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The coefficient of determination may be interpreted as the proportional reduction in error resulting from use of the regression model to predict Y. Another ...
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Regression: Model Fit Measures
1. Coefficient of Multiple Correlation R and Coefficient of Determination R 2
As previously noted one measure of model fit—how well the regression model is able to reproduce the observed scores on the dependent variable Y—is the simple Pearson’s correlation between observed Y and predicted Y'.
R = Pearson’s correlation, r, between Y and Y'
The closer R is to 1.00 the better the regression model is able to reproduce Y, the closer R is to 0.00, the worse the performance of the model in reproducing Y. While R may be negative, this is not expected or likely; one anticipates R to be positive since the regression model is designed to predict Y as well as is possible given the data.
The coefficient of determination, R^2 , is simply R squared:
R^2 = R × R = proportion of variance in Y predicted (or explained) by regression model
The coefficient of determination may be interpreted as the proportional reduction in error resulting from use of the regression model to predict Y. Another interpretation of the coefficient of determination is explained variance—the proportion of variance in Y explained, or predicted, by the regression model. The complement of this, 1−R^2 , is the amount of variance in Y that is not explained or predicted by the regression model.
1 −R^2 = proportion of variance in Y not explained by regression model
Recall the student ratings data:
Table 1: Student Ratings and Course Grades Data
Course Quarter Year Student Ratings (mean ratings for course)
Percent A's
SPSS Data File: http://www.bwgriffin.com/gsu/courses/edur8132/notes/student_ratings.sav
Recall that a residual, or error, is the difference between observed Y and predicted Y':
e = Y - Y'
One way to measure model fit is to examine variation in residuals.
From basic statistics note that variance in raw data may be calculated for the population as
σ^2 =
2
and variance for sample data may be calculated as
s^2 =
2
The difference between these formula is the degrees of freedom. In the population case the count of all observations is use, N, but in the sample formula degrees of freedom is n − 1 is used (to provide an unbiased estimate of σ^2 ).
The variance for residuals may also be calculated in the same manner taking into account regression model degrees of freedom:
2
The above produces a variance that as many names:
variance error of residuals , or variance error of estimate , or mean squared error (MSE)
and is denoted as or MSE.
σ
symbolized as
Note that as SEE, and MSE, become smaller, the fit of the model is better since the residuals are smaller.