ANCOVA - SPSS Interpretation - DVD Rating

EPS 625 – ANALYSIS OF COVARIANCE (ANCOVA) EXAMPLE USING THE GENERAL LINEAR MODEL PROGRAM

ANCOVA

One Continuous Dependent Variable (DVD Rating) – Interest Rating in DVD One Categorical/Discrete Independent Variable (Promotion) with four levels (Promotion Group 1, 2, 3, and 4) One Continuous Covariate (Age) – Actual Age of Consumer

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 DVDRating 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 four levels of the between-subjects factors used in the ANCOVA.

Between-Subjects Factors

25252525

1234

PromotionGroup

N

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 (.969) > � (.05)], so the assumption of homogeneity of regression slopes has been met.

Tests of Between-Subjects Effects

Dependent Variable: Interest Rating in DVD

1667.436a 7 238.205 5.218 .00017079.570 1 17079.570 374.163 .000

193.595 3 64.532 1.414 .244231.690 1 231.690 5.076 .027

11.363 3 3.788 .083 .9694199.564 92 45.647

126276.000 1005867.000 99

SourceCorrected ModelInterceptPromotionAge

Promotion * AgeErrorTotalCorrected Total

Type III Sumof Squares df Mean Square F Sig.

R Squared = .284 (Adjusted R Squared = .230)a.

ANCOVA EXAMPLE PAGE 2

ANCOVA Syntax to test the Assumption of Homogeneity of Variance, Linear Relationship between the Covariate and the Dependent Variable, and the Main Hypothesis UNIANOVA DVDRating BY Promotion WITH Age /METHOD = SSTYPE(3) /INTERCEPT = INCLUDE /PLOT = PROFILE( Promotion ) /EMMEANS = TABLES(Promotion) WITH(Age=MEAN) /PRINT = DESCRIPTIVE HOMOGENEITY /CRITERIA = ALPHA(.05) /DESIGN = Age Promotion . Syntax for ANCOVA to test the main hypothesis Univariate Analysis of Variance This first table identifies the four levels of the between-subjects factors used in the ANCOVA.


25252525

1234

PromotionGroup

N

The following table provides the UNADJUSTED group means and standard deviations.

Descriptive Statistics


30.68 6.713 2539.80 6.708 2531.56 6.272 2536.76 7.513 2534.70 7.698 100

Promotion Group1234Total

Mean Std. Deviation N

The following table is the Levene’s Test of Homogeneity of Variance. As we can see – this assumption is met since p (.995) > � (.05).

Levene's Test of Equality of Error Variancesa


.022 3 96 .995F df1 df2 Sig.

Tests the null hypothesis that the error variance ofthe dependent variable is equal across groups.

Design: Intercept+Age+Promotiona.


If the Assumption of Homogeneity of Variance had not be met (found significant) – this is not a major 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).

The following table actually serves two purposes… First, we use it to test if there is a linear relationship between the covariate and the dependent variable. As we can see – there is a (significant) linear relationship between the covariate (Age) and the dependent variable (DVD Rating) since p (.020) < α (.05).



1656.073a 4 414.018 9.340 .00017505.917 1 17505.917 394.940 .000

249.233 1 249.233 5.623 .0201323.306 3 441.102 9.951 .0004210.927 95 44.326

126276.000 1005867.000 99

SourceCorrected ModelIntercept

AgePromotionErrorTotalCorrected Total



The following table is the test of the main hypothesis… Here we see that the Promotion Group Main Effect is significant [p (.000) < � (.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.



1656.073a 4 414.018 9.340 .00017505.917 1 17505.917 394.940 .000

249.233 1 249.233 5.623 .020

1323.306 3 441.102 9.951 .0004210.927 95 44.326

126276.000 1005867.000 99

SourceCorrected ModelInterceptAge

PromotionErrorTotalCorrected Total



The covariate is included in the analysis to control for differences on this variable and is not the focus of the main analysis (it is used to test the linear relationship between the covariate and the dependent variable as noted above). Consequently, the results of the covariate are frequently not reported in a Results section.


Since we found a significant between-subjects main effect, we will want to calculate the measure of association, omega squared (ω2). Calculating the measure of association (omega squared) for the ANCOVA is very similar to that for the One-Way ANOVA. We only need to make a few minor adjustments to the formula – to account for the adjusted values of interest…

''

''2 )1(

WT

WB

MSSSMSKSS

+−−

=ω

For our example – we substitute into the formula and get:

201363.326.5911325.1190

326.5911978.132303.1323

326.5911326.44)3(303.1323

326.44000.5867326.44)14(306.13232 ==−=−=

+−−=ω

ω2 = .20, which means that the four levels of promotion group (the independent variable) account for approximately 20% of the total variance in the individual’s interest rating of the DVD (the dependent variable) controlling for the effect of the actual age of the individuals (the covariate).

Estimated Marginal Means The following table shows the adjusted group means… These means are adjusted for the covariate.

Promotion Group


30.883a 1.334 28.234 33.53239.882a 1.332 37.238 42.52731.695a 1.333 29.050 34.34136.339a 1.343 33.672 39.006

Promotion Group1234

Mean Std. Error Lower Bound Upper Bound95% Confidence Interval

Covariates appearing in the model are evaluated at the followingvalues: Actual Age = 36.28.

a.

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.


Profile Plots

4321

Promotion Group

40

38

36

34

32

30

Est

imat

ed M

argi

nal M

eans

Estimated Marginal Means of Interest Rating in DVD

The Profile Plot will give us a visual picture of what is going on with our study. As we can see the line represents the estimated marginal means for the interest rating in DVD at each of the levels of promotion. These values correspond to those found in the estimated marginal means table. Post hoc Analyses Because we found a significant between-subjects main effect – and there are four 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 DVDRating BY Promotion WITH Age /METHOD = SSTYPE(3) /lmatrix 'Promotion Group 1 vs Promotion Group 2' promotion 1 -1 0 0 /lmatrix 'Promotion Group 1 vs Promotion Group 3' promotion 1 0 -1 0 /lmatrix 'Promotion Group 1 vs Promotion Group 4' promotion 1 0 0 -1 /lmatrix 'Promotion Group 2 vs Promotion Group 3' promotion 0 1 -1 0 /lmatrix 'Promotion Group 2 vs Promotion Group 4' promotion 0 1 0 -1 /lmatrix 'Promotion Group 3 vs Promotion Group 4' promotion 0 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


25252525

1234

PromotionGroup

N



1656.073a 4 414.018 9.340 .00017505.917 1 17505.917 394.940 .000

249.233 1 249.233 5.623 .0201323.306 3 441.102 9.951 .0004210.927 95 44.326

126276.000 1005867.000 99

SourceCorrected ModelInterceptAgePromotionErrorTotalCorrected Total




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

LMATRIXSubcommand 1:Promotion Group 1 vsPromotion Group 2

Identity Matrix

Zero Matrix


Identity Matrix

Zero Matrix


Identity Matrix

Zero Matrix


Identity Matrix

Zero Matrix


Identity Matrix

Zero Matrix


Identity Matrix

Zero Matrix

Contrast Coefficients(L' Matrix)

TransformationCoefficients (M Matrix)Contrast Results (KMatrix)Contrast Coefficients(L' Matrix)





TransformationCoefficients (M Matrix)Contrast Results (KMatrix)

1

2

3

4

5

6


This first set of information provides the pairwise comparison of Promotion Group 1 vs. Promotion Group 2. Custom Hypothesis Tests #1

Note the -8.999 – this is the adjusted mean difference of Promotion Group 1 (M = 30.883) and Promotion Group 2 (M = 39.882). The negative value is simply because of the order (low – high = negative). Typically, we would report the absolute value (i.e., 9.00).

Contrast Results (K Matrix)a

-8.9990

-8.9991.884

.000-12.739

-5.259

Contrast EstimateHypothesized ValueDifference (Estimate - Hypothesized)

Std. ErrorSig.

Lower BoundUpper Bound

95% Confidence Intervalfor Difference

ContrastL1

InterestRating in DVD

DependentVariable

Based on the user-specified contrast coefficients (L') matrix: PromotionGroup 1 vs Promotion Group 2

a.

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, 95) = 22.821, 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 .0083 (�/3 = .05/6 = .0083). A review of the group means shows that Promotion Group 1 (M = 30.883) is significantly lower than Promotion Group 2 (M = 39.882) on their DVD interest levels controlling for age.

Test Results


1011.540 1 1011.540 22.821 .0004210.927 95 44.326

SourceContrast

Error

Sum ofSquares df Mean Square F Sig.

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 = 44.326) which we get from the above table. To calculate the effect size (adjusted Cohen’s d), we use the following formula:


'

''ˆ

error

ki

MS

XXd

−= where 6578.6326.44' ==errorMS

351648.16578.6999.8ˆ ==d = 1.35

This next set of information provides the pairwise comparison of Promotion Group 1 vs. Promotion Group 3. Custom Hypothesis Tests #2 Note the -.812 – this is the adjusted mean difference of Promotion Group 1 (M = 30.883) and Promotion Group 3 (M = 31.695). The negative is simply because of the order (low – high = negative). Typically, we would report the absolute value (i.e., .81).


-.8120

-.8121.883

.667-4.5512.927


Std. ErrorSig.



ContrastL1


DependentVariable


a.

The following table provides the necessary information to determine if the group difference is significant. In this case we see F(1, 95) =.186, p = .667 – indicating that Promotion Group 1 is not significantly different from Promotion Group 3. This is compared to our adjusted alpha level (Bonferroni adjustment) of .0083 (�/3 = .05/6 = .0083). A review of the group means shows that while Promotion Group 1 (M = 30.883) is lower than Promotion Group 3 (M = 31.695) on their DVD interest levels controlling for age, it is not significantly different.

Test Results


8.245 1 8.245 .186 .6674210.927 95 44.326

SourceContrast

Error


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 1 vs. Promotion Group 4. Custom Hypothesis Tests #3 Note the -5.456 – this is the adjusted mean difference of Promotion Group 1 (M = 30.883) and Promotion Group 4 (M = 36.339). The negative is simply because of the order (low – high = negative). Typically, we would report the absolute value (i.e., 5.46).


-5.4560

-5.4561.901

.005-9.231-1.681


Std. ErrorSig.



ContrastL1


DependentVariable


a.

The following table provides the necessary information to determine if the group difference is significant. In this case we see F(1, 95) = 8.233, p = .005 – indicating that Promotion Group 1 is significantly different from Promotion Group 4. This is compared to our adjusted alpha level (Bonferroni adjustment) of .0083 (�/3 = .05/6 = .0083). A review of the group means shows that Promotion Group 1 (M = 30.883) is significantly lower than Promotion Group 4 (M = 36.339) on their DVD interest levels controlling for age.

Test Results


364.941 1 364.941 8.233 .0054210.927 95 44.326

SourceContrast

Error


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 = 44.326) which we get from the above table.

'

''ˆ

error

ki

MS

XXd

−= 819490.6578.6456.5ˆ ==d = .82


This next set of information provides the pairwise comparison of Promotion Group 2 vs. Promotion Group 3. Custom Hypothesis Tests #4 Note the 8.187 – this is the adjusted mean difference of Promotion Group 2 (M = 39.882) and Promotion Group 3 (M = 31.695).


8.1870

8.1871.883

.0004.448

11.925


Std. ErrorSig.



ContrastL1


DependentVariable


a.

The following table provides the necessary information to determine if the group difference is significant. In this case we see F(1, 95) = 18.898, 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 .0083 (�/3 = .05/6 = .0083). A review of the group means shows that Promotion Group 2 (M = 39.882) is significantly higher than Promotion Group 3 (M = 31.695) on their DVD interest levels controlling for age.

Test Results


837.672 1 837.672 18.898 .0004210.927 95 44.326

SourceContrast

Error


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 = 44.326) which we get from the above table.

'

''ˆ

error

ki

MS

XXd

−= 229685.16578.6187.8ˆ ==d = 1.23


This next set of information provides the pairwise comparison of Promotion Group 2 vs. Promotion Group 4. Custom Hypothesis Tests #5 Note the 3.543 – this is the adjusted mean difference of Promotion Group 2 (M = 39.882) and Promotion Group 4 (M = 36.339).


3.5430

3.5431.895

.065-.2197.305


Std. ErrorSig.



ContrastL1


DependentVariable


a.

The following table provides the necessary information to determine if the group difference is significant. In this case we see F(1, 95) = 3.496, p = .065 – indicating that Promotion Group 2 is not significantly different from Promotion Group 4. This is compared to our adjusted alpha level (Bonferroni adjustment) of .0083 (�/3 = .05/6 = .0083). A review of the group means shows that while Promotion Group 2 (M = 39.882) is higher than Promotion Group 4 (M = 36.339) on their DVD interest levels controlling for age, it is not significantly different.

Test Results


154.963 1 154.963 3.496 .0654210.927 95 44.326

SourceContrast

Error




This next set of information provides the pairwise comparison of Promotion Group 3 vs. Promotion Group 4. Custom Hypothesis Tests #6 Note the -4.644 – this is the adjusted mean difference of Promotion Group 3 (M = 31.695) and Promotion Group 4 (M = 36.339). The negative is simply because of the order (low – high = negative). Typically, we would report the absolute value (i.e., 4.64).


-4.6440

-4.6441.898

.016-8.411

-.876


Std. ErrorSig.



ContrastL1


DependentVariable


a.

The following table provides the necessary information to determine if the group difference is significant. In this case we see F(1, 95) = 5.988, p = .016 – indicating that Promotion Group 3 is not significantly different from Promotion Group 4. This is compared to our adjusted alpha level (Bonferroni adjustment) of .0083 (�/3 = .05/6 = .0083). A review of the group means shows that while Promotion Group 3 (M = 31.695) is lower than Promotion Group 4 (M = 36.339) on their DVD interest levels controlling for age, it is not significantly different.

Test Results


265.410 1 265.410 5.988 .0164210.927 95 44.326

SourceContrast

Error



Documents

ANCOVA - SPSS Interpretation - DVD Rating