POLYNOMIAL REGRESSION MODELS

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

Example A study was done 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. Results: TempTherms

Scatterplot The scatter plot of this data appears to show that a quadratic model might fit better. So, hypothesize: y = β 0 + β 1 x + β 2 x 2 + ε

2.Enter Temp^2 in C1 In C2 enter = B2^2 Drag to C3:C13 1.Cut Column B (Therms) and Insert Cut Cells to Column A. 3.Tools/ Data Analysis/ Regression X Range: Columns B and C

The regression equation y = x x 2 Low p-value High R 2 and Adj. R 2

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  4 x  5 x 1 x 2 +  Solve using usual regression techniques.

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

Regression Equation: y = x x x x x 1 x 2 Low p-value High R 2 p-values for t-tests

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 may not allow us to do it –If our objective is to predict values of y – that’s okay This is a typical use of regression

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