# Two-Way MANOVA

## Two-Way MANOVA

The two-way multivariate analysis of variance (MANOVA) is an analytical technique that extends the principles of the two-way ANOVA to scenarios with multiple dependent variables. It is particularly useful in determining how two independent variables interact in their combined influence on several dependent variables.

For example, consider a study to evaluate the impact of diet type (e.g., vegetarian, keto, Mediterranean) and exercise regimen (e.g., cardio, strength training, mixed) on various health outcomes. The dependent variables, in this case, might include blood pressure, cholesterol level, and body mass index (BMI). The two-way MANOVA would enable researchers to assess how the combination of diet and exercise regimen influences these health outcomes collectively rather than just looking at each health outcome separately.

Another scenario where a two-way MANOVA could be applied is in a marketing research study analyzing the impact of advertising medium (e.g., television, online, print) and product type (e.g., consumer electronics, clothing, food items) on customer responses. The dependent variables could be customer recall, attitude toward the advertisement, and intention to purchase. This analysis would help marketers understand how the effectiveness of different advertising mediums varies with product type across several response metrics.

In both examples, the two-way MANOVA provides a more comprehensive view of the interactions between independent variables and a set of dependent variables, allowing for a more nuanced understanding of these complex relationships. This analysis is particularly valuable in fields like health science and marketing, where multiple outcome measures are often of interest.

### Assumptions of Two-Way MANOVA

10 assumptions need to be considered to run a two-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 two-way MANOVA model. These assumptions are:

• Assumption #1 is that you have two or more dependent variables 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.

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

• Assumption #2: You have two independent variables. These independent variables can be dichotomous, with only two groups, or polytomous, with three or more groups. For example, a study might examine the effects of sleep duration (with three groups: less than 6 hours, 6-8 hours, more than 8 hours) and coffee consumption (with two groups: non-drinkers and drinkers) on cognitive performance. Another example is analyzing the impact of educational level (e.g., high school, undergraduate, postgraduate) and type of employment (e.g., full-time, part-time, unemployed) on life satisfaction. In each case, the independent variables are grouped into distinct categories, and these groups are independent, allowing for a clear comparison of effects on the dependent variable(s). Understanding and adequately categorizing these variables is fundamental for conducting a valid and reliable statistical analysis.

Note. In the two-way MANOVA presented in this guide, the independent variables are referred to as fixed factors or fixed effects. This fixed factor implies that the groups of each independent variable represent all the categories of the independent variable you are interested in. For instance, if you are studying the differences in exam performance between different schools and you investigate three specific schools that interest you, the independent variable is considered a fixed factor. However, if you randomly select three schools to represent all schools, the independent variable is a random factor. A random factor requires a statistical test different from the two-way MANOVA, which is inappropriate in such cases. If you have a random factor in your study design, please contact us.

• Assumption #3 is centered on the independence of observations, a crucial statistical analysis principle. This principle mandates that there should be no relational ties among the observations in each independent variable group or across different groups. In simpler terms, this means that the participants in one group should have no connection or influence over those in another. For instance, in a study comparing dietary preferences (e.g., vegan, vegetarian, omnivore, carnivore), each individual belongs to one distinct group, and their choices or behaviors should not affect those in other groups. Similarly, in a study assessing the impact of different teaching methods (e.g., lecture-based, interactive, online) on student engagement, students are assigned one method, ensuring no crossover or influence between groups.
• The assumption of independent observations is critical in statistical terminology, where the errors in the data are assumed to be independent. Correlated errors, or non-independent errors, arise when there is a lack of independence among observations. This risk of correlated errors is high when participants are not independently assigned to groups.
• For instance, in a clinical trial examining the effectiveness of different physiotherapy techniques (e.g., manual therapy, exercise therapy, combined therapy, and a control group) on recovery from a specific injury, each patient is allocated to only one treatment group. This allocation ensures that one patient’s experiences or outcomes do not impact another’s.
• The independence of observations is a crucial aspect of study design and must be considered before conducting a two-way MANOVA. If this assumption is violated, the validity of the study’s results could be compromised, necessitating an alternative statistical method.
• Assumption #4 in a two-way MANOVA pertains to the necessity of a linear relationship between pairs of dependent variables within each independent variable group. This linear relationship means that for each combination of independent variable groups, there should be a linear correlation among the dependent variables. For instance, consider a study analyzing the effects of different types of music (e.g., classical, rock, pop, jazz) and study environments (e.g., silent, noisy, with music) on students’ comprehension and retention abilities. Comprehension and retention are the dependent variables. In this scenario, it is essential to establish a linear relationship between comprehension and retention for each music type within each study environment group. If these variables do not demonstrate a linear relationship, the effectiveness of the two-way MANOVA in detecting differences may be compromised.
• Assumption #5 states that there should be no multicollinearity. Your dependent variables should have a moderate correlation with each other. But, if the correlation between the variables is low, it’s better to run separate two-way ANOVAs for each dependent variable. On the other hand, if the correlation is too high (greater than 0.9), it may indicate multicollinearity, which is problematic for MANOVA and needs to be screened out. Although many complicated and sophisticated ways exist to detect multicollinearity, we will show you a relatively simple method of using Pearson correlation coefficients between the dependent variables to determine if any relationships are too strongly correlated.
• Assumption #6 stipulates that the data should contain neither univariate nor multivariate outliers. Univariate outliers refer to individual data points that are anomalously high or low within each group of the independent variables. These outliers are similar to those encountered in t-tests or ANOVA analyses and can significantly impact a group’s mean and standard deviation, especially in smaller sample sizes. For example, in a study assessing the effect of different study techniques (e.g., self-study, group study, online learning) and times of day (morning, afternoon, evening) on students’ test scores in mathematics and language, univariate outliers might be a student scoring exceptionally high or low in either subject compared to others in the same group. Beyond univariate outliers, multivariate outliers are also crucial in a two-way MANOVA. These are cases where an individual’s combination of scores on the dependent variables is atypical. For instance, in the abovementioned study, a multivariate outlier might be a student who scores exceptionally high in mathematics but extremely low in language. This combination is rare among other students. Tools like SPSS Statistics can calculate the Mahalanobis distance to identify such multivariate outliers. This process involves examining whether each student’s combined mathematics and language scores significantly deviate from their respective group’s norm. Detecting and addressing univariate and multivariate outliers is essential to ensure the accuracy and reliability of the MANOVA results.
• Assumption #7 requires multivariate normality in the data. Testing for multivariate normality is complex and not directly possible in SPSS Statistics. As a practical approach, the normality of each dependent variable within every combination of independent variable groups is examined as a proxy to gauge multivariate normality. In essence, multivariate normality means that for each combination of independent variable groups, all dependent variables’ residuals (or the differences between observed and predicted values) should be normally distributed. However, it’s important to note that having normally distributed residuals for each group doesn’t necessarily confirm multivariate normality. For example, if you’re studying the effects of different teaching methods (e.g., traditional, blended, online) and class sizes (small, medium, large) on students’ performance in subjects like mathematics and history, you would need to check if the scores in mathematics and history are normally distributed within each combination of teaching method and class size. While one method of testing normality is commonly used, those more versed in statistics may employ multiple methods to assess normality, such as evaluating skewness and kurtosis values or analyzing histograms. This multi-method approach can provide a more comprehensive understanding of whether the data meet the assumption of multivariate normality, which is crucial for the validity of MANOVA results.
• Assumption #8 states that an adequate sample size is necessary. The number of cases in each design cell should equal or exceed the number of dependent variables. Two or more cases per cell of the design are required.
• Assumption #9 addresses the necessity of homogeneity of variance-covariance matrices. The two-way MANOVA assumes similar variances and covariances. Box’s M test of equality of covariance can be used to test this assumption.
• Assumption #10 stipulates that homogeneity of variances is necessary. The two-way MANOVA assumes equal variances in the design’s cell for each dependent variable. Levene’s test of equality of variances can be used to test this assumption.

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### Interpret Your Two-Way MANOVA Results

Once you have run the two-way MANOVA procedure and confirmed that your data meets its assumptions, SPSS Statistics generates several tables containing all the information required to report the results.

The two main objectives of the two-way MANOVA are to determine whether there is a significant interaction effect between the two independent variables on the combined dependent variables and if so, to run follow-up tests to identify where the differences lie. Both objectives will be discussed in the following sections:

• Determining whether an interaction effect exists: To evaluate the primary results of the two-way MANOVA, you can begin by determining if there is a statistically significant interaction effect between the two independent variables on the combined dependent variables. SPSS Statistics can use four different multivariate statistics to test statistical significance (Pillai’s Trace, Wilks’ Lambda, Hotelling’s Trace, and Roy’s Largest Root). We will explain which statistics to choose and how to interpret them.
• Univariate interaction effects and simple main effects: If the interaction is statistically significant, one common approach is to determine whether there are any statistically significant univariate interaction effects for each dependent variable separately (Pituch & Stevens, 2016). SPSS Statistics will run these statistics for you (i.e., a two-way ANOVA for each dependent variable). If there are any statistically significant interaction effects, you can follow these up with simple main effects. We will explain how to interpret these follow-up tests.
• Main and univariate main effects: If your interaction effect is not statistically significant, you will follow up on the main effects instead. If you have statistically significant main effects, you can follow these up with univariate main effects. We will explain how to interpret these follow-up tests.
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