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

2. Model Candidates
Heart ~ Coffee
Heart ~ Stress
Heart ~ Coffee + Stress

3. 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)

4. Conditional Model
BetaStress =
BetaCoffee =
Marginal Models
BetaStress =
BetaCoffee =

5. 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)‏

6. 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

7. Conditional Model Coefficent for Coffee
Bonferonni 95% CI: ( , )
Scheffé 95% CI: ( , )‏
Both intervals include 0, thus not significant
Coefficient for Stress
Bonferonni 95% CI: ( , )‏
Scheffé 95% CI: ( , )
Both intervals do not include 0, thus are highly significant

8. 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.

9. 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