One-Way Repeated Measures ANOVA

The one-way repeated measures analysis of variance (ANOVA) is a statistical technique that extends the concept of the paired-samples t-test. It is utilized to identify if there are any significant differences between the means of three or more levels of a within-subjects factor, where the same cases (such as participants) are involved in each level. This test is particularly suitable for scenarios where the same individuals are tested multiple times on the same dependent variable or tested under different conditions on the same dependent variable. It’s often referred to as a within-subjects ANOVA or an ANOVA with repeated measures.

It’s important to note that while a one-way repeated measures ANOVA can be used for within-subjects factors with just two levels, it is typically reserved for situations where there are three or more levels. This is because a paired-samples t-test is generally preferred for comparisons involving only two levels.

For instance, a one-way repeated measures ANOVA could be employed to examine changes in blood pressure in patients undergoing different stress management techniques (e.g., with three techniques: meditation, counseling, and exercise). Here, “blood pressure” would be the dependent variable, and the within-subjects factor would be the “stress management technique” (having three levels). Alternatively, this ANOVA could be used to assess changes in productivity levels of employees in different office environments (e.g., with four environments: open-plan, private office, home office, and coworking space). In this case, “productivity level” is the dependent variable, and the within-subjects factor is “office environment” (with four levels).

If your study design includes two within-subjects factors (for example, both time and type of stress management technique), a two-way repeated measures ANOVA, also known as a within-within-subjects ANOVA, would be the appropriate method to use.

Assumptions of One-Way Repeated Measures ANOVA

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 measured at the 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 involves having one within-subjects factor with three or more categorical levels. Understanding the concepts of “within-subjects factor” and “levels” is key in this context. A “within-subjects factor” is essentially an independent variable in a repeated measures ANOVA. This factor involves the same cases (like participants) being measured multiple times under varying conditions or at different time points on the same dependent variable. For example, a study might assess the heart rate variability (the dependent variable) of a group of athletes under four different stress conditions (rest, low stress, moderate stress, high stress). Here, “stress condition” is the within-subjects factor with four levels. Alternatively, the study might measure students’ concentration levels (the dependent variable) at several time points (e.g., beginning of semester, mid-semester, end of semester) to see how concentration fluctuates over time. In this case, “time” is the within-subjects factor with three categorical time points, each representing a level.When we talk about “levels” in this context, we refer to the different categories or conditions under which the dependent variable is measured. These levels are either categorical (like different stress conditions) or ordinal (like different time points in a semester).It’s important to note that while a one-way repeated measures ANOVA can technically be applied with just two levels of a within-subjects factor, it is more commonly used when there are three or more levels. This is because a paired-samples t-test is typically more suitable for comparisons involving only two levels. Thus, in this guide, we focus on scenarios where the within-subjects factor encompasses three or more levels.
Assumption #4 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 an atypical result 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 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 #5 is that the dependent variable should be roughly normally distributed at each level of the within-subjects factor. This normality assumption is important for conducting valid significance testing. However, one-way repeated measures ANOVA is known for its robustness to minor violations of this assumption, meaning it can still yield valid results even if the data is not perfectly normally distributed.The flexibility of this test increases with larger sample sizes, as the Central Limit Theorem suggests that the distribution of the sample means will approach normality even if the original data is not normally distributed. Furthermore, the consistency of skewness across different levels of the within-subjects factor is less concerning than having levels with distinctly different distribution shapes.For instance, in a study evaluating the impact of different types of exercise regimes (e.g., aerobic, strength training, and flexibility exercises) on muscle flexibility, you would need to assess if muscle flexibility measurements are normally distributed for each exercise regime. In practical terms, this means checking the normality of flexibility scores post-aerobic exercise, post-strength training, and post-flexibility exercises. Even if the distributions are moderately skewed in the same way across all levels, this would not pose as much of an issue as having one exercise regime showing a different skewness or distribution shape compared to others.
Assumption #6 is sphericity. This assumption states that the variances of the differences between all combinations of levels of the within-subjects factor must be equal. However, it is often challenging to avoid violating the assumption of sphericity, which can cause the test to become too liberal. This, in turn, leads to an increase in the Type I error rate, which is the probability of detecting a statistically significant result when there isn’t one (Weinfurt, 2000). Luckily, SPSS Statistics offers a solution to this problem by providing the Mauchly’s test of sphericity. This test helps you determine whether your data has met or failed this assumption with ease.

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Interpret Your One-Way Repeated Measures ANOVA Results

After running the one-way repeated measures ANOVA procedure with either post hoc tests or planned contrasts based on our enhanced guide, SPSS Statistics generates several tables that contain all the necessary information to interpret and report the results. In this section, we will illustrate how to interpret these results.

The SPSS Statistics output provides some useful descriptive statistics that can help you get a sense of your data. These statistics include sample size, mean scores of the within-subjects factor levels, any trends, and the variation in each level. These descriptive statistics will be helpful when you report your results.

Whether your data has met or violated the assumption of sphericity determines how you interpret your results after running a one-way repeated measures ANOVA. We will show you how to interpret this critical assumption for your data, which will guide your next steps:

If your data has met the assumption of sphericity, you can interpret the standard one-way repeated measures ANOVA output in SPSS Statistics. We will guide you through interpreting the means, standard deviations, F-value, degrees of freedom, and p-value. We will also show you how to determine whether the means of the dependent variable are statistically significantly different for the different levels of the within-subjects factor, and if you can reject or fail to reject the null hypothesis. We will demonstrate how you can bring all of this together into a single paragraph that explains your results. Additionally, we will teach you how to calculate an effect size from your one-way repeated measures ANOVA results and how to include it in your write-up.
If your data has violated the assumption of sphericity, you can still continue with your analysis. However, you will need to interpret the results from a modified one-way repeated measures ANOVA, where there are adjustments to the degrees of freedom for both the within-subjects factor and error effect. This adjustment has an impact on the statistical significance (i.e., p-value) of the test. We will show you how to 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. We will also guide you through determining whether the means of the dependent variable are statistically significantly different for the different levels of the within-subjects factor, and if you can reject or fail to reject the null hypothesis. Finally, we will demonstrate how to bring all of this together into a single paragraph that explains your results. We will also teach you how to calculate an effect size from your one-way repeated measures ANOVA results and how to include it in your write-up.