2. Analysis of Covariance
Analysis of Covariance (ANCOVA) is a
statistical test related to ANOVA
It tests whether there is a significant
difference between groups after controlling
for variance explained by a covariate
A covariate is a continuous variable that
correlates with the dependent variable
3. So, what does all that mean?
This means that you can, in effect, “partial
out” a continuous variable and run an
ANOVA on the results
This is one way that you can run a
statistical test with both categorical and
continuous independent variables
4. Hypotheses for ANCOVA
H0 and H1 need to be stated slightly
differently for an ANCOVA than a regular
ANOVA
H0: the group means are equal after
controlling for the covariate
H1: the group means are not equal after
controlling for the covariate
5. Assumptions for ANCOVA
ANOVA assumptions:
Variance is normally distributed
Variance is equal between groups
All measurements are independent
Also, for ANCOVA:
Relationship between DV and covariate is
linear
The relationship between the DV and
covariate is the same for all groups
6. How does ANCOVA work?
ANCOVA works by adjusting the total SS,
group SS, and error SS of the independent
variable to remove the influence of the
covariate
However, the sums of squares must also
be calculated for the covariate. For this
reason, SSdv will be used for SS scores for
the dependent variable, and SScv will be
used for the covariate
8. Sum of Products
To control for the covariate, the sum of products
(SP) for the DV and covariate must also be
used
This is the sum of the products of the residuals
for both the DV and the covariate
In the following slides, x is the covariate, and y
is the DV. i is the individual subject, and j is the
group.
9. Total Sum of Products
))(( yyxxtotalSP ij
j i
ijxy
This is just the sum of the multiplied residuals
for all data points.
10. Group Sum of Products
)()( yyxxngroupSP j
j
jjxy
This is the sum of the products of the group
means minus the grand means times the group
size.
11. Error Sum of Products
groupSPtotalSPerrorSP
yyxxerrorSP
xyxyxy
jij
j i
jijxy
))((
This is the sum of the products of the DV and
residual minus the group means of the DV and
residual
This just happens to be the same as the difference
between the other two sum of products
12. Adjusting the Sum of Squares
Using the SS’s for the covariate and the
DV, and the SP’s, we can adjust the SS’s
for the DV
13. Sum of Squares
errorSS
errorSP
errorSSadjerrorSS
totalSS
totalSP
errorSS
errorSP
groupSSadjgroupSS
totalSS
totalSP
totalSSadjtotalSS
x
xy
yy
x
xy
x
xy
yy
x
xy
yy
2
22
2
)(
14. Now what?
Using the adjusted SS’s, we can now run
an ANOVA to see if there is a difference
between groups.
This is the exact same as a regular
ANOVA, but using the adjusted SS’s
instead of the original ones.
Degrees of freedom are not affected
15. A few more things
We can also determine whether the
covariate is significant by getting a F score
adjtotalSS
N
totalSS
totalSP
NF
y
x
xy
2
)(
)2,1(
2
16. A few more things
The group means can also be adjusted to
eliminate the effect of the covariate
xx
errorSS
errorSP
yyadj j
x
xy
jj
17. Post-hocs for ANCOVA
Post-hoc tests can be done using the
adjusted means for ANCOVA, including
LSD and Bonferroni
18. Example of ANCOVA
Imagine we gave subjects a self-esteem
test, with scores of 1 to 10
Then we primed subjects with either
positive or negative emotions.
Then we asked them to spend a few
minutes writing about themselves.
Our dependent measure is the number of
positive emotion words they used (e.g.
happy, good)
19. The null hypothesis is that the priming
doesn’t make a difference after controlling
for self-esteem
The alternative hypothesis is that the
priming does make a difference after
controlling for self-esteem
Example of ANCOVA, cont.
21. ANCOVA in SPSS
To do ANCOVA in SPSS, all you need to
do is add your covariate to the “covariate”
box in the “univariate” menu
Everything else is the exact same as it is
for ANOVA