Consider the saturated log-frequency model for the two groups of variables,
A and B,
where M is an array of expected frequencies, 
is the matrix with the
indicator variables for the variables in A such that the model is
saturated in A, 
is the matrix with the indicator variables for the
variables in B such that the model is saturated in B, and 
is the
matrix with the indicator variables for all first- and higher-order
interactions between the variables in A and B (for more detail on
log-linear modeling see, e.g., Agresti, 1996; Bishop, Fienberg, & Holland,
1975; Christensen, 1997). The 
vectors contain the parameters that
correspond to the columns of the indicator matrices.
The typical base model for CFA applications with two groups of variables
omits the last term in (1). One thus obtains
that is, a model that is saturated in the variables in A, saturated in
the variables in B, but proposes that there be no interactions among the
variables from the two different groups.
The model given in (2) is the CFA base model for both ISA and PCFA. Thus,
at the level of base models there seems to be no difference between ISA and
PCFA. Indeed, Krauth (1996, p. 138) states for PCFA that the difference
between the two groups of variables exists only at the level of substantive
interpretation
.
Thus, the base models thus far used for ISA and PCFA are
statistically the same. In the present article we propose base models
that allow one to discriminate between ISA and PCFA.
In applications of CFA one interprets members of a cell as members of a
type if 
, where i numbers the cells
in the cross-classification of all
variables, 
are the expectancies of the observed cell frequencies,
m, and 
are the expected cell frequencies,
estimated under the base model, B.
If 
the cell members belong to an antitype.
Using one of the many tests
proposed for CFA (for an overview see von Eye, 1990; von Eye & Rovine,
1988) under the appropriate measure for protection of the
experiment-wise 
, one can come to a statistical decision about
the presence of types and antitypes.