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
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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