8.5 - Unequal Slopes Model - SAS

If the data collected in the example study were instead as follows:

Females Males
Salary years Salary years
80 5 42 1
50 3 112 4
30 2 92 3
20 1 62 2
60 4 142 5

We would see in Step 2 that we do have a significant treatment × covariate interaction. Using this SAS program with the new data shown below.

data unequal_slopes;
input gender $ salary years;
datalines;
m  42  1
m  112  4
m  92  3
m  62  2
m  142  5
f  80  5
f  50  3
f  30  2
f  20  1
f  60  4
;

proc mixed data=unequal_slopes;
class gender;
model salary=gender years gender*years;
title 'Covariance Test for Equal Slopes';
/*Note that we found a significant years*gender interaction*/
/*so we add the lsmeans for comparisons*/
/*With 2 treatments levels we omitted the Turkey adjustment*/
lsmeans gender/pdiff at years=1;
lsmeans gender/pdiff at years=3;
lsmeans gender/pdiff at years=5;
run;

We get the following output:

Type 3 Test of Fixed Effects
Effect Num DF De DF F Value Pr > F
years 1 6 800.00 < .0001
gender 1 6 6.55 0.0430
years*gender 1 6 50.00 0.0004

Generating Covariate Regression Slopes and Intercepts

data unequal_slopes;
input gender $ salary years;
datalines;
m  42  1
m  112  4
m  92  3
m  62  2
m  142  5
f  80  5
f  50  3
f  30  2
f  20  1
f  60  4
;

proc mixed data=unequal_slopes;
class gender;
model salary=gender years gender*years / noint solution;
ods select SolutionF;
title 'Reparmeterized Model';
run;

Output:

Solution for Fixed Effects
Effect gender Esimate Standard Error DF t Value Pr > |t|
gender f 3.0000 3.3166 6 0.90 0.4006
gender m 15.0000 3.3166 6 4.52 0.0040
years*gender f 15.0000 1.0000 6 15.00 < .0001
years*gender m 25.0000 1.0000 6 25.00 < .0001

Here the intercepts are the Estimates for effects labeled 'gender' and the slopes are the Estimates for the effect labeled 'years*gender'. Thus, the regression equations for this unequal slopes model are:

\(\text{Females}\;\;\; y = 3.0 + 15(Years)\)

\(\text{Males}\;\;\; y = 15 + 25(Years)\)

The slopes of the regression lines differ significantly and are not parallel:

unequal models slopes plot

And here is the output:

Differences of Least Squares Means

Effect gender _gender years Estimate Standard Error DF t Value Pr > |t|
gender f m 1.00 -22.000 3.4641 6 -6.35 0.0007
gender f m 3.00 -42.000 2.0000 6 -21.00 < .0001
gender f m 5.00 -62.000 3.4641 6 -17.90 < .0001

In this case, we see a significant difference at each level of the covariate specified in the lsmeansstatement. The magnitude of the difference between males and females differs (giving rise to the interaction significance). In more realistic situations, a significant treatment × covariate interaction often results in significant treatment level differences at certain points along the covariate axis.