In this section we present an overview of possible CFA base models for cross-classifications with up to five variables. Two types of base models are distinguished. The first comprises global CFA base models, that is, models where all variables have the same status. The second type comprises two-group models. Among the two-group models we distinguish further between models of the PCFA type, that is models with predictors and criteria.
When discussing PCFA and selecting a base model for PCFA there are three aspects that need to be considered. The first aspect concerns the model for the predictors, the second aspect concerns the model for the criteria, and the third aspect concerns the model for the dependence of the criteria on the predictors. This dependence materializes typically in predictor-criterion interactions.
In the past, PCFA involved fitting a saturated model to the predictor variables even if their margins were random. There are two main reasons for this strategy. The first is that PCFA, in analogy to usual regression analysis, models the conditional distribution of the criteria, given the predictors. By fitting a saturated model to the predictors we treat them in the statistical analysis as fixed even if they are random. The second reason is that if the base model for the predictors is not saturated, types and antitypes can result that reflect predictor interactions. This would be counter to the concept of PCFA. Therefore, we retain the strategy of fitting a saturated base model to the predictors.
In contrast, we do not retain the strategy of also always fitting a saturated base model for the criterion variables. Although relationships between the criterion variables may be of secondary interest we do not see a theoretical argument for always assuming a saturated base model on the criterion side. A saturated model is trivially ``true.'' Therefore, there is no way to avoid a mis-specification of the criterion base model. In addition, one cannot always exclude the possibility that criterion interactions are caused by the predictors. These interactions will go unnoticed when the base model for the predictors is saturated.
Another important issue must be considered. Suppose a model that is more parsimonious than the saturated model fits the criteria. Then, the types and antitypes identified by PCFA can be different than the types and antitypes identified when the criterion base model is saturated. Any discrepancies between the fitting model and the saturated model may reflect predictor effects.
It is important to note that if the predictor and criterion variables are not connected in every base model, the joint distribution of all variables is collapsible onto the criterion variables (for a discussion of collapsibility see Asmussen and Edwards, 1983). Hence, we can consider a reasonable model for the criterion variables as a preliminary step to PCFA. After a predictor variable model is selected (typically the saturated model; see above), one can perform PCFA as usual. However, the fitting model for the criteria is substituted for the standard saturated model.
Table 3 presents an overview of base models for CFA and PCFA. Possible models are marked with a ``x'' and models that are excluded because of the variables’ sampling characteristics or status are marked with a ``-''. The first column in Table 3 lists the number of variables, out of 5, with fixed margins. The middle panel of Table 5 displays the order of global CFA base models. This order is determined by the level of effects taken into account when estimating expected cell frequencies (von Eye, 1990). For instance, zero order CFA, also called Configural Cluster Analysis (Lienert & von Eye, 1985) assumes in its base model a uniform distribution, and First Order CFA takes all main effects into account.
The right hand panel displays the order of CFA base models for the criterion variables in PCFA. For this panel it is assumed that all predictors have random margins. The order of base models for the predictors does not need to be displayed, because it is always the same as the number of variables if all predictors have fixed marginals, which is typically the case. The last line of the right hand panel displays n/a for each cell, because there can be no CFA of two groups of variables when all variables are predictors with fixed margins. The table does not discriminate between models that posit different relationships between predictors and criteria. We do believe it is worth discussing models where certain predictor-criterion relationships are part of some base model. However this discussion is beyond the scope of the present article and will be taken up at another occasion.
n/a indicates that a model is not conceivable because there are no
criterion variables left.
Table 3: Possible CFA Base Models for up to Five Variables in the
Cross-Classification
Table 3 suggests that global base models must be of increasingly higher order when the number of variables with fixed margins increases. The last row in the table shows that CFA cannot be performed when all variables have fixed margins because, as was explained before, base models must be saturated in variables with fixed margins.
To illustrate the use of Table 3 we now list sample CFA base models. The listed models are not necessarily the only possible base models. However, they can be used to derive other possible models. Table 4 contains sample base models for the left panel of Table 3. The variables used in the examples are the Predictors, P1 and P2, and the Criteria, C1, C2, C3, C4 and C5. Table 5 displays base models for PCFA.
| Global CFA Base Model | Model Specification in Bracket Notation |
| Zero order Global Model | no effects, no fixed margins |
| First Order Global Model | [P1][P2][C1][C2][C3] |
| Second Order Global Model | [P1, P2][C1, C2][P1, C1][P1, C2][P2, C1][P2, C2] |
| Third Order Global Model | [P1, P2][C1, C2, C3] |
| Fourth Order Global Model | [P1][C1, C2, C3, C4] |
| Fifth Order Global Model | [C1, C2, C3, C4, C5] (Saturated) |
| Base Model for PCFA Criterion Variables | Model Specification in Bracket Notation |
| Zero Order Fixed-Effect PCFA Base Model | [P1][P2] |
| First Order Fixed-Effect PCFA Base Model | [P1][P2][C1][C2][C3] |
| Second Order Fixed-Effect PCFA Base Model | [P1, P2][C1, C2][C1, C3][C2, C3] |
| Third Order Fixed-Effect PCFA Base Model | [P1][P2][C1, C2, C3] |
| Fourth Order Fixed-Effect PCFA Base Model | [P1][C1, C2, C3, C4] |
| Fifth Order Fixed-Effect PCFA Base Model | n/a = not applicable because there are no |
| criterion variables left |