Ancova SPSS Interpretation

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

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

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

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

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)

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 1? 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).

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.

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.

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.

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 WITH age

/METHOD = SSTYPE(3)

/lmatrix 'Promotion Group 1 vs Promotion Group 2'

promotion 1 -1 0

/lmatrix 'Promotion Group 1 vs Promotion Group 3'

promotion 1 0 -1

/lmatrix 'Promotion Group 2 vs Promotion Group 3'

promotion 0 1 -1.

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

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.

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).

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.

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.

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).

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.

Because no significant difference was found for these two groups – no Effect Size needs to be calculated.

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).

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.

ANCOVA Example

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