Methods and formulas for random factors and mixed models in Fit General Linear Model

Select the method or formula of your choice.

Variance components

Minitab calculates variance components only for random factors. A model with two random factors is used to present the formulas.

where, αi, βj , (αβ)ij, and εijk are independent random variables. The variables are normally distributed with mean zero and variances given by these formulas:

These variances are the variance components. In this case, test the hypothesis that the variance components are equal to zero.

For an unrestricted mixed model with a fixed factor, A, and a random factor, B, this formula describes the model:

where αi are fixed effects and βj, (αβ)ij and εijk are uncorrelated random variables having zero means and these variances:

These variances are the variance components. The Σα i = 0.

This information is for balanced models. For information on unbalanced or more complex models, see Montgomery1 and Neter2.

  1. D.C. Montgomery (1991). Design and Analysis of Experiments, Third Edition. John Wiley & Sons.
  2. J. Neter, W. Wasserman and M.H. Kutner (1985). Applied Linear Statistical Models, Second Edition. Irwin, Inc.

Expected mean squares

The formulas for the expected mean squares for a random effects model with two factors, A and B are:

The formulas for the expected mean squares for an unrestricted mixed model with a fixed factor, A, and a random factor, B, are:

For the general rules on calculating expected mean squares, and for information on unbalanced or more complex models, see Montgomery1 and Neter2.

  1. D.C. Montgomery (1991). Design and Analysis of Experiments, Third Edition. John Wiley & Sons.
  2. J. Neter, W. Wasserman and M.H. Kutner (1985). Applied Linear Statistical Models, Second Edition. Irwin, Inc.

Notation

TermDescription
bnumber of levels in factor B
anumber of levels in factor A
nnumber of observations
σ2estimated variance of the model
estimated variance of A
estimated variance of B
estimated variance of AB
fixed effects of A

F-statistic for models with random factors

How the F-statistics in the ANOVA output are calculated

Each F-statistic is a ratio of mean squares. The numerator is the mean square for the term. The denominator is chosen such that the expected value of the numerator mean square differs from the expected value of the denominator mean square only by the effect of interest. The effect for a random term is represented by the variance component of the term. The effect for a fixed term is represented by the sum of squares of the model components associated with that term divided by its degrees of freedom. Therefore, a high F-statistic indicates a significant effect.

When all the terms in the model are fixed, the denominator for each F-statistic is the mean square of the error (MSE). However, for models that include random terms, the MSE is not always the correct mean square. The expected mean squares (EMS) can be used to determine which is appropriate for the denominator.

Example

Suppose you performed an ANOVA with the fixed factor Screen and the random factor Tech, and get the following output for the EMS:
Source Expected Mean Square for Each Term
(1) Screen (4) + 2.0000(3) + Q[1]
(2) Tech (4) + 2.0000(3) + 4.0000(2)
(3) Screen*Tech (4) + 2.0000(3)
(4) Error (4)

A number with parentheses indicates a random effect associated with the term listed beside the source number. (2) represents the random effect of Tech, (3) represents the random effect of the Screen*Tech interaction, and (4) represents the random effect of Error. The EMS for Error is the effect of the error term. In addition, the EMS for Screen*Tech is the effect of the error term plus two times the effect of the Screen*Tech interaction.

To calculate the F-statistic for Screen*Tech, the mean square for Screen*Tech is divided by the mean square of the error so that the expected value of the numerator (EMS for Screen*Tech = (4) + 2.0000(3)) differs from the expected value of the denominator (EMS for Error = (4)) only by the effect of the interaction (2.0000(3)). Therefore, a high F-statistic indicates a significant Screen*Tech interaction.

A number with Q[ ] indicates the fixed effect associated with the term listed beside the source number. For example, Q[1] is the fixed effect of Screen. The EMS for Screen is the effect of the error term plus two times the effect of the Screen*Tech interaction plus a constant times the effect of Screen. Q[1] equals (b*n * sum((coefficients for levels of Screen)**2)) divided by (a - 1), where a and b are the number of levels of Screen and Tech, respectively, and n is the number of replicates.

To calculate the F-statistic for Screen, the mean square for Screen is divided by the mean square for Screen*Tech so that the expected value of the numerator (EMS for Screen = (4) + 2.0000(3) + Q[1] ) differs from the expected value of the denominator (EMS for Screen*Tech = (4) + 2.0000(3) ) only by the effect due to the Screen (Q[1]). Therefore, a high F-statistic indicates a significant Screen effect.

Why does my ANOVA output include an "x" beside a p-value in the ANOVA table and the label "Not an exact F-test"?

An exact F-test for a term is one in which the expected value of the numerator mean squares differs from the expected value of the denominator mean squares only by the variance component or the fixed factor of interest.

Sometimes, however, such a mean square cannot be calculated. In this case, Minitab uses a mean square that results in an approximate F-test and displays an "x" beside the p-value to identify that the F-test is not exact.

For example, suppose you performed an ANOVA with the fixed factor Supplement and the random factor Lake, and the got following output for the expected mean squares (EMS):
Source Expected Mean Square for Each Term
(1) Supplement (4) + 1.7500(3) + Q[1]
(2) Lake (4) + 1.7143(3) + 5.1429(2)
(3) Supplement*Lake (4) + 1.7500(3)
(4) Error (4)

The F-statistic for Supplement is the mean square for Supplement divided by the mean square for the Supplement*Lake interaction. If the effect for Supplement is very small, the expected value of the numerator equals the expected value of the denominator. This is an example of an exact F-test.

Notice, however, that for a very small Lake effect, there are no mean squares such that the expected value of the numerator equals the expected value of the denominator. Therefore, Minitab uses an approximate F-test. In this example, the mean square for Lake is divided by the mean square for the Supplement*Lake interaction. This results in an expected value of the numerator being approximately equal to that of the denominator if the Lake effect is very small.

About the "Denominator of F-test is zero or undefined" message

Minitab will display an error that the denominator of the F-test is zero or undefined for one of the following reasons:
  • There is not at least one degree of freedom for error.
  • The adjusted MS values are very small, and thus there is not enough precision to display the F and p-values. As a workaround, multiply the response column by 10. Then perform the same regression model, but instead use this new response column for the response.

    Note

    Multiplying the response values by 10 will not affect the F and p-values that Minitab displays the output. However, decimal position will be affected in the remaining output, specifically, the sequential sums of squares, Adj SS, Adj MS, Fit, standard error of the fits, and residual columns.

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