Variance Decomposition for Index Variable

[willfla29]

Hi all,

I have three variables that have been combined into an index (simply the average of the score on three variables). A reviewer has asked what the contribution of one of these three variables to the variance of the index. This seems like an ANOVA problem, but I am not entirely sure. Is there a good STATA command for doing this?

Thanks!

[Oded Mcdossi]

You choose to combine your three variables by a simple average of the three, which means you assume equal share of the three to the total variance. Try the factor approach and test this assumption. First, do a factor analysis and check how much factors you get, if it's only one, then check the proportion of each variable's variance that can be explained by the factors using the communalities (h2). In Stata the communalities defined as 1-uniqueness.
Type help -factor- for more information.

[Clyde Schechter]

The contribution of the three variables to the variance in the index need not be equal just because they are equally weighted in its calculation. That would be true if each of the constituent variables had the same variance. But if they differ, then the contributions to total variance will differ, even in an equally weighted sum.

If the three constituent variables that were combined into the index are independent, then the contribution to variance can be estimated simply by correlating the constituent variable with the index and squaring the correlation coefficient. However, if the constituent variables are not independent, then the "contribution to variance" depends on the order in which the variables are considered. It is hard to know exactly what the reviewer is asking for without seeing the full and exact request. But typically they mean the contribution when the variable is entered last. Probably the simplest way to get this is to dig out the old -nestreg- command (one of the few estimation commands that does not support factor variable notation--not that it matters here):

nestreg: regress index var1 var2 var3

The change in R2 for entering var3 last will probably be what the reviewer wants.

[Joseph Luchman]

Another way, which is less simple conceptually, is to obtain the Shapley value decomposition of the R² for each variable (available using several commands on SSC - shapley, shapley2, and domin [with the general dominance weights]).

The Shapley value/general dominance weight is, basically, the average marginal contribution each variable makes to the R² (of the variables explaining variance in the index) across all possible combinations of predictor variables. Might be overkill in your case - but I thought it was worth suggesting as you may find the procedure useful.

- joe