## One-Way MANOVA

The one-way multivariate analysis of variance (MANOVA) is a statistical method that extends the one-way ANOVA by accommodating two or more dependent variables instead of one. The one-way MANOVA assesses the differences in a combined set of dependent variables, known as a ‘linear composite’ or vector, across groups defined by an independent variable. This approach effectively creates a new, composite dependent variable that encapsulates the multiple dependent variables, maximizing the observable differences between the groups.

However, like the one-way ANOVA, the one-way MANOVA is an omnibus test, meaning that while it can indicate the presence of differences between groups, it does not specify which specific groups differ. This analysis necessitates further analysis when the study involves more than two groups.

For instance, a one-way MANOVA could be applied to investigate whether dietary habits (measured as nutrient intake and caloric consumption) differ among individuals with different exercise routines (e.g., “no exercise,” “moderate exercise,” “intense exercise”). Here, “nutrient intake” and “caloric consumption” are the dependent variables, and “exercise routine” is the independent variable with three groups.

In another scenario, a one-way MANOVA might be used to evaluate job satisfaction and work-life balance among employees in various sectors, such as “technology,” “healthcare,” “education,” “finance,” and “retail.” The dependent variables in this case would be “job satisfaction” and “work-life balance,” while the independent variable is the “employment sector.”

In both examples, the one-way MANOVA allows for a comprehensive analysis of how the combination of multiple dependent variables varies across different groups of a single independent variable.

**Assumptions of One-Way MANOVA**

10 assumptions need to be considered to run a one-way MANOVA. The first three assumptions relate to your choice of study design and the measurements you chose to make, while 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. 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 for a one-way MANOVA requires the presence of an independent variable composed of two or more categorical, independent groups. 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). If you’re unfamiliar with these terms or concepts, consult a guide on the different types of variables for a clearer understanding. Additionally, if your study involves two independent variables instead of just one, a two-way MANOVA would be more appropriate than a one-way MANOVA, as it allows for examining the interaction effects between the two independent variables.
- Assumption #3 emphasizes the need for independence of observations. This assumption means there should be no relationship among the observations within each group defined by the independent variable and no relationship between the different groups. This independence is typically achieved by having different individuals in each group. For instance, consider a study dividing participants into groups based on their preferred learning style: visual, auditory, or kinesthetic. Each participant would belong to only one of these groups, ensuring no overlap; a visual learner, for instance, wouldn’t be simultaneously categorized as an auditory or kinesthetic learner. In another scenario, you might assign participants to different dietary groups for a nutrition study, such as vegetarian, vegan, or omnivorous diets. Again, each participant would be part of only one dietary group. Additionally, the ‘no relationship’ criterion implies that individuals in different groups should be unrelated. For example, family members or close friends should not be placed in the same experimental groups, as their relationships might influence the study’s outcomes. Also, the actions or experiences of participants in one group should not affect those in another group. This independence is more a matter of study design and cannot be directly tested using statistical software. Nevertheless, it’s a critical consideration for ensuring the validity of a one-way MANOVA. If this assumption is not met, the MANOVA’s results could be biased or misleading, necessitating a different analytical approach.
- Assumption #4 is the absence of both univariate and multivariate outliers. This assumption means that there should be no univariate outliers for any dependent variable within each group defined by the independent variable. Univariate outliers are individual data points significantly different from the rest within a group, either exceptionally high or low. This assumption is similar to what is considered in one-way ANOVA but applies to each dependent variable in the MANOVA context. Identifying these outliers is crucial because they can significantly impact the mean and standard deviation of the group, potentially skewing the statistical test results. This effect is especially pronounced in smaller sample sizes.
- Assumption #5 is the requirement of multivariate normality in the data. Testing for this assumption can be challenging, as it’s not directly assessable in software like SPSS Statistics. As a workaround, researchers often examine the normality of each dependent variable within every independent variable group. This check serves as an approximate indicator of whether the data exhibits multivariate normality.
- Assumption #6 states that the dependent variables should not be highly correlated with each other, which is called multicollinearity. If the correlation between dependent variables is too low, it might be better to run separate one-way ANOVAs instead of a one-way MANOVA. On the other hand, if the correlation is too high (greater than 0.9), it can result in multicollinearity, which can negatively impact the MANOVA. You can use the simple method of detecting multicollinearity using Pearson correlation coefficients between the dependent variables to determine if any relationships are too strongly correlated.
- Assumption #7 requires a linear relationship between dependent variables for each independent variable group. The one-way MANOVA requires a linear relationship between each pair of dependent variables for each group of the independent variable. You can test whether a linear relationship exists by visually inspecting a scatterplot matrix for each group of the independent variable, School. If the relationship approximately follows a straight line, you have a linear relationship.
- Assumption #8 is about the sample size. You need to have an adequate sample size, and at a bare minimum, there needs to be as many cases in each independent variable group as there are the number of dependent variables.
- Assumption #9 requires homogeneity of variance-covariance matrices, which can be tested using Box’s M test of equality of covariance. If your data fails to meet this assumption, there are ways to proceed.
- Assumption #10 is about the homogeneity of variances between groups of the independent variable for each dependent variable, which can be tested using Levene’s test of equality of variances. Suppose there is a violation of the assumption of homogeneity of variance-covariance matrices. In that case, the results from Levene’s test of equality of variances can inform which dependent variable might be causing the problem.

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**Interpret Your One-Way MANOVA Results**

After completing the one-way MANOVA procedure and ensuring that your dataset aligns with its assumptions, SPSS Statistics will generate tables containing all the necessary information to report the results. We will guide you through the interpretation of these outcomes.

The one-way MANOVA serves two primary objectives: (a) determining the statistical significance of the independent variable groups concerning the dependent variables and (b) identifying the specific differences among the independent variable groups. Both of these objectives will be addressed in the sections that follow:

- Exploratory Data Analysis: You can begin your analysis by understanding your data through descriptive statistics. This analysis involves calculating the mean, standard deviation, and case count for each dependent variable (e.g., Science_Score and History_Score) separately for each category of the independent variable, such as School. Alternatively, you can utilize measures like the standard error of the mean or 95% confidence intervals. Regardless of the outcome of the one-way MANOVA test, you will likely need to present your descriptive statistics. While doing so, refrain from drawing conclusions or inferences about the data (save this for your discussion section). However, it can be beneficial to highlight trends or disparities between groups (e.g., which independent variable groups exhibit higher or lower mean scores and whether the variation in the dependent variable is consistent across all independent variable groups).
- One-way MANOVA Findings: When assessing the primary outcomes of the one-way MANOVA, begin by ascertaining whether there exists a statistically significant distinction between the groups for both dependent variables (e.g., Science_Score and History_Score). SPSS Statistics provides four different multivariate statistics for testing the significance of group differences (e.g., Pillai’s Trace, Wilks’ Lambda, Hotelling’s Trace, and Roy’s Largest Root). We can explain which one to select and how to interpret these statistics if needed.
- Individual one-way ANOVAs and Post Hoc Comparisons: If the MANOVA outcome indicates statistical significance, you may contemplate conducting a post hoc examination. There is ongoing debate regarding the appropriate approach for following up on a one-way MANOVA. However, this guide adopts a straightforward strategy – the default action in SPSS Statistics – which involves conducting individual one-way ANOVAs for each dependent variable (e.g., a one-way ANOVA to assess the impact of School on Science_Score and another for School on History_Score) in the presence of a statistically significant result. For any one-way ANOVAs that prove statistically significant, you can proceed with Tukey post hoc tests or another preferred multiple comparison procedure. Feel free to use our enhanced guide to interpreting these follow-up results.