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

Select the method or formula of your choice.

## Tukey method for a 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.

###### Note

Calculations for the test statistic, adjusted p-value, individual confidence level, and grouping information table match the calculations for general linear models. For more information, go to the Methods and Formulas for comparisons for general linear models.

### 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:
TermDescription
the quantile from the studentized range distribution with df degrees of freedom
the simultaneous probability of making a Type I error
mthe number of levels in the term
dfthe 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.

## Fisher method for pairwise comparisons in a 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.

###### Note

Calculations for the test statistic, adjusted p-value, individual confidence level, and grouping information table match the calculations for general linear models. For more information, go to the Methods and Formulas for comparisons for general linear models.

### Critical value

The critical value has the following expression:
TermDescription
the upper percentile of the student's t distribution with df degrees of freedom
αthe individual probability of making a Type I error
dfthe 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.

## Bonferroni method for pairwise comparisons in a 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.

###### Note

Calculations for the test statistic, adjusted p-value, individual confidence level, and grouping information table match the calculations for general linear models. For more information, go to the Methods and Formulas for comparisons for general linear models.

### Critical value

The critical value has the following expression:
TermDescription
the upper percentile of the student's t distribution with df degrees of freedom
αthe simultaneous probability of making a Type I error
c
kthe number of levels in the fixed effect term or the random term
dfthe 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.

## Sidak method for pairwise comparisons in a 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.

###### Note

Calculations for the test statistic, adjusted p-value, individual confidence level, and grouping information table match the calculations for general linear models. For more information, go to the Methods and Formulas for comparisons for general linear models.

### Critical value

The critical value has the following expression:
TermDescription
the upper percentile from the student's t distribution with df degrees of freedom
αthe simultaneous probability of making a Type I error
c
kthe number of levels in the fixed effect term or the random term
dfthe 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.

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