Normal Q Q. Residuals vs Fitted. Standardized residuals. Theoretical Quantiles. Fitted values. Scale Location 26. Residuals vs Leverage

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1 Residuals Residuals vs Fitted Standardized residuals Normal Q Q Fitted values Theoretical Quantiles Standardized residuals Scale Location Standardized residuals Residuals vs Leverage Cook s distance Fitted values Leverage Figure 8: Plots for assessing the model. Multiple Regression Data Descriptions (1) (Dalgaard, 2002). Dalgaard presents an analysis related to a study concerning human lungs functionality in patients with cystic fibrosis. Data are in the ISwR package which can be downloaded from the web. After connecting to the web, all you need to do is to type the following in R: > cyst=read.table(" > cyst age sex height weight bmp fev1 rv frc tlc pemax [1,] [2,] [3,] [4,] [5,]

2 [6,] [7,] [8,] [9,] [10,] [11,] [12,] [13,] [14,] [15,] [16,] [17,] [18,] [19,] [20,] [21,] [22,] [23,] [24,] [25,] The description for the data set is also given at page 228 of the textbook (Appendix B): The cystfibr data frame has 25 rows and 10 columns. It contains lung function data for for cystic fibrosis patients (7-23 years old). Format: This data frame contains the following columns: age a numeric vector: Age in years. sex a numeric vector code. 0:male, 1:female. height a numeric vector. Height (cm). weight a numeric vector. Weight (kg). bmp a numeric vector. Body mass (% of normal). fev 1 a numeric vector. Forced expiratory volume. rv a numeric vector. Residual volume. frc a numeric vector. Functional residual capacity. tlc a numeric vector. Total lung capacity. pemax a numeric vector. Maximum expiratory pressure. 19

3 Exploratory Analysis First of all, by typing: > plot(cyst) you can obtain a matrix for all pairwise scatterplots associated with the data set (figure 1). This is an extremely powerful tool in visualizing the multivariate data. secondly, by typing: > attach(cyst) the default data set of in R is going to be cystfibr. Also, we can get a matrix for all the pairwise correlations by typing: > cor(cyst) age sex height weight bmp age sex height weight bmp fev rv frc tlc pemax fev1 rv frc tlc pemax age sex height weight bmp fev rv frc tlc pemax note that this will enable us to browse through all the pairwise linear associations. 20

4 Modeling Let s take pemax as the response variable. All other variables are considered as the explanatory ones. One may consider starting the analysis with the most complete model, as follows. pemax = β 0 +β 1 age+β 2 sex+β 3 height+β 4 weight+β 5 bmp+β 6 fev1+β 7 rv+β 8 frc+β 9 tlc here is the output for the full model: > summary(lm(pemax~age+sex+height+weight+bmp+fev1+rv+frc+tlc)) Call: lm(formula = pemax ~ age + sex + height + weight + bmp + fev1 + rv + frc + tlc) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) age sex height weight bmp fev rv frc tlc Residual standard error: on 15 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 9 and 15 DF, p-value: > test=lm(pemax~age+sex+height+weight+bmp+fev1+rv+frc+tlc) 21

5 > step(test) Start: AIC= pemax ~ age + sex + height + weight + bmp + fev1 + rv + frc + tlc - sex tlc height age frc fev rv <none> weight bmp Step: AIC=167.2 pemax ~ age + height + weight + bmp + fev1 + rv + frc + tlc - tlc height age frc rv <none> weight bmp fev Step: AIC=165.5 pemax ~ age + height + weight + bmp + fev1 + rv + frc - frc height age rv <none> fev

6 - weight bmp Step: AIC= pemax ~ age + height + weight + bmp + fev1 + rv - age height rv <none> weight bmp fev Step: AIC= pemax ~ height + weight + bmp + fev1 + rv - height rv <none> weight bmp fev Step: AIC= pemax ~ weight + bmp + fev1 + rv <none> rv bmp fev weight Call: lm(formula = pemax ~ weight + bmp + fev1 + rv) Coefficients: (Intercept) weight bmp fev1 rv

7 The Final Model > summary(lm(formula = pemax ~ weight + bmp + fev1 + rv)) Call: lm(formula = pemax ~ weight + bmp + fev1 + rv) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) weight *** bmp * fev * rv Signif. codes: 0 *** ** 0.01 * Residual standard error: on 20 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 4 and 20 DF, p-value: > anova(lm(formula = pemax ~ weight + bmp + fev1 + rv)) Analysis of Variance Table Response: pemax Df Sum Sq Mean Sq F value Pr(>F) weight *** bmp fev * rv Residuals Signif. codes: 0 *** ** 0.01 *

8 A Naive Bootstrap Approach to the Final Model >cyst.resid=residuals(lm(formula = pemax ~ weight + bmp + fev1 +rv)) >cyst.predict=predict(lm(formula = pemax ~ weight + bmp + fev1 +rv)) >B=1000 >Tboot=matrix(0,nrow=B,ncol=5) >for(i in 1:B) >{ >e.star=sample(cyst.resid,25,replace=t) >y.star=cyst.predict+e.star >Tboot[i,]=summary(lm(y.star~weight + bmp + fev1 + rv))$coefficients[,1] >print(i) } >par(mfrow=c(2,3)) >for(i in 1:5) { hist(tboot[,i]) } >mean(tboot[,1]) [1] > sd(tboot[,1]) [1] > > mean(tboot[,2]) [1] > sd(tboot[,2]) [1] > > mean(tboot[,3]) [1] > sd(tboot[,3]) [1] > > mean(tboot[,4]) [1] > sd(tboot[,4]) [1] > mean(tboot[,5]) [1] > sd(tboot[,5]) [1]

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10 age sex height weight bmp fev rv frc tlc pemax Figure 9: Plots for assessing the model.

11 Histogram of Tboot[, i] Histogram of Tboot[, i] Histogram of Tboot[, i] Frequency Frequency Frequency Tboot[, i] Tboot[, i] Tboot[, i] Histogram of Tboot[, i] Histogram of Tboot[, i] Frequency Frequency Tboot[, i] Tboot[, i] Figure 10: Bootstrapping the final model. 28