothesis. When there are multiple rows in the table, the test is adjusted for multiple comparisons, using Holm's correction by default; this can be changed using the argument adjust (see the help page of the function p.ajust for further information).

Why do I obtain p-values from a Chi-squared distribution instead of an F distribution?

testInteractions provides whatever test is given by the function linearHypothesis from package car for the given type of model. As explained in the manual of that package, that is an F-test for univariate tests if it is possible to determine the degrees of freedom of the residuals, or a Chi-squared test otherwise. It can even be other type of test (Pillai, etc.) in the case of multivariate models (with vector outputs).

Can I analyse with "phia" models fitted by other methods (GAM, RLM...)?

The package "phia" is not restricted by design to a particular set of models, but it has only been tested with linear models (univariate and multivariate), generalised linear models, and mixed models fitted by the functions of package nlme and lme4.

Contributions are welcome from any user that may provide data and expected results to validate the outcomes. But please consider that other models should be supported by the function linearHypothesis of package car, which is used by the functions coded in "phia" to make statistical tests.