The three-way repeated measures ANOVA is used to determine if there is a statistically significant interaction effect between three within-subjects factors on a continuous dependent variable (i.e., if a three-way interaction exists). As such, it extends the two-way repeated measures ANOVA, which is used to determine if such an interaction exists between just two within-subjects factors (i.e., rather than three within-subjects factors).

**Note: **It is quite common for “within-subjects factors” to be called “independent variables”, but we will continue to refer to them as “within-subjects factors” (or simply “factors”) in this guide. Furthermore, it is worth noting that the three-way repeated measures ANOVA is also referred to more generally as a “factorial repeated measures ANOVA” or more specifically as a “three-way within-subjects ANOVA”.

A three-way repeated measures ANOVA can be used in a number of situations. For example, you might be interested in the effect of two different types of ski goggle (i.e., blue-tinted or gold-tinted ski goggles) for improving ski performance (i.e., time to complete a ski run). However, you are concerned that the effect of the different lens colours on ski performance might be different depending on the snow condition (i.e., whether there has been recent snow fall or not), as well as whether it is overcast or sunny (i.e., current weather conditions). Indeed, you suspect that the effect of the type of lens colour on ski performance will depend on both the snow conditions and the current weather conditions. As such, you want to determine if a three-way interaction effect exists between lens colour, snow conditions and the current weather conditions (i.e., the three within-subjects factors) in explaining ski performance. A three-way repeated measures ANOVA can be used to examine whether such a three-way interaction exists.

## Assumptions

In order to run a three-way repeated measures ANOVA, there are five assumptions that need to be considered. The first two relate to your choice of study design, whilst the other three reflect the nature of your data:

- Assumption #1: You have
**one dependent variable**that is measured at the**continuous**level (i.e., it is measured at the**interval**or**ratio**level). Examples of continuous variables include revision time (measured in hours), intelligence (measured using IQ score), exam performance (measured from 0 to 100), weight (measured in kg), and so forth.

- Assumption #2: You have
**three within-subjects factors**where each within-subjects factor consists of**two or more categorical levels**. These are two particularly important terms that you will need to understand in order to work through this guide; that is, a “within-subjects factor” and “levels”. Both terms are explained below: A**factor**is another name for an**independent variable**. However, we use the term “factor” instead of “independent variable” throughout this guide because in a repeated measures ANOVA, the independent variable is often referred to as the**within-subjects factor**. The “within-subjects” part simply means that the same cases (e.g., participants) are either: (a) measured on the same dependent variable at the same “time points”; or (b) measured on the same dependent variable whilst undergoing the same “conditions” (also known as “treatments”). For example, you might have measured 10 individuals’ 100m sprint times (the dependent variable) on five occasions (i.e., five time points) during the athletics season to determine whether their sprint performance improved. Alternately, you may have measured 20 individuals’ task performance (the dependent variable) when working in three different lighting conditions (e.g., red, blue and natural lighting) to determine whether task performance was affected by the colour lighting in the room. For now, all you need to know is that a within-subjects factor is another name for an independent variable in a three-way repeated measures ANOVA where the same cases (e.g., participants) are measured on the same dependent variable on two or more occasions.When referring to a within-subjects factor, we also talk about it having “levels”. More specifically, a within-subjects factor has “categorical” levels, which means that it is measured on a**nominal**,**ordinal**or**discrete-time**scale. Such ordinal or discrete-time variables in a three-way repeated measures ANOVA are typically two or more “time points” (e.g., two time points where the dependent variable is measured “pre-intervention” and “post-intervention”; three time points where the dependent variable is measured: “pre-intervention”, “post-intervention” and “6-month follow-up”; or four time points where the dependent variable is measured: at “10 secs”, “20 secs”, “30 secs” and “40 secs”). Such nominal variables in a two-way repeated measures ANOVA are typically two or more “conditions” (e.g., two conditions where the dependent variable is measured: a “control” and an “intervention”; three conditions where the dependent variable is measured: a “control”, “intervention A” and “intervention B”; or four conditions where the dependent variable is measured: in a room with “red lighting”, “blue lighting”, yellow lighting” and “natural lighting”). The number of time points or conditions are referred to as “levels” of the ordinal, nominal or discrete-time variable (e.g., three time points reflects three levels). Therefore, when we refer to a “level” of a within-subjects factor in the guide, we are only referring to “one” level (e.g., the room with “red lighting” or the room with “blue lighting”). However, when we refer to “levels” of a within-subjects factor, we are referring to “two or more” levels (e.g., “red and blue” lighting, or “red, blue and yellow” lighting).

- Assumption #3: There should be no significant outliers in any cell of the design. Outliers are simply data points within your data that do not follow the usual pattern (e.g., in a study of 100 students’ IQ scores, where the mean score was 108 with only a small variation between students, one student had a score of 156, which is very unusual, and may even put her in the top 1% of IQ scores globally). The problem with outliers is that they can have a negative impact on the three-way repeated measures ANOVA by: (a) distorting the differences between cells of the design; and (b) causing problems when generalizing the results (of the sample) to the population. Due to the effect that outliers can have on your results, you have to choose whether you want to: (a) keep them in your data; (b) remove them; or (c) alter their value in some way.

- Assumption #4: Your dependent variable should be approximately normally distributed for each cell of the design. The assumption of normality is necessary for statistical significance testing using a three-way repeated measures ANOVA. However, the three-way repeated measures ANOVA is considered somewhat “robust” to violations of normality. This means that some violation of this assumption can be tolerated and the test will still provide valid results. Therefore, you will often hear of this test only requiring
*approximately*normally distributed data. Furthermore, as sample size increases, the distribution can be very non-normal and, thanks to the Central Limit Theorem, the three-way repeated measures ANOVA can still provide valid results. Also, it should be noted that if the distributions are all skewed in a similar manner (e.g., all moderately negatively skewed), this is not as troublesome when compared to the situation where you have combinations of levels of the three within-subjects factors that have differently-shaped distributions (e.g., not all combinations of levels of the three within-subjects factors are moderately negatively skewed). Therefore, in this example, you need to investigate 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: The variance of the differences between groups should be equalThis assumption is referred to as the assumption of sphericity. It is sometimes described as the repeated measures equivalent of the homogeneity of variances and refers to the variances of the differences between the levels rather than the variances within each level. This assumption is necessary for statistical significance testing in the three-way repeated measures ANOVA. This assumption is very important and violation of sphericity can lead to invalid results.

## Interpreting Results

After running the three-way repeated measures ANOVA procedure and testing for the assumptions of the three-way repeated measures ANOVA, SPSS Statistics will have generated a number of tables and graphs that provide the starting point to interpret your results. We show you how to interpret these results and follow them up. We also show how to write up this output as you work through the section.

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

- STEP #1:

Do you have a statistically significant three-way interaction?A three-way interaction is when one or more simple two-way interactions are different (at the level of a third factor) on the dependent variable. Which of your three factors make up the simple two-way interaction and which acts as the third factor will depend of your study design.For example, we have a three-way interaction between cardio, weights and time (cardio*weights*time), but are interested in the simple two-way interaction between the two factors (weights*time) at the different levels of the factor, cardio. In other words, is the effect of the interaction between weights and time on strength scores affected by whether cardiovascular training is present?If**yes**– you have a statistically significant three-way interaction – go to STEP 2A.

If**no**– you**do not**have a statistically significant three-way interaction – go to STEP 2B.

- STEP #2A:

You have a statistically significant three-way interaction. Do you have any statistically significant simple two-way interactions?You know that there is a statistically significant difference in the simple two-way interactions for one or more levels of a third factor on the dependent variable. This informs you that you need to investigate each simple two-way interaction for statistical significance. For example, if there was a simple two-way interaction between factor 1 and factor 2 (factor 1*factor 2) at one or more levels of factor 3, which let’s say has two levels (level A and level B), simple two-way interactions will tell you whether the dependent variable differs based on this interaction (i.e., between factor 1 and factor 2) at just one level (e.g., level A or level B) of factor 3 or both levels of factor 3.For example, let’s say that we know there is a statistically significant difference in strength score based on the interaction between weight training and time (i.e., the two factors: weights and time), based on whether cardiovascular training is present (i.e., the third factor, cardio). Determining whether there are any statistically significant simple two-way interactions will tell us whether strength scores are affected by a weights*time effect at one or both of the levels of our third factor, cardio. However, you should note that it is possible to have a statistically significant three-way interaction, but not have any statistically significant simple two-way interactions.If**no**– you**do not**have any statistically significant simple two-way interactions – end analysis and write up.

If**yes**– you have statistically significant simple two-way interactions – go to STEP 3A.

- STEP #2B:

You**do not**have a statistically significant three-way interaction. Do you have any statistically significant two-way interactions?A two-way interaction ‘ignores’ the influence of a third factor. Since you are running a three-way repeated measures ANOVA, meaning that you have a total of three factors, there are three possible two-way interactions (i.e., factor 1*factor 2, factor 1*factor 3 and factor 2*factor 3). Therefore, a two-way interaction is when there is a difference in the dependent variable based on an interaction between two factors.For example, since our three factors are weights, cardio and time, the three possible two-way interactions are: cardio*weights, cardio*time and weights*time. Therefore, if there was a two-way interaction between cardio*time on strength score, this would mean that strength score differs based on some combination of cardio (i.e., whether cardiovascular training is present) and time (i.e., when the strength score was taken during the intervention).If**no**– you**do not**have any statistically significant two-way interactions – end analysis and write up.

If**yes**– you have statistically significant two-way interactions – go to STEP 3B

- Step #3A

You have any statistically significant simple two-way interactions. Do you have any statistically significant simple simple main effects?After you know which of the levels of factor 3 affect the scores on the dependent variable based on the interaction between factor 1 and factor 2 (factor 1*factor 2), simple simple main effects determine the effect of factor 1 on the dependent variable at each level of factor 2 and vice versa. For example, let’s imagine that there was a statistically significant simple two-way interaction between factor 1 and factor 2 on the dependent variable at level B of factor 3, but not at level A (e.g. assuming that factor 3 had two levels: A and B).You could choose to investigate: (a) the effects of factor 1 on the scores on the dependent variable at each level of factor 2; (b) the effects of factor 2 on the scores on the dependent variable at each level of factor 1; or (c) both. Let’s imagine that you were interested in (a): the effects of factor 1 on the scores on the dependent variable at each level of factor 2. Also, let’s imagine that factor 2 had three levels: A, B and C. Simple simple main effects would tell you if factor 1 led to different scores on the dependent variable depending on which of the three levels of factor 2 were present.For example, let’s imagine that there was a statistically significant simple two-way interaction between weight and time on strength score with no cardiovascular training, but not with cardiovascular training. We could now investigate: (a) the effects of time on strength score at each level of weight with no cardiovascular training; (b) the effects of weights on strength score at each level of time with no cardiovascular training; or (c) both. Let’s imagine that we were interested in (a): the effects of time on strength score at each level of weight with no cardiovascular training. Simple simple main effects would tell us if time leads to different strength scores depending on whether weight training was performed where no cardiovascular training was performed for either intervention.If**no**– you**do not**have any statistically significant simple simple main effects – end analysis and write up.

If**yes**– you have statistically significant simple simple main effects – go to STEP 4A.

- Step #3B

You have statistically significant two-way interactions. Are there any statistically significant simple main effects?After you know which of the three possible two-way interactions are statistically significant, simple main effects determine whether the effect of one of the factors on the dependent variable differs based on the values of the other factor and vice versa (these two factors being the factors involved in the statistically significant interaction). However, if the effect of the factor in question has more than two groups/levels, you will not be able to determine which specific groups differ.For example, if strength scores differed based on a two-way interaction between weights and time (i.e., weights*time), simple main effects would tell us the effect of time on strength scores at each level of cardio and vice versa. However, simple main effects would not tell us where any differences lie, only that there was a difference in strength score over all levels of time, for example.If**no**– there are**no**statistically significant simple main effects – end analysis and write up.

If**yes**– there are statistically significant simple main effects – go to STEP 4B.

- Step #4A

You have statistically significant simple simple main effects. Are there any statistically significant simple simple comparisons?Finally, after you know that there are simple simple main effects assuming a factor 1*factor 2 interaction at one or more levels of factor 3, and have one or more statistically significant simple simple main effects, you can now use simple simple comparisons to determine: (a) exactly where the differences were (e.g., factor 1 led to different scores on the dependent variable when level B of factor 2 was present, but not when level A was present); and (b) the direction and magnitude of the difference in the dependent variable (e.g., that the dependent variable was higher by X amount when level B of factor 2 was present compared with level A).Therefore, let’s imagine that the simple simple main effect of time was statistically significant for weight training only. We can now use simple simple comparisons to determine: (a) exactly where any differences were in strength score between all possible combinations of time (pre, mid and post) for weight training only; and (b) the direction and magnitude of any differences in strength score based on these different time points (e.g., strength score was 11.3 kg higher mid- weight training only trial compared to pre- weight training only trial).If**no**– there are**no**statistically significant simple simple comparisons – end analysis and write up.

If**yes**– there are statistically significant simple simple comparisons – interpret findings and write up.

- Step #4B

You have statistically significant simple main effects. Are there any statistically significant pairwise comparisons?Finally, after you know that there are simple main effects, pairwise comparisons determine where these differences lie. For example, if the scores on the dependent variable differed based on some combination of factor 2 and factor 3 (i.e., factor 2*factor 3), pairwise comparisons would tell you: (a) exactly where the differences were (e.g., when the second level of factor 2 and first level of factor 3 were present compared to when the second level of factor 2 and third level of factor 3 were present); and (b) the direction and magnitude of the difference in the dependent variable (e.g., that the dependent variable was higher by X amount when the second level of factor 2 and first level of factor 3 were present compared to when the second level of factor 2 and third level of factor 3 were present).For example, if simple main effects showed us that strength scores differed based on some combination of the different levels of weights and time, pairwise comparisons could tell us that strength score was, for example, 19.2 kg higher between pre- and mid- time points when weight training was performed.If**no**– there are**no**statistically significant pairwise comparisons – end analysis and write up.

If**yes**– there are statistically significant pairwise comparisons – interpret findings and write up.

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