1. POLYNOMIAL REGRESSION MODELS

2. One-Variable Polynomial Models Recall with one variable the first step is to plot the y values vs. x to assist in hypothesizing the form of the model. Polynomial model in one variable is: y =  0 +  1 x +  2 x 2 +  3 x 3 + …  p x p +  Usually p  3

3. Example A study is being performed to determine the relationship between average daily gas usage (as measured in therms) for a 3000 sq. ft. home and average daily temperature in Santa Ana, Ca. The scatterplot on the next slide shows that a quadratic model (involving both x and x 2 ) appears to give a better fit than the line model Hypothesize: y = β 0 + β 1 x + β 2 x 2 + ε

5. = B2^2 Drag to A3:A13 X Range: Columns A and B

6. The regression equation y = 59.093-1.595x+.011x 2 Low p-value High R 2 and Adj. R 2

7. Polynomial Models With More Than One Variable Could have interaction –Interaction between x 1 and x 2 is represented by an x 1 x 2 term. Example:Example: Therms may be a quadratic function of Temperature, Rainfall, and include interaction Two variable quadratic model with interaction: y =  0 +  1 x 1 +  2 x 2 +  3 x 1 2 +  4 x 2 2 +  5 x 1 x 2 +  Solve using usual regression techniques.

8. =B2^2 =C2^2 =B2*C2 Drag D2:F2 to D13:F13 Enter contiguous X-Range

9. Regression Equation: y = 152.31 - 4.21x 1 - 37.97x 2 +.03x 1 2 + 2.51x 2 2 +.55x 1 x 2 Low p-value High R 2 p-values for t-tests

10. Interpretation of t-tests in this model.Because x 1 is correlated with x 1 2, x 2 is correlated with x 2 2, and both x 1 and x 2 are correlated with x 1 x 2, the t-tests may not show that these terms, independently, are significant in this model. –Multicollinearity –If we need to interpret the meaning of each coefficient (each  ) – this multicollinearity will not allow us to do it –If our objective to to predict values of y – that’s okay This is a typical use of regression

11. Review One variable models –Graph first to help in hypothesizing model More than one variable models –May or may not wish to hypothesize interaction (x 1 x 2 ) Perform regression analysis in usual way –Difficult to interpret estimates for  ’s –Low Significance F – use model for prediction