1. Interactions
Interaction: Does the relationship between two variables depend on a third variable?
Does the relationship of age to BP depend on gender
Does a certain BP-lowering drug work as well in blacks than in non-blacks
Does the relationship between education and income differ by region of the country
Sometimes called “effect modification”

2. Model for FEV Example Y = b0 + b1X1 + b2X2
X1 = smoking status (1=smoker, 0=nonsmoker)
X2 = age
Smokers
FEV = b0 + b1 + b2age
Non Smokers
FEV = b0 + b2age
FEV (smokers) – FEV (non-smokers) = b1
Assumes the slope of age is same for smokers and non-smokers

3. Non-smokers
FEV
Smokers b1 b2 b1 b2
AGE

4. Modeling Interaction for FEV Example
Y = b0 + b1X1 + b2X2 + b3X3
X1 = smoking status (1=smoker, 0=nonsmoker)
X2 = age
X3 = age x smoking status
Smokers: FEV =
Non Smokers: FEV =
FEV (Smokers) – FEV (Non-smokers) =
Ho: b3 = 0
b0 + b1 + (b2 + b3) age
b0 + b2 age
b1 + b3age

5. Non-smokers FEV b1 + b3age smokers b2 b2 + b3 AGE
Note: Difference in slopes implies smoker/nonsmoker difference depends on age (and vice versa)
Non-smokers
FEV
b1 + b3age
smokers b2
b2 + b3
AGE

8. DATA fev;
INFILE DATALINES;
INPUT age smk fev;
agesmk = age*smk;
DATALINES;

9. PROC REG;
MODEL fev = age;
PLOT fev*age;
WHERE smk=0;
TITLE 'Non-smokers';
RUN;
WHERE smk=1;
TITLE 'Smokers';

10. SMOKERS
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept <.0001
age <.0001
NON SMOKERS
Intercept <.0001
age
B1 for smokers =
B1 for non-smk =
Are these statistically significant?

11. MODEL fev = age smk agesmk; RUN;
PROC REG;
MODEL fev = age smk agesmk;
RUN;
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept <.0001
age
smk
agesmk
Interpretation:
B(agesmk) = is difference in slopes between smk/nonsmk
B(age) = is slope for non-smokers (smk=0)
SMOKERS
Intercept <.0001
age <.0001
NON-SMOKERS
Intercept <.0001
age

12. Polynomial Regression: Adding Quadratic Term
Y = bo + b1X + b2X2
Can be used if linear relationship does not hold
Example: alcohol intake and mortality
Example: cholesterol and mortality
Add a quadratic (squared) term
Can test hypothesis that quadratic term in needed
Ho: b2 = 0
Ha: b2 ≠ 0

14. Linear Regression Does not Fit Well

15. Adding Quadratic Term
Plot mvo2kg*ffbw predicted.*ffbw/overlay

16. PROC REG DATA = physfit ; MODEL mvo2kg = ffbw;
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model
Error
Corrected Total
Root MSE R-Square
Dependent Mean Adj R-Sq
Coeff Var
Variable DF Estimate SE t Value Pr > |t|
Intercept <.0001
ffbw

17. PROC REG DATA = physfit ; MODEL mvo2kg = ffbw;
MODEL mvo2kg = ffbw ffbw2;
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model
Error
Corrected Total
Root MSE R-Square
Dependent Mean Adj R-Sq
Coeff Var
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept <.0001
ffbw
ffbw
ffbw2 = ffbw * ffbw
Computed in datastep

18. Model Selection
Measure many predictors; how do you decide which to include in your model?
Depends on reason for fitting model
Prediction? Examine specific effects?
Statistical criteria do exist, should not be used in place of scientific criteria
Best used in exploratory context

19. Statistical principles to use
Forward, backward, and stepwise selection
Compare p-values of terms; add/remove based on  = 0.05 or 0.10
R2 methods
Look for models with highest R2
Other methods exist

20. Possible Uses for Using Statistical Criteria
Outcome: Measure of Teenage Drinking
Many Possible Predictors
Questionnaire on relationships, friends, family, church support etc.
Outcome: Echocardographic determined hypertrophy of the heart
Many Possible ECG predictors
Computer measurements from ECG

21. Backward selection procedure
Removes worst variable, then second worst, etc
PROC REG DATA = physfit;
MODEL mvo2kg = male age hgt wgt ffbw rhr / selection=backward;
RUN;
Final model:
Parameter Standard
Variable Estimate Error Type II SS F Value Pr > F
Intercept <.0001
male <.0001
age
wgt <.0001
ffbw <.0001
rhr

22. Forward selection procedure
Start with best single variable, adds next best, etc
PROC REG DATA = physfit;
MODEL mvo2kg = male age hgt wgt ffbw rhr / selection=forward;
RUN;
This example - ends up including all terms except height
Exactly same model as one picked by backward selection

23. “MAXR” method PROC REG DATA = physfit;
Select several models based on maximal R2
PROC REG DATA = physfit;
MODEL mvo2kg = male age hgt wgt ffbw rhr / selection=maxr;
RUN;
Will give “best” models with 1, 2, 3... Terms
You choose best overall among the “best”

24. Final models by MAXR method

25. Two general principles to use
Parsimony - less is more
Common sense
Don’t use social security number to predict height!
Cautionary Note
Models with several variables are not as good at predicting as model might suggest.