Hi Margaret,

first a few clarifications about terms:

1) "construct" is a very ambiguous term, as it can mean everything--from simple ontological denotations like "chair", or "bike" to denotations of specific/ singular latent entities (like electrons up to job satisfaction) to broad categories like "sport" or "performance". The first (theoretical) goal is hence always to PRECISELY clarify what one's "construct" means. Here, unfortunately many approaches fail already.

2)) A latent variable is the same as a ("one") dimension. These are things in statistical models (like SEM or CFA) and these "things" like the overall model again attempts to reflect reality: That is, the latent variable attempts to reflect an unobserved singular variable (one dimension) and the model (=structure consisting of effects and restrictions/non-effects) attempts to reflect a true underlying "data generating process" or causal system.

3) A factor is a specific latent variable, namely it is proposed to causally underlie *several* observed variables. Hence, the factor is always connected with a specific model structure: the "common factor model". Essential implications of this very specific and restrictive model structure are a) that the factor reflects something real, b) that the factor causes the observed variables, c) that these effects generate correlations among the observed variables (like any other common cause or confounder model!) and, consequently, d) that these correlations disappear when the factor is statistically controlled (local independability).

Your question about a possible tautology is in principle already answered by point 3. By specifying your three-factor model with A, B and C as separate but correlating factors, you also specify a very restrictive causal structure with a ton of implied local independabilities. That is, that there are no residual correlations left. Your model chisquare test is a joined statistical test of these implications. If the test is significant, then there are deviations from this omnibus hypothesis beyond those expected by chance.

If the test is significant, you are in a dilemma: Having many indicators could possibly lead to one or some deviations while the main structure still is valid and correctly specified. Unfortunately, you cannot presume this as the model structure could be simply wrong and the number of specified factors is wrong. In this case, your modeled factor(s) will be best biased or in the worst case complete nonsense. Hence, the essential conditions 3a and 3b fail. This is a bad thing. Hence, I regard the test as an alarm signal that something may be wrong.

Overall, models with latent variables are not tautological as they reflect a claim that refers to something hidden behind the data and implies testable implications that have to hold in the data set. Think of it within the following example: You think there is a giant walking on a beach but all you have are footprints. If the depth of the footprints is very shallow: would this provide evidence for your claim? And: Are the footprints the same (tautological) as the giant?

Hope that helps

Best,

Holger

Hi Margaret,

first a few clarifications about terms:

1) "construct" is a very ambiguous term, as it can mean everything--from simple ontological denotations like "chair", or "bike" to denotations of specific/ singular latent entities (like electrons up to job satisfaction) to broad categories like "sport" or "performance". The first (theoretical) goal is hence always to PRECISELY clarify what one's "construct" means. Here, unfortunately many approaches fail already.

2)) A latent variable is the same as a ("one") dimension. These are things in statistical models (like SEM or CFA) and these "things" like the overall model again attempts to reflect reality: That is, the latent variable attempts to reflect an unobserved singular variable (one dimension) and the model (=structure consisting of effects and restrictions/non-effects) attempts to reflect a true underlying "data generating process" or causal system.

3) A factor is a specific latent variable, namely it is proposed to causally underlie *several* observed variables. Hence, the factor is always connected with a specific model structure: the "common factor model". Essential implications of this very specific and restrictive model structure are a) that the factor reflects something real, b) that the factor causes the observed variables, c) that these effects generate correlations among the observed variables (like any other common cause or confounder model!) and, consequently, d) that these correlations disappear when the factor is statistically controlled (local independability).

Your question about a possible tautology is in principle already answered by point 3. By specifying your three-factor model with A, B and C as separate but correlating factors, you also specify a very restrictive causal structure with a ton of implied local independabilities. That is, that there are no residual correlations left. Your model chisquare test is a joined statistical test of these implications. If the test is significant, then there are deviations from this omnibus hypothesis beyond those expected by chance.

If the test is significant, you are in a dilemma: Having many indicators could possibly lead to one or some deviations while the main structure still is valid and correctly specified. Unfortunately, you cannot presume this as the model structure could be simply wrong and the number of specified factors is wrong. In this case, your modeled factor(s) will be best biased or in the worst case complete nonsense. Hence, the essential conditions 3a and 3b fail. This is a bad thing. Hence, I regard the test as an alarm signal that something may be wrong.

Overall, models with latent variables are not tautological as they reflect a claim that refers to something hidden behind the data and implies testable implications that have to hold in the data set. Think of it within the following example: You think there is a giant walking on a beach but all you have are footprints. If the depth of the footprints is very shallow: would this provide evidence for your claim? And: Are the footprints the same (tautological) as the giant?

Hope that helps

Best,

Holger

I agree with Holger Steinmetz. A simple example to resolve the tautology problem is that you might well have found that a single factor model was an adequate fit to the data, and was not statistically different from the three-factor model. You had no way of knowing that in advance, until you tested it. In particular, you might have assumed that your latent concepts were very different (a proposed three-factor model), but your respondents might have seen the differences as too small to matter (an observed one-factor solution).

Confirmatory factor analysis might be tautological if you defined it as an impossibility for you to be wrong. But hypothesizing items fall into three distinct constructs, even if you're pretty sure you're right, is not the same as statistical test where a logically possible outcome really is another best fit to the data. A nice illustration similar to yours from Social Psychology is, "attitude." Is an attitude better understood as a single construct (i.e., a valence toward a target) or is it better understood with the ABC Model of Attitudes - three correlated but distinct valences toward the same target (affect, behavior, & cognition). It's one of the first papers to use structural equation modeling (SEM) to test a hypothesis about the nature of an underlying theoretical construct (citation below)! Hope this helps some, Margaret. ~ Kevin

Breckler, S. J. (1984). Empirical Validation of affect, behavior, and cognition as distinct components of attitude. Journal of Personality and Social Psychology, 47(6), 1191-1205.