Luke, Cyrus, et al.
There seems to be a flurry of questions on this model, an ANCOVA model with a factor-by-covariate interaction. At the risk of being horribly didatic, I've spelled out the (nearly) complete taxonomy of models, finally ending with the model you have asked about. See below.
Let me know if this helps.
-Tom
____________________________________________
Thomas Nichols, PhD
Director, Modelling & Genetics
GlaxoSmithKline Clinical Imaging Centre
Senior Research Fellow
Oxford University FMRIB Centre
Notation - General Linear Model
Y = X beta + epsilon
Y is response (data), X is design matrix of predictors, beta is vector of
coefficients of predictors, epsilon is random error.
Two Sample T Model
Two groups of subjects, 3 in group I, 2 in group II.
Design Matrix Parameterization 1a
1 0
1 0
1 0
0 1
0 1
Coefficient Interpretation
beta1: Mean of Group I
beta2: Mean of Group II
Contrast Interpretation
[ -1 1 ] Difference of group means, Group II > Group I
[ 1 0 ] Mean of Group I -- ONLY meaningful group level fMRI
[ 0 1 ] Mean of Group II -- ONLY meaningful group level fMRI
Design Matrix Parameterization 1b
-1 1
-1 1
-1 1
1 1
1 1
Coefficient Interpretation
beta1: Twice the group mean difference, Group II > Group I
beta2: Overall mean WHEN no group difference
Contrast Interpretation
[ 1 0 ] Twice difference of group means, Group II > Group I
[ 0.5 0 ] Difference of group means, Group II > Group I
Both contrasts will give identical T and P-values.
Simple Correlation Model
One group of 5 subjects, with age covariate with values:
20 25 30 35 40
Design Matrix Parameterization 2a (no centering)
1 20
1 25
1 30
1 35
1 40
Coefficient Interpretation
beta1: Expected response for Age of 0
beta2: Expected change in response with increase of 1 year (no
longitudinal interpretation, only cross-sectional)
Contrast Interpretation
[ 0 1 ] Increase in response with increasing age
[ 1 0 ] NOT MEANINGFUL
Design Matrix Parameterization 2b (covariate centering)
1 -10
1 -5
1 0
1 5
1 10
Coefficient Interpretation
beta1: Expected response for (Age-Average(Age)) of 0, i.e. Age = Average(Age)
beta2: Expected change in response with increase of 1 year (no
longitudinal interpretation, only cross-sectional)
Contrast Interpretation
[ 0 1 ] Increase in response with increasing age (identical to
previous parameterization).
[ 1 0 ] Expected response for an individual with average age,
while accounting for (removing from error the) linear
effect of age -- ONLY meaningful group level fMRI
ANCOVA Model - No Interaction
Two groups of subjects, 3 in group I, 2 in group II.
Age covariate with values: 20 25 30 35 40
Design Matrix Parameterization 3a
1 0 20
1 0 25
1 0 30
0 1 35
0 1 40
Coefficient Interpretation
beta1: Expected response of Group I for Age of 0
beta2: Expected response of Group II for Age of 0
beta3: Expected change in response with increase of 1 year (no
longitudinal interpretation, only cross-sectional)
Contrast Interpretation
[ 1 0 0 ] NOT MEANINGFUL
[ 0 1 0 ] NOT MEANINGFUL
[-1 1 0 ] Difference of group means, Group II > Group I,
while accounting for (removing from error the) linear
effect of age. (NOTE: No interaction, so differential
group effect doesn't vary with age).
[ 0 0 1 ] Increase in response with increasing age. (NOTE: No
interaction, so age effect is same for both groups).
Design Matrix Parameterization 3b - Centering
1 0 -10
1 0 -5
1 0 0
0 1 5
0 1 10
Coefficient Interpretation
beta1: Expected response of Group I for Age = Average(Age)
beta2: Expected response of Group II for Age = Average(Age)
beta3: Expected change in response with increase of 1 year (no
longitudinal interpretation, only cross-sectional)
Contrast Interpretation
[ 1 0 0 ] Expected Group I response at Age = average age,
while accounting for linear effect of age
-- ONLY meaningful group level fMRI
[ 0 1 0 ] Expected Group II ... " ...
[-1 1 0 ] (same as previous parameterization)
[ 0 0 1 ] (same as previous parameterization)
ANCOVA Model - With Interaction
Two groups of subjects, 3 in group I, 2 in group II.
Age covariate with values: 20 25 30 35 40, with group-dependent age
effects.
Design Matrix Parameterization 4a
1 0 20 0
1 0 25 0
1 0 30 0
0 1 0 35
0 1 0 40
Coefficient Interpretation
beta1: Expected response of Group I for Age of 0
beta2: Expected response of Group II for Age of 0
beta3: Expected change in response with increase of 1 year for Group I
beta4: Expected change in response with increase of 1 year for Group II
Contrast Interpretation
[ 1 0 0 0 ] NOT MEANINGFUL
[ 0 1 0 0 ] NOT MEANINGFUL
[-1 1 0 0 ] NOT MEANINGFUL
(Difference of group means, Group II > Group I, WHEN
Age is 0. Interaction means differential group effect
varies with age.)
[ 0 0 1 0 ] Increase in response in Group I with increasing age.
[ 0 0 0 1 ] Increase in response in Group II with increasing age.
[ 0 0 -1 1 ] Different in age effects,
Group II Age Slope > Group I Age Slope
Design Matrix Parameterization 4b - With Centering
1 0 -10 0
1 0 -5 0
1 0 0 0
0 1 0 5
0 1 0 10
NOTE: Centering of covariate is done BEFORE splitting into two
covariates; the split covariates are *not* subsequently centered.
Coefficient Interpretation
beta1: Expected response of Group I for Age = Average(Age)
beta2: Expected response of Group II for Age = Average(Age)
beta3: Expected change in response with increase of 1 year for Group I
beta4: Expected change in response with increase of 1 year for Group II
Contrast Interpretation
[ 1 0 0 0 ] Average Group I response at Age = average age,
while accounting for group-specific linear effect
of age -- ONLY meaningful group level fMRI
[ 0 1 0 0 ] Average Group II ... " ...
[-1 1 0 0 ] NOT MEANINGFUL, probably.
(Difference of group means, Group II > Group I, WHEN
Age = Average(Age). Dangerous to test/interpret
as, again, interaction means differential group effect
varies with age.)
[ 0 0 1 0 ] (same as previous parameterization)
[ 0 0 0 1 ] (same as previous parameterization)
[ 0 0 -1 1 ] (same as previous parameterization)
