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Statistics Lecture 18
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Today Last Day: Finish Chapter 6 Today: R2 and Start Chapter 7
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Coefficient of Determination (Sections 2.9 and 6.4)
Have discussed several aspects of regression (estimation, prediction, hypothesis testing, …) Have not mentioned most common statistics for describing a linear association These are the coefficient of determination (R2) and correlation coefficient (R)
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Coefficient of Determination (Sections 2.9 and 6.4)
Recall, SSTO measures: SSE measure: For simple linear regression, measure of the effect of X in reducing the variation in Y: As a proportion of the total variation:
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Coefficient of Determination (Sections 2.9 and 6.4)
Notes : Since SSTO>SSE, R2 Interpretation of R2 for Simple Linear Regression: When all points fall exactly on the line (or plane) R2= For simple linear regression, b1=0 (i.e., a horizontal line) R2=
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Coefficient of Determination (Sections 2.9 and 6.4)
When an additional explanatory variable is added to the model, the R2 goes Adjusted Coefficient of determination:
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Correlation Coefficient (Sections 2.9 and 6.4)
Positive square root of R2 times Interpretation for simple linear regression In general,
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Extra Sum of Squares (Chapter 7)
Consider Example on page 257 Y = Percent Body Fat X1= Triceps Skinfold Thickness X2 = Thigh Circumference X3 = Midarm Circumference Suppose only consider first two explanatory variables What model would we fit?
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Extra Sum of Squares (Chapter 7)
What hypotheses do the individual actually help assess?
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Extra Sum of Squares (Chapter 7)
Implication of t-tests: Important issue:
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Coefficient of Determination (Sections 2.9 and 6.4)
Limitations: No single measure will be useful to describe the usefulness for all applications A large R2 does not imply that useful predictions can always be made A large R2 does not imply that the linear model is a good fit An R2 near zero does not imply that the explanatory variables are unrelated to Y
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