lecture 13 analysis of covariance and covariance interaction
DESCRIPTION
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 : y ij = y + i ij + c ij + e ij - PowerPoint PPT PresentationTRANSCRIPT
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