Regression Coefficient Interpretation (OLS)

April 8, 2010 4:11 PM

In an OLS regression the coefficient is b = cov(X,Y)/var(X). What is the plain english interpretation of this coefficient in terms of variations (ie. b is the variation in Y that can be explained by variation in X)? How does this relate to r^2?
posted by bucksox to Science & Nature (2 answers total)

No. b is not the variation in Y that can be explained by variation in X. That is in fact the R^2 -- or more generally/accurately it reflects the fraction of total variance in Y accounted for by your model (the variance in X for a univariable linear model). In plain English, the regression coefficient is the best fit slope of the association between Y and X. In terms of variation, it is a best estimate of how much Y is expected to vary with respect to a given change in X.

Remember, R^2 is always between 0 and 1. The regression coefficient is not, and depends on the units of X and Y. Mathematically in simple model (i.e. y = a + b x), b and R^2 are mathematically related as follows:

b = R (SD_y / SD_x) where SD is standard deviation.
posted by drpynchon at 5:09 PM on April 8, 2010 [1 favorite]

The plainest English interpretation of b is that it is the expected change in Y if you increase X by one.
posted by ROU_Xenophobe at 5:32 PM on April 8, 2010