Restricted vs Unrestricted Models
Which is the Best One?
For a mixed effect model, when an interaction term includes both fixed and random factors, the interaction term must be classified either fixed or random in determining the correct divisor (the associated error) for the F-Statistics. An Unrestricted Model treats a mixed interaction term as a random factor, while a Restricted Model does not treat the mixed interaction term as a complete random factor. In a restricted model, the mixed interaction term with both fixed and random factors treats individual factor as it is; meaning that the fixed is treated as fixed and random is treated as random. While the detailed distinction can be seen in the earlier sections for the development of Expected Mean Square for mixed models, Table 8, Table 9, and Table 10 provide some quick comparisons between the restricted vs unrestricted mixed models.
Table 8
Unrestricted vs. Restricted Mixed Models
Table 9
Unrestricted vs. Restricted Mixed Models (A B Fixed & C Random)
Table 10
Unrestricted vs. Restricted Mixed Models (A B Fixed & C Random and Nested in B)
Which Model to Use Unrestricted or Restricted?
In the expected mean square table for the mixed models, we see that the restricted model is more generalized than the unrestricted one. Therefore, some authors have suggested/preferred the restricted model (Montgomery, 2019). The error structures associated with the effects are less messy/complex for the restricted model, resulting in more exact F-Statistics as compared to the more pseudo/approximate F-Statistics for the unrestricted models (Table 9 & Table 10). Nevertheless, the choice between models should be guided by the practical experimental situations/scenarios, rather than the convenience of getting more exact F-tests. Therefore, to determine the best model, we need to understand the mixed interaction term in practice.
Is there enough reason to consider the interaction term as purely a random factor? If the exact experiments are conducted again with the same levels of the fixed factor and the same levels of the random factor, is there a different result for the mixed interaction term? If the result is observed to be different for the mixed interaction term every time the exact same experiment is replicated, the term can be assumed as a pure random effect factor. In other words, the variances for the within-factor-level and between-factor-level are the same. A different or random result for the mixed interaction term is possible only if the term is, in fact, a random factor, which calls for an unrestricted model. Therefore, the random mixed interaction term should be included in the devisor for the F-Statistics as a random error.
If the result is observed to be the same, some fixed effect for the mixed interaction term is suspected over the levels of the random factor. The sum of the mixed interaction effect over the levels of the random factor is hypothesized as zero. Therefore, the mixed interaction term should be not be assumed purely random anymore, and a restricted model is recommended.
Therefore, the choice between the restricted vs unrestricted models depends on the experimental situations. Moreover, software packages such as SAS, SPSS, JMP, Minitab, and Design Experts can be used to analyze either model easily. However, the default in most software is the unrestricted model.
The final decision on which model to use can be made at the data analysis stage of the design of the experiment. In most experimental scenarios, statistical decisions may not change between the restricted vs unrestricted model; the interpretations/meaning of the results would change for the mixed interaction term though!