Data Example

In this paragraph we re-analyze a data set that has been repeatedly used for illustration of PCFA and its characteristics (Lienert, 1978; Mellenbergh, 1996; von Eye et al. 1996). The data describe suicide attempts in a sample of n = 482 individuals. The following three categorical variables are analyzed: Gender (G; 1 = male, 2 = female); Motive for Suicide (M; 1 = illness; 2 = psychiatric disorder; 3 = alcoholism); and Outcome of Suicide Attempt (O; 1 = survived; 2 = dead). Table 6 displays the observed cell frequencies for the 2 x 3 x 2 cross-classification of these three variables.

 

Cell Indices Observed Frequencies for Outcome of Suicide Attempt

GM Survived Died

11 64 18

12 76 47

13 7 8

21 86 16

22 61 25

23 47 27

For the following analyses we assume that G and M are the predictors and O is the criterion variable. To illustrate various assumptions that can be investigated in regard to a possibly emerging pattern of types and antitypes, we apply the following CFA base models:

1. First order CFA [G][M][O]. This model is included here only to provide readers with results from classical first order CFA. A comparison with results from PCFA is not possible because the model [G][M][O] does not discriminate between predictors and criteria. This model of variable independence is of interest when researchers are interested in types and antitypes that can be traced back to local interactions of variables but not to main effects. Only one variable can be fixed by design or sampling characteristics. Types and antitypes can result from any interaction in these three variables, but not from main effects.
2. Mixed-sampling PCFA [GM]. This model considers the margins of the predictor variables, Gender and Motive as fixed. The margins of the outcome variable, Outcome of Suicide Attempt, are considered random. The predictors may interact because it seems plausible that males and females differ in motive for suicide (Chipuer & von Eye, 1989). This model is, on the criterion side, a zero order CFA. Types and antitypes can result from (1) the main effect of the criterion, and (2) interactions among the predictors and the criterion.
3. First order fixed-effect PCFA [GM][O]. In the present example, this is the standard PCFA (and ISA) base model. It considers both the margins of the predictor variables and the margins of the criterion as fixed. Types and antitypes can only result from interactions between the predictors and the criterion.

Table 7 displays the results from these three CFA base models. For each model we adjusted = 0.05 using the Bonferroni method which resulted in an adjusted = 0.0042. The table presents the deviance residuals, , and their one-sided tail probabilities.

Table 7 suggests that first order CFA identifies two types and two antitypes. The first type is constituted by Configuration 122. These are men that suffer from some psychiatric disorder and succeed in committing suicide. The second type is constituted by Configuration 232. These are alcoholic women that succeed in committing suicide. Each of these two patterns occurs more often than expected from the base model of variable independence. The first antitype, constituted by Configuration 131, describes alcoholic men that fail to succeed in committing suicide. The second antitype, constituted by Configuration 212, describes women that suffer from some illness and succeed in committing suicide. Each of these two patterns occurs less often than expected from the base model.

 

Cell Indices GMO [G][M][O] [GM] [GM][O]

111 64 .587 .279 3.150 .0008 T .773 .220

112 18 -1.392 .082 -4.182 -1.280 .100

121 76 1.015 .155 3.769 -1.208 .114

122 47 3.289 .0005 T -.101 .460 1.753 .040

131 7 -4.869 -3.422 .0003 A -1.183 .118

132 8 -1.198 .116 -3.117 .0009 A 1.544 .061

211 86 1.752 .040 4.612 1.581 .057

212 16 -2.864 .002 A -5.617 -2.781 .002 A

221 61 -2.258 .012 .550 .291 .020 .492

222 25 -1.494 .068 -4.751 -.031 .488

231 47 2.065 .020 4.101 -.753 .226

232 27 3.031 .001 T .561 .287 1.107 .134

Tail probabilities with four or more leading zeros are displayed as ``''; types are indicated by T; antitypes are indicated by A.

When describing and interpreting these types and antitypes it is important to stay at a purely descriptive level. The CFA base model does not allow one to discriminate between predictors and criteria, nor does it support notions of causality (see von Eye & Brandtstädter, 1998). The sampling scheme that underlies this analysis is either multinomial or product multinomial, with no effect on the magnitude of the estimated expected cell frequencies or the model parameters (which are typically not of interest in CFA applications). In contrast, the second base model, [GM] does support the distinction between predictors and criteria. This model assumes that Gender and Motive for suicide attempt allow one to predict the outcome of a suicide attempt. In addition, the model [GM] is based on a mixed sampling scheme where the uni- and bivariate marginals of G and M are fixed and the marginals of O are random. As was indicated in Section 3, not considering the main effect of the criterion implies the assumption of a uniform marginal distribution for the criterion. If this assumption is of no substantive interest, criterion marginals must be made part of the base model (see below).

The model [GM] identifies four types and five antitypes. These types and antitypes overlap with the ones identified by first order standard CFA only in part. Specifically, the two types found by first order CFA are not found here.

When interpreting the results from the model [GM] one has to take into account the nature of the base model which treats G and M as predictors and O as the outcome variable. For instance, the pattern ``male, suffering from some illness'' allows one to predict that a suicide attempt is survived (Configuration 111). This pattern constitutes a type, indicating that it occurs more often than expected from the assumption that G and M are not predictors of (1) main effects of O and (2) interactions between G and M on the one hand side and O on the other. Whether (1) or (2) is the case cannot be determined from the information provided by CFA. Accordingly, the first antitype suggests that it occurs less often than expected from the base model that men that suffer from some illness succeed in committing suicide.

Rather than interpreting the three types and four antitypes in detail we now inspect the results from the third model, [GM][O]. This model can be interpreted in two ways. It is either a standard PCFA, or it is a first order fixed-effect PCFA where researchers posit that the effects of the predictors manifest only in interactions between the two predictors and the criterion, and the marginals of the criterion are fixed by design or sampling characteristics. In either case, this model takes more information into account than the other two models when estimating the expected cell frequencies. Therefore, there are fewer ways to deviate from the expected cell frequencies and it does not come as a surprise that the number of types and antitypes is smaller than for the other two models. (It should be noted, however that this is not necessarily always the case.) The model identifies no type and only one antitype. It is Configuration 212 which had emerged as an antitype in the other two analyses also.

In sum, this example illustrates again that Mellenbergh's (1996) statement is correct. Different models of CFA can lead to dramatically different patterns of types and antitypes.