# One-way MANCOVA in SPSS Statistics

## Introduction

The one-way multivariate analysis of covariance (MANCOVA) can be thought of as an extension of the one-way MANOVA to incorporate a covariate or an extension of the one-way ANCOVA to incorporate multiple dependent variables. This covariate is linearly related to the dependent variables and its inclusion into the analysis can increase the ability to detect differences between groups of an independent variable. A one-way MANCOVA is used to determine whether there are any statistically significant differences between the adjusted means of three or more independent (unrelated) groups, having controlled for a continuous covariate.

For example, you could use a one-way MANCOVA to determine whether a number of different exam performances differed based on test anxiety levels amongst students, whilst controlling for revision time (i.e., your dependent variables would be "humanities exam performance", "science exam performance" and "mathematics exam performance", all measured from 0-100, your independent variable would be "test anxiety level", which has three groups – "low-stressed students", "moderately-stressed students" and "highly-stressed students" – and your covariate would be "revision time", measured in hours). You want to control for revision time because you believe that the effect of test anxiety levels on overall exam performance will depend, to some degree, on the amount of time students spend revising.

The one-way MANCOVA is very useful, but it is important to realize that the one-way MANCOVA is an *omnibus* test statistic. It will tell you whether the groups of the independent variable statistically significantly differed based on the combined dependent variables, after adjusting for the covariate, but it will not explain the result further. In other words, the one-way MANCOVA will not tell you about the differences between specific groups. Using the example above, a statistically significant one-way MANCOVA would indicate that there is a difference in test anxiety levels on the combined scores from the three exams (i.e., the humanities, science and mathematics exams). However, it will not indicate whether low-stressed students scored higher on the combined exam scores than higher stressed students, or even more specifically, whether low-stressed students scored higher on a specific exam (e.g., the science exam) compared to highly-stressed students. That said, there are follow-up tests (known as **post hoc tests**) that can be used to determine where these differences between groups are, which we discuss at the end of this guide.

In this "quick start" guide, we show you how to carry out a one-way MANCOVA 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 one-way MANCOVA 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 MANCOVA, a critical part of the process involves checking to make sure that the data you want to analyse can actually be analysed using a one-way MANCOVA. You need to do this because it is only appropriate to use a one-way MANCOVA if your data "passes" 11 assumptions that are required for a one-way MANCOVA 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 11 assumptions is likely to be the the most time consuming part of your analysis, requiring you to work through additional procedures in SPSS Statistics when performing your analysis, as well as spending time thinking about your data and what to do if your data violates different assumptions. In the section below, these 11 assumptions are briefly set out:

**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**one independent variable**should consist of**two or more categorical**,**independent groups**(i.e., you have a**nominal variable**or an**ordinal variable**). Examples of**nominal variables**include gender (e.g., 2 groups: male and female), ethnicity (e.g., 3 groups: Caucasian, African American and Hispanic) and profession (e.g., 4 groups: surgeon, doctor, nurse and dentist). Examples of**ordinal variables**include cholesterol concentration (e.g., 2 groups: 5 mmol/L or below and above 5 mmol/L), physical activity level (e.g., 3 groups: low, moderate and high) and body mass index (e.g., 4 groups: underweight, normal, overweight, obese).**Assumption #3:**Your**one or more covariates**are all**continuous variables**(see Assumption #1 for examples of continuous variables). A covariate is simply a continuous independent variable that is added to a MANOVA model to produce a MANCOVA model. This covariate is used to adjust the means of the groups of the categorical independent variable. In a MANCOVA the covariate is generally only there to provide a better assessment of the differences between the groups of the categorical independent variable on the dependent variables.**Assumption #4:**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. For example, there must be different participants in each group of the independent variable 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 one-way MANCOVA.**Assumption #5:**There should be**a linear relationship between each pair of dependent variables within each 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 with loess lines**of the dependent variables for each group of the independent variable. In order to do this, you will need to split your data file in SPSS Statistics before generating the scatterplot matrices.**Assumption #6:**There should be**a linear relationship between the covariate and each dependent variable within each group of the independent variable**. Similarly to Assumption #5 above, you can test for this assumption by plotting a**scatterplot matrix with loess lines**of the covariate for each of the dependent variables, for each group of the independent variable. You will again need to make sure that your data file is split in SPSS Statistics before generating the scatterplot matrices.**Assumption #7:**There should be**homogeneity of regression slopes**. This assumption states that the relationship between the covariate and each separate dependent variable, as assessed by the regression slope, is the same in each group of the independent variable. Simply put, Assumption #6 assessed whether the relationships were linear; this assumption now checks that these linear relationships are**the same**. You can test this assumption is SPSS Statistics.**Assumption #8:**There should be**homogeneity of variances and covariances**. In other words, the one-way MANCOVA assumes that the variances and covariances of the dependent variables are equal in all groups of the independent variable. You can test this assumption in SPSS Statistics using**Box's M Test of Equality of Covariance Matrices**.**Assumption #9:**There should be**no significant univariate outliers in the groups of your independent variable in terms of each dependent variable**. If there are any dependent variable 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, these scores are called**univariate outliers**. Univariate outliers can have a large negative effect on your results because they can exert a large influence (i.e., can cause a large change) on the mean for that group, which can affect the statistical test results. Univariate outliers are more important to consider when you have smaller sample sizes, as the effect of the outlier will be greater. Univariate outliers can be detected by inspecting the**standardized residuals**that can be producing using SPSS Statistics.**Assumption #10:**There should be**no significant multivariate outliers in the groups of your independent variable in terms of each dependent variable**. Multivariate outliers are cases (e.g., participants in our example) that have an unusual combination of scores on the dependent variables within each group of the independent variable. SPSS Statistics can calculate a measure called**Mahalanobis distance**that can be used to determine whether a particular case might be a multivariate outlier.**Assumption #11:**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**residuals**for each group 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 this using the**Shapiro-Wilk test of normality**in SPSS Statistics.

You can check assumptions #5, #6, #7, #8, #9, #10 and #11 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 the one-way MANCOVA might not be valid.

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