The one-way repeated measures analysis of variance (ANOVA) is an extension of the paired-samples t-test and is used to determine whether there are any statistically significant differences between the means of three or more levels of a within-subjects factor. The levels are related because they contain the same cases (e.g., participants) in each level. The participants are either the same individuals tested on three or more occasions on the same dependent variable or the same individuals tested under three or more different conditions on the same dependent variable. This test is also referred to as a within-subjects ANOVA or ANOVA with repeated measures.

Note: Whilst a one-way repeated measures ANOVA can be used when your within-subjects factor has just two levels, it is typically only used when the within-subjects factor has three or more levels. The reason for this is that when there are only two levels, a paired-samples t-test is more commonly used. This is why we refer to the one-way repeated measures ANOVA having three or more levels in this guide.

For example, you could use a one-way repeated measures ANOVA to understand whether there is a difference in cigarette consumption amongst heavy smokers after a hypnotherapy program (e.g., with three-time points: cigarette consumption immediately before, 1 month after and 6 months after the hypnotherapy program). In this example, “cigarette consumption” is your dependent variable, whilst your within-subjects factor is “time” (i.e., with three levels, where each of the three-time points is considered a level). Alternately, you could use a one-way repeated measures ANOVA to understand whether there is a difference in braking distance in a car based on four different colored tints of windscreen (e.g., braking distance under four conditions: no tint, low tint, medium tint, and dark tint). In this example, “braking distance” is your dependent variable, whilst your within-subjects factor is “condition” (i.e., with four levels, where each of the four conditions is considered a level).

Note: Whilst the repeated measures ANOVA is used when you have just one within-subjects factor, if you have two within-subjects factors (e.g., you measured time and condition), you will need to use a two-way repeated measures ANOVA, also known as a within-within-subjects ANOVA.

## Assumptions

In order to run a one-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. These assumptions are:

• 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 one within-subjects factor that consists of three 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’ 100 m 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 one-way repeated measures ANOVA where the same cases (e.g., participants) are measured on the same dependent variable on three 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 or ordinal scale. Such ordinal variables in a one-way repeated measures ANOVA are typically three or more “time points” (e.g., 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 sec” and “40 secs”). Such nominal variables in a one-way repeated measures ANOVA are typically three or more “conditions” (e.g., 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 is referred to as “levels” of the ordinal or nominal 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: Whilst a one-way repeated measures ANOVA can be used when your within-subjects factor has just two categorical levels, it is typically only used when the within-subjects factor has three or more categorical levels. The reason for this is that when there are only two categorical levels, a paired-samples t-test is more commonly used. This is why we refer to the one-way repeated measures ANOVA having three or more levels in this guide.

• Assumption #3: There should be no significant outliers in any level of the within-subjects factorOutliers 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 impact on the one-way repeated measures ANOVA by (a) distorting the differences between the levels of the within-subjects factor (whether increasing or decreasing the scores on the dependent variable), and (b) causing problems when generalizing the results (of the sample) to the population.
• Assumption #4: Your dependent variable should be approximately normally distributed for each level of the within-subjects factorThe assumption of normality is necessary for statistical significance testing using a one-way repeated measures ANOVA. However, the one-way repeated measures ANOVA is considered “robust” to violations of normality. This means that some violations 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 normal data. Furthermore, as sample size increases, the distribution can be very non-normal and, thanks to the Central Limit Theorem, the one-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 as when compared to the situation where you have levels that have differently-shaped distributions (e.g., not all levels of a within-subjects factor are moderately negatively skewed). Therefore, in this example, you need to investigate whether CRP is normally distributed for each level of the within-subjects factor, time. In other words, whether crp_pre, crp_mid and crp_post are normally distributed.

Note: Technically, it is the residuals (errors) that need to be normally distributed. However, for a repeated measures one-way ANOVA the distribution of the scores (observations) in each level of the within-subjects factor will be the same as the distribution of the residuals in each level.

• Assumption #5: Known as sphericity, the variances of the differences between all combinations of levels of the within-subjects factor must be equal. Unfortunately, it is considered difficult not to violate the assumption of sphericity (e.g., Weinfurt, 2000), which causes the test to become too liberal (i.e., leads to an increase in the Type I error rate; that is, the probability of detecting a statistically significant result when there isn’t one). Fortunately, SPSS Statistics makes it easy to test whether your data has met or failed this assumption using Mauchly’s test of sphericity.

## Interpreting Results

After running the one-way repeated measures ANOVA procedure with either post hoc tests or planned contrasts in the previous section, SPSS Statistics will have generated a number of tables that contain all the information you need to interpret and report your results. we show you how to interpret these results.

You will get some useful descriptive statistics from the SPSS Statistics output that will help you get a “feel” for your data (and will also be used when you report your results). This includes information on sample size, which levels of the within-subjects factor had the higher/lower mean score and if there are any trends, and if the variation in each level is similar.

If you have been following this guide from the very beginning, you’ll know that how you interpret your results after running a one-way repeated measures ANOVA depends on whether your data met or violated the assumption of sphericity.  We show you how to interpret this critical assumption for your data, which will determine what you do next:

• Sphericity was met: If your data has met the assumption of sphericity, you simply need to interpret the ‘standard’ one-way repeated measures ANOVA output in SPSS Statistics. We will (a) interpret the SPSS Statistics output for the one-way repeated measures ANOVA, including the means, standard deviations, F-value, degrees of freedom and p-value; (b) determine whether the means of the dependent variable are statistically significantly different for the different levels of the within-subjects factor; (c) determine if we can reject, or fail to reject, the null hypothesis; and (d) how you can bring all of this together into a single paragraph that explains your results. You can also add an effect size to your analysis, which is becoming a more common method of expressing your results. We will teach you how to calculate an effect size from your one-way repeated measures ANOVA results, and how to add this to your write-up.
• Sphericity was violated: If your data has violated the assumption of sphericity, you can still continue with your analysis. However, you will have to interpret the results from a modified one-way repeated measures ANOVA where there have been adjustments to the degrees of freedom for both the within-subjects factor and error effect (Greenhouse & Geisser, 1959), which has an impact on the statistical significance (i.e., p-value) of the test. Therefore, we will (a) interpret the SPSS Statistics output for the Greenhouse and Geisser (1959) adjusted one-way repeated measures ANOVA, explaining the means, standard deviations, F-value, degrees of freedom and p-value; (b) determine whether the means of the dependent variable are statistically significantly different for the different levels of the within-subjects factor; (c) determine if we can reject, or fail to reject, the null hypothesis; and (d) bring all of this together into a single paragraph that explains your results. We can also add an effect size to our analysis, which is becoming a more common method of expressing your results. Therefore, we will teach you how to calculate an effect size from your one-way repeated measures ANOVA results, and how to add this to your write-up.