Stat479 Assignment #6 Solution Key Fall 2013

Transcription

1 Stat479 Assignment #6 Solution Key Fall 2013 Problem 1 (a) Source d.f. SS MS F p-value Regression <.0001 Error Corrected Total (b) β 0= , s.e.( β 0) = ; β 1= , s.e.( β 1) = y = x (c) Expected loss in mean muscle mass, E(y) for 1 year increase in age= Thus expected loss in mean muscle mass, for 5-year increase in age= 5 X = (d) R 2 = = % This means that 67.88% of the variability in muscle mass is explained by the predicted value from a linear regression model using age as the explanatory variable. (e) 95% C.I. for β 1 : ( , ) We have 95% confidence that the expected increase in mean muscle mass, E(y) for 1-year) increase in age lies inside the above interval. (f) A t-test for H 0 : β 1 = 0 against H 1 : β 1 0 is: t-value= 5.44 for which the p-value is <.0001; Thus we reject the null hypothesis at α =.05 (g) From the SAS output the point estimate E(y) at x= 60 i.e. μ(60) is (h) A 95% confidence interval for the mean muscle mass, E(y) at x= 60 is ( , ) I. See SAS Output attached. II. See attached plot: Assumption of constant variance as x increases appear to be satisfied as the residuals are evenly spread around the zero line as x increases. III. See attached plots: The above is also true of the plot of residuals against the predicted values. The normal probability plot of the studentized residuals does not show a pattern to indicate that the distribution of the errors deviates from a normal distribution.

2 Simple Linear Regression of Horsepower on Speed Dependent Variable: y Muscle Mass 05:47 Monday, December 02, Number of Observations Read 17 Number of Observations Used 16 Number of Observations with Missing Values 1 Source DF Analysis of Variance Sum of Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Variable Label DF Parameter Estimate Parameter Estimates Standard Error t Value Pr > t 95% Confidence Limits Intercept Intercept < x Age <

3 Simple Linear Regression of Horsepower on Speed Dependent Variable: y Muscle Mass 05:47 Monday, December 02, Obs Dependent Variable Predicted Value Output Statistics Std Error Mean Predict 95% CL Mean Residual Std Error Residual Student Residual Cook's D * ** * ** ** * *** *** * * * ** Sum of Residuals 0 Sum of Squared Residuals Predicted Residual SS (PRESS)

4 Simple Linear Regression of Horsepower on Speed 05:47 Monday, December 02,

5 Simple Linear Regression of Horsepower on Speed 05:47 Monday, December 02,

6 Simple Linear Regression of Horsepower on Speed 05:47 Monday, December 02,

7 Simple Linear Regression of Horsepower on Speed 05:47 Monday, December 02,

8 Problem 2 The case statistics and the plots shown (see attached SAS outputs for this part) show clearly that (a) Car O is an x-outlier. The Hat Diag for this case is 0.27 which is markedly larger than the other hats (as well as it is larger than the cutoff 4/16=.25). It stands well away from the other cars in the x- direction in the plots MPG vs. Weight. Clearly, several plots shown in the diagnostics panel are affected by this case. (b) Car A is a possible y-outlier. Its observed value is much smaller than the value predicted by the fitted line. The RStudent for case A is 3.91 which is larger than the 5% critical value of 3.62 from Table B.10. (c) The two largest Cooks D values are the case A and O above. For Car O, this statistic is large primarily because it is a high leverage case (i.e. the Hat Diag is large) and not because it is a y-outlier. Thus it fits the model well but has very high influence. For Car A, Cooks D is large clearly because it is a y-outlier, and therefore does not fit the model very well at all. (d) The following is a summary of statistics resulting from fitting the model to three different data sets: Model Estimated β 0 Estimated β 1 MSE R 2 All data A deleted O deleted The case statistics for model fitted with A deleted improves the model significantly. indicate the Car A is still influential but not a y-outlier and the plots also support this. The case statistics for model fitted with Car O deleted does not give a better fitting model overall. (e) Clearly, the case statistics in Parts a), b) and c) for the model fitted for the complete data set indicated the outcome of part d). That is, removing a case that is highly influential affects the fit of the model. If the influential case is a y-outlier, the model fit is expected to improve. Thus instead of refitting models with cases deleted, the user can use the case statistics from the original fit to make similar conclusions. This is the way these statistics are meant to be used. Also the other statistics like DFFITS and DFBETAS can be used to determine how each of the suspected cases affect the overall model fit.

9 The SAS System 05:51 Monday, December 02, Dependent Variable: y MPG Number of Observations Read 16 Number of Observations Used 16 Source Analysis of Variance DF Sum of Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Variable Label Parameter Estimates DF Parameter Estimate Standard Error t Value Pr > t Intercept Intercept <.0001 x Weight(lbs.) <.0001

10 Obs Car Dependent Variable Predicted Value Output Statistics The SAS System Std Error Std Error Student Mean Predict Residual Residual Residual Dependent Variable: y MPG 05:51 Monday, December 02, Cook's D RStudent 1 A ***** B C ** D * E F * G ** H * I * J K L M ** N O ** P * Obs Car Hat Diag H Output Statistics Cov Ratio DFFITS DFBETAS Intercept 1 A B C D E F G H I J K L M N O P x

11 The SAS System 05:51 Monday, December 02, Dependent Variable: y MPG

12 05:51 Monday, December 02,

13 The SAS System 02:44 Wednesday, November 06, Dependent Variable: y MPG Number of Observations Read 15 Number of Observations Used 15 Source Analysis of Variance DF Sum of Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Variable Label Parameter Estimates DF Parameter Estimate Standard Error t Value Pr > t Intercept Intercept <.0001 x Weight(lbs.) <.0001

14 02:44 Wednesday, November 06,

15 The SAS System 02:46 Wednesday, November 06, Dependent Variable: y MPG Number of Observations Read 15 Number of Observations Used 15 Source Analysis of Variance DF Sum of Squares Mean Square F Value Pr > F Model Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Variable Label Parameter Estimates DF Parameter Estimate Standard Error t Value Pr > t Intercept Intercept <.0001 x Weight(lbs.)

16 02:46 Wednesday, November 06,