EPS 625 – ANALYSIS OF COVARIANCE (ANCOVA) EXAMPLE
USING THE GENERAL LINEAR MODEL PROGRAM
ANCOVA
One Dependent Variable – Interest Rating in DVD
One Independent Variable with three levels (Promotion Group 1, 2, and 3)
One Covariate – actual age of individuals
Research Question: Is there a difference in interest ratings of a DVD depending on which type of
promotion is provided controlling for differences in the actual age of the consumer?
ANCOVA Syntax to test the Assumption of Regression (Slopes)
UNIANOVA
dvdscore BY promotion WITH age
/METHOD = SSTYPE(3)
/INTERCEPT = INCLUDE
/CRITERIA = ALPHA(.05)
/DESIGN = promotion age age*promotion .
Univariate Analysis of Variance
This first table identifies the three levels of the between-subjects factors used in the ANCOVA
Between-Subjects Factors
N
Promotion
Group
1
2
3
24
23
23
This analysis is done to check the assumption of homogeneity of regression slopes, not to test the main
hypothesis. The factor (Promotion Group) and covariate (Actual Age) do not interact [p (.913) > α
(.05)], so the assumption of homogeneity of regression slopes has been met
Te sts of Be twe en-Subjects Effects
Dependent Variable: Total DVD assessment
Source
Correc ted Model
Int ercept
promot ion
age
promotion * age
Error
Total
Correc ted Total
Ty pe III Sum
of Squares
1366.407a
11250. 870
172.092
199.896
df
5
1
2
1
Mean S quare
273.281
11250. 870
86.046
199.896
F
6.083
250.422
1.915
4.449
Sig.
.000
.000
.156
.039
8.192
2
4.096
.091
.913
2875.365
84890. 000
4241.771
64
70
69
44.928
a. R Squared = .322 (Adjusted R S quared = .269)
Syntax for ANCOVA to test the main hypothesis
UNIANOVA
dvdscore BY promotion WITH age
/METHOD = SSTYPE(3)
/INTERCEPT = INCLUDE
/EMMEANS = TABLES(promotion) WITH(age=MEAN)
/PRINT = DESCRIPTIVE HOMOGENEITY
/CRITERIA = ALPHA(.05)
/DESIGN = age promotion .
Univariate Analysis of Variance
Between-Subjects Factors
N
Promotion
Group
1
2
3
24
23
23
The following table provides the UNADJUSTED group means. However, they do provide an initial
indication that Promotion Group 2 has a higher interest rating than the other two groups. The question
then becomes – is that difference significant – and more so – different when we control for age
De scri ptive Statistics
Dependent Variable: Tot al DVD assessment
Promotion Group
1
2
3
Total
Mean
30.67
39.70
31.61
33.94
St d. Deviat ion
6.857
6.990
6.542
7.841
N
24
23
23
70
The following table is the Levene’s Test of Homogeneity of Variance. As we can see – this assumption
is met since p (.981) > α (.05)
Levene's Test of Equality of Error Variancesa
Dependent Variable: Total DVD asses sment
F
.019
df1
df2
2
Sig.
67
.981
Tests the null hypothesis that the error variance of
the dependent variable is equal acros s groups .
a. Design: Intercept+age+promotion
If the Assumption of Homogeneity of Variance had not be met (found significant) – this is not a
BIG problem if the cell sizes are equal (i.e., the largest group size is not more than 1½ times
greater than the smallest group size). This is the case for two reasons, first, the ANCOVA
statistic is a robust statistic and second, because of the way SPSS calculates the ANCOVA
(Leech, Barrett, & Morgan, 2005).
ANCOVA EXAMPLE
PAGE 2
The following table is the test of the main hypothesis… Here we see that the Promotion Group Main
Effect is significant [p (.000) < α (.05)] controlling for the effect of age. Because we found a significant
main effect – and there are more than two levels for the independent variable – we will need to conduct
follow-up procedures (i.e., post hoc procedures or multiple comparisons tests) to determine significant
pairwise differences.
Te sts of Betw een-Subjects Effects
Dependent Variable: Total DVD ass ess ment
Source
Correc ted Model
Int ercept
age
promotion
Error
Total
Correc ted Total
Ty pe III Sum
of Squares
1358.214a
11518. 662
214.124
df
3
1
1
Mean Square
452.738
11518. 662
214.124
F
10.362
263.644
4.901
Sig.
.000
.000
.030
1151.025
2
575.512
13.173
.000
2883.557
84890. 000
4241.771
66
70
69
43.690
a. R Squared = .320 (Adjusted R Squared = .289)
The covariate is included in the analysis to control for differences on this variable and is not the
focus of the analysis. Consequently, the results of the covariate are frequently not reported in a
Results section.
Estimated Marginal Means
The following table shows the adjusted group means… These means are adjusted for the covariate.
Promotion Group
Dependent Variable: Total DVD as ses sment
Promotion Group
1
2
3
Mean
30.642 a
39.711 a
31.619 a
Std. Error
1.349
1.378
1.378
95% Confidence Interval
Lower Bound Upper Bound
27.948
33.336
36.960
42.463
28.867
34.371
a. Covariates appearing in the model are evaluated at the following
values: Actual Age = 38.27.
Note the difference between the unadjusted and the adjusted means… For this example – they
are relatively the same – however, depending on the effect (influence) of the covariate – these
means can be notably different.
ANCOVA EXAMPLE
PAGE 3
Because we found a significant between-subjects main effect – and there are three levels to our
independent variable – we will need to conduct a follow-up test to determine where any significant
pairwise differences are.
One option is to use the lmatrix syntax command which uses the appropriate error term to make pairwise
comparisons. We will still need to control for Type I error. While there are several methods from which
to choose – we will use the Bonferroni adjustment (alpha divided by the number of comparisons).
Syntax for the lmatrix command
UNIANOVA
dvdscore BY promotion
/METHOD = SSTYPE(3)
/lmatrix 'Promotion Group
promotion 1 -1 0
/lmatrix 'Promotion Group
promotion 1 0 -1
/lmatrix 'Promotion Group
promotion 0 1 -1.
WITH age
1 vs Promotion Group 2'
1 vs Promotion Group 3'
2 vs Promotion Group 3'
Because we use the top three lines of the ANCOVA syntax – we will get a few redundant tables… i.e.,
the Between-Subjects Factors and the Tests of Between-Subjects Effects. These can be ignored here.
Univariate Analysis of Variance
Between-Subjects Factors
N
Promotion
Group
1
2
3
24
23
23
Te sts of Betw een-Subjects Effects
Dependent Variable: Total DVD ass ess ment
Source
Correc ted Model
Int ercept
age
promotion
Error
Total
Correc ted Total
Ty pe III Sum
of Squares
1358.214a
11518. 662
214.124
1151.025
2883.557
84890. 000
4241.771
df
3
1
1
2
66
70
69
Mean Square
452.738
11518. 662
214.124
575.512
43.690
F
10.362
263.644
4.901
13.173
Sig.
.000
.000
.030
.000
a. R Squared = .320 (Adjusted R Squared = .289)
ANCOVA EXAMPLE
PAGE 4
The following table provides a summary of the lmatrix syntax that we just requested. For this analysis –
there is no pertinent information contained in this table – as such, it too can be ignored.
Custom Hypothesis Tests Index
1
2
3
Contrast Coefficients
(L' Matrix)
Transformation
Coefficients (M Matrix)
Contrast Results (K
Matrix)
Contrast Coefficients
(L' Matrix)
Transformation
Coefficients (M Matrix)
Contrast Results (K
Matrix)
Contrast Coefficients
(L' Matrix)
Transformation
Coefficients (M Matrix)
Contrast Results (K
Matrix)
LMATRIX
Subcom
mand 1:
Promotio
n Group 1
vs
Promotio
n Group 2
Identity
Matrix
Zero
Matrix
LMATRIX
Subcom
mand 2:
Promotio
n Group 1
vs
Promotio
n Group 3
Identity
Matrix
Zero
Matrix
LMATRIX
Subcom
mand 3:
Promotio
n Group 2
vs
Promotio
n Group 3
Identity
Matrix
Zero
Matrix
ANCOVA EXAMPLE
PAGE 5
This first set of information provides the pairwise comparison of Promotion Group 1 vs. Promotion
Group 2.
Custom Hypothesis Tests #1
Note the -9.069 – this is the adjusted mean difference of Promotion Group 1 (M = 30.642) and
Promotion Group 2 (M = 39.711). The negative is simply because of the order (low – high = negative).
Typically, we would report the absolute value (i.e., 9.069).
a
Contra st Results (K Matri x)
Contrast
L1
Dependent
Variable
Total DVD
as sess ment
Contrast E stimate
-9.069
Hy pothesiz ed Value
Difference (Est imate - Hypothesized)
St d. E rror
Sig.
95% Confidenc e Interval
for Difference
0
-9. 069
1.929
.000
-12.920
-5. 218
Lower Bound
Upper Bound
a. Based on the user-specified contrast coeffic ient s (L') matrix : Promotion
Group 1 vs Promot ion Group 2
Note the footnote (a) provides a reminder of which groups are being compared… that is, provided we
indicated that in the lmatrix syntax. While the above table also indicates significance – it does not
provide us with the F values needed to put into a report.
The following table provides the necessary information to determine if the group difference is
significant. In this case we see F(1, 66) = 22.109, p < .001 – indicating that Promotion Group 1 is
significantly different from Promotion Group 2. This is compared to our adjusted alpha level
(Bonferroni adjustment) of .017 (α/3 = .05/3 = .017). A review of the group means shows that
Promotion Group 1 (M = 30.642) is significantly lower than Promotion Group 2 (M = 39.711) on their
DVD interest levels controlling for age.
Test Results
Dependent Variable: Total DVD as sess ment
Source
Contrast
Sum of
Squares
965.962
Error
2883.557
df
1
Mean Square
965.962
66
43.690
F
22.109
Sig.
.000
Because we found a significant difference – we will need to follow this up with the calculation of an
Effect Size. Don’t forget to use the appropriate error term (MS’W = 43.690) which we get from the
above table.
ES 
X'i  X'k
MS 'W

9.069
43.690

9.069
 1.372045076  1.37
6.609841148
ANCOVA EXAMPLE
PAGE 6
This next set of information provides the pairwise comparison of Promotion Group 1 vs. Promotion
Group 3.
Custom Hypothesis Tests #2
Note the -.977 – this is the adjusted mean difference of Promotion Group 1 (M = 30.642) and Promotion
Group 3 (M = 31.619). The negative is simply because of the order (low – high = negative). Typically,
we would report the absolute value (i.e., .977).
a
Contra st Results (K Matri x)
Contrast
L1
Dependent
Variable
Total DVD
as sess ment
Contrast E stimate
-.977
Hy pothesiz ed Value
Difference (Est imate - Hypothesized)
St d. E rror
Sig.
95% Confidenc e Interval
for Difference
0
-.977
1.929
.614
-4. 828
2.874
Lower Bound
Upper Bound
a. Based on the user-specified contrast coeffic ient s (L') matrix : Promotion
Group 1 vs Promot ion Group 3
The following table provides the necessary information to determine if the group difference is
significant. In this case we see F(1, 66) =.256, p = 0.614 – indicating that Promotion Group 1 is not
significantly different from Promotion Group 3. This is compared to our adjusted alpha level
(Bonferroni adjustment) of .017 (α/3 = .05/3 = .017). A review of the group means shows that while
Promotion Group 1 (M = 30.642) is lower than Promotion Group 3 (M = 31.619) on their DVD interest
levels controlling for age, it is not significantly lower.
Test Results
Dependent Variable: Total DVD as sess ment
Source
Contrast
Sum of
Squares
11.204
Error
2883.557
df
1
Mean Square
11.204
66
43.690
F
.256
Sig.
.614
Because no significant difference was found for these two groups – no Effect Size needs to be
calculated.
ANCOVA EXAMPLE
PAGE 7
This next set of information provides the pairwise comparison of Promotion Group 2 vs. Promotion
Group 3.
Custom Hypothesis Tests #3
Note the 8.093 – this is the adjusted mean difference of Promotion Group 2 (M = 39.711) and Promotion
Group 3 (M = 31.619).
a
Contra st Results (K Matri x)
Contrast
L1
Dependent
Variable
Total DVD
as sess ment
Contrast E stimate
8.093
Hy pothesiz ed Value
Difference (Est imate - Hypothesized)
St d. E rror
Sig.
95% Confidenc e Interval
for Difference
0
8.093
1.949
.000
4.201
11.984
Lower Bound
Upper Bound
a. Based on the user-specified contrast coeffic ient s (L') matrix : Promotion
Group 2 vs Promot ion Group 3
The following table provides the necessary information to determine if the group difference is
significant. In this case we see F(1, 66) = 17.238, p < .001 – indicating that Promotion Group 2 is
significantly different from Promotion Group 3. This is compared to our adjusted alpha level
(Bonferroni adjustment) of .017 (α/3 = .05/3 = .017). A review of the group means shows that
Promotion Group 2 (M = 39.711) is significantly higher than Promotion Group 3 (M = 31.619) on their
DVD interest levels controlling for age.
Test Results
Dependent Variable: Total DVD as sess ment
Source
Contrast
Sum of
Squares
753.147
Error
2883.557
df
1
Mean Square
753.147
66
43.690
F
17.238
Sig.
.000
Because we found a significant difference – we will need to follow this up with the calculation of an
Effect Size. Don’t forget to use the appropriate error term (MS’W = 43.690) which we get from the
above table.
ES 
X'i  X'k
MS
'
W

8.093
43.690

8.093
 1.224386459  1.22
6.609841148
ANCOVA EXAMPLE
PAGE 8