Chapter 24 – ANOVA for repeated measures designs
This chapter deals with analyses using ANOVA when at least one of the factors is related – that is, a repeated measures or matched pairs factor (independent variable) has been used.
Exercises
The data sets used to calculate the repeated measures examples in this chapter are provided below.
Exercise 24.1
Calculating a one-way repeated measures ANOVA example
You will need the following data set to complete this exercise:
The file repeated measures 1-way.sav (SPSS) or repeated measures 1-way.xls (Excel) contains data for a fictitious study in which new employees were assessed for efficiency in their similar jobs after one month, six months and twelve months. Calculate the one-way repeated measures results and compare with the answer given below. The repeated measures variable is contained in the columns entitled efficency1, efficiency6 and efficiency12. In SPSS use the General linear model menu item. Don’t forget to check and report Mauchly’s test for sphericity.
Show answer
The means (and standard deviations) of the efficiency scores after 1 month, 6 months and 12 months respectively were M = 38.3 (3.91), M = 41.5 (6.51) and M = 46.1 (6.92). The means differed significantly with F (2,30) = 8.247, p = .001, effect size (eta squared ) = .355. Mauchly’s test was not significant, p = .349.
Exercise 24.2
Calculating a two-way mixed design ANOVA example
You will need this data set to carry out this exercise:
In this exercise you can tackle a two-way mixed design where there is one repeated measures factor (efficiency from the last exercise) and one between groups factor. This new factor is one of training.
Imagine the new employees in the last exercise were randomly divided into a group that received no training, one that received training and one that received training and some team building exercises early on in their employment at the company. You need the file repeated measures mixed.sav (SPSS) or repeated measures mixed.xls (Excel).
Conduct the two-way analysis for efficiency and training ignoring the column headed ‘graduate’ for now. Compare your results with those under ‘Show Answer’ below. Select Descriptives under Options to get the means and SDs for the efficiency conditions. To get the means and SDs for the training groups on each efficiency level you can create a new variable which is the mean of the three efficiency measures then use Analyze/Compare Means/Means. EM Means will give means but inappropriate SD estimates.
Show answer
There was a main effect for efficiency with the means rising from M = 43.2, SD = 7.33 at one month, through M = 45.8, SD = 7.68 at six months to M = 49.06, SD = 6.09 at twelve months. F2,90 =12.454, p < .001, effect size (partial eta2) = .217
There was a main effect for training with means of M = 41.96, SD = 3.86 for the untrained group, M = 46.313, SD = 4.48 for the trained group and M = 49.813, SD = 4.30 for the trained and team building group. F2,45 = 13.879, p < .001, effect size (partial eta2 ) = .382.
The interaction was not significant. Sphericity was at an acceptable level (p = .213). Levene’s test for homogeneity of variance was not significant for any level of efficiency so equality of variances was assumed.
If you’re really feeling adventurous you could try the three-way mixed ANOVA that is produced by including the factor of graduate. This tells us whether the participant was a graduate or not. I have only provided brief details of results below but enough to let you see you’ve performed the analysis correctly.
Show answer
Main effect efficiency F2,84 = 13.034, p <.001, eta2 = .237
Main effect training F2,42 = 17.598, p < .001, eta2 = .456
Main effect graduate F1,42 = 4.21, p = 046, eta2 = .091
Interaction efficiency x training not significant
Interaction efficiency x graduate not significant
Interaction training x graduate significant F2,42 = 5.424, p = .008, eta2 = .205
Three-way interaction efficiency x training x graduate approaching significance p = .054
Exercise 24.3
Calculation of a two-way repeated measures ANOVA
The data set these files are based on is an experiment where participants undergo the Stroop experience. Stroop was the psychologist responsible for demonstrating the dramatic effect that occurs when people are asked to name the colour of the ink in which words are written – there is a big problem if the word whose colour you are naming is a different colour word (e.g., red written in green – an ‘incongruent’ colour word)! People take much longer to name the ink colour of a set of such words than they do to name the colours of ‘congruent’ words (colour words written in the ink colour of the word they spell).
A further refinement of the experiment, based on a theory of sub-vocal speech when reading, is the prediction that words that sound like colour words (such as ‘shack’ or ‘crown’) should also produce some interference, if incongruent, thus lengthening times to name ink colours. The Stroop factor of this experiment then involves three conditions: naming the ink colour of congruent words; naming the ink colour of incongruent words sounding like colour words; and naming the ink colour of incongruent colour words.
In the imaginary experiment here we have introduced a second factor, which is that people perform the three Stroop tasks both alone and then in front of an audience. The data are presented as a 2 x 3 repeated measures design so there are six columns of raw data, the numbers being number of seconds to read the list of words. Control is naming the ink colour of congruent words, rhyme uses words sounding like incongruent colour words and colour uses incongruent colour words. The end part of each variable refers to the audience conditions, alone if no audience and aud with an audience observing.
Remember that in SPSS you have to name the two repeated measures factors then carefully select columns when asked to define the levels of each variable. If you enter the repeated measures variable names as first ‘stroop’, then ‘audience’ you will be asked to identify variables in the order stroop 1, audience 1, stroop 1, audience 2 and so on, so that’s controlalone, controlaud, rhymealone … and so on. To get the true means and SDs for the overall Stroop conditions you’ll need to compute a new variable which is the mean of the two columns for each level. For control the new variable will be (controlalone + controlaud)/2 and so on for rhyme and colour.
Carry out the two-way analysis, remembering to check Mauchly’s sphericity statistic and to ask for descriptive statistics so you can see the mean of each level of each variable.
Show answer
The main effect for Stroop is basically massive (as it nearly always is). The overall means for the three conditions were control M = 44.5 (SD = 14.54), rhyme M = 58.9 (SD = 14.82) and colour M = 97.7 (SD = 20.47). F2,18 = 34.873, p < .001, partial eta2 = .795
There was no effect for audience and the interaction stroop x audience was not significant. Sphericity was not a problem.
Exercise 24.4
Questions on SPSS results for a two-way ANOVA
The table below shows part of the SPSS output for a two-way ANOVA calculation. Extroverts and introverts (factor extint) have been asked to perform a task more than once during the day to see whether extroverts improve through the day and introverts worsen.
| df | F | p | Effect size = eta2 | |
| Performance | 2 | 3.795 | .026 | .073 |
| Performance x extint | 2 | 23.225 | P<.001 | .326 |
| Error (performance) | 96 | |||
| Extint | 1 | .026 | .872 | .001 |
| Error | 48 |
Weblinks
ANOVA for repeated measures designs
From Laerd Statistics – a complete run through one-way repeated measures ANOVA analysis with full visuals of each step plus links to results output and explanation.
And here’s the complementary two-way analysis:
How to perform a two-way repeated measures ANOVA in SPSS Statistics | Laerd Statistics
A YouTube video – using SPSS for a two-way repeated measures ANOVA analysis:
Two-way repeated-measures ANOVA in SPSS – YouTube
The Laerd Statistics pages on mixed ANOVA:
How to perform a Mixed ANOVA in SPSS Statistics | Laerd Statistics
Further Information
The Page trend test
In Chapter 24 of the book there is a short description of the Page trend test, which is used when you have three or more related sets of data and you want to see whether they follow a trend across conditions. We’ll use the (very minimal) data below as an example. Imagine children have been tested for reading improvement on three successive occasions. We want to see if there is significant improvement.
| Reading scores for four children tested three times | |||||
| Score | Rank | Score | Rank | Score | Rank |
| 3 | 2 | 2 | 1 | 10 | 3 |
| 5 | 1 | 7 | 2 | 8 | 3 |
| 6 | 1 | 9 | 3 | 7 | 2 |
| 3 | 1 | 8 | 2 | 11 | 3 |
| Totals: | Ra = 5 | Rb = 8 | Rc = 11 | ||