The Lesser–Unsworth data was to estimate a bivariate regression equation relating seed weight to cumulative solar radiation and level of ozone pollution. This exercise continues with the analysis of that model using the centered independent variables.
(a) The more complex model used in Exercise 3.9 included the independent variables cumulative solar radiation, ozone level, and the product of cumulative solar radiation and ozone level (plus an intercept).
(i) Construct the analysis of variance for this model showing sums of squares, degrees of freedom, and mean squares. What is the estimate of σ2?
(ii) Compute the standard errors for each regression coefficient. Use a joint confidence coefficient of 90% and construct the Bonferroni confidence intervals for the four regression coefficients. Use the confidence intervals to draw conclusions about which regression coefficients are clearly different from zero.
(iii) Construct a test of the null hypothesis that the regression coefficient for the product term is zero (use α = .05). Does your conclusion from this test agree with your conclusion based on the Bonferroni confidence intervals? Explain why they need not agree.
(b) The simpler model in Exercise 3.9 did not use the product term. Construct the analysis of variance for the model using only the two independent variables cumulative solar radiation and ozone level.
(i) Use the residual sums of squares from the two analyses to test the null hypothesis that the regression coefficient on the product term is zero (use α = .05). Does your conclusion agree with that obtained in Part (a)?
(ii) Compute the standard errors of the regression coefficients for this reduced model. Explain why they differ from those computed in Part (a).
(iii) Compute the estimated mean seed weight for the mean level of cumulative solar radiation and .025 ppm ozone. Compute the estimated mean seed weight for the mean level of radiation and .07 ppm ozone. Use these two results to compute the estimated mean loss in seed weight if ozone changes from .025 to .07 ppm. Define a matrix of coefficients K such that these three linear functions of
can be written as K
. Use this matrix form to compute their variances and covariances.
(iv) Compute and plot the 90% joint confidence region for β1
and β2
ignoring β0. (This joint confidence region will be an ellipse in the two dimensions β1
and β2.)
Exercise 3.9
This exercise uses the Lesser–Unsworth data in which seed weight is related to cumulative solar radiation for two levels of exposure to ozone. Assume that “low ozone” is an exposure of .025 ppm and that “high ozone” is an exposure of .07 ppm.
(a) Set up X and β for the regression of seed weight on cumulative solar radiation and ozone level. Center the independent variables and include an intercept in the model. Estimate the regression equation and interpret the result.
(b) Extend the model to include an independent variable that is the product term between centered cumulative solar radiation and centered ozone level. Estimate the regression equation for this model and interpret the result. What does the presence of the product term contribute to the regression equation?