I've spent several months thinking about the issue of whether or not it is appropriate to apply Rasch models to formative measurement models, and I'm looking to see whether anybody else has considered this issue, and what the answer is. People frequently apply the Rasch model with little consideration given to whether it's appropriate to the measurement model they are working with.
I want to begin by stating a few, I would consider, relevant features of a formative model versus a reflective model.
- In a reflective model, indicators are caused by a latent variable
- In a formative model, indicators are the cause of the "latent variable"
- In a reflective model, indicators should all correlate with each other
- In a formative model, this need not be the case
- Removing an item from a reflective model may reduce reliability of the final score, but not qualitatively change its interpretation (items are interchangeable)
- Removing an item from a formative model will change the meaning of the final score (removing earnings from a formative model of SES, will result in a measure of SES that is substantively different)
From my thinking about the subject, I've identified at least three issues with using Rasch analyses for formative models.
One goal of a Rasch analysis appears to me to be to demonstrate that a measurement model is unidimensional. As the indicators of a formative model need not correlate, and we may in fact want to avoid strongly correlating items, I'd suggest that a formative model may be almost always multidimensional. Since a formative model need not be unidimensional, then why might we care if it is or isnt'?
One property of a measurement model that adequately fits the Rasch model is that the estimate of person abilities that are obtained should remain similar if a different set of items are used. Because what is being measured is a consequence of the items used in a formative model though, this property could never be met. If I change the items then I'm measuring something different and it would be extremely unlikely that people's abilities on that new variable will be the same.
Dealing with misfit
In a Rasch analysis of a measurement tool, it is not uncommon that if items are found to exhibit particularly harmful misfit then they are removed from the tool. Indeed, the results of Rasch analyses may inform tool development. However, for reasons already suggested this would be dangerous with a formative model (because the variable being measured would now be different).
From my survey of the literature, proponents of Rasch only seem to have discussed whether or not a Rasch analysis is sensitive to the causal differences between a formative and reflective model (e.g., here). Not whether there's a reason to be applying Rasch to a formative model in the first place.
tl:dr Given that formative models may have properties that differ from the sorts of measurement models that Rasch analyses were originally intended for, and given that it's not clear that one could do anything to respond to issues flagged up by a Rasch analysis, is there any reason to apply a Rasch model to a formative measurement model?