LECTURE 13
ANALYSIS OF COVARIANCE
AND COVARIANCE
INTERACTION
and ATI (Aptitude-Treatment
Interaction)
ANCOVA
ANCOVA model
The simplest ANCOVA model includes a covariate C, an exogenous treatment
variable X, and an outcome Y:
yij = y + iij + cij + eij
This is a regression equation relating the exogenous variables to the
endogenous outcome. In classical ANOVA terms, the model is written as
yij = y + ii + (cij - c.. ) + eij
In this formulation the grand mean y plays the same role as in ANOVA, the
mean performance of all populations. The term ii is the effect of the
treatment, and the term (cij - xi. ) is the regression effect of the covariate
deviation from the covariate grand mean on the outcome. This equation can
be rewritten as
yij = (- C ) + ii + cij + eij
y
X1 data
swarm
 = average slope
y1.
 = average slope
y1.
1
X2 data
swarm
y
2
y2.
y2.
COVARIATE
_
_
c2. c..
_
c1.
Fig. 12.1: Graph of relationships between treatment, covariate, and outcome in ANCOVA
y
Group 1
data
swarm
 = average slope
y1.
 = average slope
y
Difference between
groups for y scores
predicted from mean
of covariate
y2.
Group 2
data
swarm
_
_
_
c2.
c..
c1.
Fig. 12.2: Graph of treatment effect in ANCOVA
COVARIATE
y
X1 data
swarm
Average slope 
X2 data
swarm
COVARIATE
Fig. 12.3: Representation of the slope parameter in ANCOVA as the average of group slopes
SOURCE
Covariate
df
Sum of Squares
1
R2(cij – c..)2
Treatment…k-1
error
total
n(ŷi. – y..)2
n(k-1)-1 (ŷij - ŷi.)2
kn-1(ŷij – y..)2
Mean Square
F
SSc
SSc/MSe
SStreat / k-1
MStreat/MSe
SSe / [n(k-1)-1]
-
SSy.c / (n-1)
-
Table 12.1: Analysis of Covariance table
SSc
SSy
SSc
SSy
SS
SSCovariate
Covariate
e
e
,
sstreat
SS
SS
e
Type III
e
sstreat
a. Randomized design
b. Nonrandomized design
Fig. 12.4: Venn diagram for ANCOVA with covariate, k treatments and outcome




c
c
y
y
cx

Randomized design



Nonrandomized design
Fig. 12.5: Path model representation of ANCOVA

1
2

Fig. 12.6: ANCOVA average slope and interaction slope components
y
XY1 data
swarm
Difference
between
treatment
groups
XY2 data
swarm
No differences
among treatment
groups
Ca
Cb
COVARIATE
Fig. 12.7: Treatment effects dependent on covariate prediction values C a and Cb
D(c)
D(c)
D(c)
0
0
0
Covariate c
D(y) = B2 + B4c
Covariate c
D(y) = B2 + 0c
Covariate c
D(y) = 0 + 0c
Fig. 12.8: ATI represented as a difference function D , three cases: a) treatment and
interaction, b) treatment only, and c) no treatment or interaction
D(C)
D(C) + [2F2,N-4 s2D(C)
D(C) - [2F2,N-4 s2D(C)
0
b
Covariate C
RC Region of significance: D(c)  0
Fig. 12.9c: Single region of significance R C for significant ATI
D(C)
D(C) + [2F2,N-4 s2D(C)
D(C) - [2F2,N-4 s2D(C)
RC Region of significance: D(c)  0
Covariate C
0
a
b
RC Region of significance: D(c)  0
Fig. 12.9b: Dual region of significance RC for significant ATI
Externalizing behavior (Dep. Var.)
81.8
Males
B3(Males) = -.655257
69.5
Females
B3(Females) = -.437531
28.1
19.9
Internalizing behavior (Covariate)
94.6
Region of
significance
D(C)
D(C) + [2F2,N-4 s2D(C)
RC Region of significance: a  D(c)  b
a
b
Covariate C
D(C) - [2F2,N-4 s2D(C)
Fig. 12.9a: Single region of significance R C for significant ATI
HLM Issues
• Random Intercepts and Slopes:
– Suppose we assume the regressions for the various
groups are NOT based on fixed covariate values but
that these are samples from the population (the real
situation). Then the intercepts and slopes are not fixed
but can vary randomly from sample to sample
– This means that the covariate is a RANDOM factor, not
a fixed factor; either or both intercept and slope could
be random.
Random Covariate Parameters
• Y = b0j + b1jXij + eij [student i in cluster j first
level model]
• b0j = g00 + g01Zj + u0j [intercept regression
equation depends on cluster j second level
value Z]
• b1j = g10 + g11Zj + u1j [slope depends on
cluster j second level value Z]
Random Covariate Parameters
Example: students in a classroom:
achievement Y is a function of expectation
for mastery X
Classrooms have a teacher-defined learning
climate Z, and the level (intercept) of
achievement Y depends on this climate as
well as the relationship of achievement to
expectation for mastery (slope)
Group 4
Random Covariate Parameters
b1j = g10 + g11Zj + u1j
Group 3
Y
Random slopes
Group 2
b0j = g00 + g01Zj + u0j
Random
intercepts
Group 1
Covariate X
Mixed Models procedures
• Fixed Effects ANOVA Table
Source df MS F sig.
• Random Effects Variance-Covariance Table
Source Variance S.E. sig.
Sources
Covariance S.E.
sig.
SAS approach
proc mixed noclprint covtest noitprint ; class
cls ;
model mnrat1=OVAG gen eth eth*gen
gen*OVAG eth*OVAG gen*eth*OVAG
/solution ddfm=bw ;
random intercept OVAG/sub=cls type=un;
Covariance Parameter Estimates RANDOM EFFECTS
Standard
Z
Cov Parm Subject Estimate
Error Value Pr Z
intercept UN(1,1)
cls 0.1050 0.01486
7.06
<.0001
corr(i,s)UN(2,1) cls
0.02269 0.02523
0.90 0.3685
slope UN(2,2)
cls
0.2211 0.08588
2.57
0.0050
Residual
0.3361 0.009478 35.46
<.0001
Type 3 Tests of Fixed Effects
Num Den
Effect
DF DF F Value Pr > F
OVAG
1
gen
1
eth
1
gen*eth
1
OVAG*gen
1
OVAG*eth
1
OVAG*gen*eth 1
2650
152
164
152
2650
2650
2650
435.46
18.43
18.99
7.38
9.15
5.28
0.03
<.0001
<.0001
<.0001
0.0074
0.0025
0.0217
0.8609