The one-way multivariate analysis of variance (MANOVA) is an extension of the one-way ANOVA to incorporate two or more dependent variables (i.e., the one-way ANOVA investigates just one dependent variable). Unlike the one-way ANOVA, which tests for differences in the mean values of the dependent variable between the groups of the independent variable, the one-way MANOVA tests for the ‘linear composite’ or vector of the means between the groups of the independent variable. Essentially, it combines the two or more dependent variables to form a ‘new’ dependent variable in such a way as to maximize the differences between the groups of the independent variable. It is between this new, composite variable that you test for statistically significant differences between the groups.

Also, like the one-way ANOVA, it is important to realize that the one-way MANOVA is an *omnibus* test statistic and cannot tell you which specific groups were significantly different from each other; it only tells you that at least two groups were different. Since you may have three, four, five or more groups in your study design, determining which of these groups differ from each other is important.

For example, you could use a one-way MANOVA to determine whether exam performance in maths and English differed based on test anxiety levels amongst students (i.e., your dependent variables would be “maths exam performance” and “English exam performance”, both measured from 0-100, and your independent variable would be “test anxiety level”, which has three groups: “low-stressed students”, “moderately-stressed students” and “highly-stressed students”). As another example, a one-way MANOVA could be used to understand whether there is a difference in enjoyment and fulfilment amongst first year sales graduates based on degree subject (i.e., your dependent variables would be “salary” and “bonuses” and your independent variable would be “degree subject”, which has five groups: “business studies”, “psychology”, “biological sciences”, “engineering” and “law”).

## Assumptions

In order to run a one-way MANOVA, there are 10 assumptions that need to be considered. The first three assumptions relate to your choice of study design and the measurements you chose to make, whilst the remaining seven assumptions relate to how your data fits the one-way MANOVA model. These assumptions are:

- Assumption #1: You have
**two or more dependent variables**that are measured at the**continuous**level. Examples of**continuous variables**include height (measured in centimeters), temperature (measured in °C), salary (measured in US dollars), revision time (measured in hours), intelligence (measured using IQ score), firm size (measured in terms of the number of employees), age (measured in years), reaction time (measured in milliseconds), grip strength (measured in kg), power output (measured in watts), test performance (measured from 0 to 100), sales (measured in number of transactions per month), academic achievement (measured in terms of GMAT score), and so forth.

- Assumption #2: You have
**one independent variable**that consists of**two or more categorical**,**independent groups**. Example independent variables that meet this criterion include ethnicity (e.g., three groups: Caucasian, African American, and Hispanic), physical activity level (e.g., four groups: sedentary, low, moderate and high), profession (e.g., five groups: surgeon, doctor, nurse, dentist, therapist), and so forth. If you are unfamiliar with any of the terms above, you might want to read our Types of Variables guide. Also, if you have two independent variables rather than just one, you should consider a**two-way MANOVA**instead of a one-way MANOVA.

**Note 1:** The “groups” of the independent variable are also referred to as “categories” or “levels”, but the term “levels” is usually reserved for groups that have an order (e.g., fitness level, with three levels: “low”, “moderate” and “high”).

- Assumption #3: You should have
**independence of observations**, which means that there is no relationship between the observations in each group of the independent variable or between the groups themselves. Indeed, an important distinction is made in statistics when comparing values from either different individuals or from the same individuals. Independent groups (in a one-way MANOVA) are groups where there is no relationship between the participants in any of the groups. Most often, this occurs simply by having different participants in each group. For example, if you split a group of individuals into four groups based on their physical activity level (e.g., a “sedentary” group, “low” group, “moderate” group and “high” group), no one in the sedentary group can also be in the high group, no one in the moderate group can also be in the high group, and so forth. As another example, you might randomly assign participants to either a control trial or one of two interventions. Again, no participant can be in more than one group (e.g., a participant in the the control group cannot be in either of the intervention groups). This will be true of any independent groups you form (i.e., a participant cannot be a member of more than one group). In actual fact, the ‘no relationship’ part extends a little further and requires that participants in different groups are considered unrelated, not just different people (e.g., participants might be considered related if they are husband and wife, or twins). Furthermore, participants in one group cannot influence any of the participants in any other group. Independence of observations is largely a study design issue rather than something you can test for, but it is an important assumption of the one-way MANOVA.

**Note:** If you are using the same participants in each group or they are otherwise related, a **one-way repeated measures MANOVA** may be a more appropriate test. If this is the case, please feel free to contact us at info@amstatisticalconsulting.com. We can let you know when we will be able to add an SPSS Statistics guide for the one-way repeated measures MANOVA.

- Assumption #4: There should be no univariate or multivariate outliers. There should be no
**univariate outliers**in each group of the independent variable for any of the dependent variables. 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. In fact, this is a similar assumption to the one-way ANOVA, but for each dependent variable that you have in your MANOVA analysis. We refer to them as**univariate**in this guide to distinguish them from**multivariate outliers**, which you also have to test for. Univariate outliers are scores that are unusual in any group of the independent variable in that their value is extremely small or large compared to the other scores (e.g., 8 participants in a group scored between 60-75 out of 100 in a difficult maths test, but one participant scored 98 out of 100). Outliers can have a large negative effect on your results because they can exert a large influence (i.e., change) on the mean and standard deviation for that group, which can affect the statistical test results. Outliers are more important to consider when you have smaller sample sizes, as the effect of the outlier will be greater. Therefore, in this example, you need to investigate whether the dependent variables, English_Score and Maths_Score, have any univariate outliers for each group of School (i.e., you are testing whether English score and maths score are outlier free for “School A,” “School B” and “School C”, separately).

- Assumption #5: There needs to be multivariate normality. The MANOVA needs the data to be multivariate normal. 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 each of the groups of the independent variable is often used in its place as a best ‘guess’ as to whether there is multivariate normality.

**Explanation:** If there is multivariate normality, there will be normally distributed data (residuals) for each of the groups of the independent variable for all the dependent variables. However, the opposite is not true; normally distributed group residuals do not guarantee multivariate normality.

Therefore, in this example, you need to investigate whether English_Score and Maths_Score are normally distributed for each group of School (i.e., for “School A,” “School B” and “School C”, separately).

- Assumption #6: There should be no multicollinearityIdeally, 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 – one for each dependent variable – rather than a one-way MANOVA. Alternately, if the correlation(s) are too high (greater than 0.9), you could have multicollinearity. This is problematic for MANOVA and needs to be screened out. Whilst there are a great deal of complicated, but sophisticated methods of detecting multicollinearity, we show you the relatively simple method of detecting multicollinearity using Pearson correlation coefficients between the dependent variables to determine if there are any relationships that are too strongly correlated.

- Assumption #7: There should be a linear relationship between the dependent variables for each group of the independent variable. In a one-way MANOVA, there needs to be a linear relationship between each pair of dependent variables for each group of the independent variable. In this example, there is only one pair of dependent variables because there are only two dependent variables, English_Score and Maths_Score. If the variables are not linearly related, the power of the test is reduced (i.e., it can lead to a loss of power to detect differences). You can test whether a linear relationship exists by plotting and visually inspecting a scatterplot matrix for each group of the independent variable, School, to see if a linear relationship exists. If the relationship approximately follows a straight line, you have a linear relationship. However, if you have something other than a straight line, for example, a curved line, you do not have a linear relationship.

- Assumption #8: You should have an adequate sample size. Although the larger your sample size, the better, at a bare minimum, there needs to be as many cases (e.g., pupils) in each group of the independent variable as there are number of dependent variables. In this example, this means that there needs to be more than two cases per group (e.g., at least two pupils in School A, at least two pupils in School B and at least two pupils in School C).Whilst you will probably know whether your data meets this assumption without having to formally test for it, the one-way MANOVA procedure produces the
**Between-Subjects Factors**table that shows the number of cases per group.

- Assumption #9: There should be homogeneity of variance-covariance matrices. A further assumption of the one-way MANOVA is that there are similar variances and covariances. This assumption can be tested using
**Box’s M test of equality of covariance**, which is run as part of the one-way MANOVA procedure. If you contact us, we will show you how to interpret the results of this test, and if your data fails to meet this assumption, ways to proceed.

- Assumption #10: There should be homogeneity of variances. The one-way MANOVA assumes that there are equal variances between the groups of the independent variable, School, for each dependent variable: English_Score and Maths_Score. This can be tested using
**Levene’s test of equality of variances**.

**Note:** If you have violated the assumption of homogeneity of variance-covariance matrices (see **Assumption #9** above), the results from Levene’s test of equality of variances can inform you which dependent variable might be causing the problem (i.e., the dependent variable(s) that have unequal variances).

## Interpreting Results

After running the one-way MANOVA procedure and testing that your data meet the assumptions of a one-way MANOVA in the previous sections, SPSS Statistics will have generated a number of tables that contain all the information you need to report the results of your one-way MANOVA. We show you how to interpret these results.

The one-way MANOVA has two main objectives: (a) to determine whether the groups of the independent variable are statistically significant in terms of the dependent variables; and (b) if so, determine where any differences in the groups of the independent variable lie. Both of these objectives will be answered in the following sections:

- Descriptive statistics: You can start your analysis by getting an overall impression of what your data is showing using descriptive statistics. This includes the mean, standard deviation, and number of cases for all the dependent variables (English_Score and Maths_Score) separately for each group of the independent variable, School, as well as the overall score. Another method is to use the standard error (of the mean) or confidence intervals (usually 95% confidence intervals). Whatever the result of the one-way MANOVA statistical test, you will most likely need to report your descriptive statistics. When doing this, you should refrain from making any conclusions or inferences about the data (save this for your discussion section), but it is helpful to try to provide trends or highlight differences between groups (i.e., which groups of the independent variable had the higher or lower mean scores and if the variation of the dependent variable is similar in each group of the independent variable).
- One-way MANOVA results: In evaluating the main one-way MANOVA results, you can start by determining if there is a statistically significant difference between the groups on both dependent variables (English_Score and Maths_Score). There are four different multivariate statistics that can be used to test the statistical significance of the differences between groups when using SPSS Statistics (i.e.,
**Pillai’s Trace**,**Wilks’ Lambda**,**Hotelling’s Trace**and**Roy’s Largest Root**). If you contact us, we will explain which to choose and how to interpret these statistics. - Univariate one-way ANOVAs and multiple comparisons: If the MANOVA result is statistically significant, you can consider running a post hoc test. There is quite a bit of controversy over how you should follow up a one-way MANOVA. However, this guide is going to take the most straightforward approach – and the default action of SPSS Statistics – of following up the statistically significant result with a univariate one-way ANOVA for each dependent variable (i.e., a one-way ANOVA for the effect of School on English_Score and a one-way ANOVA for the effect of School on Maths_Score). For any of your univariate one-way ANOVAs that are statistically significant, you can follow them up with a Tukey post hoc test (or another multiple comparison procedure of your choosing). If you contact us, we will explain how to interpret these follow-up tests.

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