Methods and formulas for pairwise comparison for mixed effects models in Comparisons

The two-sided 100(1 − α) confidence interval for the difference of means has the following expression:

For more information on how to calculate the fitted means and the standard error of the difference, go to Methods and formulas for fitted means in Fit Mixed Effects Model.

Critical value

The critical value is from the Studentized Range Distribution with tail probability α, m levels of the fixed effect term or the random term, and df degrees of freedom:

Term	Description
	the quantile from the studentized range distribution with df degrees of freedom
	the simultaneous probability of making a Type I error
m	the number of levels in the term
df	the degrees of freedom

The degrees of freedom depend on whether the comparison is for a fixed effect term or a random term.

Degrees of freedom (df)

For a fixed effect term, the degrees of freedom (df) are the same as the degrees of freedom for testing the corresponding fixed effect term. For a random term, the degrees of freedom use Satterthwaites approximation method.

For more information on the calculation of the degrees of freedom, go to Methods and formulas for tests of fixed effects in Fit Mixed Effects Model.

The two-sided 100(1 − α) confidence interval for the difference of means has the following expression:

For more information on how to calculate the fitted means and the standard error of the difference, go to Methods and formulas for fitted means in Fit Mixed Effects Model.

Critical value

The critical value has the following expression:

Term	Description
	the upper percentile of the student's t distribution with df degrees of freedom
α	the individual probability of making a Type I error
df	the degrees of freedom

The degrees of freedom depend on whether the comparison is for a fixed effect term or a random term.

Degrees of freedom (df)

For a fixed effect term, the degrees of freedom (df) are the same as the degrees of freedom for testing the corresponding fixed effect term. For a random term, the degrees of freedom use Satterthwaites approximation method.

For more information on the calculation of the degrees of freedom, go to Methods and formulas for tests of fixed effects in Fit Mixed Effects Model.

The two-sided 100(1 − α) confidence interval for the difference of means has the following expression:

For more information on how to calculate the fitted means and the standard error of the difference, go to Methods and formulas for fitted means in Fit Mixed Effects Model.

Critical value

The critical value has the following expression:

Term	Description
	the upper percentile of the student's t distribution with df degrees of freedom
α	the simultaneous probability of making a Type I error
c
k	the number of levels in the fixed effect term or the random term
df	the degrees of freedom

The degrees of freedom depend on whether the comparison is for a fixed effect term or a random term.

Degrees of freedom (df)

For a fixed effect term, the degrees of freedom (df) are the same as the degrees of freedom for testing the corresponding fixed effect term. For a random term, the degrees of freedom use Satterthwaites approximation method.

For more information on the calculation of the degrees of freedom, go to Methods and formulas for tests of fixed effects in Fit Mixed Effects Model.

The two-sided 100(1 − α) confidence interval for the difference of means has the following expression:

For more information on how to calculate the fitted means and the standard error of the difference, go to Methods and formulas for fitted means in Fit Mixed Effects Model.

Critical value

The critical value has the following expression:

Term	Description
	the upper percentile from the student's t distribution with df degrees of freedom
α	the simultaneous probability of making a Type I error
c
k	the number of levels in the fixed effect term or the random term
df	the degrees of freedom

The degrees of freedom depend on whether the comparison is for a fixed effect term or a random term.

Degrees of freedom (df)

For a fixed effect term, the degrees of freedom (df) are the same as the degrees of freedom for testing the corresponding fixed effect term. For a random term, the degrees of freedom use Satterthwaites approximation method.

For more information on the calculation of the degrees of freedom, go to Methods and formulas for tests of fixed effects in Fit Mixed Effects Model.