The one-way multivariate analysis of covariance (one-way MANCOVA) can be thought of as an extension of the one-way MANOVA to incorporate a continuous 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 a categorical 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.

Note 1: Whilst a one-way MANCOVA can be used with a nominal or ordinal independent variable, it treats the independent variable as nominal (i.e., it will not take into account the ordered nature of an ordinal variable). Furthermore, whilst the covariate does not have to be measured on a continuous scale, if your covariate is ordinal or nominal, please contact us because you will need to use a different statistical test.

Note 2: If you have two or more continuous covariates, there are some additional considerations when carrying out and interpreting the one-way MANCOVA. Therefore, we will be adding a separate guide for a one-way MANCOVA with multiple continuous variables. If this is of interest, please contact us and we will let you know when the guide becomes available.

It is important to realize that the one-way MANCOVA is an omnibus test statistic. Therefore, it will tell you whether the groups of the independent variable statistically significantly differed based on the combined dependent variables, after accounting for the covariate, but it will not explain the result further. For example, it will not tell you which groups of your independent variable differed in terms of each dependent variable (rather than the combined dependent variables), and how these groups differed (after accounting for the covariate). Therefore, you can follow up a statistically significant one-way MANCOVA by carrying out multiple univariate one-way ANCOVAs (one for each dependent variable), and then multiple comparisons as a post hoc test. Whilst there are other methods to follow up a statistically significant one-way MANCOVA, univariate one-way ANCOVAs and multiple comparisons are the default in SPSS Statistics.

Note: If you would like to follow up a statistically significant one-way MANCOVA using a different method (i.e., not the default method in SPSS Statistics), please contact us, letting us know the method you would like to use.

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 continuous dependent variables would be “humanities exam performance”, “science exam performance” and “mathematics exam performance”, all measured from 0-100, your ordinal independent variable would be “test anxiety level”, which has three groups – “low-stressed students”, “moderately-stressed students” and “highly-stressed students” – and your continuous 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. If there is a statistically significant difference in test anxiety levels on the combined dependent variables, after controlling for revision time, you can use univariate one-way ANCOVAs (one for each of the three dependent variables) to determine whether test anxiety levels differed for each of these three exams (i.e., for humanities, science and mathematics exams separately), after controlling for the effect of revision time. If any of the univariate one-way ANCOVAs is statistically significant, you can follow these up using multiple comparisons to determine how exam performance (for those exam subjects where the one-way ANCOVA was statistically significant), differed between the test anxiety groups, after adjusting for revision time (e.g., if the one-way ANCOVA for science exam performance was statistically significant, you could use multiple comparisons to determine how science exam performance differed between the low-stressed students and moderately-stressed students, the low-stressed students and highly-stressed students, and the moderately-stressed students and the highly-stressed students).

## Assumptions of one-way MANCOVA

In order to run a one-way MANCOVA, there are 11 assumptions that need to be considered. The first four assumptions relate to your choice of study design and the measurements you chose to make, whilst the second seven assumptions relate to how your data fits the one-way MANCOVA 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 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.

Note: You should note that SPSS Statistics refers to continuous variables as Scale variables.

• Assumption #2: You have one independent variable that consists of two or more categoricalindependent groups (i.e., you have a categorical variable). A categorical variable can be either a nominal variable or an ordinal variable, but the one-way MANCOVA does not take into account the ordered nature of an ordinal variable. Examples of nominal variables include gender (with two groups: “male” and “female”), ethnicity (with three groups: “African American”, “Caucasian” and “Hispanic”), transport type (four groups: “cycle”, “bus”, “car” and “train”) and profession (five groups: “consultant”, “doctor”, “engineer”, “pilot” and “scientist”). Examples of ordinal variables include educational level (e.g., with three groups: “high school”, “college” and “university”), physical activity level (e.g., with four groups: “sedentary”, “low”, “moderate” and “high”), revision time (e.g., with five groups: “0-5 hours”, “6-10 hours”, “11-15 hours”, “16-20 hours” and “21-25 hours”), Likert items (e.g., a 7-point scale from “strongly agree” through to “strongly disagree”), amongst other ways of ranking categories (e.g., a 5-point scale explaining how much a customer liked a product, ranging from “Not very much” to “Yes, a lot”).

Explanation 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”). However, these three terms – “groups”, “categories” and “levels” – can be used interchangeably. We will mostly refer to them as groups, but in some cases we will refer to them as levels. The only reason we do this is for clarity (i.e., it sometimes sounds more appropriate in a sentence to use levels instead of groups, and vice versa).

Explanation 2: The categorical independent variable in any type of MANCOVA is also commonly referred to as a factor. For example, a one-way MANCOVA is a MANCOVA analysis involving one factor (i.e., one categorical independent variable). Furthermore, when an independent variable/factor has independent groups (i.e., unrelated groups), it is further classified as a between-subjects factor because you are concerned with the differences in the dependent variables between different cases (e.g., participants). However, for clarity we will simply refer to them as independent variables in this guide.

Explanation 3: For the one-way MANCOVA demonstrated in this guide, the independent variables are referred to as fixed factors or fixed effects. This means that the groups of each independent variable represent all the categories of the independent variable you are interested in. For example, you might be interested in exam performance differences between schools. If you investigated three different schools and it was only these three schools that you were interested in, the independent variable is a fixed factor. However, if you picked the three schools at random and they were meant to represent all schools, the independent variable is a random factor. This requires a different statistical test because the one-way MANCOVA is the incorrect statistical test in these circumstances. If you have a random factor in your study design, please contact us and we will look to add an SPSS Statistics guide to help with this.

• Assumption #3: You have one covariate that is measured at the continuous level (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 and is usually of less direct importance than the categorical independent variable. In a MANCOVA analysis 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.

Important: You can have more than one continuous covariate in a one-way MANCOVA, but we only show you how to analyze a design with one continuous covariate in this guide. If you would like us to add an SPSS Statistics guide for the one-way MANCOVA with multiple continuous covariates, please contact us.

Note 1: If your covariate is not measured on a continuous scale (e.g., your covariate is a dichotomous, ordinal or nominal variable), you will need to use a different statistical test. If you would like us to add an SPSS Statistics guide to help with this situation, please contact us, letting us know whether your covariate was measured on a dichotomous, ordinal or nominal scale.

Note 2: If you have two independent variables rather than just one and this second independent variable is not another covariate, you should consider a two-way MANCOVA instead of a one-way MANCOVA. If you would like us to add an SPSS Statistics guide for the two-way MANCOVA to the site, please contact us.

• 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. 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 MANCOVA) 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. This is generally considered the most important assumption (Hair et al., 2014). Violation of this assumption is very serious (Stevens, 2009; Pituch & Stevens, 2016).

Note: When we talk about the observations being independent, this means that the observations (e.g., participants) are not related. More specifically, it is the errors that are assumed to be independent. In statistics, errors that are not independent are often referred to as correlated errors. This can lead to some confusion because of the similarity of the name to that of tests of correlation (e.g., Pearson’s correlation), but correlated errors simply means that the errors are not independent. The errors are at high risk of not being independent if the observations are not independent.

For example, if you split a group of individuals into four groups based on their physical activity level (e.g., a “sedentary” group, a “low” group, a “moderate” group and a “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. 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 using SPSS Statistics, but it is an important assumption of the one-way MANCOVA. If your study fails this assumption, you will need to use another statistical test instead 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. The first assumption you need to test is whether there is a linear relationship between each pair of dependent variables, chol, crp and sbp, within each group of the independent variable, group (i.e., the “Low”, “Moderate” and “High” physical activity groups).
• Assumption #6: There should be a linear relationship between the covariate and each dependent variable within each group of the independent variableThe second assumption you need to test is whether there is a linear relationship between the covariate, weight, and dependent variables, chol, crp and sbp, within each group of the independent variable, group (i.e., the “Low”, “Moderate” and “High” physical activity groups).
• Assumption #7: You should have homogeneity of regression slopes. This assumption states that the relationship between the covariate, weight, and each separate dependent variable (i.e., chol, crp and sbp), as assessed by the regression slope, is the same in each group of the independent variable (i.e., the “Low”, “Moderate” and “High” physical activity groups). Simply put, Assumption #6 assessed whether the relationships were linear; this assumption now checks that these linear relationships are the same. In order to test whether the slopes are different, an interaction between the covariate and the independent variable must be added to the one-way MANCOVA model. If the interaction term (i.e., group*weight) is not statistically significant, you have met this assumption. Please contact us. We will show how to test the assumption of homogeneity of regression slopes and what to do if you fail it.

## Interpreting Results of one-way MANCOVA

After running the one-way MANCOVA procedure and testing that your data meet the assumptions of the one-way MANCOVA 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 MANCOVA. We show you how to interpret these results.

The main objective of a one-way MANCOVA is to determine whether the independent variable is statistically significant in terms of the dependent variables, whist controlling for a continuous covariate. However, the one-way MANCOVA does not indicate which groups of the independent variable differ. Therefore, if the one-way MANCOVA is statistically significant, a secondary objective is to follow up with a post hoc analysis to determine where differences between the groups of the independent variable lie. Both of these objectives will be answered in the following sections:

• Descriptive statistics and estimates: You can start your analysis by getting an overall impression of what your data is showing through the Descriptive Statistics and Estimates tables. The Descriptive Statistics table presents the means of the dependent variables for each group of the independent variable, group (i.e., for the “Low”, “Moderate” and “High” physical activity groups). Although you will want to report means, the one-way MANCOVA analysis is based on differences in adjusted means. These are the means of the dependent variables – chol, crp and sbp – for each group of the independent variable, group, after they have been adjusted for the covariate, weight. The adjusted means are contained within the Estimates table. It is highly unlikely that the means and adjusted means will be equal. Ideally, you should report both, but adjusted means should be the priority.
• One-way MANCOVA result: In evaluating the main one-way MANCOVA result, you can start by determining if there is a statistically significant difference between the groups of the independent variable, group (i.e., between the “Low”, “Moderate” and “High” physical activity groups), on the combined dependent variables (chol, crp and sbp), whilst controlling for the covariate, weight. 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 TraceWilks’ LambdaHotelling’s Trace and Roy’s Largest Root). Please contact us. We will explain how to interpret the most popular statistic (i.e., Wilks’ Lambda).
• Post hoc tests: If the one-way MANCOVA result is statistically significant, you can consider running one-way ANCOVAs and multiple comparisons as post hoc tests. There is quite a bit of controversy over how you should follow up a one-way MANCOVA. 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 one-way ANCOVA for each dependent variable (i.e., a one-way ANCOVA for each of the dependent variables, chol, crp and sbp). For any of your one-way ANCOVAs that are statistically significant, you can follow them up using pairwise comparisons with a Bonferroni correction. Please contact us. We will explain how to interpret these follow-up tests.