In matrix terms, the vector of estimated coefficients follows:
For further details on the notation, go to the Methods section.
Standard errors of the coefficients in mixed models
The standard errors of the coefficients depend on the test method for the fixed effects. For further details on the notation, go to the Methods section and the Tests of Fixed Effects section.
Kenward-Roger's approximation
The standard errors of the estimated coefficients are the square roots of the dialgonal elements of the matrix .
where
Satterthwaite Approximation
The standard errors are the square roots of the diagonal elements of the matrix, .
where
Degrees of freedom for coefficients
The following hypotheses are for the tests of the coefficients:
The following equations give the degrees of freedom for the coefficient:
where
For further details on the notation, go to the Methods section and the Tests of Fixed Effects section.
Confidence intervals for coefficients
The (1 − α)% confidence limits for the coefficient has the following equation:
Notation
Term
Description
the estimated coefficient
the 1 − α/2 percentile from the t distribution with df degrees of freedom
the standard error of the estimated coefficient
T-value
Notation
Term
Description
test statistic for the coefficient
estimated coefficient
standard error of the estimated coefficient
p-value (P)
The following equation gives the two-sided p-value for the null hypothesis that a coefficient equals 0:
Notation
Term
Description
The probability that under the null hypothesis T is less than the absolute value of the calculated . Here, T follows a t-distribution with df degrees of freedom.
.
.
coefficient:
coefficient has the following equation:
coefficient
estimated coefficient
estimated coefficient
. Here, T follows a t-distribution with df degrees of freedom.
coefficient.