# Finding a Common Thread in Disparate Indicators

All of the theory I know about factor analysis, scales, etc revolves around separating out indicators into groups that each have (ideally) one underlying dimension. I have a student for whom we'd like to do sort of the opposite of that, and I'm a bit stuck.

He has a whole range of disparate measures of acting-under-social-norms (from alcohol abuse, to registering with a doctor, etc). We don't want to split these into separate factors, but rather to reveal an underlying latent variable that (we hope) is common to all of them, even if it isn't the driving force for any one of them, or for any particular group of them.

Is there a way of doing this? So far, I tried looking at the first principal component, but that seems tilted towards capturing the alcohol use (clearly, a bunch of alcohol abuse indicators clump together to explain a large chunk of the overall variance) and anyway the scree plot does not have as strong an elbow as we'd normally see in factor analysis (reflective of the wide variety of indicators we have gathered). Clearly, we don't want to do a "rotation" because that is doing the exact opposite of what we want: to separate them into cohesive clusters of indicators, whereas we want actually something that (to some extent) can underly all of them.

Is there any reasonable way of doing this?

Have you considered the bifactor measurement model? From what I can glean from your question, it looks as though you are interested in a single social norms dimension with an item bank that could reasonably be used to measure very different dimensions (e.g., alcohol abuse). If I am correct, the bifactor model may be the solution you are looking for. It assumes a single primary dimension (i.e., the social norms dimension) that all items are related to, as well as multiple secondary dimensions (e.g., an alcohol abuse dimension) that account for variation in an item cluster (e.g., alcohol abuse items) orthogonal to the primary dimension of interest.

Such a setup purifies the primary dimension from aspects of the item cluster (e.g., drinking behavior unrelated to social norms) unrelated to what is shared among all items that load onto the primary dimension.

Most applications of the bifactor are confirmatory, though you can still run an exploratory bifactor analysis$$^1$$. Additionally, you can estimate parameters using factor analysis or item response theory (e.g., DeMars, 2013; Reise et al., 2007; Wirth & Edwards, 2013).

$$^1$$ See Jennrich and Bentler (2011) and/or Mansolf and Reise (2016) for more technical discussions of exploratory bifactor models. Also, see Lorenzo-Seva and Ferrando (2019) a general guide to fitting exploratory bifactor models.

References

DeMars, C. E. (2013). A tutorial on interpreting bifactor model scores. International Journal of Testing, 13(4), 354-378.

Jennrich, R. I., & Bentler, P. M. (2011). Exploratory bi-factor analysis. Psychometrika, 76(4), 537-549.

Lorenzo-Seva, U., & Ferrando, P. J. (2019). A general approach for fitting pure exploratory bifactor models. Multivariate behavioral research, 54(1), 15-30.

Reise, S. P., Morizot, J., & Hays, R. D. (2007). The role of the bifactor model in resolving dimensionality issues in health outcomes measures. Quality of Life Research, 16(1), 19-31.

Mansolf, M., & Reise, S. P. (2016). Exploratory bifactor analysis: The Schmid-Leiman orthogonalization and Jennrich-Bentler analytic rotations. Multivariate Behavioral Research, 51(5), 698-717.

Wirth, R. J., & Edwards, M. C. (2007). Item factor analysis: current approaches and future directions. Psychological methods, 12(1), 58.

• This is exactly what I was looking for. Thank you!! Commented Oct 12, 2021 at 19:09
• Great, glad to help! Commented Oct 12, 2021 at 19:11

This adds a bit more to Preston's answer. I approve of the Reise citation.

However, I'd urge you to consider how closely related the traits you're trying to measure are.

He has a whole range of disparate measures of acting-under-social-norms (from alcohol abuse, to registering with a doctor, etc). We don't want to split these into separate factors, but rather to reveal an underlying latent variable that (we hope) is common to all of them...

You only named two, but alcohol abuse and registering with a doctor don't seem strongly related to me. By my reading of Reise, if you have a bunch of closely related but not totally identical latent traits, it's acceptable to use a bifactor model's primary factor. From my field, cognitive, affect, and somatic symptoms of depression are probably closely related enough that you could use a bifactor structure to produce a summary depression score. If you were to administer a bunch of items for those 3 traits and estimate 3 different scores, the scores would probably be highly correlated.

If you have a bunch of latent traits that are weakly related, maybe even close to independent, then I am not sure what a bifactor model gains you. I bet you can still estimate one, I'm just not sure what the gain is relative to calculating separate factor scores.

Clearly, we don't want to do a "rotation" because that is doing the exact opposite of what we want: to separate them into cohesive clusters of indicators, whereas we want actually something that (to some extent) can underly all of them.

Actually, I think that with an EFA you do want to rotate the factor solution while exploring the data structure. This will show how correlated the identified factors are. If the factors are reasonably correlated and you can make a theoretical case that they're linked through an overarching construct, then you can go ahead - how much is "reasonably" is going to be a judgment call, as with many things in statistics, and the same applies for making that theoretical case. In subsequent analysis, you'd use a bifactor structure if it were justified.

I haven't read in depth, but I believe that it is possible to perform a bifactor rotation in an exploratory model; this is possibly what some of the sources in the other answer refer to.

• Thank you! This is definitely food for thought. Admittedly, I chose the two least related variables, but the variables are quite wide ranging across substance abuses and a variety of other behavioural things, where the only thing they should really share is the social disorder aspect, so indeed this matches your description of what not to do... I need to think through what you're saying. I don't agree that a standard rotation makes sense since that is all about trying to separate the different components. Also, the hope is not that they are correlated by that they share a small... Commented Oct 23, 2021 at 17:11
• @justme I've clarified the answer to emphasize: I'd recommend using a standard orthogonal rotation to explore the data structure. You'd then decide if it fit a bifactor model. If it does, you can use a bifactor structure in CFA/SEM modeling, or you can use a bifactor rotation in EFA to calculate factor scores. Commented Oct 23, 2021 at 17:13
• ...common covariance, despite being quite disparate otherwise. Then, the hope was that that small shared covariance component would combine to stretch the shape of the data to give a first principal component (or first bifactor factor) that brings in a bunch of them. But maybe I'm thinking about this wrong? We have indeed been looking at bifactor as a rotation post-PCA. Commented Oct 23, 2021 at 17:13
• Thank you! What I don't quite understand is how an orthogonal rotation would tell us whether a bifactor solution makes sense? When the orthogonal rotation would try to force the variables to separate, whereas the bifactor rotation is still looking for orthogonal secondary factors, but allows the primary one still to combine all of them? Isn't the point of the bifactor method that it allows us to pull out a common thread from actually disparate indicators (and the secondary factors hoover up what remains?) Commented Oct 23, 2021 at 17:16