Analysis of Covariance
David Markham
djmarkham@bsu.edu
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
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
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
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
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
Sum of Squares
SS xtotal 



SS xgroup 
SS xerror 
( xij  x)
n j ( x j  x)
2
2
( xij  x j )
2
SS x error  SS x total  SS x group
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.
Total Sum of Products
SP xytotal 

( xij  x)( yij  y)
j

i
This is just the sum of the multiplied residuals
for all data points.
Group Sum of Products
SP xygroup 

nj( xj  x)( yj  y)
j

This is the sum of the products of the group
means minus the grand means times the group
size.
Error Sum of Products
SP xyerror 

( xij  xj)( yij  yj)
j
i
SP xyerror  SP xytotal  SP xygroup

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
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
Sum of Squares

SP
SS adjtotal  SS total 

xy total
y
y
SS yadjgroup  SS y
2
SS x total

SP
group 
SS y adjerror  SS y error 
xy error
SS x error
(SPxy error )
SS x error
  SP
2
SS x total
2

xy total
2
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
A few more things
 We
can also determine whether the
covariate is significant by getting a F score
 (SPxy total) 2 

N  2 
 SS x total 


F(1, N  2) 
SS y adjtotal
A few more things
 The
group means can also be adjusted to
eliminate the effect of the covariate
 SPxy error
adj y j  y j  
 SS x error


 xj  x



Post-hocs for ANCOVA
 Post-hoc
tests can be done using the
adjusted means for ANCOVA, including
LSD and Bonferroni
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)
Example of ANCOVA, cont.
 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
Data
Subject # Priming
Self-Esteem Positive Words
1
Positive
1
7
2
Positive
5
10
3
Positive
7
11
4
Negative
8
7
5
Negative
3
4
6
Negative
6
5
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