The structural model for the two random factor model is similar to that for the two fixed factor model.
We assume that for all i, j and k, εijk, αi, βj and (αβ)ij are pairwise independent and
Thus we have
The null and alternative hypotheses for the main effects and interaction are as follows:
The definitions of SSA, SSB, SAB, SSE and similarly for the MS and df terms are exactly the same as for the two fixed factor model described in Two Factor ANOVA. We also have (using a and b in place of r and c):
In order to conduct the proper F tests, we need to modify the approach used for two fixed factor ANOVA. As there we want a ratio which is 1 when the null hypothesis is true and greater than 1 when the alternative hypothesis is true. For the A and B main factors we therefore need to use MAB as the denominator when calculating F. This can be done as follows:
|Test Desired||Null Hyp H0||Alt Hyp H1||Statistical Test|
|Factor A Effect|
|Factor B Effect|
Estimates of Variances
With a balanced model, we can estimate and construct a confidence interval based on
Real Statistics Data Analysis Tool: The Real Statistics Resource Pack provides an option to the Two Factor ANOVA data analysis tool which supports random models.
This analysis can be done as described in Two Mixed Factors ANOVA, where the row factor is a random factor instead of a fixed factor. This corresponds to performing Example 1 of Two Mixed Factors ANOVA where the three anti-terrorism training courses being analyzed are randomly chosen from a large number of possible training courses. In Figure 5 of Two Mixed Factors ANOVA you would choose the Anova – Random instead of Anova – Mixed as the Analysis Type.