Can I regress an index value with variables used to create the index?

Question

I developed an index value (vulnerability score scale of 0 to 1) using a series of variables. I would like to regress these variables with the index value to determine the relative predictive power of each variable. Can I do this?

I ran the regression and came up with standardized B coefficients. I then interpreted those as relative contribution of the variable towards predicting the index value (vulnerability score).

I know one cannot regress a variable against itself but I am essentially doing this but primarily am just looking at determining to what degree each variable predicts the indexed value.

Any insights would be helpful if this is a proper use of regression or if there is an alternative method to assess this. Thanks!

Answer 1

No. You can't. In fact, since you created the index, you already know the contribution of each value to the index. You shouldn't do regression here and you don't need an alternate.

Answer 2

This sounds like a job for confirmatory factor analysis (CFA). If you're willing to consider vulnerability score a latent variable that affects the other variables used to measure it, CFA can produce factor scores based on an empirically derived set of linear weights applied to your indicator variables. Factor loadings are correlations between individual items and the latent factor estimated from their covariance structure. Items' unique variance is often considered measurement error, while their common variance determines their loadings. Loadings reflect the contributions of measured variables to the common factor. This should tell you everything that it sounds like you want to learn from your proposed analysis, whereas that analysis would tell you nothing, as @PeterFlom has explained.

An additional perk of CFA is that overall model fit statistics can tell you how well your latent factor explains the covariance among your measured variables. Modification indices can then tell you how your model might be improved. For instance, if you have very many items, they're likely to be multidimensional. If some of those additional dimensions aren't useful to you, you can control the variance they explain and use the remaining variance to estimate the general factor of primary interest with bifactor analysis ^{(Reise, Moore, & Haviland, 2010)}. I've said a little more about this method in my answer to "Factor analysis of questionnaires composed of Likert items", which may have more info of use to you if your data aren't continuous and normally distributed.

^{Reference
Reise, S. P., Moore, T. M., & Haviland, M. G. (2010). Bifactor models and rotations: Exploring the extent to which multidimensional data yield univocal scale scores. Journal of Personality Assessment, 92(6), 544–559. Retrieved from http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2981404/.}