Table 1: Fish Biomass data set on 26 streams

Transcription

1 Math 221: Multiple Regression S. K. Hyde Chapter 27 (Moore, 5th Ed.) The following data set contains observations on the fish biomass of 26 streams. The potential regressors from which we wish to explain the biomass (y) are the average depth of the stream (x 1 ), the area of instream cover (x 2 ) and the percent of canopy cover (x 3 ). Obs no. x 1 x 2 x 3 y Table 1: Fish Biomass data set on 26 streams

2 Multiple Regression, page 2 Dependent Variable: Y fish biomass Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model Error C Total Root MSE R-square Dep Mean Adj R-sq C.V Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > T INTERCEP X X X Figure 1: SAS Multiple Regression printout

3 Multiple Regression, page 3 1. Find a 99% confidence interval for the coefficient of the canopy cover variable. 2. Test the hypothesis of no multiple linear relationship (e.g. The overall significance of the multiple linear regression equation). F = 3. How well does the multiple linear regression equation predict the data? 4. Test the hypothesis that the coefficient of x 1 is zero. t =

4 Multiple Regression, page 4 5. Test the hypothesis that the coefficient of x 2 is zero. t = 6. Test the hypothesis that the coefficient of x 3 is zero. t = 7. Predict the fish biomass for a depth of the stream of 5, an area of 3 for the instream cover and the 42 percent canopy cover.

5 Multiple Regression, page 5 1. Find a 99% confidence interval for the coefficient of the canopy cover variable. The degrees of freedom for the critical value are found by either finding the degrees of freedom for the ERROR on the printout (df = 22) or by computing df = n k 1 = = 22, where k is the number of independent variables. The critical value found in the t-table is The estimate and standard error of the estimate for canopy cover can be found on the printout. They are and respectively. Hence, a 99% confidence interval for the coefficient of canopy cover is b 3 ± m = b 3 ± t SE b3 = ± 2.819( ) = ( , ) 2. Test the hypothesis of no multiple linear relationship (e.g. The overall significance of the multiple linear regression equation). The test statistic and P -value for this test can be found on the printout under the Analysis of Variance (ANOVA) section. The f stat is and its P -value is given to the right of it on the table. The model is not significant The model is significant 0.05 f = P =.0001 Decision: Reject H 0 There is a significant multiple regression model between the depth of the stream, percent of canopy cover, and area of instream cover. Jointly, the three variables explain a significant amount of variability in the fish biomass. 3. How well does the multiple linear regression equation predict the data? Since r 2 =.9373, then it can be concluded that it fits well. In fact the adjusted r 2 value is not very much different. 4. Test the hypothesis that the coefficient of x 1 is zero. The test statistic and P -value for this test can be found on the line labeled X1 in the Parameter Estimates table of the printout. The test statistic is with a P -value of The test of hypothesis proceeds as follows: β 1 = 0 β t = P =.0001 Decision: Reject H 0 There is sufficient evidence to indicate that the coefficient for depth of the stream is different than zero.

6 Multiple Regression, page 6 5. Test the hypothesis that the coefficient of x 2 is zero. The test statistic and P -value for this test can be found on the line labeled X2 in the Parameter Estimates table of the printout. The test statistic is with a P -value of The test of hypothesis proceeds as follows: β 2 = 0 β t = P =.0001 Decision: Reject H 0 There is sufficient evidence to indicate that the coefficient for area of instream cover is different than zero. 6. Test the hypothesis that the coefficient of x 3 is zero. The test statistic and P -value for this test can be found on the line labeled X3 in the Parameter Estimates table of the printout. The test statistic is with a P -value of The test of hypothesis proceeds as follows: β 3 = 0 β t = P =.0741 Decision: Fail to Reject H 0 There is not sufficient evidence to indicate that the coefficient for percent of canopy cover is different than zero. 7. Predict the fish biomass for a depth of the stream of 5, an area of 3 for the instream cover and the 42 percent canopy cover. ŷ = (5) (3) (42) =