1. Applied Epidemiologic Analysis - P8400 Fall 2002
Lab 4
Intercept, Variable Centered
& Interaction
Henian Chen, M.D., Ph.D.
Applied Epidemiologic Analysis - P Fall 2002

2. Description of the Life Satisfaction Data% N
Mean
Range
LIFESAT
751 74
10 ~ 100
Age 22
17 ~ 28
Sex
Female=0
50.2
377
Male=1
49.8
374
Income
Low=0
57.0
428
High=1
43.0
323
Applied Epidemiologic Analysis - P Fall 2002

3. Applied Epidemiologic Analysis - P8400 Fall 2002
Intercept
Y intercept is the estimated Y when all Xi=0
Interaction
The circumstance in which the impact of one variable on Y is conditional on (varies across) the values of another predictor.
Applied Epidemiologic Analysis - P Fall 2002

4. Applied Epidemiologic Analysis - P8400 Fall 2002
Centering
Subtracting the sample mean on a variable X from each subject’s score on X. x = X - Mx
Center X if X doesn’t have a meaningful zero. With centered variable x, the mean is zero. Thus the regression of Y on x at x=0 (intercept) becomes meaningful.
Y = α + β1 age
Y = α + β1 age + β2 X + β3 age*X
β1: regression of Y on age at X=0
β2: regression of Y on X at age=0
Applied Epidemiologic Analysis - P Fall 2002

5. Applied Epidemiologic Analysis - P8400 Fall 2002
SAS Program
proc import datafile='a:life-satisfaction751.txt' out=lifesat dbms=tab replace;
getnames=yes;
run;
data lifesat1;
set lifesat;
agec=age-22;
age17=age-17;
age28=age-28;
proc reg data=lifesat1;
model lifesat= ; /* model 1 */
model lifesat=sex; /* model 2 */
model lifesat=income; /* model 3 */
model lifesat=age; /* model 4 */
model lifesat=age17; /* model 5 */
model lifesat=age28; /* model 6 */
model lifesat=agec; /* model 7 */
model lifesat=sex income /* model 8 */
model lifesat=sex income sex_inco; /* model 9 with interaction */
run; 
Applied Epidemiologic Analysis - P Fall 2002

6. Dependent Variable: LIFESAT
Model 2
proc reg data=lifesat;
model lifesat=sex;
run;
Dependent Variable: LIFESAT
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept <.0001
SEX
is the average life satisfaction score for Females.
– is the difference on life satisfaction score between male and female. Male is less than Female. Male’s average score= – =
Applied Epidemiologic Analysis - P Fall 2002

7. Dependent Variable: LIFESAT
Model 4
proc reg data=lifesat;
model lifesat=age;
run;
Dependent Variable: LIFESAT
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept <.0001
AGE
is the average life satisfaction score for subjects at age=0. This intercept does not make sense for us because we do not have a age=0 in our data.
– means, on the average, each additional year from 17 to 28 is associated with a decrease in life satisfaction score of
Applied Epidemiologic Analysis - P Fall 2002

8. Dependent Variable: LIFESAT
Model 5
proc reg data=lifesat;
model lifesat=age17; /* age17 = age - 17 */
run;
Dependent Variable: LIFESAT
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept <.0001
AGE
is the average life satisfaction score for subjects at age=17.
The regression coefficient is the same ( ). Only the regression intercept has changed.
Applied Epidemiologic Analysis - P Fall 2002

9. Applied Epidemiologic Analysis - P8400 Fall 2002
Two-way Interaction
Lifesat = α + β1 sex + β2 income + β3 sex*income
For female (0) and low Income (0):
Life Satisfaction score = α + β1*0 + β2*0 + β3*0*0
= α
For female (0) and high Income (1):
Life Satisfaction score = α + β1*0 + β2*1 + β3*0*1
  = α + β2
Male (1) and low Income (0):
Life Satisfaction score = α + β1*1 + β2*0 + β3*1*0
  = α + β1
Male (1) and high Income (1):
Life Satisfaction score = α + β1*1 + β2*1 + β3*1*1
= α + β1 + β2 + β3
Applied Epidemiologic Analysis - P Fall 2002

10. Three-way Interaction
Three independent variables: A, B, C
Y = α + β1A + β2B + β3C + β4AB + β5AC + β6BC + β7ABC
All lower order terms must be included in the regression model for the β7 coefficient to represent the effect of the three-way interaction on Y.
To test d-way interaction, the model must be included:
all main effect variables
all two-way interaction
all three-way interaction
all (d-1)-way interaction
even though some of them are not significant
Applied Epidemiologic Analysis - P Fall 2002