Supplementary Exercises 10.17 and 10.26 of IPS7e ------------------------------------------------ Measurement of heart rate (hr) and oxygen uptake (vo2) for one individual under different exercise conditions. Both variables are response variables but the interest is in predicting vo2 from hr because the heart rate is much easier to measure. 19 measurements were taken. The statistical model of interest is a linear regression model: vo2_i = beta0 + beta1 * hr_i + eps_i where the errors (eps_i) are i.i.d. from N(0,sigma). 10.17: ------ Minitab commands for (a)+(b): MTB > WOpen "R:\Chapter 10\ex10_017.mtw". Retrieving worksheet from file: ‘R:\Chapter 10\ex10_017.mtw’ Worksheet was saved on 07/11/2014 MTB > Plot 'VO2'*'HR'; SUBC> Symbol. Scatterplot of VO2 vs HR MTB > Fitline 'VO2' 'HR'; SUBC> Confidence 95.0. Regression Analysis: VO2 versus HR The regression equation is VO2 = - 2.804 + 0.03865 HR S = 0.120456 R-Sq = 93.8% R-Sq(adj) = 93.5% Analysis of Variance Source DF SS MS F P Regression 1 3.76185 3.76185 259.27 0.000 Error 17 0.24666 0.01451 Total 18 4.00852 Fitted Line: VO2 versus HR Comments: --------- All points are pretty close to the line, and none of them seem unusual or outlying. There is a small cluster of points around HR=95, but this is not a problem with model assumptions and could in this case correspond to the experimentor having deliberately taken several measurements in this range, by setting the exercise conditions the subject was exposed to. There also appears to be a wider spread of the points around the line in this cluster, but this is difficult to assess visually because more points will naturally scatter more than a single point. In summary, no obvious problems with any of the points can be identified. The estimated regression line is given in the Minitab listing, VO2 = - 2.804 + 0.03865 HR (c) The Minitab listing also gives an ANOVA table, which can be used for testing the hypothesis H0: beta1=0, as follows: F=259.27, DF=(1,17), P<0.0005. There is very strong evidence against H0. The data show, beyond any reasonable doubt, that a (linear) relation between HR and VO2 exists. We could also have tested H0 by a t-test: the value is t=16.10 (in the listing below), with the same P-value and conclusion. (d) We rerun the model in the Regression menu in order to do predictions for heart rates of 96 and 115. MTB > Regress; SUBC> Response 'VO2'; SUBC> Nodefault; SUBC> Continuous 'HR'; SUBC> Terms HR; SUBC> Constant; SUBC> Unstandardized; SUBC> Tmethod; SUBC> Tanova; SUBC> Tsummary; SUBC> Tcoefficients; SUBC> Tequation; SUBC> TDiagnostics 0. Regression Analysis: VO2 versus HR Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 1 3.76185 3.76185 259.27 0.000 HR 1 3.76185 3.76185 259.27 0.000 Error 17 0.24666 0.01451 Lack-of-Fit 11 0.07331 0.00666 0.23 0.983 Pure Error 6 0.17335 0.02889 Total 18 4.00852 Model Summary S R-sq R-sq(adj) R-sq(pred) 0.120456 93.85% 93.48% 92.13% Coefficients Term Coef SE Coef T-Value P-Value VIF Constant -2.804 0.258 -10.86 0.000 HR 0.03865 0.00240 16.10 0.000 1.00 Regression Equation VO2 = -2.804 + 0.03865 HR Fits and Diagnostics for Unusual Observations Obs VO2 Fit Resid Std Resid 1 0.4730 0.8289 -0.3559 -3.15 R R Large residual MTB > Predict 'VO2'; SUBC> Nodefault; SUBC> KPredictors 96; SUBC> KPredictors 115; SUBC> TEquation; SUBC> TPrediction. Prediction for VO2 Variable Setting HR 96 Fit SE Fit 95% CI 95% PI 0.906248 0.0382219 (0.825607, 0.986889) (0.639621, 1.17288) Variable Setting HR 115 Fit SE Fit 95% CI 95% PI 1.64064 0.0336520 (1.56964, 1.71164) (1.37677, 1.90451) Comments: --------- From the listing we read off the estimates and intervals for observations 2 and 16: HR=96: yhat=0.906, 95% prediction interval: (0.640, 1.173) HR=115: yhat=1.641, 95% prediction interval: (1.377, 1.905) We use prediction intervals because we want to assess the plausible range for a new observation. (e) It depends on how accurately the researchers want to know VO2. The regression equation predicts only the subject's mean VO2 for a given heart rate, and the prediction intervals show that there is considerable variation about the line so that the actual VO2 is quite variable around that estimate. 10.26: ------ Answers are given to additional questions. (a) The ANOVA table was already shown in the listings above. (b) H0: beta1=0, that is, a horizontal line, or no linear relation between heart rate and oxygen uptake. (c) Already above: F(1,17) and P<0.0005. (d) t=16.10, so t^2=259.21, which apart from round-off error equals the F=259.27 of the ANOVA table. (e) R^2=93.8%, a high percentage of the variance is explained.