The marginal fitted values are calculated using the following equation.
Notation
Term
Description
the vector of the new settings for the fixed effect terms
the estimated coefficients for the fixed effect terms
Standard error of the marginal fitted value (SE Fit)
The standard error of the marginal fitted values in the mixed model depend on the test method for the fixed effects. For both methods, the standard error equals the square root of the variance for the fitted value.
Kenward-Roger's method
where
Satterthwaites approximation
where
Confidence intervals for marginal means
The confidence interval for the marginal mean response at the new setting is:
is the marginal fit. is the standard error of the fit.
The degrees of freedom use this formula for the marginal case:
where
Marginal prediction interval
The marginal prediction interval is:
=
=
The degrees of freedom use this formula for the marginal case:
where
Notation
Term
Description
W
the asymptotic variance-covariance matrix of the variance component estimates
the estimated coefficients for the fixed effect terms
the vector of the new settings for the fixed effect terms
the vector of the new settings for the random terms
the variance component for the ith random effect term
c
the number of random terms in the model
X
the n x p design matrix for the fixed effects terms,
the inverse of the variance-covariance matrix
the n x mi design matrix for the random term in the model
is the marginal fit. 
is the standard error of the fit.
= 
= 
random term in the model