CFA: Fix one loading per factor, or fix the variance of the factors?

Question

I often see it noted that to help obtain an identifiable CFA solution you can either fix one loading per factor (usually fixing those loadings to 1), or else you can fix the variance of each latent factor (usually fixing those variances to 1).

Hoyle (2014) notes that the former approach- which he calls the "marker indicator" approach - is more commonly used.

I've been trying to think of situations in which one approach would be preferable to the other.

Hoyle (2014) p366 notes in relation to the marker indicator approach that

"the absence of an unstandardized solution often contraindicates this approach, (e.g. when the researcher is interested in evaluating the measurement invariance of a test instrument)."

Meanwhile, Jeremy Miles suggests here that fixing the variances of the factors was potentially a sensible approach in relation to a situation where assessing the statistical significance of the loadings was considered important.

Is there some general rule I can apply to work out when I should be using one approach rather than another? Also, is it fair to say that it usually doesn't matter which approach is taken?

Hoyle, R. H. (Ed.). (2012). Handbook of structural equation modeling. Guilford Press.

Answer 1

Steiger wrote a paper on this in 2002: http://psycnet.apa.org/journals/met/7/2/210/

I find the Hoyle writing to be confusing (I think this will take you to the page: https://books.google.com/books?id=4s7SAgAAQBAJ&pg=PA366&lpg=PA366&dq=%22%22the+absence+of+an+unstandardized+solution+often+contraindicates+this+approach,+(e.g.+when+the+researcher+is+interested+in+evaluating+the+measurement+invariance+of+a+test+instrument).%22%22&source=bl&ots=1hTQFhcPjt&sig=qN3bITbraSWheRqIYfuaKtcH37w&hl=en&sa=X&ved=0ahUKEwido6bEha_TAhUOz2MKHT2QBsIQ6AEIKjAB#v=onepage&q=%22%22the%20absence%20of%20an%20unstandardized%20solution%20often%20contraindicates%20this%20approach%2C%20(e.g.%20when%20the%20researcher%20is%20interested%20in%20evaluating%20the%20measurement%20invariance%20of%20a%20test%20instrument).%22%22&f=false -0 there is (IMHO) too many uses of "this" when I'm not sure what "this" refers to.

Here's the gist of the issue (or some of the issues). Sometimes you want the variance to be meaningful. This is the case when you are looking at measurement invariance - you want to know if factor loadings (and means) vary over time or between groups. In this case the "marker indicator" approach is necessary to give the correct answer.