Transcript Slide 1

Analysis of Covariance (ANCOVA)
Analysis of variance conducted after
removing the relationship of some
extraneous variable (covariate) with the
dependent variable (Y)
What variability in DV can be explained by
IV, AFTER removing variability
explained by the covariate?
Reasons for Using ANCOVA
1. reduce ‘error’ (MSW) by removing
effects of extraneous variable
2. adjust DV scores, what would they be in
the absence of the covariate
(estimated through regression)
Ideally
small number of covariates
each correlated with the DV
uncorrelated with each other
Analysis of Covariance (ANCOVA)
Example – even after ‘randomly assigning’
participants to levels of the IV, some
differences still exist before IV is
introduced.
See example on next slide of ‘anxiety’
differences before any exposure to
anxiety stimulus or any ‘treatment’
manipulation
Regression is used to adjust the DV scores
Adjusted scores reflect the removal of the
variability in the DV explained
by the Covariate
what would Anxiety at time 3 be, when Anxiety at time 1 is
held constant?
The regression residuals reflect variability
not explained by the covariate
– it becomes the ‘error’ (MSW) term
and will be lower than the MSW would have been
Differences among the adjusted means
(MSB) can be evaluated relative to the
unexplained variance that remains
after removal of the relationship of the
DV with the covariate
What is IV – DV relationship after removing
the Covariate – DV relationship
The adjusted scores of the DV
are based on a regression equation
using the pooled regression coefficient
pooled across the separate regression
equations for each group in the design.
Thus, for each group the regression coefficient is the same, but
the intercepts will vary (graph to follow)
1
1
1
Original
mean
2
1
2
1
Adjusted
mean
2
DV (Y)
1
2
3
2
2
3
3
Slopes the same,
intercepts differ
3
3
Mean of
Covariate
3
Covariate
Group means are adjusted to what
they would be at mean of covariate
Assumptions for ANCOVA
Same as those for ANOVA, plus
Homogeneity of regression coefficients
since use ‘pooled’ estimate of regression coefficient
Linear relationship of DV with Covariate
since using linear regression to adjust scores on DV
What to report
Original Means and Standard Deviations
Adjusted Means
For multilevel variable use pairwise
comparisons option, not post hoc
Adjusted Means and SEs
Original Means and SDs
Estimates
Descriptive Statistics
Dependent Variable: Anxiety after treatment
Dependent Variable: Anxiety after treatment
Type of Treatment
Humor
Neutral Info
Wait
Total
Mean
32.6000
35.3667
39.6333
35.8667
Std. Deviation
7.00049
7.54519
8.56812
8.17945
N
30
30
30
90
Type of Treatment
Humor
Neutral Info
Wait
Mean
33.190a
34.835a
39.575a
Std. Error
1.273
1.272
1.267
95% Confidence Interval
Lower Bound
Upper Bound
30.659
35.721
32.306
37.364
37.056
42.094
a. Evaluated at covariates appeared in the model: Anxiety at baseline =
37.2444.
ANOVA Without Covariate
Tests of Between-Subj ects Effects
Dependent Variable: Anxiety after treatment
Source
Corrected Model
Intercept
TRTMNT
Error
Total
Corrected Total
Type III Sum
of Squares
753.267a
115777.600
753.267
5201.133
121732.000
5954.400
df
2
1
2
87
90
89
Mean Square
376.633
115777.600
376.633
59.783
F
6.300
1936.626
6.300
Sig.
.003
.000
.003
Partial Eta
Squared
.127
.957
.127
ANOVA With Covariate
a. R Squared = .127 (Adjusted R Squared = .106)
Tests of Between-Subj ects Effects
Dependent Variable: Anxiety after treatment
MSW reduced due to
covariate
MSB reduced due to
covariate
Source
Corrected Model
Intercept
ANXIETY1
TRTMNT
Error
Total
Corrected Total
Type III Sum
of Squares
1812.535a
2481.979
1059.268
658.206
4141.865
121732.000
5954.400
df
3
1
1
2
86
90
89
a. R Squared = .304 (Adjusted R Squared = .280)
Mean Square
604.178
2481.979
1059.268
329.103
48.161
F
12.545
51.535
21.994
6.833
Sig.
.000
.000
.000
.002
Partial Eta
Squared
.304
.375
.204
.137