Three-Way Repeated Measures ANOVA

The three-way repeated measures ANOVA is a robust statistical test used in experimental psychology and other scientific fields. The three-way repeated measures ANOVA enables researchers to explore complex interactions among three within-subject factors on a continuous outcome, thus extending the capabilities of the two-way repeated measures ANOVA by incorporating an additional variable into the analysis.

For instance, let’s consider a study designed to evaluate the effects of music genre (classical, jazz, or pop), volume level (low, medium, or high), and time of day (morning, afternoon, or evening) on cognitive task performance. Researchers can analyze the interaction between these three factors and their collective influence on task performance by employing a three-way repeated measures ANOVA. This method enables them to discern whether the impact of music genre on cognitive performance is consistent across various volume levels and times of day or if certain combinations of these factors are more conducive to cognitive enhancement. Insights derived from analyzing such complex interactions are crucial in understanding how environmental factors affect human performance.

Assumptions of Three-Way Repeated Measures ANOVA

Five assumptions need to be considered to run a three-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 states that you have three within-subject factors, each consisting of two or more categorical levels. In this guide, you will come across two essential terms: “within-subjects factor” and “levels.” A factor is another term for an independent variable, but in repeated measures ANOVA, it is usually called the within-subjects factor. This term indicates that the same individuals or cases are either measured on the same dependent variable at the same “time points” or measured on the same dependent variable while undergoing the same “conditions” or “treatments.” For instance, you might have recorded the blood pressure of 40 individuals at five different times throughout the day to examine the impact of daily activities on their cardiovascular health. In contrast, you could have measured the productivity of 25 employees while they were exposed to different environmental conditions to determine if environmental conditions affect their work efficiency. In both cases, the continuous nature of the variables allows for a detailed analysis of how they fluctuate under different conditions, providing valuable insights into the effects of daily routines on health and the influence of environmental factors on work performance. In a three-way repeated measures ANOVA, a within-subjects factor is another name for an independent variable, and the same cases are measured on the same dependent variable on two or more occasions. When referring to a within-subjects factor, we also discuss it as having “levels.” A within-subjects factor can have “categorical” levels, which means it is measured on a nominal, ordinal, or discrete-time scale. In a three-way repeated measures ANOVA, ordinal or discrete-time variables typically have two or more “time points,” while nominal variables usually have two or more “conditions.” The number of time points or conditions are called “levels” of the ordinal, nominal, or discrete-time variable. Therefore, when we refer to a “level” of a within-subjects factor in the guide, we only refer to “one” level. However, when we refer to “levels” of a within-subjects factor, we refer to “two or more” levels.
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 would be an atypical result if one individual shows a drastic deviation by exhibiting a reduction of 70 mg/dL while the rest show minimal variation with an average reduction of 20 mg/dL. 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 three-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.

Note: Technically, it is the residuals (errors) that need to be normally distributed, but the observations can act in their place (i.e., as surrogates).

Assumption #5 asserts that the variance of the differences between groups must be uniform. This assumption is known as 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 the context of 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 Three-Way Repeated Measures ANOVA Results

After running the three-way repeated measures ANOVA procedure and testing its assumptions, SPSS Statistics generates tables and graphs that provide the starting point for interpreting your results. We show you how to interpret these results, follow up on them, and write this output.

To interpret the results for your three-way repeated measures ANOVA, you can follow four steps, although whether you will need to follow all four steps (or just two or three steps) will depend on your results. First, determine whether a statistically significant three-way interaction exists (STEP #1). This step starts the process of interpreting your results.

**STEP #1:**

First, check for a statistically significant three-way interaction. This interaction indicates whether the simple two-way interactions differ across levels of the third factor. For instance, in a study with diet type (e.g., vegetarian, keto, balanced), exercise frequency (e.g., daily, biweekly, none), and hydration level (e.g., high, moderate, low), you’d examine if the interaction between diet type and exercise frequency on energy levels varies across hydration levels. If you find a significant three-way interaction, proceed to STEP 2A. Otherwise, go to STEP 2B.

**STEP #2A:**

If a significant three-way interaction exists, investigate the simple two-way interactions. Are there differences in these interactions at different levels of the third factor? Using the previous example, you might explore how diet and exercise frequency interact to affect energy levels at different hydration levels. A significant two-way interaction indicates you should proceed to STEP 3A. If there are none, your analysis may end here.

**STEP #2B:**

In the absence of a significant three-way interaction, assess for any significant two-way interactions. These interactions ignore the influence of the third factor. With our example, this could involve examining interactions like diet type* exercise frequency, diet type* hydration level, and exercise frequency* hydration level. If no significant two-way interactions are found, your analysis may conclude. Otherwise, proceed to STEP 3B.

**Step #3A:**

After identifying significant two-way interactions, look for significant simple main effects. These show how one factor affects the dependent variable at each level of another factor. For instance, if diet and exercise frequency significantly interact at a high hydration level but not at low or moderate levels, examine the effect of diet type on energy levels at each exercise frequency only at the high hydration level. If there are significant simple main effects, move to STEP 4A; if not, your analysis might end.

**Step #3B:**

With significant two-way interactions, determine if there are significant simple main effects. This step explores if the effect of one factor on the dependent variable differs based on the other factor’s levels. If no simple main effects are found, the analysis may conclude. If there are, proceed to STEP 4B.

**Step #4A:**

Conduct simple comparisons to pinpoint where differences lie upon finding significant simple main effects. For example, suppose diet type significantly affects energy levels at different exercise frequencies under high hydration. In that case, simple comparisons can reveal the specific conditions under which energy levels differ and by how much. If no significant simple comparisons are found, end your analysis. If there are, interpret and write up the findings.

**Step #4B:**

If there are significant simple main effects, use pairwise comparisons to understand where these differences occur. For example, if energy levels vary based on diet type and exercise frequency combinations, pairwise comparisons can specify which particular combinations lead to significant differences in energy levels. If no significant pairwise comparisons are found, the analysis concludes. If there are, interpret the results and prepare your report.