Fraction of variance unexplained
Suppose we are given a regression function
yielding for each
is the vector of the ith
observations on all the explanatory variables.:181
We define the fraction of variance unexplained (FVU) as:
Alternatively, the fraction of variance unexplained can be defined as follows:
It is useful to consider the second definition to understand FVU. When trying to predict Y
, the most naïve regression function that we can think of is the constant function predicting the mean of Y
. It follows that the MSE of this function equals the variance of Y
; that is, SSerr
, and SSreg
= 0. In this case, no variation in Y
can be accounted for, and the FVU then has its maximum value of 1.
More generally, the FVU will be 1 if the explanatory variables X
tell us nothing about Y
in the sense that the predicted values of Y
do not covary
. But as prediction gets better and the MSE can be reduced, the FVU goes down. In the case of perfect prediction where
for all i
, the MSE is 0, SSerr
= 0, SSreg
, and the FVU is 0.
^ Achen, C. H. (1990). "'
What Does "Explained Variance" Explain?: Reply". Political Analysis
(1): 173–184. doi
Last edited on 15 April 2021, at 18:12
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