# Two-Way RM ANOVA

## Two-Way Repeated Measures ANOVA

The two-way repeated measures ANOVA is a statistical test used to identify whether there is a significant interaction effect between two within-subjects factors on a continuous dependent variable. This type of ANOVA extends the one-way repeated measures ANOVA, which considers only one within-subjects factor. In this guide, we will refer to “within-subjects factors” as “factors” or “independent variables.”

The two-way repeated measures ANOVA, known as a “two-way within-subjects ANOVA” or “within-within-subjects ANOVA,” is used in various situations. For instance, let’s say you experimented to determine the effect of two types of ski goggles (blue-tinted and gold-tinted) on ski performance (i.e., time to complete a ski run), and you suspect that the effect of the different lens colors on ski performance might vary depending on the weather conditions; sunny or overcast. You believe that ski performance depends on both ski goggle lens color and weather conditions, and you want to know if a two-way interaction effect exists between the two within-subjects factors in explaining ski performance. A two-way repeated measures ANOVA can help determine whether such an interaction exists.

### Assumptions of Two-Way Repeated Measures ANOVA

Five assumptions need to be considered to run a two-way repeated measures ANOVA. The first two relate to your choice of study design, while the other three reflect the nature of your data:

• Assumption #1 is that you have a dependent variable measured at a continuous level (i.e., the interval or ratio 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 is that you have two within-subjects factors, each consisting of two or more categorical levels. It is essential to have clarity on two key terms: “within-subjects factor” and “levels,” to understand the structure of a two-way repeated measures ANOVA. These concepts are crucial in analyzing how different conditions or time points affect a dependent variable within the same subjects. In repeated measures ANOVA, a “factor” is an independent variable. However, we refer to it as a “within-subjects factor” because the same participants are repeatedly measured under varying conditions or at different time points. For instance, you may evaluate the memory recall ability of 15 students at three distinct time points to determine how memory retention changes over time. Similarly, you could assess how environmental noise levels affect the concentration of 30 office workers. The same participants would be subjected to different noise conditions to assess the impact of noise levels on their ability to concentrate. “Levels” refer to the distinct categories or time points within a within-subjects factor. In the memory recall example, the three levels are the different time points (pre-learning, post-learning, and one-week follow-up). In the noise level study, the levels are the various environmental noise conditions (silence, moderate, loud). Therefore, when discussing a within-subjects factor in a two-way repeated measures ANOVA, “levels” signify the different categorical states or discrete time points the participants experience. Each level represents a unique condition or time point, providing a basis for comparing the effects of these variables on the dependent variable.

Note: If you have three within-subjects factors rather than just two, you must run a three-way repeated measures ANOVA. This extension of the two-way repeated measures ANOVA determines if there is an interaction effect between three within-subjects factors on a continuous dependent variable (i.e., if a three-way interaction exists).

• Assumption #3 emphasizes eliminating significant outliers within any treatment group in a rigorous experimental setup. It is crucial to do so to ensure that the results are not skewed and accurately represent the treatment’s effectiveness. For instance, consider a study involving 150 individuals examining the impact of different diets on cholesterol levels. This study will be atypical if one individual shows a drastic deviation by exhibiting a 70 mg/dL reduction. In contrast, the rest show minimal variation with an average 20 mg/dL reduction. Such outliers might suggest unique physiological responses or external factors affecting that individual, which are not representative of the diet’s effectiveness. These outliers can have a negative impact on the two-way repeated measures ANOVA by distorting the differences between cells of the design and causing problems when generalizing the results to the population. Therefore, it is essential to identify and understand outliers in data analysis and choose whether to keep them, remove them, or alter their value in some way, given their potential impact on the results.
• Assumption #4 states that for each cell of the design, your dependent variable should be approximately normally distributed. It’s necessary for conducting statistical significance testing using a three-way repeated measures ANOVA. However, the test is “robust” to violations of normality, which means that even if there is some violation of this assumption, the test will still provide valid results. Therefore, it’s often acceptable to have approximately normally distributed data for this test. Also, as sample size increases, the distribution can be very non-normal, and the three-way repeated measures ANOVA can still provide valid results, thanks to the Central Limit Theorem. However, it’s important to note that if the distributions are all skewed similarly (e.g., all moderately negatively skewed), this is less of an issue than if you have different-shaped distributions for combinations of levels of the three within-subjects factors. Therefore, in this case, it’s crucial to examine whether strength scores are normally distributed for each cell of the design.
• Assumption #5 asserts that the variance of the differences between groups must be uniform. This assumption is sphericity, equivalent to the homogeneity of variances in the repeated measures. It is focused on the differences between levels rather than the variances within each level. This condition is vital for statistical significance testing in three-way repeated measures ANOVA. The violation of sphericity can result in inaccurate results, underscoring the importance of this assumption.

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### Interpret Your Two-Way Repeated Measures ANOVA Results

Following a structured approach is essential for accurate interpretation when analyzing the results from a two-way repeated measures ANOVA. Here’s how you can proceed with two steps, using an example involving factors like exercise type (e.g., cardio, strength training) and hydration level (e.g., well-hydrated, dehydrated) on physical performance.

STEP ONE: Determine if a Statistically Significant Two-Way Interaction Exists The primary objective in this step is to ascertain whether there’s a significant interaction between your two within-subjects factors, such as between exercise type and hydration level, on physical performance. You begin by examining profile plots and descriptive statistics to get a preliminary view of potential interactions. However, more than visual inspection is needed to confirm interactions, as these plots are based on sample data, and we need to understand the effect in the population. You’ll use a statistical test to formally test for an interaction effect, checking for sphericity with Mauchly’s test. If sphericity is assumed, the results for the two-way interaction will be unbiased. If sphericity is violated, an adjustment is needed to correct this bias. In SPSS, you’ll get results for both scenarios: when sphericity is met and when it’s not.

STEP TWO – OPTION A: Investigating Statistically Significant Simple Main Effects: If your analysis reveals a significant two-way interaction between exercise type and hydration level, the next step is to explore the simple main effects. This step means examining how one factor, like exercise type, affects performance at each level of the other factor, hydration level, and vice versa. In SPSS, this can be done by analyzing subsets of your data. For instance, you might explore how cardio and strength training affect performance differently when participants are well-hydrated versus when they are dehydrated. This analysis is akin to conducting separate one-way repeated measures ANOVAs for each level of the other factor.

STEP TWO – OPTION B: Determining Statistically Significant Main Effects: In cases without a significant two-way interaction, let’s focus shifts to the main effects of each within-subjects factor. For example, if the interaction between exercise type and hydration level isn’t significant, you’d separately examine the main effect of exercise type and hydration level on physical performance. If you find a significant main effect, you can conduct post hoc tests or pairwise comparisons to pinpoint where differences lie. This post hoc test could involve identifying how physical performance differs across various hydration levels, regardless of the exercise type, or how different types of exercise impact performance, irrespective of hydration status.

Each step in this process provides insight into how multiple factors interact and influence a dependent variable, facilitating a comprehensive understanding of your study’s outcomes.

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