# One-Way MANCOVA

## One-Way MANCOVA

The one-way multivariate analysis of covariance (one-way MANCOVA) extends the capabilities of the one-way MANOVA and one-way ANCOVA by incorporating either a continuous covariate or multiple dependent variables. This addition enhances the sensitivity of the analysis to detect differences among groups of a categorical independent variable. The one-way MANCOVA is employed to determine whether there are any statistically significant variations in the adjusted means among three or more unrelated groups, all while controlling for a continuous covariate.

Note 1: While the one-way MANCOVA can accommodate a nominal or ordinal independent variable, it treats this variable as nominal, meaning it does not consider an ordinal variable’s ordered nature. Additionally, if your covariate is ordinal or nominal, it is advisable to employ a different statistical test.

Note 2: Handling two or more continuous covariates introduces additional complexities in conducting and interpreting the one-way MANCOVA. As a result, a separate guide for one-way MANCOVA with multiple continuous variables will be made available for those interested.

Understanding that the one-way MANCOVA is an omnibus test statistic is crucial. It provides information about whether the groups of the independent variable significantly differ when considering the combined dependent variables after adjusting for the covariate. However, it doesn’t offer further insights into which independent variable groups differ concerning each dependent variable and how these differences manifest after accounting for the covariate. To address this, you can follow up a statistically significant one-way MANCOVA by conducting multiple univariate one-way ANCOVAs, one for each dependent variable, followed by multiple comparison tests as post hoc analysis. While alternative methods exist to follow up a significant one-way MANCOVA, univariate one-way ANCOVAs and multiple comparisons are the default in SPSS Statistics.

Note: If you prefer to employ a different follow-up method for a statistically significant one-way MANCOVA (other than the default method in SPSS Statistics), you can reach out to request your preferred approach.

### Assumptions of one-way MANCOVA

To run a one-way MANCOVA, 11 assumptions need to be considered. The first four assumptions relate to your choice of study design and the measurements you chose to make, while the second seven assumptions relate to how your data fits the one-way MANCOVA model. These assumptions are:

• Assumption #1 is that two or more dependent variables should be measured at the continuous level.  Continuous variables can take on infinite values within a given range. For instance, temperature, time, height, weight, distance, age, blood pressure, speed, electricity consumption, and sound level are typical continuous variables. For instance, the temperature in a room can be any value within the limits of the thermometer, such as 22.5°C, 22.51°C, and so on. Similarly, time can be measured to any level of precision, like seconds, milliseconds, or even smaller units. Height and weight can vary infinitely within their possible range, measured in units like meters or feet, and can include fractions (like 1.75 meters). Distance between two points, age measured in years, months, days, and even smaller units, blood pressure measured in millimeters of mercury (mmHg), speed measured in units like kilometers per hour or miles per hour, electricity consumption measured in kilowatt-hours or other units, and sound level measured in decibels are other examples of continuous variables that can take on a range of continuous values.
• Assumption #2: You have one independent variable that consists of two or more categoricalindependent groups (i.e., you have a categorical variable).  This structure is essential for the analysis to compare the different groups effectively. Various independent variables can fulfill this criterion, such as dietary preference (e.g., with groups like omnivore, vegetarian, vegan, and pescatarian) or educational level (e.g., with groups like high school, undergraduate, graduate, and postgraduate). Another example could be the types of transportation used (e.g., with groups like car, public transport, bicycle, and walking)..
• Assumption #3: You have one covariate measured at the continuous level (see Assumption #1 for examples of continuous variables). A covariate is a continuous independent variable added to an ANOVA model to produce an ANCOVA model. This covariate is used to adjust the means of the groups of the categorical independent variable. It acts the same as in a normal multiple regression but is usually of less direct importance (i.e., the coefficient and other attributes are often of secondary importance or not at all). In an ANCOVA, the covariate is generally only there to better assess the differences between the groups of the categorical independent variable on the dependent variable.

Assumption #4 stresses the importance of maintaining independence among observations. This assumption means that the observations should not be connected in any way, within or across different groups associated with an independent variable. For instance, in a study measuring the effectiveness of different medications on patients with a certain condition, the response of one patient mustn’t affect another within the same group or across different groups. This assumption is important in statistical tests like a t-test, where the groups must be distinct and unrelated regarding participant characteristics. Unique participants should be included in each group to achieve this. Regarding independence in observations, the focus is on their lack of connection. This independence refers to the errors in measurement, which should be independent. If these errors are not independent, they are called correlated errors. Correlated errors indicate a violation of this independence assumption, which often occurs when the observations themselves are not independent. The independence of observations is a crucial assumption in statistical analysis, and it cannot be tested using certain software. However, it is a necessary assumption in many statistical tests, including the one-way ANOVA. If this assumption is violated, it is necessary to use another statistical test instead of the one-way ANOVA.

• Assumption #5: There should be a linear relationship between each pair of dependent variables within each independent variable group. The first assumption you need to test is whether there is a linear relationship between each pair of dependent variables within each independent variable group, such as the relationship between height and weight in different age groups.
• Assumption #6: There should be a linear relationship between the covariate and each dependent variable within each independent variable group. The second assumption you need to test is whether there is a linear relationship between the covariate and dependent variables within each independent variable group, such as the relationship between age and reaction time in different groups.
• Assumption #7: You should have 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 independent variable group.

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### Interpret Your One-Way MANCOVA Results

Once you have executed the one-way MANCOVA process and verified that your data adheres to its assumptions as covered in earlier sections, SPSS Statistics will have produced several tables with the necessary information for reporting your one-way MANCOVA results. This section guides you through understanding these results.

The primary goal of a one-way MANCOVA is to assess if there is a significant statistical relationship between the independent and dependent variables while accounting for a continuous covariate. However, the one-way MANCOVA doesn’t specify which groups within the independent variable are different. If your one-way MANCOVA shows statistical significance, a secondary goal is to conduct a post hoc analysis to identify the specific differences among the independent variable groups. This and other objectives will be addressed in the sections below:

• Descriptive statistics and estimates: Begin your analysis by examining the Descriptive Statistics and Estimates tables for an overview of your data. The Descriptive Statistics table displays the mean values of the dependent variables for each group within the independent variable (e.g., groups based on different levels of a particular behavior like “Low”, “Moderate”, and “High”). While you should report these means, the one-way MANCOVA focuses on the differences in adjusted means, which are the means of the dependent variables after accounting for the covariate. These adjusted means are found in the Estimates table. Typically, the raw means and adjusted means are different. Both should be reported, with emphasis on the adjusted means.
• One-way MANCOVA result: To evaluate the main one-way MANCOVA result, check if there is a statistically significant difference among the independent variable groups on the combined dependent variables while controlling for the covariate. SPSS Statistics uses four multivariate tests (Pillai’s Trace, Wilks’ Lambda, Hotelling’s Trace, and Roy’s Largest Root) to determine this significance. Wilks’ Lambda is often the most commonly interpreted statistic.
• Post hoc tests: Should the one-way MANCOVA indicate statistical significance, you might proceed with one-way ANCOVAs and multiple comparisons for post hoc analysis. Despite debate over the best follow-up methods for a one-way MANCOVA, a straightforward approach is to perform a one-way ANCOVA for each dependent variable. Then, for any statistically significant one-way ANCOVAs, compare pairwise with Bonferroni correction. Further guidance on interpreting these follow-up tests is available upon request.

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