The tests of the fixed effect terms are F tests. The null hypothesis for the test depends on whether the test is for a fixed factor term or a covariate term. For a fixed factor term, the null hypothesis is that the term does not significantly affect the response. For a covariate term, the null hypothesis is that no association exists between the response and the covariate term.
Minitab offers 2 methods to test fixed effect terms: Kenward-Roger approximation and Satterthwaite approximation. For more information on Kenward-Roger approximation, see Kenward and Roger.^{1} For more information on Satterthwaite approximation, see Giesbrecht and Burns ^{2} as well as Fai and Cornelius. ^{3}
The calculation of the denominator degrees of freedom for the F-statistic and the calculation of the F-statistic differ. The calculation of the numerator degrees of freedom and the determination of a p-value for a given F-statistic are the same for both methods.
Kenward-Roger approximation is one method to test the statistical significance of fixed effect terms.
where
Term | Description |
---|---|
l | the numerator degrees of freedom, which is the number of parameters in the term to test |
0 | the matrix with 0 components |
I_{l} | the identity matrix with l dimension |
c + 1 | the number of variance components |
w_{rs} | (r, s)^{th} component of the asymptotic variance-covariance matrix of |
V^{−1} | the inverse of the variance-covariance matrix |
For further details on the notation, go to the Methods section.
where
If either condition is not true, then λ = 1.
Under the null hypothesis, lambda × F is asymptotically F distributed with degrees of freedom DF Num, and DF Den. The calculation of the P-value uses this property.
Satterthwaite approximation is one method to test the statistical significance of fixed effect terms.
where L and have the same definitions as in Kenward-Roger approximation.
The process for the determination of the degrees of freedom has multiple steps.
Perform the spectral decomposition on the variance of the fixed effect parameter vector estimate:
where P is an orthogonal matrix of eigenvectors and D is a diagonal matrix of eigenvalues, both of dimension l × l.
Define l_{r} to be the r^{th} row of P'L, r = 1, ..., l and let
where
i = 1, …, c, and
Let
where is an indicator function that eliminates terms with
The denominator degrees of freedom depend on the value of E.
Effect | DF |
---|---|
Fixed Factor | |
Covariate | 1 |
Interactions that involve fixed factors |
Term | Description |
---|---|
k | the number of levels in the fixed factor term |
m | the number of factors in the interaction |
Term | Description |
---|---|
the cumulative distribution function of the F-distribution with degrees of freedom equal to DF Num and DF Den, respectively | |
the calculated F-value for a term |