In this lesson, we will learn how to perform one-way ANOVA for repeated measures in SPSS. To keep things interesting, we will spice our analysis and include between-subject and within-subject factors to ensure we cover most scenarios.

Since practice is the best teacher, you can download the sample SPSS dataset I will use in this lesson below and follow along. When prompted for the password type: **uedufy**

## Lesson Outcomes

In a nutshell, these are the topics I expect you to master once complete this lesson:

- Know when to use repeated measures ANOVA for one-way and two-way.
- Understand the assumptions of repeated measures ANOVA.
- Perform one-way ANOVA for repeated measures in SPSS using the between-subjects and within-subject factors.
- Interpret the repeated measures ANOVA results in SPSS

Get a cup of coffee or tea and let’s learn something new today!

## When To Use One-Way Repeated Measures ANOVA?

ANOVA for repeated measures is used when we want to compare the mean scores between three or more groups on different observations, unlike the **T-Test** analysis which is usually used to compare the mean of one group to another.

Here is what your study should have to require one-way ANOVA for repeated measures analysis:

- one group of respondents measured using the same scale on three or more different periods OR each participant measured using at least three different items (e.g., questions) using the same response scale.
- One categorical independent variable (also known as a
*repeated factor*). - One continuous dependent variable is used to repeat observations also known as a
*repeated measure*.

*Confused? Let’s fix that by looking to a repeated measure design case study next.*

Assuming we want to test the efficacy of two weight-loss diet programs (e.g., low-carb and low-fat), and 3 repeated periods: **1st Period** (before starting the diet); **2nd Period** (after 30 days), and **3rd Period **(after 90 days).

We can design this study to include **50 participants**, divided into **two groups of 25 participants each**. A participant can be part of one group only and only one diet program will be applied to each group.

Here how the dataset for this case study looks like in the SPSS **Data View** tab:

And here is the **Variable View** tab in SPSS:

The independent variable in this example is the [diet] **Program** which consists of two levels: *low-carb* and *low-fat* *diets*. The dependent variable is **1st Period**, **2nd Period**, and **3rd Period**. We also have the **ID** where each respondent in the study is associated with a unique number.

Since this study has one independent variable and we plan to measure the diet effects for the same participants in two different groups over three-period points, the ** one-way ANOVA for repeated measures** would be the right approach.

Before preceding with the ANOVA for repeated measures analysis, it is important to make sure we align with the required assumption for the ANOVA test:

**Independence:**all observations must be independent therefore each group in the study must have unique participants. A participant can only be part of one group and there is no way the subject can influence members of another group.**Sphericity:**the variance between groups must be equal.**Normality:**data must have a multivariate normal distribution in the population under study.

*Alright. With that out of the way, it is time to launch SPSS, and finally, have some fun! *

## Perform One-Way ANOVA For Repeated Measures in SPSS [Practice]

Import the dataset you downloaded earlier in SPSS by navigating to **Open → Data** and selecting the ** .sav** file or simply double-click on the

**file to automatically open it in SPSS.**

*.sav*Here are the steps to conducting one-way repeated measures ANOVA analysis in SPSS:

- Navigate to
**Analyze → General Linear Model → Repeated Measures**in the SPSS top menu. - On the Repeated Measure Define Factor(s) window, specify:

- The
**Within-Subject Factor Name**, e,g.,*Period*or*Time Frame.*You can rename the name with something appropriate in the case is needed. - The
**Number of Levels**respectively the number of dependent variables in the study. In our case, this is three. - Click the
**Add**button to move the factor and the defined variables into the respective box.

- Click the
**Define**button.

- In the
window we have to specify in the right order a dependent variable for each*Repeated Measures***Within-Subjects Variables**.

For instance, In our SPSS sample dataset, we have three dependent variables representing three periods: 1st Period, 2nd Period, and 3rd Period. In your study, these

The **1st Period** will be moved to the **_?_(1)** position, the **2nd Period** to the **_?_(2) **position, the **3rd Period** to **_?_(3)**, and so on. You can select each variable and use the arrow to move it in the appropriate box or even easier, use the mouse to drag and drop each dependent variable to its appropriate position.

4. Add the independent variable of interest to the **Between-Subject Factor(s) **box. In our case, the independent variable is the **Program**.

Before we proceed with the one-way ANOVA for repeated measures in SPSS, we need to change some settings for the analysis. Click on the **Plots** button.

- On the
**Profile Plots**window, move the independent factor (*Program*) to the**Horizontal Axis**box, and the dependent factor (*Period*) to the**Separate Lines**box. Click the**Add**button.

The **Plots** box will get populated. For instance, for our dataset, the plot should be *Program*Period* as in the picture below.

Click **Continue**. You should be back at the *Repeated Measures: Options* window.

Since we only have two levels for the independent variable (

Low CarbsandLow Fat Diets) we do not need to run thePost Hoctest (also known as the multiple comparison test). However, if the ANOVA for repeated measures analysis consists of three or more groups, aPost Hoctest is necessary to investigate which groups’ mean is different and control the familywise error rate. This is only the case when the ANOVA P-value is significant for the means of all groups.

- Next, click on the
**EM Means**(Estimated Marginal Means) button. Here we need to adjust the following settings:

- We want to see the means for the dependent variable so move the
**Period**to the**Display Means**box. - Check the
**Compare main effects**checkbox. - Select the
**Bonferroni****Confidence interval adjustment**drop-down menu.

The

Bonferroni correctionis the simplest way to control the risk of encountering the type I error (false-positive) and rejecting thenull hypothesiswhen is actually true.

Click **Continue** to save the settings and exit the *EM Means* window.

- [Optional] Once on the
, click on the*Repeated Mesures: Options***Options**button and make sure the following checkboxes are checked:

- Descriptive statistics
- Estimates of effect size
- Homogeneity tests

Click **Continue** to save the settings and exit the *Options* window.

- Finally, on the
window, click*Repeated Measures***OK**to proceed with the ANOVA for repeated measures analysis in SPSS.

## Interpret ANOVA for Repeated Measures Output in SPSS

- The first table in the ANOVA repeated measures output is
**Within-Subjects Factors**which is represented by three periods, respectively*1st Period*,*2nd Period*, and*3rd Period*in our example.

- The
**Between-Subjects Factors**table shows the treatments (conditions) applied to the subjects. In our examples, these conditions are*Low Carbs Diet*and*Low Fat Diet*. We can also observe that the population N is equal for both treatments (N = 25 subjects).

- In the
**Descriptive Statistics**table we can see that the mean is overall higher in the 1st Period (Total Mean =**54.76**), followed by the 2nd Period (Total Mean =**45.74**) and 3rd Period (**42.16**) which can indicate a link between the diet programs and weight loss of the participants.

Moreover, when we look at the mean between the groups for each diet program, we can observe that the mean for ** Low Carbs Diet** group is lower than that of the

*Low Fat Diet**group which may indicate that a low carbs diet program is more efficient – with the exception of 1st Period where the means between the two programs are the closest.*

- The
**Box’s Test of Equality of Covariance Matrices**test also known as the*Box’s M**Test*is a parametric test for comparing variability in multivariate samples. This test explicitly checks to see whether two or more covariance matrices are homogeneous (equal).

As seen in the output below, the *Box’s Test* result is **14.961**, and the Significance P-value of **0.030**. It is important to keep in mind that the Box’s Test **α level is 0.01** (Significant if **<0.01**). Therefore we have a Box’s Test that has a non-statistical significant output of **0.030**. Therefore, we fail to reject the null hypothesis and meet the equal population covariance matrices assumption.

- The
**Multivariate Test**shows us the overall mean difference in the repeated measures. The multivariate test implies observational independence and multivariate normality. One advantage of using the*Multivariate Test*is that it does not imply sphericity, as the univariate approach does – as the case in this example.

The *Multivariate Test *is conducted using four different test statistics (*Pillai’s Trace*, *Wilks’ Lambda*, *Hotelling’s Trace*, and *Roy’s Largest Root*) which provide basically the same information. The most important metric here is the **P-value** (Sig. column) with values across all statistical tests for *Period* and *Period*Program* of **0.000 **therefore statistically significant (Significant if **<0.05**)

**Mauchly’s Test of Sphericity**test determines if the variances between all differences between all possible pairs of groups are equal (whether the sphericity assumption is violated or not).

The *Mauchly’s** **Test* has an **α level of** **0.05** and to meet the sphericity assumption we need to have a value greater than that. In our case, the calculated P-value is** 0.02** therefore we are not meeting the sphericity assumption.

- The
**Tests of Within-Subjects Effects**test determines if there were any significant differences between means at any point in time. As with the previous tests, the*Sphericity Assumed*has an**α level of 0.05**, and a P-value**<0.05**shows statistical significance.

In our case, we can assume sphericity for both ** Period** and

**.**

*Period * Program*- The
**Tests of Within-Subjects Contrasts**test determines if there was a statistically significant change between the means at various time points and is useful when performing trend analysis. The*Tests of Within-Subjects Contrasts*has an**α = 0.05**. In our case study, we can see thatand*Period*are statistically significant with P-values <0.05 as highlighted in the table below.*Period * Program*

**Levene’s Test of Equality of Error Variances**tests the homogeneity of variance by comparing the variances of two or more groups given a variable.*Levene’s Test of Equality of Error Variances*has an**α level of 0.05**where any P-value**lower than**0.05 violates the homogeneity of variance assumption.

Looking at Levene’s Test of Equality of Error Variance output for our case study, we can see that all P-values are >0.05 therefore we do not violate the assumption of homogeneity of variance.

- The
**Tests of Between-Subjects Effects**investigates the differences between respondents. The*Tests of Between-Subjects Effects have an***α level of 0.05**. Here we are looking at the effect of the*Program*variable which shows no statistical significance at P-value =**0.253**as only**0.027**(*Partial Eta Squared*) of the variance in the dependent variable can be explained by the*Program*variable.

- The
**Estimated Margin Means**table shows us an estimate summary for the mean and**standard error**for each period as well as the lower and upper bound values at a 95% confidence interval. You can observe that the**Mean**in the*Estimates*table is actually the equivalent of the**Total Mean**in the*Descriptive Statistics*table (3).

- The
**Pairwise Comparison**test determines if there are any statistical differences between the pairs – in our case the three*Periods*of the study. You can see that we have statistical significance between each pair (P-value <0.05).

12. Finally, the **Multivariate Test** table shows us a similar output as the Multivariate Test we discussed earlier (5) and shows statistical significance for all the different test statistics used in the ANOVA for repeated measures analysis.

## Key Takeaways

Before you decide to proceed with the next lesson, here are some important things to remember:

- ANOVA for repeated measures is useful when looking to
*compare the mean scores between three or more groups on different observations*. For comparing the means between two groups, the sample T-Test is sufficient. - One-way repeated measures ANOVA test is used when a repeated measure design study consists of one independent variable while with the
*two-way repeated measures ANOVA*we use two predictors. - To be able to conduct the repeated measures ANOVA test, the study repeated measure design should consist of at least one categorical independent variable and one continuous dependent variable.

## References

Cohen, J. (1988). Statistical power for the behavioral sciences (2nd edition). Lawrence Earlbaum: Hillsdale, NJ.

Field, A. (2018). Discovering statistics using IBM SPSS statistics (5th edition). Sage: Thousand Oaks, CA.

Pallant, J. (2010). *SPSS survival manual: A step-by-step guide to data analysis using SPSS*. Maidenhead: Open University Press/McGraw-Hill.

Pituch, K.A. and Stevens, J.P. (2016) Applied Multivariate Statistics for the Social Sciences.