1. Pearson Group Assignment 3 Coffee, Stress and Heart Example

2. Candidate Models Heart ~ Coffee Heart ~ Coffee Heart ~ Stress Heart ~ Stress Heart ~ Coffee + Stress Heart ~ Coffee + Stress

3. Data Ellipses…

4. Data Ellipses A data ellipse shows the spread of data (good visualization tool) and informs us about the variance and covariance structure of the data. Simply put, the distribution of our data. Red ellipses for r=1 (1 standard deviation) Green ellipses for r=2 (2 standard deviations)

5. Conditional Model BetaStress = 1.1993 BetaCoffee = -0.4091 Marginal Models BetaStress = 0.89317 BetaCoffee = 1.1082

6. 95% Confidence Ellipses and Intervals for theCoefficients for Stress and Coffee Green Ellipse: Scheffé Red Ellipse and Its 'Shadow' Intervals: Bonferroni Blue Interval: Bonferroni (marginal model)‏

7. Scheff Scheffé vs. Bonferroni Scheff CI Scheffé CI Larger than Bonferroni CI with minimal number of parameters Larger than Bonferroni CI with minimal number of parameters Allows for data snooping Allows for data snooping Based on F-statistic Based on F-statistic Bonferroni CI Incorporates a penalty for examining several coefficients simultaneously Incorporates a penalty for examining several coefficients simultaneously Based on t-statistic Based on t-statistic

8. Conditional Model Coefficent for Coffee Bonferonni 95% CI: (-1.02494, 0.20646) Bonferonni 95% CI: (-1.02494, 0.20646) Scheffé 95% CI: (-1.19090, 0.37288)‏ Scheffé 95% CI: (-1.19090, 0.37288)‏ –Both intervals include 0, thus not significant Coefficient for Stress Bonferonni 95% CI: (0.72586, 1.67264)‏ Bonferonni 95% CI: (0.72586, 1.67264)‏ Scheffé 95% CI: (0.59790, 1.80061) Scheffé 95% CI: (0.59790, 1.80061) –Both intervals do not include 0, thus are highly significant

9. Conditional Model In this case, either confidence interval we use will give us the same conclusion In this case, either confidence interval we use will give us the same conclusion If we move the ellipses upward introducing measurement error on Stress, we can make the coefficient of Coffee significant according to Bonferroni’s method. If we move the ellipses upward introducing measurement error on Stress, we can make the coefficient of Coffee significant according to Bonferroni’s method. Although, it is still insignificant according to Scheffé method. Although, it is still insignificant according to Scheffé method.

10. Conclusions Both Coffee and Stress alone are excellent predictors of Heart for this data set (positive linear relationship)‏ Both Coffee and Stress alone are excellent predictors of Heart for this data set (positive linear relationship)‏ Scenario 1: Coffee (consumption) may be a cause of Heart (condition) with (occupational) Stress as a mediating factor: use Heart ~ Coffee Scenario 1: Coffee (consumption) may be a cause of Heart (condition) with (occupational) Stress as a mediating factor: use Heart ~ Coffee Scenario 2: Stress may be a common cause for both Coffee and Heart: use Heart ~ Coffee + Stress Scenario 2: Stress may be a common cause for both Coffee and Heart: use Heart ~ Coffee + Stress Scenarios 1 and 2 and other possible scenarios cannot be decided from this small data set with highly confounded variables Scenarios 1 and 2 and other possible scenarios cannot be decided from this small data set with highly confounded variables To determine the proper scenario, we need to collect more data, less confounded if possible, and consult the relevant medical literature and health researchers for helpful insights To determine the proper scenario, we need to collect more data, less confounded if possible, and consult the relevant medical literature and health researchers for helpful insights Statistician take into consideration the significance of all possible models, and then decide on an appropriate model Statistician take into consideration the significance of all possible models, and then decide on an appropriate model Statistician must try to insure all important factors and predictors are considered in the model fitting process Statistician must try to insure all important factors and predictors are considered in the model fitting process