1. G89.2228 Lecture 10b Example: Recognition Memory
Regression: Expected Y given X
Variance accounted for

2. Example: Recognition for Studied Words
Suppose we ask students if they recognize certain words as having been on a study list. The distribution of n correct is below:

3. The variance of Y may be affected by experimental condition
Subjects were randomly assigned to groups that were given 10, 15, 20, 25, and 30 seconds of study time

4. Regression: Expected Y given X
In this example, E(Y|X) is not constant for different choices of X.
Instead of estimating many means of Y|X, we estimate a* and b*, the y-intercept and the slope of the line.

5. Estimating regression statistics
b* and a* can be estimated using ordinary least squares methods. The resulting estimates are:
They minimize the sum of squared residuals, , where is the predicted value of
Note that a is not always interpretable. Centering X can help.

6. Partitioning the Variance
Notice that the residual variability is less than sY.
Thus, r2 may be considered the fraction of the variance accounted for by the regression prediction. This is approximate when considering s2, but is exact for the sums of squares:

7. Example calculations S2Y= 118.61 S2X= 50.85 SXY= 66.27
r = b= a= -2.28
SSfit=
SSresidual=
SSY=
r2= 5096/ = .728
ANOVA test of fit F(1,58)=(5096/[1902.2/58])=155.4