Multiple Analysis of Variance – MANOVA Joe F. Hair, Jr. Kennesaw State University For more details, see Chapter 7, Hair, et al, Multivariate Data Analysis, 7e, 2010. What is MANOVA? Concepts • A statistical method for testing whether the vector of means (variate) • • • • • across groups on multiple variables are equal (i.e., the probability that any differences in the variate means across several groups are due solely to sampling error). Variables in ANOVA (Analysis of Variance): Dependent variables are metric. Independent variable(s) is nominal with two or more levels – also called treatment, manipulation, or factor. One-way MANOVA: only one independent variable with two or more levels. Two-way MANOVA: two independent variables each with two or more levels (factorial design). One or more could be control variables. With MANOVA (Multiple Analysis of Variance), two or more metric dependent variables are tested as the outcome of a treatment(s). With MANOVA there are two variates – one for the dependent variables and another for the independent variables. The variate optimally combines the multiple dependent measures into a single value that maximizes the differences across the groups. MANOVA Concepts continued . . . • Factorial design – a research design that has two or more non-metric • • • independent variables (factors or treatments). Control (blocking) variables – an independent variable not of interest to the study that may be a source of differences that may obscure some results of interest to the study. Metric control variables – are independent variables you wish to control for in the analysis. They are included a covariates (ANCOVA & MANCOVA). Any time your analysis includes more than one independent variable interaction effects are created. Null and Alternative Hypotheses H0: The vectors of means on multiple dependent variables across groups are the same (equal). Ha: The vectors of means on multiple dependent variables for at least one pair of groups are different. How do you determine which means are significantly different? • The Hotelling’s T2 assesses whether you can conclude that statistical differences are present somewhere between the variates of means across the groups. • To identify where the differences are between each of the groups (when there are three or more groups) you must use follow-up (post-hoc) tests called “multiple comparison tests”. Many multiple comparison tests are available in SPSS. MANOVA Terms Main Effect = the impact any single experimental variable has on a response (dependent) variable. Interaction Effect = the combined impact of multiple independent variables on a response variable; i.e., is the difference in the mean ratings of the ads (response variable) the same when we compare males and females? Blocking Variable = a grouping variable the researcher doesn’t manipulate or control in any way, such as gender. What multiple comparison tests are available in SPSS? Games-Howell recommended if equal variances not assumed. Scheffe recommended if equal variances assumed. What assumptions need to be considered? • Samples are independent (independence of observations). • Dependent variables are normally distributed for each of the • • • • • samples – with larger sample sizes ( > 20/group) not a serious problem should this be violated somewhat. Dependent variables must exhibit multicollinearity. Whether the sample sizes for the groups are very different (a ratio of 1.5 or higher across groups may be a problem). The variances for the different populations from which the samples are drawn are equal (homoscedasticity = homogeneity of the variance-covariance matrices among the groups) – possibly a problem if they are not equal or at least comparable. Outliers need to be identified and removed from the sample, or other adjustments undertaken. In general MANOVA is a fairly robust procedure. HBAT DATABASE VARIABLES Example 1: Two Group MANOVA HBAT 200 Database (MDA, pp. 383-388) • • • Independent variables are: X5 (Distribution System), and X1 (Customer Type). Dependent variables are: X19 (Satisfaction), X20 (Likelihood of Recommending HBAT), and X21 (Likelihood of Future Purchases) Research questions are: 1. What differences are present in customer satisfaction and other purchase outcomes between the two channels in the distribution system? 2. Is HBAT establishing better relationships with its customers over time, as reflected in customer satisfaction and other purchase outcomes? 3. What is the relationship between the distribution system and these relationships with customers in terms of the purchase outcomes? USING SPSS TO DO A Two Group MANOVA The metric dependent variables for this hypothesis are X19 (Satisfaction), X20 (Likelihood of Recommending HBAT), and X21 (Likelihood of Future Purchases). The nonmetric independent variable is X5 (Distribution System). The click-through sequence to run the two group MANOVA is: ANALYZE GENERAL LINEAR MODEL UNIVARIATE. Click on X19, X20 and X21 to highlight them and then on the arrow box to move them into the Dependent Variable box. Click on X5 to highlight it and then on the arrow box to move it to the box labeled “Fixed Factors.” Click on the Options tab and highlight the (OVERALL) and X5 in the Factors and Factor Interactions window, and move them into the Display Means for: window. Next check the Compare main effects box and then in the Display window below click on Descriptive statistics, Observed power and Homogeneity Tests (Levene test of equal variances) and then Continue. Now go to the Plots tab and highlight X5 and move it to the Horizontal Axis window and then under Plots below click Add. Finally, click on Continue and then OK to execute the program. 10 XXX Differences between Two Independent Groups X5 (Distribution System) XXX Differences between Two Independent Groups X5 (Distribution System) 0 = Indirect through Broker 1 = Direct to Customer XXX XXX Differences between Two Independent Groups X5 (Distribution System) 0 = Indirect through Broker 1 = Direct to Customer Higher means = more favorable ratings (0-10 scale). Direct to Customer consistently rated higher. Assessing Assumptions – Homoscedasticity Levene’s Test is a univariate assessment of homoscedasticity. All three metric dependent variables are not significant, indicating univariate homogeneity of variance across the two groups. Assessing Assumptions – Homoscedasticity Box’s M is a multivariate assessment of homoscedasticity. The non-significant results indicate multivariate homogeneity of variance across the two groups. Assessing Assumptions – Outliers The three dependent variables were examined for outliers using the SPSS Explore and Box Plots options. These three plots identify several outliers, but none was an outlier for all three variables. Respondent 38 was an outlier twice for the Direct to Customer group and was considered for removal, but a decision was made to retain them. The outliers are at the high end of the distribution, individuals that are very favorable about HBAT. But overall the observations are about equally above and below the mode. Assessing Assumptions – Normality Histograms The histograms below show the data for X19 – Satisfaction divided by the two distribution groups (X5). There is some deviation from normal but for social science empirical data this type of deviations is typical. Similar charts were completed for variables X20 and X21 and the overall conclusion was the deviations from normal were not sufficient to justify data transformations. Assessing Assumptions – Normality Normal Q-Q Plot A Q-Q plot displays observed values against a known distribution, in this case a normal distribution. If the sample distribution is normal, the plot will have observations distributed closely around the straight line. In the above chart, the expected normal distribution is the straight line angled at 45 degrees and the line of little circles is the observed values from the HBAT sample data for X19 – Satisfaction (i.e., the actual sample distribution shown as it deviates from the straight line) . The plot shows the distribution is pretty much normal. Assessing Assumptions – Normality Detrended Normal Q-Q Plot The Detrended Normal Q Q plot, shows the differences between the observed and expected values of a normal distribution. If the distribution is normal, the points should cluster in a horizontal band around zero with no pattern. The charts above suggest some deviation from normal, particularly the left chart that has an up and down pattern . Our overall conclusion is that this distribution is close enough to treat as a normal distribution, and similar to most empirical data in social science research. Assessing Assumptions – Normality Statistical Tests The above tests of normality can be obtained from the Explore option. Both tests calculate the level of significance for the differences from a normal distribution. The tests are not useful for very small samples (N = <30), quite sensitive to larger samples (N = 1,000+), and interpreted cautiously with samples that range from N = 200 to 1,000. The tests for variables X20 and X21 were both not significant, indicating no issues with normality. Both tests for X19 are significant but based on other information the deviations from normality for this variable are acceptable and do not require transformation. XXX Running Explore – SPSS Steps XXX Statistics and Plots tabs What to check? Multivariate Statistical Testing Question: Do the two groups exhibit statistically significant differences on the three purchase outcome variables? Yes The four multivariate tests all confirm statistically significant differences between the two types of distribution. These four tests assess the set of three variables combined. The univariate tests below assess each variable separately and all indicate statistically significant differences. See means on slide 12 of this presentation. The power for the statistical tests was 1.0, indicating the sample sizes and effect size were sufficient to support significant differences if detected. Overall Interpretation of MANOVA Results Differences between Two Independent Groups X5 (Distribution System) 0 = Indirect through Broker 1 = Direct to Customer The results summarized in the previous slides on variable X5 confirm that the type of distribution channel does affect customer perceptions with regard to the three purchase outcomes. The statistically significant differences, that are of sufficient magnitude to base managerial decisions on, and the consistently more favorable evaluations (higher means) on the purchase outcomes variables (X19, X20 & X21), indicate that the direct distribution channel is more effective in creating positive customer perceptions on a wide range of purchase outcomes. Differences between Three Independent Groups X1 (Length of Time a Customer) 1 = Less than 1 Year 2 = 1 to 5 Years 3 = Over 5 Years XXX XXX The sample sizes are almost equally split between the groups. The group means are larger for longer term customers indicating more favorable purchase outcome perceptions. The multivariate test of homoscedasticity above (Box’s M) indicates that heteroscedasticity is not present (lack of significance = .069). The Levene’s univariate test for equality of error variances indicates two variables (X20 & X21) are not significant = homoscedasticity, but X19 is highly significant (.001) indicating that heteroscedasticity is likely present in that variable. Given the relatively large sample sizes in all the groups and homoscedasticity for variables X20 and X21, corrective remedies are not necessary for X19. MANOVA Results Differences between Three Independent Groups X1 (Length of Time a Customer) 1 = Less than 1 Year 2 = 1 to 5 Years 3 = Over 5 Years Interpreting MANOVA results when an independent variable has three or more levels requires a two-step process: 1.Examination of the main effect of the independent variable on the dependent variables. The two tests on the next slide (#28) – Multivariate Tests and Univariate Tests indicate there is a significant main effect across the three groups on the dependent variables. 2.Identifying differences, if any, between the three dependent variables for the three or more individual groups of the independent variable. The post hoc tests on slide #29 indicate significant differences between all three groups on variable X19, but for variables X20 and X21 the differences between the 1 to 5 Years and More than 5 Years groups are not statistically different. This second level of analysis is necessary when an independent variable has three or more groups. Main Effects Tests The four multivariate tests all confirm statistically significant differences on the three dependent variables combined (as a variate) between the three groups based on length of time a customer. The four tests assess the set of three dependent variables combined. The univariate tests above assess each variable separately across all groups. The results indicate statistically significant differences between the three groups. See means on previous slide. The post hoc tests below assess each variable separately and between all combinations of the three groups. Results indicate statistically significant differences between all three groups for variable X19, but the 1 – 5 Years and Over 5 Years groups for variables X20 and X21 are not significantly different. MANOVA Results Differences between Three Independent Groups X1 (Length of Time a Customer) 1 = Less than 1 Year 2 = 1 to 5 Years 3 = Over 5 Years The results summarized in the previous slides on variable X1 confirm that the Length of Time a firm is a customer does affect customer perceptions with regard to the three purchase outcomes. The overall main effects measured using both multivariate and univariate tests are statistically different. Moreover, the consistently more favorable evaluations (higher means) on the three purchase outcomes variables (X19, X20 & X21) indicate that longer term customers are more favorable. But when examined individually we see that for X20 and X21 the differences are not statistically significantly different between the 1 – 5 and More than 5 Years customer groups. Factorial Design – MANOVA with Two Independent Variables X1 (Length of Time a Customer) and X5 (Distribution System) XXX MANOVA Example – 2 x 3 Design Factorial Design with Two Independent variables X1 and X5 Research Questions: 1. What are the main effects of the two independent variables? 2. What are the interaction effects? Sample Size Considerations The sample sizes are generally OK with the exception of Direct to Customer for Less than 1 year (N = 16). This means statistical significance related to this group must be interpreted cautiously. Homoscedasticity Tests As with prior MANOVA examples, the assumption of greatest importance is the homogeneity of variance-covariance matrices across groups. The multivariate test of homoscedasticity (Box’s M) indicates that heteroscedasticity is not present (lack of significance = .153). The Levene’s univariate test for equality of error variances indicates two variables (X20 & X21) are not significant (.113 & .425) = homoscedasticity. Variable X19 is not significant (.059) either indicating that heteroscedasticity is likely not present in any of the three variables. The MANOVA model for a factorial design tests not only for the main effects of both independent variables, but also their interaction or joint effect on the dependent variables. The first step is to examine the interaction effect and determine whether it is statistically significant. If it is significant, then the researcher must confirm that the interaction effect is ordinal. If it is found to be disordinal, the statistical tests of main effects are not valid. But assuming a significant ordinal or a nonsignificant interaction effect, the main effects can be interpreted directly without adjustment. Interaction effects can be identified both graphically and statistically. The most common graphical means is to create line charts depicting pairs of independent variables. As illustrated in the graphs on the following slides, significant interaction effects are represented by nonparallel lines (with parallel lines denoting no interaction effect). If the lines depart from parallel but never cross in a significant amount, then the interaction is deemed ordinal. If the lines do cross to the degree that in at least one instance the relative ordering of the lines is reversed, then the interaction is deemed disordinal. The next slide portrays each dependent variable across the six groups, indicating by the nonparallel pattern that an interaction may exist. As we can see in each graph, the middle level of X1 (1 to 5 years with HBAT) has a substantially smaller difference between the two lines (representing the two distribution channels) than the other two levels of X1. We can confirm this observation by examining the group means from slide #34. Using X19 (Satisfaction) as an example, we see that the difference between direct and indirect distribution channels is 1.138 for customers of less than 1 year, which is quite similar to the difference between channels (1.325) for customers of greater than 5 years. However, for customers served by HBAT from 1 to 5 years, the difference between customers of the two channels is only (.285). Thus, the differences between the two distribution channels, although found to be significant in earlier examples, can be shown to differ (interact) based on how long the customer has been with HBAT. The interaction is deemed ordinal because in all instances the direct distribution channel has higher satisfaction scores. Graphical Displays of Interaction Effects of Purchase Outcomes (X19, X20 & X21) Across Groups (X1 & X5) Testing Interaction and Main Effects Effect size is much smaller for interaction than for independent variables (X1 & X5). The above table contains the MANOVA results for testing both the interaction and main effects. To test for a significant interaction effect you first examine the multivariate effects and in this case all four tests are statistically significant. Next (see next slide) , univariate tests for each dependent variable are examined – results show the interaction effect is significant for each of the three dependent variables. The statistical tests confirm what was shown in the graphs: A significant ordinal interaction effect occurs between X5 and X1. Note: normality and presence of outliers for the variables included in this example were examined and determined to be within acceptable ranges. Testing Main Effects – Univariate η2 Effect Overall Interaction Effect All relationships tested above were significant. Effect size is smaller for interaction than for independent variables (X1 & X5). Estimating Main Effects If the interaction effect is deemed nonsignificant or even significant and ordinal, then the next step examines the significance of the main effects for their differences across the groups. If a disordinal interaction effect (non-parallel lines) is found, the main effects are confounded by the disordinal interaction and tests for differences should not be performed. With a significant ordinal interaction (parallel lines), the next step is to determine whether both independent variables still have significant main effects when considered simultaneously. The previous slides show the MANOVA results for the main effects of X1 and X5 and the tests for the interaction effect. X1 (Customer Type) and X5 (Distribution System) have a significant impact (main effect) on the three purchase outcome variables, both as a set (multivariate) and separately (univariate). The impact of the two independent variables can be compared by examining the relative effect sizes as shown by η2 (eta squared). The effect sizes for each variable are somewhat higher for X1 when compared to X5 on either the multivariate or univariate tests. For example, with the multivariate tests the eta squared values for X1 range from .244 to .488, but they are lower (all equal to .285) for X5. Similar patterns can be seen on the univariate tests. This comparison gives an evaluation of practical significance separate from the statistical significance tests. When compared to either independent variable, however, the effect size attributable to the interaction effect is much smaller (e.g., multivariate eta squared values ranging from .062 to .101). Interpretation of Results Interaction of X1 by X5 The non-parallel lines for each dependent measure are shown by the marked narrowing of the differences in distribution channels for customers of one to five years (see slide #38). While the effects of X1 and X5 are still present, there are marked differences in these impacts depending on which specific sets of customers we examine – direct distribution versus broker. Main Effect of X1 This is illustrated for all three purchase outcomes by the upward sloping lines across the three levels of X1 on the x-axis. The effects are consistent with earlier findings in that all three purchase outcomes increase favorably as the length of the relationship with HBAT increases. Main Effect of X5 Shown by the separation of the two lines representing the two distribution channels, with the direct distribution curve being consistently higher. Thus, the direct distribution channel consistently generates more favorable purchase outcomes.