By Ruben Geert van den Berg on September 16, 2014 under Repeated Measures ANOVA Tutorials.

# SPSS Repeated Measures ANOVA

SPSS repeated measures ANOVA is a procedure for testing whether the means of 3 or more metric variables are equal in some population.ANOVA is short for “**An**alysis **o**f **Va**riance. This is a family of statistical procedures for testing whether means for groups of cases and/or variables are equal. These variables have been measured on the same cases. Whatever distinguishes these variables (sometimes just the time of measurement) is known as the within-subjects factor. The simplest possible scenario examines a single within-subjects factor with three levels.

## Repeated Measures ANOVA Example

A marketeer wants to launch a new commercial and has four concept versions. She shows the four concepts to 40 participants and asks them to rate each one of them on an 11-point scale, resulting in commercial_ratings.sav.Although such ratings are strictly ordinal variables, we'll treat them as metric variables under the assumption of equal intervals. Part of these data are shown below.

The research question is which -if any- concept is the most appealling. We'll estimate the appeal of each concept by computing its mean rating. We'll then try and generalize this sample outcome to the population by testing the **null hypothesis** that
the 4 population mean scores are all equal.

## 1. Quick Data Check

Before jumping blindly into statistical tests, let's first get a rough idea of what the data look like. Do the frequency distributions look plausible? Are there any system missing values or user missing values that we need to define? For quickly answering such questions, we'll open the data and run histograms with the syntax below.

***1. Set default directory.**

cd 'd:downloaded'. /*or wherever data file is located.

***2. Open data.**

get file 'commercial_appeal.sav'.

***3. Quick check.**

freq com_1 to com_4

/format notable

/hist.

## Result

The mean ratings vary between 4.2 and 6.3, a finding that certainly has practical significance. Note that the first concept receives a score of 10 by one respondent while none of the other respondents scores higher than 6. This is slightly odd but we don't see this pattern in the other histograms. We don't remove this respondent from the data.

## 2. Assumptions Repeated Measures ANOVA

Running a statistical test doesn't always make sense; results reflect reality only insofar as relevant assumptions are met. For a (single factor) repeated measures ANOVA these are

**Independent observations**(or, more precisely, independent and identically distributed variables).- The test variables follow a
**multivariate normal distribution**in the target population. **Sphericity**. This means that the population variances of all possible difference scores (com_1 - com_2, com_1 - com_3 and so on) are equal.

Assumption 1 is mostly theoretical. Violation of assumption 2 hardly affects test results for reasonable sample sizes (say n >30). Sphericity is tested with **Mauchly’s test** which is always included in SPSS repeated-measures ANOVA output. If this test indicates a serious violation of sphericity, we'll report corrected test results as we'll see in a minute.

## 3. Run SPSS Repeated Measures ANOVA

We may freely choose a name for our within-subjects factor. We went with “commercial” because it's the commercial that differs between the four ratings made by each respondent.

We may also choose a name for whatever each of the four variables is supposed to reflect. In this case we went with “rating”

We now select all four variables and move them to the

Under we'll select .

Clicking results in the syntax below.

***Run repeated measures ANOVA.**

GLM com_1 com_2 com_3 com_4

/WSFACTOR=commercial 4 Polynomial

/MEASURE=rating

/METHOD=SSTYPE(3)

/PRINT=DESCRIPTIVE

/CRITERIA=ALPHA(.05)

/WSDESIGN=commercial.

## 4. SPSS Repeated Measures ANOVA Output

We'll ignore some of the output tables (most notably Multivariate Tests) but the **Descriptive Statistics** are essential. If the null hypothesis (all means equal) is rejected, then the observed means indicate which commercial may be most appealling - which is exactly what we want to know.

These means can be obtained by DESCRIPTIVES as well but make sure you use listwise deletion in this case. This is because SPSS Repeated Measures ANOVA also uses only cases without any missing values on all four variables. Since the significance test is based only on complete cases, we also report the means from complete cases only.

By default, SPSS tests whether the assumption of sphericity is met. It does so by using **Mauchly's test** as shown above.

If the data are perfectly spherical in the population, there's a 54% chance of finding the deviation from sphericity that is observed in our sample. We therefore assume that the **sphericity assumption has been met** by our data.

If this hadn't been the case, we'd have to use a correction factor called Epsilon. Epsilon is the Greek letter ‘e’ and written as ε. There are different ways for estimating it, including the Greenhouse-Geisser, the Huynh-Feldt and lower bound methods. Since our data seem spherical, however, we'll just ignore all these corrected results. We'll simply interpret the uncorrected results denoted as “**Sphericity Assumed**” in the table below.

The **Tests of Within-Subjects Effects** is our core output. Since our main question is whether the within-subjects factor “commercial” affects the mean ratings, we focus on the top half of this table. Since we assumed sphericity, we only need the first row.

The p-value is .000; if the means are perfectly equal in the population, there's a 0% chance of finding the differences between the means that we observe in the sample. We therefore reject the null hypothesis of equal means.

The F-value is not really interesting but we'll report it anyway. The same goes for the effect degrees of freedom (df1) and the error degrees of freedom (df2).

## 5. Reporting the Measures ANOVA Result

When reporting a basic repeated-measures ANOVA, we usually report some **descriptive statistics**, the outcome of **Mauchly's test** and the outcome of the **within-subjects tests**. The latter includes the 4 numbers we mentioned just previously:

- the F-value (15.37 in our example);
- df1 (3 in our example);
- df2 (117 in our example);
- the p-value (0.000 in our example).

Finally, the official way to combine these four numbers could be something like
*“The four commercials were not rated equally, F(3,117) = 15.4, p = .000.”*
Thank you for reading.

## This Tutorial has 12 Comments

## By Dr. Linus Mhomga on November 3rd, 2015

I found this tutorial helpful and handy as it assisted my understanding of analysis of data by repeated measures ANOVA

## By Ruben Geert van den Berg on September 30th, 2015

Thanks for your comment! And, indeed, in the absence of an interaction effect, we recommend reporting the main effects (if any). Keep in mind here that small p-values are often not really interesting: they simply say it's highly unlikely that your population effects are zero. Well, if they are not zero, then what's a more plausible indication? The answer is the sample effects, mainly reflected in patterns of means.

So that's a main reason we find the actual means usually more interesting than p-values. Hope that makes some sense.

## By Mervin on September 30th, 2015

Hello and thank you for providing this tutorial. If we don't find a significant interaction, do we still have to provide the means?

## By Arthur on July 15th, 2015

Beautiful tutorial!

## By Andy Safranski on June 26th, 2015

Excellent tutorial! I was looking for some refresher help on some statistical work and this did the trick. The examples are clear and simple. I really appreciate the end of each tutorial where you lay out the "official way to report".

Thanks much.