LECTURE 13ANALYSIS OF COVARIANCE

AND COVARIANCE INTERACTION

and ATI (Aptitude-Treatment Interaction)

ANCOVAANCOVA model

The simplest ANCOVA model includes a covariate C, an exogenous treatment variable X, and an outcome Y:

yij = y + iij + 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 + ii + (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 ii 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 ) + ii + cij + eij

y

COVARIATE

X1 dataswarm

= average slope

X2 dataswarm

= average slope

_ _ _ c2. c.. c1.

1

2

y

Fig. 12.1: Graph of relationships between treatment, covariate, and outcome in ANCOVA

y1.

y2.

y2.

y1.

y Group 1dataswarm

= average slope

Group 2dataswarm

y

y1.

= average slope

Difference betweengroups for y scorespredicted from meanof covariate

y2.

COVARIATE _ _ _ c2. c.. c1.

Fig. 12.2: Graph of treatment effect in ANCOVA

yX1 dataswarm

X2 dataswarm

COVARIATE

Average slope

Fig. 12.3: Representation of the slope parameter in ANCOVA as the average of group slopes

SOURCE df Sum of Squares Mean Square F

Covariate 1 R2(cij – c..)2 SSc SSc/MSe

Treatment…k-1 n(ŷi. – y..)2 SStreat / k-1 MStreat/MSe

error n(k-1)-1 (ŷij - ŷi.)2 SSe / [n(k-1)-1] -

total kn-1(ŷij – y..)2 SSy.c / (n-1) -

Table 12.1: Analysis of Covariance table

sstreat

SSy

Fig. 12.4: Venn diagram for ANCOVA with covariate, k treatments and outcome

SSe

SSCovariatee

a. Randomized design

SSCovariatee

sstreat,Type III

SSe

SSy

b. Nonrandomized design

SScSSc

c

y

Fig. 12.5: Path model representation of ANCOVA

Randomized design

c

y

Nonrandomized design

cx

1

2

Fig. 12.6: ANCOVA average slope and interaction slope components

y XY1 dataswarm

XY2 dataswarm

Ca Cb

COVARIATE

Fig. 12.7: Treatment effects dependent on covariate prediction values Ca and Cb

No differencesamong treatmentgroups

Differencebetweentreatmentgroups

Covariate c

D(c)

D(y) = B2 + B4c

Covariate c

D(c)

D(y) = B2 + 0c

Covariate c

D(c)

D(y) = 0 + 0c

0 00

Fig. 12.8: ATI represented as a difference function D , three cases: a) treatment andinteraction, b) treatment only, and c) no treatment or interaction

Covariate C

RC Region of significance: D(c) 0

0

D(C)

D(C) + [2F2,N-4 s2

D(C)

D(C) - [2F2,N-4 s2

D(C)

Fig. 12.9c: Single region of significance RC for significant ATI

b

0 a b

Covariate C

RC Region of significance: D(c) 0

D(C)

D(C) + [2F2,N-4 s2

D(C)

D(C) - [2F2,N-4 s2

D(C)

Fig. 12.9b: Dual region of significance RC for significant ATI

RC Region of significance: D(c) 0

81.8

69.5

94.6

19.9

Males

B3(Males) = -.655257

28.1

Females

B3(Females) = -.437531

Externalizing behavior (Dep. Var.)

Internalizing behavior (Covariate)Region ofsignificance

Covariate Cb

RC Region of significance: a D(c) b

D(C)

D(C) + [2F2,N-4 s2

D(C)

D(C) - [2F2,N-4 s2

D(C)

Fig. 12.9a: Single region of significance RC for significant ATI

a

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)

Random Covariate Parameters

Random intercepts

Random slopes

Covariate X

Yb1j = g10 + g11Zj + u1j

b0j = g00 + g01Zj + u0j

Group 1

Group 2

Group 3

Group 4

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 0.3685slope 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 2650 435.46 <.0001 gen 1 152 18.43 <.0001 eth 1 164 18.99 <.0001 gen*eth 1 152 7.38 0.0074 OVAG*gen 1 2650 9.15 0.0025 OVAG*eth 1 2650 5.28 0.0217 OVAG*gen*eth 1 2650 0.03 0.8609