# One-way repeated measures MANOVA in SPSS Statistics

## Introduction

A **one-way repeated measures multivariate analysis of variance** (i.e., the **one-way repeated measures MANOVA**), also referred to as a **doubly multivariate MANOVA**, is used to determine whether there are any **differences** in **multiple dependent variables** over **time** or between **treatments**, where participants have been measured at **all** time points or taken part in **all** treatments. To learn more about these two types of **study design** where the one-way repeated measures MANOVA is commonly used (i.e., examining "differences over time" and "differences between treatments"), see the examples below:

Note: The one-way repeated measures MANOVA can be thought of as an **extension** to the one-way repeated measures ANOVA, which is used when you only have one dependent variable or are interested in analysing only one dependent variable at a time, or as the **within-subjects** (i.e., repeated measures) version of the **between-subjects** one-way MANOVA, which is used when you are interested in differences between groups that are **independent/unrelated** rather than groups that are **related**.

- Differences over
**TIME** - Differences between
**TREATMENTS**

Imagine that a broadband provider wants to determine whether introducing a new bonus scheme for its sales staff will increase their levels of organisational commitment, where increased organisational commitment generally results in employees staying at an organisation longer and being relatively more productive. Therefore, 30 sales staff are randomly selected to take part in a study where the new bonus scheme is applied to their employment contract. All other things being equal, management at the broadband provider consider the new bonus scheme to be more generous overall, and hope this will increase levels of organisational commitment.

The organisational commitment of the 30 sales staff is measured at three time points: (1) just before the new bonus scheme is introduced; (2) one month after the new bonus scheme is introduced, when the sales staff will receive their first bonuses based on their sales results; and (3) six months after the new bonus scheme is introduced, when sales staff receive their bi-annual reviews. These three time points are designed to allow management at the broadband provider to assess how the new bonus scheme might affect organisational commitment amongst sales staff in the short- and medium-term. At each of these three time points – "pre-bonus", "short-term review" (at Month 1) and "medium-term review" (at Month 6) – three types of organisational commitment are measured using a questionnaire, known as "affective commitment", "continuance commitment" and normative commitment", which each assess different aspects of organisational commitment and are seen a good measures of an employee's overall organisational commitment. The questionnaire provides three scores for each of the 30 sales staff: one score showing their "affective commitment", one score showing their "continuance commitment", and another score showing their "normative commitment".

Therefore, in this experiment "organisational commitment" is assessed using three dependent variables, "affective commitment", "continuance commitment" and normative commitment", and the independent variable is "time", with three related groups: "pre-bonus", "short-term review" and "medium-term review". The three groups are related because all 30 sales staff were measured in terms of each dependent variable at all three time points, so each employee has nine scores: one for each of the three dependent variables at each time point (i.e., 3 x 3 = 9).

The broadband provider analyses the data collected using a **one-way repeated measures MANOVA** to determine whether there is a statistically significant **difference** in organisational commitment – measured in terms of affective commitment, continuance commitment and normative commitment **combined** – over the three time points: pre-bonus, short-term review and medium-term review.

Imagine that large chain of opticians want to determine whether a new type of lens used in glasses improves patients' vision. Therefore, 80 patients are randomly selected to take part in a study where they are asked to wear glasses with two different types of lens: (a) the "standard" lens that they sell; and (b) the "new" lens they are considering whether to introduce. When wearing each of the two types of lens, the 80 participants complete a "visual acuity test" and a "depth perception test", which together are used to assess the performance of each lens in improving patients' vision.

Therefore, in this experiment "lens performance" is assessed using two dependent variables, "visual acuity" and "depth perception", and the independent variable is "lens type", with two related groups: the "standard lens" and the "new lens". The two groups are related because all 80 participants were tested using both lenses, so each participant has four scores: a "visual acuity" score and a "depth perception" score for the "standard" lens and the "new" lens (i.e., 2 x 2 = 4).

The optician analyses the data collected using a **one-way repeated measures MANOVA** to determine whether there is a statistically significant **difference** in lens performance – measured in terms of visual acuity and depth perception **combined** – based on whether the standard lens or new lens was used.

If a one-way repeated measures MANOVA **is** statistically significant, this would suggest that there is a **difference** in the combined dependent variables between the two or more related groups. Taking the first example above, a statistically significant one-way repeated measures MANOVA would suggest that there was a difference in the three combined types of organisational commitment – that is, the three dependent variables: "affective commitment", "continuance commitment" and normative commitment" – between the "pre-bonus", "short-term review" and "medium-term review" time points (i.e., before, 1 month after and 6 months after the new bonus scheme was introduced), which represent the three related groups.

However, the one-way repeated measures MANOVA is an **omnibus test**, which means that it **cannot** tell us **where** the differences are when you have three or more related groups. For example, it cannot tell us that these three combined types of organisational commitment were different at the "short-term review" time point **compared to** the "pre-bonus" time point. Similarly, the one-way repeated measures MANOVA cannot tell us that there was a difference in the combined organisational commitment score between the "short-term review" and "medium-term review". It can **only** tell us that **at least two** related groups (i.e., time points) were different.

Since you may have three, four, five or more time points (or treatments) in your study design, determining which of these related groups differ from each other is important. Therefore, you can determine **where** these differences are by carrying out a **follow-up analysis** (also known as **post hoc testing**). If you are interested in understanding which related groups are different at the **multivariate level** (i.e., where the differences are between the **combined** dependent variables), you can consider **multivariate contrasts**. However, if you are interested in understanding which related groups are different at the **univariate level** (i.e., where the differences are between **each** dependent variable **separately**), you can consider a **univariate analysis**. Alternatively, if you are interested in **how** the **combined** dependent variables **change over time** between each related group (e.g., **how** the combined scores of the dependent variables change from the "pre-bonus", to the "short-term review", and then to the "medium-term review"), you can assess whether there are any **polynomial trends**.

Note: If you would like us to expand this section to explain these three different types of follow-up analysis (i.e., **multivariate contrasts**, **univariate analysis** and **polynomial trends**), please contact us.

In this "quick start" guide, we show you how to carry out a one-way repeated measures MANOVA using SPSS Statistics and which table you need to use to determine whether you have a statistically significant result. However, before running the one-way repeated measures MANOVA you need to understand the different assumptions that your data must meet in order for a one-way repeated measures MANOVA to give you a valid result. We discuss these assumptions next.

###### SPSS Statistics

## Assumptions

When you choose to analyse your data using a one-way repeated measures MANOVA, part of the process involves checking to make sure that the data you want to analyse can actually be analysed using a one-way repeated measures MANOVA. You need to do this because it is only appropriate to use a one-way repeated measures MANOVA if your data passes **seven assumptions** that are required for a one-way repeated measures 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 seven 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 seven 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**variables). 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**independent variable**should consist of**two or more categorical**,**related groups**. Groups can be considered to be**related**when a participant is measured at**all**time points or receives**all**treatments. Alternatively, participants can be**matched**(e.g., twins, family members). Whilst groups should be related, participants should be independent, such that one participant should not be able to influence the scores of another participant.**Assumption #3:**You should have an**adequate sample size**. Although the larger your sample size the better, for a one-way repeated measures MANOVA to run, you need to have more cases (e.g., participants) in each related group than the number of dependent variables you are analysing. For example, if you had six dependent variables where participants were measured over five time points (i.e., you have five related groups), there must be at least six participants in each of the five related groups for the one-way repeated measures MANOVA to run.**Assumption #4:**There should be**no univariate or multivariate outliers**. First, there should be**no (univariate) outliers**in each related group of the independent variable for any of the dependent variables. This is a similar assumption to the one-way repeated measures ANOVA, but for each dependent variable that you have in your one-way repeated measures MANOVA analysis. Univariate outliers are often just called**outliers**and are the same type of outliers you will have come across if you have ever conducted t-tests or ANOVAs. We refer to them as**univariate**in this guide to distinguish them from**multivariate outliers**, which are cases (e.g., participants) which have an**unusual combination**of scores on the dependent variables.**Assumption #5:**There should be**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 each of the related groups of the independent variable is often used in its place as a best 'guess' as to whether there is multivariate normality. You can test for normality in SPSS Statistics using**numerical methods**such as the**Shapiro-Wilk test of normality**and**graphical methods**such as**histograms**and**Normal Q-Q plots**.**Assumption #6:**There should be a**linear relationship between each pair of dependent variables for each related group of the independent variable**. 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 each related group of the independent variable.**Assumption #7:**There should be**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 repeated measures ANOVAs**, and if the correlations are**too high**(generally considered greater than 0.9), you could have**multicollinearity**. This is problematic for the one-way repeated measures MANOVA and needs to be screened out.

You can check assumptions #4 through #7 using SPSS Statistics. Before doing this, you should make sure that your data meets assumptions #1 through #3, 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 one-way repeated measures MANOVA might **not** be valid.

Note: When we launch our premium one-way repeated measures MANOVA guide, we will show you how to check assumptions #4 through #7 in SPSS Statistics to make sure that your data has passed these assumptions, and if it fails any, explain what you can do in order to continue with your analysis. If you would like us to email you when this guide becomes available, please contact us.

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