What is "Adjusted R^2" in Multiple Regression

[poster1234us]

Hi there.

When performing multiple regressions, Excel 2000 reports a regression
statistic that I am not familiar with. I know what R^2 is, but does
anyone know what Adjusted R^2 is?

 

[Guest]

R squared is the pearson product squared. It refers to goodness of fit of the
line to the actual points of data. The value will be between 1 and 0 with
higher numbers being better. Note that regression will always generate a
line. Whether that line indicates a trend or not is another matter.

 

[Mike Middleton]

poster1234us -

R-Square (R^2) is the proportion of variation in the dependent variable (Y)
that can be explained by the predictors (X variables) in the regression
model.

As predictors (X variables) are added to the model, each predictor will
explain some of the variance in the dependent variable (Y) simply due to
chance. One could continue to add predictors to the model which would
continue to improve the ability of the predictors to explain the dependent
variable, although some of this increase in R-Square would be simply due to
chance variation. The adjusted R-Square attempts to yield a more honest
value to estimate R-Square.

Adjusted R-Square is computed using the formula 1-((1-R^2)*(N-1)/(N-k-1)).

When the number of observations (N) is small and the number of predictors
(k) is large, there will be a much greater difference between R-Square and
adjusted R-Square (because the ratio of (N-1)/(N-k-1) will be much less than
1).

By contrast, when the number of observations is very large compared to the
number of predictors, the value of R-Square and adjusted R-Square will be
much closer because the ratio of (N-1)/(N-k-1) will approach 1.

- Mike
www.mikemiddleton.com

 

[poster1234us]

Thanks, Mike. That was a big help.