Two-way MANOVA in SPSS Statistics

Introduction

The two-way multivariate analysis of variance (two-way MANOVA) is often considered as an extension of the two-way ANOVA for situations where there is two or more dependent variables. The primary purpose of the two-way MANOVA is to understand if there is an interaction between the two independent variables on the two or more dependent variables.

For example, you could use a two-way MANOVA to understand whether there were differences in students' short-term and long-term recall of facts based on lecture duration and fact type (i.e., the two dependent variables are "short-term memory recall" and "long-term memory recall", whilst the two independent variables are "lecture duration", which has four groups – "30 minutes", "60 minutes", "90 minutes" and "120 minutes" – and "fact type", which has two groups: "quantitative (numerical) facts" and "qualitative (textual/contextual) facts"). Alternately, you could use a two-way MANOVA to understand whether there were differences in the effectiveness of male and female police officers in dealing with violent crimes and crimes of a sexual nature taking into account a citizen's gender (i.e., the two dependent variables are "perceived effectiveness in dealing with violent crimes" and "perceived effectiveness in dealing with sexual crimes", whilst the two independent variables are "police officer gender", which has two categories – "male police officers" and "female police offices" – and "citizen gender", which also has two categories: "male citizens" and "female citizens").

As mentioned earlier, a two-way MANOVA has generally one primary aim: to understand whether the effect of one independent variable on the dependent variables (collectively) is dependent on the value of the other independent variable. This is called an "interaction effect". However, if no interaction effect is present (usually assessed as whether the interaction effect is statistically significant), you would normally be interested in the "main effects" of each independent variable instead. This is somewhat akin to assessing the effect that an independent variable has on the dependent variables collectively when "ignoring" the value of the other independent variable. On the other hand, if a statistically significant interaction is found, you need to consider an method of following up the result (i.e., what follow-up analyses you may want to run).

In this "quick start" guide, we show you how to carry out a two-way MANOVA using SPSS Statistics, as well as interpret and report the results from this test. However, before we introduce you to this procedure, you need to understand the different assumptions that your data must meet in order for a two-way MANOVA to give you a valid result. We discuss these assumptions next.

SPSS Statistics

Assumptions

When you choose to analyse your data using a two-way MANOVA, part of the process involves checking to make sure that the data you want to analyse can actually be analysed using a two-way MANOVA. You need to do this because it is only appropriate to use a two-way MANOVA if your data "passes" nine assumptions that are required for a two-way MANOVA to give you a valid result. Do not be surprised if, when analysing your own data using SPSS Statistics, one or more of these assumptions is violated (i.e., is not met). This is not uncommon when working with real-world data. However, even when your data fails certain assumptions, there is often a solution to overcome this.

In practice, checking for these nine assumptions adds some more time to your analysis, requiring you to work through additional procedures in SPSS Statistics when performing your analysis, as well as thinking a little bit more about your data. These nine assumptions are presented below:

• Assumption #1: Your two or more dependent variables should be measured at the interval or ratio level (i.e., they are continuous). Examples of continuous variables include revision time (measured in hours), intelligence (measured using IQ score), exam performance (measured from 0 to 100), weight (measured in kg), and so forth. You can learn more about interval and ratio variables in our article: Types of Variable.
• Assumption #2: Your two independent variables should consist of two or more categorical, independent groups. Example independent variables that meet this criterion include ethnicity (e.g., 3 groups: Caucasian, African American and Hispanic), physical activity level (e.g., 4 groups: sedentary, low, moderate and high), profession (e.g., 5 groups: surgeon, doctor, nurse, dentist, therapist), and so forth.
• Assumption #3: You should have independence of observations, which means that there is no relationship between the observations in each group or between the groups themselves. For example, there must be different participants in each group with no participant being in more than one group. This is more of a study design issue than something you can test for, but it is an important assumption of the two-way MANOVA.
• Assumption #4: You should have an adequate sample size. Although the larger your sample size, the better; for MANOVA, you need to have more cases in each group than the number of dependent variables you are analysing.
• Assumption #5: There are no univariate or multivariate outliers. First, there can be no (univariate) outliers in all combinations of groups of your two independent variables for any of the dependent variables. This is a similar assumption to the two-way ANOVA, but for each dependent variable that you have in your MANOVA analysis. Univariate outliers are often just called outliers and are the same type of outliers you will have come across if you have conducted t-tests or ANOVAs. We refer to them as univariate in this guide to distinguish them from multivariate outliers. Multivariate outliers are cases which have an unusual combination of scores on the dependent variables. You can detect univariate outliers in SPSS Statistics using boxplots and check for multivariate outliers using a measure called Mahalanobis distance, which you can also do using SPSS Statistics.
• Assumption #6: There is multivariate normality. Unfortunately, multivariate normality is a particularly tricky assumption to test for and cannot be directly tested in SPSS Statistics. Instead, normality of each of the dependent variables for all combinations of the groups of your two independent variables is often used in its place as a best 'guess' as to whether there is multivariate normality. You can test for this using the Shapiro-Wilk test of normality, which is easily tested for using SPSS Statistics.
• Assumption #7: There is a linear relationship between each pair of dependent variables for all combinations of groups of your two independent variables. If the variables are not linearly related, the power of the test is reduced. You can test for this assumption by plotting a scatterplot matrix for all combinations of group of your two independent variables. In order to do this, you will need to split your data file in SPSS Statistics before generating the scatterplot matrices.
• Assumption #8: There is homogeneity of variance-covariance matrices. You can test this assumption in SPSS Statistics using Box's M Test of Equality of Covariance Matrices. Despite the importance of this assumption for a two-way MANOVA, if your data fails this assumption, there are a number of ways to proceed with your analysis.
• Assumption #9: There is no multicollinearity. Ideally, you want your dependent variables to be moderately correlated with each other. If the correlations are low, you might be better off running separate one-way ANOVAs, and if the correlation(s) are too high (greater than 0.9), you could have a multicollinearity problem. This is problematic for MANOVA and needs to be screened out.

You can check assumptions #5, #6, #7, #8 and #9 using SPSS Statistics. Before doing this, you should make sure that your data meets assumptions #1, #2, #3 and #4, although you don't need SPSS Statistics to do this. Just remember that if you do not run the statistical tests on these assumptions correctly, the results you get when running a two-way MANOVA might not be valid.

In the section, Test Procedure in SPSS Statistics, we illustrate the SPSS Statistics procedure to perform a two-way MANOVA assuming that no assumptions have been violated. First, we set out the example we use to explain the two-way MANOVA procedure in SPSS Statistics.