The PCFA model given in Equation 5 and the example in Section 3 are extreme in the sense that not a single effect of the criterion variables is part of the model. This may seem implausible because researchers typically assume that main effects that are caused by independent variables manifest only in variable interactions. The following hypothetical example illustrates that this is not necessarily the case. Consider an evaluation experiment in which researchers compare two training methods. A sample of N = 22 participants was randomly drawn from a pool of athletes who still had to take the qualification test for the next intergalactical games. 11 randomly selected athletes went to the camp where training method A was used, the remaining 11 went to the camp where training method B was used. After completion of the training program, all athletes took the qualification test. Only one athlete from each camp failed the test, all others passed.
We now analyze these frequencies using two PCFA base models. The first base
model is that of standard PCFA which, in this case, is identical to the base
model for first order CFA. This model takes the main effects of both the
predictor and the criterion into account. Notice that this is also the
model for the Pearson 
-test of variable independence. The second base
model employed for analysis of these data is that of mixed-sampling PCFA
which considers only the predictor effects, thus hypothesizing a joint
frequency distribution that is fully determined by the predictor in this
cross-classification, that is, Training Program (P with categories A and B).
The Result of Test variable (T with categories pass and fail) is supposed
to have no effect whatsoever. This model can yield types and antitypes if
(1) the criterion has main effects, and/or (2) there exist
predictor-criterion interactions. Results of these two analyses are
summarized in Table 2. We used the Anscombe z-test and Bonferroni-adjustment
of 
which led to 
= 0.0125.
| Cells | Observed frequencies | Fixed-effect PCFA | Mixed-sampling PCFA | ||||
| PT | Exp. freq. | z | p(z) | Exp. Freq. | z | p(z) | |
| Ap | 10 | 10 | 0 | .5 | 5.5 | 1.79 | .036 |
| Af | 1 | 1 | 0 | .5 | 5.5 | 2.32 | .010 A |
| Bp | 10 | 10 | 0 | .5 | 5.5 | 1.79 | .036 |
| Bf | 1 | 1 | 0 | .5 | 5.5 | 2.32 | .010 A |
The mixed-sampling PCFA yields two antitypes, whereas standard fixed-effect
PCFA indicates neither types nor antitypes. In fact, the goodness-of-fit
Pearson = 0.0 indicates for this artificial data example that there
is no association between Training Program and Success in the qualification
test. In other words, the model of variable independence applies. In
contrast, the mixed-sampling PCFA suggests that it is less likely than
expected from the model that assumes no main effects for the Result of
Test variable that participants in either training program fail the test.
The goodness-of-fit for this base model is poor, as the Pearson
= 14.72 (df = 2; 
= 0.006) indicates. This discrepancy between
observed and estimated expected cell frequencies is solely due to the main
effect of the criterion variable. This main effect is considered as caused
by the independent variable, as is explained in the following paragraph.
To correctly interpret and apply this last statement one has to consider the assumptions made when applying the base model of mixed-sampling CFA. The model posits that the entire variability in a table can be accounted for by only considering the main effects and interactions of a subset of variables, that is, the predictors. Otherwise, and because there is no additional information that is being used, there is the assumption of a uniform frequency distribution. For the data example in Table 2 this assumption translates into a uniform distribution for the entire table because the experiment used the same number of athletes for each training program. This is equivalent to the assumption that, without the training, the selected athletes have a 50% chance of qualifying for the intergalactic games. Researchers who are unable or not willing to make this assumption may possess information that can be made part of the CFA base model (see Gutiérrez-Peña & von Eye, in preparation). If, for example, the chances of qualifying differ from 50%, one must take this information into account. Failure to take into account this information can render the types and antitypes detected by CFA invalid. In addition, a control group of athletes who do not participate in the training programs, may be considered for valid conclusions.
Thus, this example shows that the extreme base model of mixed-sampling PCFA in which no effects of the criterion are taken into account, can lead to meaningful results if the hidden assumption that the marginal frequencies on the criterion side are uniformly distributed can be defended. If this assumption cannot be defended, we recommend using the fixed-effect PCFA base model that is saturated in the predictors and considers the main effects of the criteria. Alternatively, Bayesian approaches to CFA (Wood, Sher, & von Eye, 1994; Gutiérrez-Peña & von Eye, in preparation) and parametric CFA (Spiel & von Eye, 1993) may be considered. More complex models may be needed depending on substantive assumptions.