- #1
- 8,520
- 16
Often in empirical studies you see statements that factor X explains some fraction of the variance in some other variable V, and thinking about what this means intuitively made me curious about the following question. Suppose you have a model where the values of some set of factors X1, X2, ..., Xn in a given member of the population, taken together, completely determine that member's value for V with probability 1. And say we look at how much of the variance in V is explained by each factor individually in the population as a whole, and find X1 accounts for a fraction F1 of the variance in V, X2 accounts for F2 of the variance in V, and so forth. Is it necessarily the case here that all the individual variances add to 1, i.e. F1 + F2 + ... + Fn = 1? Or would this depend on the exact nature of the function that takes the values of X1, X2, ..., XN as input and gives you the value for V as output? (and if so, are there some types of simple functions--like if the value of V is just a linear sum of the values of X1, X2, ..., Xn--where it would be the case that the individual variances would add to 1?)