Statistics 350 Lecture 18

Today Last Day: Finish Chapter 6 Today: R2 and Start Chapter 7

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)

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:

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=

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:

Correlation Coefficient (Sections 2.9 and 6.4) Positive square root of R2 times Interpretation for simple linear regression In general,

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?

Extra Sum of Squares (Chapter 7) What hypotheses do the individual actually help assess?

Extra Sum of Squares (Chapter 7) Implication of t-tests: Important issue:

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