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

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1 EPS 6 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 () with four levels ( Group,,, and ) 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 WITH Age /METHOD = SSTYPE() /INTERCEPT = INCLUDE /CRITERIA = ALPHA(.05) /DESIGN = Age Age*. Univariate Analysis of Variance This first table identifies the four levels of the between-subjects factors used in the ANCOVA. Between-Subjects Factors Group N This analysis is done to check the assumption of homogeneity of regression slopes, not to test the main hypothesis. The factor ( Group) and covariate (Actual Age) do not interact [p (.969) > (.05)], so the assumption of homogeneity of regression slopes has been met. Corrected Model Intercept Age * Age Total Corrected Total Tests of Between-Subjects Effects Type III Sum of Squares df Mean Square F Sig a a. R Squared =.8 (Adjusted R Squared =.0)

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 WITH Age /METHOD = SSTYPE() /INTERCEPT = INCLUDE /PLOT = PROFILE( ) /EMMEANS = TABLES() WITH(Age=MEAN) /PRINT = DESCRIPTIVE HOMOGENEITY /CRITERIA = ALPHA(.05) /DESIGN = Age. 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. Between-Subjects Factors Group N The following table provides the UNADJUSTED group means and standard deviations. Descriptive Statistics Group Total 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 Variances a F df df Sig Tests the null hypothesis that the error variance of the dependent variable is equal across groups. a. Design: Intercept+Age+ PAGE

3 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 ½ 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, 005). 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 (.00) < α (.05). Corrected Model Intercept Age Total Corrected Total Tests of Between-Subjects Effects Type III Sum of Squares df Mean Square F Sig a a. R Squared =.8 (Adjusted R Squared =.) The following table is the test of the main hypothesis Here we see that the 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. Corrected Model Intercept Age Total Corrected Total Tests of Between-Subjects Effects Type III Sum of Squares df Mean Square F Sig a a. R Squared =.8 (Adjusted R Squared =.) 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. PAGE

4 Since we found a significant between-subjects main effect, we will want to calculate the measure of association, omega squared (ω ). 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 ω ' SS B ( K ) = ' ' SST + MSW For our example we substitute into the formula and get:.06 ( ).6.0 () ω = = = = MS ' W =.06 ω =.0, which means that the four levels of promotion group (the independent variable) account for approximately 0% 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. Group Group Mean Std. Lower Bound Upper Bound 0.88 a a a a a. Covariates appearing in the model are evaluated at the following values: Actual Age = 6.8. 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. PAGE

5 Profile Plots Estimated Marginal Means of Interest Rating in DVD 0 Estimated Marginal Means Group 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). PAGE 5

6 Syntax for the lmatrix command UNIANOVA DVDRating BY WITH Age /METHOD = SSTYPE() /lmatrix ' Group vs Group ' promotion /lmatrix ' Group vs Group ' promotion 0-0 /lmatrix ' Group vs Group ' promotion /lmatrix ' Group vs Group ' promotion 0-0 /lmatrix ' Group vs Group ' promotion /lmatrix ' Group vs Group ' promotion 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 Group N Corrected Model Intercept Age Total Corrected Total Tests of Between-Subjects Effects Type III Sum of Squares df Mean Square F Sig a a. R Squared =.8 (Adjusted R Squared =.) PAGE 6

7 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 5 6 Coefficients (L' Matrix) Transformation Coefficients (M Matrix) Results (K Matrix) Coefficients (L' Matrix) Transformation Coefficients (M Matrix) Results (K Matrix) Coefficients (L' Matrix) Transformation Coefficients (M Matrix) Results (K Matrix) Coefficients (L' Matrix) Transformation Coefficients (M Matrix) Results (K Matrix) Coefficients (L' Matrix) Transformation Coefficients (M Matrix) Results (K Matrix) Coefficients (L' Matrix) Transformation Coefficients (M Matrix) Results (K Matrix) LMATRIX Subcommand : Group vs Group Identity Matrix Zero Matrix LMATRIX Subcommand : Group vs Group Identity Matrix Zero Matrix LMATRIX Subcommand : Group vs Group Identity Matrix Zero Matrix LMATRIX Subcommand : Group vs Group Identity Matrix Zero Matrix LMATRIX Subcommand 5: Group vs Group Identity Matrix Zero Matrix LMATRIX Subcommand 6: Group vs Group Identity Matrix Zero Matrix PAGE 7

8 This first set of information provides the pairwise comparison of Group vs. Group. Custom Hypothesis Tests # Note the this is the adjusted mean difference of Group (M = 0.88) and Group (M = 9.88). The negative value is simply because of the order (low high = negative). Typically, we would report the absolute value (i.e., 9.00). L Estimate Hypothesized Value Results (K Matrix) a Difference (Estimate - Hypothesized) Std. Sig. for Difference Lower Bound Upper Bound Dependent Variable Interest Rating in DVD a. Based on the user-specified contrast coefficients (L') matrix: Group vs Group 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(, 95) =.8, p <.00 indicating that Group is significantly different from Group. This is compared to our adjusted alpha level (Bonferroni adjustment) of.008 (/ =.05/6 =.008). A review of the group means shows that Group (M = 0.88) is significantly lower than Group (M = 9.88) on their DVD interest levels controlling for age. Test Results Sum of Squares 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 =.6) which we get from the above table. To calculate the effect size (adjusted Cohen s d), we use the following formula: PAGE 8

9 ' ' ˆ X i X k ' d = where MS error =.6 = MS ' error d ˆ = =.568 = This next set of information provides the pairwise comparison of Group vs. Group. Custom Hypothesis Tests # Note the -.8 this is the adjusted mean difference of Group (M = 0.88) and Group (M =.695). The negative is simply because of the order (low high = negative). Typically, we would report the absolute value (i.e.,.8). L Estimate Hypothesized Value Results (K Matrix) a Difference (Estimate - Hypothesized) Std. Sig. for Difference Lower Bound Upper Bound Dependent Variable Interest Rating in DVD a. Based on the user-specified contrast coefficients (L') matrix: Group vs Group.97 The following table provides the necessary information to determine if the group difference is significant. In this case we see F(, 95) =.86, p =.667 indicating that Group is not significantly different from Group. This is compared to our adjusted alpha level (Bonferroni adjustment) of.008 (/ =.05/6 =.008). A review of the group means shows that while Group (M = 0.88) is lower than Group (M =.695) on their DVD interest levels controlling for age, it is not significantly different. Test Results Sum of Squares df Mean Square F Sig Because no significant difference was found for these two groups no Effect Size needs to be calculated. PAGE 9

10 This next set of information provides the pairwise comparison of Group vs. Group. Custom Hypothesis Tests # Note the this is the adjusted mean difference of Group (M = 0.88) and Group (M = 6.9). The negative is simply because of the order (low high = negative). Typically, we would report the absolute value (i.e., 5.6). L Estimate Hypothesized Value Results (K Matrix) a Difference (Estimate - Hypothesized) Std. Sig. for Difference Lower Bound Upper Bound Dependent Variable Interest Rating in DVD a. Based on the user-specified contrast coefficients (L') matrix: Group vs Group The following table provides the necessary information to determine if the group difference is significant. In this case we see F(, 95) = 8., p =.005 indicating that Group is significantly different from Group. This is compared to our adjusted alpha level (Bonferroni adjustment) of.008 (/ =.05/6 =.008). A review of the group means shows that Group (M = 0.88) is significantly lower than Group (M = 6.9) on their DVD interest levels controlling for age. Test Results Sum of Squares 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 =.6) which we get from the above table. ' ' ˆ X i X k 5.56 d = d ˆ = = =.8 MS ' error PAGE 0

11 This next set of information provides the pairwise comparison of Group vs. Group. Custom Hypothesis Tests # Note the 8.87 this is the adjusted mean difference of Group (M = 9.88) and Group (M =.695). L Estimate Hypothesized Value Results (K Matrix) a Difference (Estimate - Hypothesized) Std. Sig. for Difference Lower Bound Upper Bound Dependent Variable Interest Rating in DVD a. Based on the user-specified contrast coefficients (L') matrix: Group vs Group The following table provides the necessary information to determine if the group difference is significant. In this case we see F(, 95) = 8.898, p <.00 indicating that Group is significantly different from Group. This is compared to our adjusted alpha level (Bonferroni adjustment) of.008 (/ =.05/6 =.008). A review of the group means shows that Group (M = 9.88) is significantly higher than Group (M =.695) on their DVD interest levels controlling for age. Test Results Sum of Squares 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 =.6) which we get from the above table. ' ' ˆ X i X k 8.87 d = d ˆ = = =. MS ' error PAGE

12 This next set of information provides the pairwise comparison of Group vs. Group. Custom Hypothesis Tests #5 Note the.5 this is the adjusted mean difference of Group (M = 9.88) and Group (M = 6.9). L Estimate Hypothesized Value Results (K Matrix) a Difference (Estimate - Hypothesized) Std. Sig. for Difference Lower Bound Upper Bound Dependent Variable Interest Rating in DVD a. Based on the user-specified contrast coefficients (L') matrix: Group vs Group The following table provides the necessary information to determine if the group difference is significant. In this case we see F(, 95) =.96, p =.065 indicating that Group is not significantly different from Group. This is compared to our adjusted alpha level (Bonferroni adjustment) of.008 (/ =.05/6 =.008). A review of the group means shows that while Group (M = 9.88) is higher than Group (M = 6.9) on their DVD interest levels controlling for age, it is not significantly different. Test Results Sum of Squares df Mean Square F Sig Because no significant difference was found for these two groups no Effect Size needs to be calculated. PAGE

13 This next set of information provides the pairwise comparison of Group vs. Group. Custom Hypothesis Tests #6 Note the -.6 this is the adjusted mean difference of Group (M =.695) and Group (M = 6.9). The negative is simply because of the order (low high = negative). Typically, we would report the absolute value (i.e.,.6). L Estimate Hypothesized Value Results (K Matrix) a Difference (Estimate - Hypothesized) Std. Sig. for Difference Lower Bound Upper Bound Dependent Variable Interest Rating in DVD a. Based on the user-specified contrast coefficients (L') matrix: Group vs Group The following table provides the necessary information to determine if the group difference is significant. In this case we see F(, 95) = 5.988, p =.06 indicating that Group is not significantly different from Group. This is compared to our adjusted alpha level (Bonferroni adjustment) of.008 (/ =.05/6 =.008). A review of the group means shows that while Group (M =.695) is lower than Group (M = 6.9) on their DVD interest levels controlling for age, it is not significantly different. Test Results Sum of Squares df Mean Square F Sig Because no significant difference was found for these two groups no Effect Size needs to be calculated. PAGE

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