MATH ASSIGNMENT 2: SOLUTIONS

Transcription

1 MATH ASSIGNMENT 2: SOLUTIONS (a) Fitting the simple linear regression model to each of the variables in turn yields the following results: we look at t-tests for the individual coefficients, and the -F test statistics. Variable T-test Conclusion β s β t p F p age INFLUENTIAL height INFLUENTIAL weight INFLUENTIAL bmp NOT INFLUENTIAL fev MARGINALLY INFLUENTIAL rv NOT INFLUENTIAL frc MARGINALLY INFLUENTIAL tlc NOT INFLUENTIAL sex NOT INFLUENTIAL The variables declared as Marginally Influential are not significant at the α = 0.05 significance level once a multiple testing correction is introduced; we are performing 9 tests, so we should test each hypothesis at the α = 0.05/9 = significance level. Note that the coefficient estimates for sex are baseline level (in the subgroup with sex=), and difference from baseline (in the subgroup with sex=0). The fact that baseline subgroup coefficient is significantly different from zero does not mean that this variable is influential in predicting y. It is possible to fit the model to the log-transformed response variable. This was not asked for, but is an acceptable approach. For the log-transformed data, the results are as follows; Variable T-test Conclusion β s β t p F p age INFLUENTIAL height INFLUENTIAL weight INFLUENTIAL bmp NOT INFLUENTIAL fev MARGINALLY INFLUENTIAL rv NOT INFLUENTIAL frc MARGINALLY INFLUENTIAL tlc NOT INFLUENTIAL sex NOT INFLUENTIAL Overall, inspecting the R 2 values, it seems that the prediction of y rather than log(y) is better, but the difference is marginal. Thus, on the basis of simple linear regression one variable at a time, only age, height and weight are influential variables in the model predicting y. 0 Marks MATH 204 ASSIGNMENT 2 SOLUTIONS Page of 2

2 (b) The results of the multiple regression analysis, with all variables included in linear form without interaction, as specified by the model can be found on pages 9-22 of the printout. y = β 0 + β x + β 2 x β 9 x 9 + ɛ The following results can be discerned (only the first is needed for full marks) No variables are individually influential - all the p-values for the coefficients and in the componentwise tests are greater than which contradicts the results from (a). 5 Marks However, we can go further. Using the table, we can construct a global test of the hypothesis H 0 : β = β 2 =... = β 8 = β (0) 9 = 0 H a : At least one β j 0 to assess whether the variables are worth including at all, or whether the null model described by H 0 fits as well. Here β (0) 9 is the coefficient for the sex=0 subgroup. We construct the test statistic F by F = MSR MSE = (SS SSE)/k SSE/(n k ) where k = 9 is the total number of predictors, and MSR is the mean square due to the regression - recall that SS = SSR + SSE. Here, for the original scale data F = ( )/ /5 = 3.05 In the usual way for, we compare this with the Fisher-F(9, 5) distribution. We find that F 0.95 (9, 5) = so H 0 is rejected; hence it is worth fitting the collection of variables, as this model fits better than the model with all variables excluded. This type of analysis is sometimes called ANCOVA, with covariates. We also have R 2 = SSE SS = = Adjusted R 2 = SSE/(n k ) SS/(n ) so the overall explanatory power is not very good. = / /24 = (c) The correlation table (page 23) indicates that a lot of the variables are strongly correlated. In particular, age, height and weight have correlations above 0.9, so are strongly positively correlated. 3 Marks (d) The results in (c) partly explain those in (a) and (b) as the strong positive correlation between the apparently influential variables age, height and weight makes the results of the multiple regression hard to interpret. In the multiple regression, we are not truly observing the effect of each variable manifested through the estimated coefficient. The variables are borrowing strength from each other in the prediction of the response; when they are fitted together, the influence of each covariate is diluted. 2 Marks MATH 204 ASSIGNMENT 2 SOLUTIONS Page 2 of 2

3 Regression for pemax Regression Residual Total for B B Std. Error Beta t Sig. Lower Bound Upper Bound (Constant) Age in years Regression Residual Total for B B Std. Error Beta t Sig. Lower Bound Upper Bound (Constant) Height (cm)

4 Regression Residual Total for B B Std. Error Beta t Sig. Lower Bound Upper Bound (Constant) Weight (kg) Regression Residual Total for B B Std. Error Beta t Sig. Lower Bound Upper Bound (Constant) Body mass percentage (% of normal)

5 Regression Residual Total for B B Std. Error Beta t Sig. Lower Bound Upper Bound (Constant) Forced expiratory volume Regression Residual Total for B B Std. Error Beta t Sig. Lower Bound Upper Bound (Constant) Residual volume

6 Regression Residual Total for B B Std. Error Beta t Sig. Lower Bound Upper Bound (Constant) Functional residual capacity Regression Residual Total for B B Std. Error Beta t Sig. Lower Bound Upper Bound (Constant) Total lung capacity

7 5 Regression for log(pemax) Regression Residual Total for B B Std. Error Beta t Sig. Lower Bound Upper Bound (Constant) Age in years Regression Residual Total for B B Std. Error Beta t Sig. Lower Bound Upper Bound (Constant) Height (cm)

8 Regression Residual Total for B B Std. Error Beta t Sig. Lower Bound Upper Bound (Constant) Weight (kg) Regression Residual Total for B B Std. Error Beta t Sig. Lower Bound Upper Bound (Constant) Body mass percentage (% of normal)

9 Regression Residual Total for B B Std. Error Beta t Sig. Lower Bound Upper Bound (Constant) Forced expiratory volume Regression Residual Total for B B Std. Error Beta t Sig. Lower Bound Upper Bound (Constant) Residual volume

10 Regression Residual Total for B B Std. Error Beta t Sig. Lower Bound Upper Bound (Constant) Functional residual capacity Regression Residual Total for B B Std. Error Beta t Sig. Lower Bound Upper Bound (Constant) Total lung capacity

11 9 General Linear Analysis for pemax Corrected Intercept age Error Total Corrected Total Intercept age Corrected Intercept height Error Total Corrected Total Intercept height

12 0 Corrected Intercept weight Error Total Corrected Total Intercept weight Corrected Intercept bmp Error Total Corrected Total Intercept bmp

13 Corrected Intercept fev Error Total Corrected Total Intercept fev Corrected Intercept rv Error Total Corrected Total Intercept rv

14 2 Corrected Intercept frc Error Total Corrected Total Intercept frc Corrected Intercept tlc Error Total Corrected Total Intercept tlc

15 3 Between-Subjects Factors Sex of patient Value Label N 0 Male 4 Female Corrected Intercept sex Error Total Corrected Total Intercept [sex=0] [sex=]

16 4 General Linear Analysis for log(pemax) Corrected Intercept age Error Total Corrected Total Intercept age Corrected Intercept height Error Total Corrected Total Intercept height

17 5 Corrected Intercept weight Error Total Corrected Total Intercept weight Corrected Intercept bmp Error Total Corrected Total Intercept bmp

18 6 Corrected Intercept fev Error Total Corrected Total Intercept fev Corrected Intercept rv Error Total Corrected Total Intercept rv

19 7 Corrected Intercept frc Error Total Corrected Total Intercept frc Corrected Intercept tlc Error Total Corrected Total Intercept tlc

20 8 Between-Subjects Factors Sex of patient Value Label N 0 Male 4 Female Corrected Intercept sex Error Total Corrected Total Intercept [sex=0] [sex=]

21 9 Multiple regression for pemax Between-Subjects Factors Sex of patient Value Label N 0 Male 4 Female Corrected Intercept age height weight bmp fev rv frc tlc sex Error Total Corrected Total Intercept age height weight bmp fev rv frc tlc [sex=0] [sex=]

22 20 Std. Residual Predicted Observed Observed Predicted Std. Residual : Intercept + age + height + weight + bmp + fev + rv + frc + tlc + sex

23 2 Multiple regression for log(pemax) Corrected Intercept age height weight bmp fev rv frc tlc sex Error Total Corrected Total Intercept age height weight bmp fev rv frc tlc [sex=0] [sex=]

24 22 Std. Residual Predicted Observed Observed Predicted Std. Residual : Intercept + age + height + weight + bmp + fev + rv + frc + tlc + sex

25 (c) Correlation matrix for continuous covariates, response and log response Age in years Height (cm) Weight (kg) Body mass percentage (% of normal) Forced expiratory volume Residual volume Functional residual capacity Total lung capacity Maximum expiratory pressure (cm water) log(pemax) Pearson Correlation Sig. (2-tailed) N Pearson Correlation Sig. (2-tailed) N Pearson Correlation Sig. (2-tailed) N Pearson Correlation Sig. (2-tailed) N Pearson Correlation Sig. (2-tailed) N Pearson Correlation Sig. (2-tailed) N Pearson Correlation Sig. (2-tailed) N Pearson Correlation Sig. (2-tailed) N Pearson Correlation Sig. (2-tailed) N Pearson Correlation Sig. (2-tailed) N **. Correlation is significant at the 0.0 level (2-tailed). *. Correlation is significant at the 0.05 level (2-tailed). Correlations Body mass percentage Forced expiratory Residual Functional residual Total lung Maximum expiratory pressure Age in years Height (cm) Weight (kg) (% of normal) volume volume capacity capacity (cm water) log(pemax).926**.906** ** -.639** -.469*.63**.59** **.92**.44* ** -.624** -.457*.599**.589** **.92**.673**.449* -.622** -.67** -.48*.635**.606** *.673**.546** -.582** -.434* *.546** -.666** -.665** -.443*.453*.48* ** -.570** -.622** -.582** -.666**.9**.589** ** -.624** -.67** -.434* -.665**.9**.704** -.47* -.409* * -.457* -.48* *.589**.704** **.599**.635** * * ** **.589**.606**.93.48* * **