Supplementary Exercise 2.53 of IPS7e ------------------------------------ Measurements of blood flow in the stomach of dogs by two methods: a catheter inserted into a vein and mildly radioactive microsphreres injected into the blood stream. The first method may be considered as the reference method, and the second method is a new method that is simpler and less costly to employ. Both methods give values for blood flow in milliliters of blood per minute. Ten pairs of measurements were taken; we will refer to these as from 10 different dogs although that is not clearly stated in the problem. Both variables are response variables, and we are interested in predicting blood flow in the vein from a microsphere measurement. The statistical model is a linear regression model: vein_i = beta0 + beta1 * spheres_i + eps_i where the errors (eps_i) are i.i.d. from N(0,sigma). (a)+(b) Minitab commands: MTB > WOpen "R:\Chapter 2\ex02_053.mtw". Retrieving worksheet from file: ‘R:\Chapter 2\ex02_053.mtw’ Worksheet was saved on 07/11/2014 MTB > Plot 'vein'*'spheres'; SUBC> Symbol. Scatterplot of vein vs spheres MTB > Fitline 'vein' 'spheres'; SUBC> Confidence 95.0. Regression Analysis: vein versus spheres The regression equation is vein = 1.031 + 0.9020 spheres S = 1.75664 R-Sq = 93.6% R-Sq(adj) = 92.8% Analysis of Variance Source DF SS MS F P Regression 1 360.570 360.570 116.85 0.000 Error 8 24.686 3.086 Total 9 385.256 Fitted Line: vein versus spheres Comments: --------- The relation is indeed strong and appears approximately linear, with the points scattered narrowly around the line. The estimated regression line is: vein = 1.031 + 0.9020*spheres, corresponding to an intercept of 1.031 and a slope of 0.9020. The estimated spread about the line is 1.757. There is very strong significance of the line (or evidence against H0: slope=0, or no association between the two variables). In this instance, this is little surprising because the two variables are supposed to measure the same physiological parameter. Although not directly asked for, in this situation it is of interest to compute confidence intervals for the two parameters of the line and compare those to the values of the "perfect line": intercept=0 and slope=1. Using the formula for a 95% CI: estimate +- t(.975,DFE) * SE and the values in the Minitab listing below, we obtain 95% CI for intercept: 1.031 +- 2.306*1.224 = 1.031 +- 2.823 slope: 0.902 +- 2.306*0.083 = 0.902 +- 0.191 where we used tstar=2.306 from the t(8)-distribution. It is seen that both confidence intervals include the "perfect" values. That still does not mean that there is no evidence against both intercept=0 and slope=1, because looking at the confidence intervals assesses them separately but the two parameters are *not* estimated independently from each other. One approach is to refit the model without the intercept (Minitab allows this), and then test that the slope equals 1. It turns it there is indeed no evidence against this (results not shown). (c) Minitab commands: MTB > Regress; SUBC> Response 'vein'; SUBC> Nodefault; SUBC> Continuous 'spheres'; SUBC> Terms spheres; SUBC> Constant; SUBC> Unstandardized; SUBC> Tmethod; SUBC> Tanova; SUBC> Tsummary; SUBC> Tcoefficients; SUBC> Tequation; SUBC> TDiagnostics 0. Regression Analysis: vein versus spheres Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 1 360.57 360.570 116.85 0.000 spheres 1 360.57 360.570 116.85 0.000 Error 8 24.69 3.086 Total 9 385.26 Model Summary S R-sq R-sq(adj) R-sq(pred) 1.75664 93.59% 92.79% 89.26% Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 1.03 1.22 0.84 0.424 spheres 0.9020 0.0834 10.81 0.000 1.00 Regression Equation vein = 1.03 + 0.9020 spheres Fits and Diagnostics for Unusual Observations Std Obs vein Fit Resid Resid 2 8.300 5.270 3.030 2.00 R R Large residual MTB > Predict 'vein'; SUBC> Nodefault; SUBC> KPredictors 6; SUBC> KPredictors 12; SUBC> KPredictors 18; SUBC> TEquation; SUBC> TPrediction. Prediction for vein Variable Setting spheres 6 Fit SE Fit 95% CI 95% PI 6.44304 0.810307 (4.57447, 8.31161) (1.98202, 10.9041) Variable Setting spheres 12 Fit SE Fit 95% CI 95% PI 11.8549 0.562628 (10.5575, 13.1523) (7.60137, 16.1084) Variable Setting spheres 18 Fit SE Fit 95% CI 95% PI 17.2667 0.691232 (15.6728, 18.8607) (12.9136, 21.6199) Comments: --------- The predicted venous measurements are 6.44, 11.86, and 17.27, respectively. It is seen that all these are within 10% of the microspheres values used for the prediction, thereby meeting the criterion stated in the problem. It should be noted that such a 10% rule is by no means universal. There are other ways of quantifying agreement (in particular a statistic called the concordance correlation coefficient), and rules can be set based on such statistics. It would also seem natural to consider the prediction intervals, which in this case are seen to be quite wide. The intervals take into account how much the points scatter about the line, and that is important for assessing the possible range of the true blood flow when estimated from a microshpere measurement.