Pearson Group Assignment 3 Coffee, Stress and Heart Example

Model Candidates Heart ~ Coffee Heart ~ Stress Heart ~ Coffee + Stress

Data Ellipses Data ellipses show areas of concentration of the data and relationships between variables Red ellipses for r=1 (1 standard deviation) Green ellipses for r=2 (2 standard deviations)

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

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

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

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

Conditional Model 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. Although, it is still insignificant according to Scheffé method.

Conclusions 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 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 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 must try to insure all important factors and predictors are considered in the model fitting process