The two-way repeated measures ANOVA is used to determine if there is a statistically significant interaction effect between two within-subjects factors on a continuous dependent variable (i.e., if a two-way interaction exists). It is an extension of the one-way repeated measures ANOVA, which only includes a single within-subjects factor.
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 two-way repeated measures ANOVA is also referred to as a “within-within-subjects ANOVA” or “two-way within-subjects ANOVA”.
A two-way repeated measures ANOVA can be used in a number of situations. For example, imagine you are interested in the effect of two different types of ski goggle (i.e., blue-tinted or gold-tinted ski goggles) on ski performance (i.e., time to complete a ski run). In particular, you are concerned that the effect of the different lens colours on ski performance might be different depending on whether it is overcast or sunny (i.e., under different weather conditions). You suspect that ski performance will depend on both ski goggle lens colour and the weather conditions. As such, you want to determine if a two-way interaction effect exists between ski goggle lens colour and weather conditions (i.e., the two within-subjects factors) in explaining ski performance. A two-way repeated measures ANOVA can be used to examine whether such a two-way interaction exists.
In order to run a two-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. You can learn more about continuous variables in our article: Types of Variable.
- Assumption #2: You have two 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 color 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 two-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 two-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).
Note: If you have three within-subjects factors rather than just two, you will need to run a three-way repeated measures ANOVA. The three-way repeated measures ANOVA is used to determine if there is an interaction effect between three within-subjects factors on a continuous dependent variable (i.e., if a three-way interaction exists). It is an extension of the two-way repeated measures ANOVA.
- Assumption #3: There should be no significant outliers in any combination of the related groups. Outliers are simply single 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 effect on the two-way repeated measures ANOVA, distorting the differences between the related groups (whether increasing or decreasing the scores on the dependent variable), which reduces the accuracy of your results. Fortunately, when using SPSS Statistics to run a two-way repeated measures ANOVA on your data, you can easily detect possible outliers. In our enhanced two-way repeated measures ANOVA guide, we: (a) show you how to detect outliers using SPSS Statistics; and (b) discuss some of the options you have in order to deal with outliers.
- Assumption #4: The distribution of the dependent variable in each combination of the related groups should be approximately normally distributed. We talk about the two-way repeated measures ANOVA only requiring approximately normal data because it is quite “robust” to violations of normality, meaning that assumption can be a little violated and still provide valid results. You can test for normality using the Shapiro-Wilk test of normality (using residuals), which is easily tested for using SPSS Statistics. In addition to showing you how to do this in our enhanced two-way repeated measures ANOVA guide, we also explain what you can do if your data fails this assumption (i.e., if it fails it more than a little bit).
- Assumption #5: Known as sphericity, the variances of the differences between all combinations of related groups must be equal. Fortunately, SPSS Statistics makes it easy to test whether your data has met or failed this assumption. Therefore, in our enhanced two-way repeated measures ANOVA guide, we (a) show you how to perform Mauchly’s Test of Sphericity in SPSS Statistics, (b) explain some of the things you will need to consider when interpreting your data, and (c) present possible ways to continue with your analysis if your data fails to meet this assumption.
After running the two-way repeated measures ANOVA procedure, SPSS Statistics will have generated a number of tables and graphs that provide the starting point to interpret your results.
There are two steps you can follow to interpret the results for your two-way repeated measures ANOVA. First, you need to determine whether a statistically significant two-way interaction exists (STEP ONE). Next, if you have a statistically significant two-way interaction, you need to determine whether you have any statistically significant simple main effects (STEP TWO – OPTION A), but if you do not have a statistically significant two-way interaction, you need to determine whether you have any statistically significant main effects (STEP TWO – OPTION B). These two steps are explained below:
- STEP ONE:
Determine whether a statistically significant two-way interaction exists: The primary goal of running a two-way repeated measures ANOVA is to determine whether there is a statistically significant two-way interaction between the two within-subjects factors (i.e., a treatment*time interaction). We can gain an initial impression of whether we have an interaction between the two within-subjects factors by visually: (a) inspecting the profile plots that have been produced, and (b) consulting the descriptive statistics for the dependent variables based on the levels of the two within-subjects factors, which helps to verify any of the trends you identify in the profile plot. However, despite the usefulness of profile plots in understanding your data, you cannot determine an interaction effect from them because the profile plot is based on the sample data and we are interested in determining whether there is an interaction effect in the population (Fox, 2008). Therefore, a formal statistical test is required to test for the presence of an interaction effect (i.e., via statistical significance testing). Before you can find out the result of the two-way treatment*time interaction (i.e., whether the two-way interaction effect is statistically significant), you need to establish if the assumption of sphericity has been violated (specifically for the interaction term) using Mauchly’s test of sphericity. Accepting the assumption of sphericity indicates that the statistical result of the two-way interaction will not be biased (with regard to this particular assumption) and no adjustment to the test is needed. On the other hand, if the assumption of sphericity is violated, this means that the result is biased in that it too easily returns a statistically significant result. However, a correction can be made to correct for this bias. SPSS Statistics will produce four different test results for the two-way repeated measures ANOVA. The first result is for when the assumption of sphericity is met and the other three results are for when the assumption is violated.
- STEP TWO – OPTION A:
If you have a statistically significant two-way interaction, determine whether you have any statistically significant simple main effects: When you have a statistically significant two-way interaction, reporting the main effects can be misleading and you will want to determine the difference between trials at each level of time and vice versa, called simple main effects. Unfortunately, it is not possible to run simple main effects using syntax in SPSS Statistics, but it is possible to analyze the data using the GLM: Repeated Measures procedure on different subsets of the variables int_1 through con_3. Essentially, for a two-way repeated measures ANOVA, running simple effects is the same as running separate one-way repeated measures ANOVAs.As with any two-way interaction, you can investigate the effects of within-subjects factor A at every level of within-subjects factor B and/or the effects of within-subjects factor B at every level of within-subjects factor A. In this example, you could, therefore, investigate the effect of treatment on CRP concentration at every time point (i.e., at every level of time) or investigate the effect of time for the control trial and the exercise intervention trial (i.e., at every level of treatment). Often, you would choose one or the other based on theoretical reasons. For example, you might consider one of the within-subjects factors to be a moderating variable. In this example, we will consider both options.
- STEP TWO – OPTION B:
If you do not have a statistically significant two-way interaction, determine whether you have any statistically significant main effects: If you do not have a statistically significant two-way interaction, you need to interpret the main effects for the two within-subjects factors (i.e., the main effect for treatment and the main effect for time). If a main effect is statistically significant you can follow up that main effect with pairwise comparisons; that is, with a post hoc test. This will inform you where the differences in CRP concentration between time points lie.