| Interpreting repeated measures one-way ANOVA
How repeated measures ANOVA works
Repeated measures one-way ANOVA compares three or more matched groups, based on the assumption that the differences between matched values are Gaussian. For example, one-way ANOVA may compare measurements made before, during and after an intervention, when each subject was assessed three times.The P value answers this question: If the populations really have the same mean, what is the chance that random sampling would result in means as far apart (or more so) as observed in this experiment?
ANOVA table
The P value is calculated from the ANOVA table. With repeated measures ANOVA, there are three sources of variability: between columns (treatments), between rows (individuals) and random (residual). The ANOVA table partitions the total sum-of-squares into those three components. It then adjusts for the number of groups and number of subjects (expressed as degrees of freedom) to compute two F ratios. The main F ratio tests the null hypothesis that the column means are identical. The other F ratio tests the null hypothesis that the row means are identical (this is the test for effective matching). In each case, the F ratio is expected to be near 1.0 if the null hypothesis is true. If F is large, the P value will be small.
The circularity assumption
Repeated measures ANOVA assumes that the random error truly is random. A random factor that causes a measurement in one subject to be a bit high (or low) should have no affect on the next measurement in the same subject. This assumption is called circularity or sphericity. It is closely related to another term you may encounter, compound symmetry.
Repeated measures ANOVA is quite sensitive to violations of the assumption of circularity. If the assumption is violated, the P value will be too low. You'll violate this assumption when the repeated measurements are made too close together so that random factors that cause a particular value to be high (or low) don't wash away or dissipate before the next measurement. To avoid violating the assumption, wait long enough between treatments so the subject is essentially the same as before the treatment. When possible, also randomize the order of treatments.
You only have to worry about the assumption of circularity when you perform a repeated measures experiment, where each row of data represents repeated measurements from a single subject. It is impossible to violate the assumption with randomized block experiments, where each row of data represents data from a matched set of subjects.
Was the matching effective? (repeated measures one-way ANOVA)
A repeated measures experimental design can be very powerful, as it controls for factors that cause variability between subjects. If the matching is effective, the repeated measures test will yield a smaller P value than an ordinary ANOVA. The repeated measures test is more powerful because it separates between-subject variability from within-subject variability. If the pairing is ineffective, however, the repeated measures test can be less powerful because it has fewer degrees of freedom.
Prism tests whether the matching was effective and reports a P value that tests the null hypothesis that the population row means are all equal. If this P value is low, you can conclude that the matching is effective. If the P value is high, you can conclude that the matching was not effective and should consider using ordinary ANOVA rather than repeated measures ANOVA.
How to think about results from repeated measures one-way ANOVA
Repeated measures ANOVA compares the means of three or more matched groups. The term repeated measures strictly applies only when you give treatments repeatedly to each subject, and the term randomized block is used when you randomly assign treatments within each group (block) of matched subjects. The analyses are identical for repeated measures and randomized block experiments, and Prism always uses the term repeated measures.
Your approach to interpreting repeated measures ANOVA results will be the same as interpreting the results of ordinary one-way ANOVA. See How to think about results from one-way ANOVA.
Checklist. Is repeated measures one way ANOVA the right test for these data?
Before accepting the results of any statistical test, first think carefully about whether you chose an appropriate test. Before accepting results from repeated measures one-way ANOVA, ask yourself the questions listed below. Prism can help you answer the first question. You must answer the remaining questions based on experimental design.
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Discussion |
| Was the matching effective? |
The whole point of using a repeated measures test is to control for experimental variability. Some factors you don't control in the experiment will affect all the measurements from one subject equally, so will not affect the difference between the measurements in that subject. By analyzing only the differences, therefore, a matched test controls for some of the sources of scatter.
The matching should be part of the experimental design and not something you do after collecting data. Prism tests the effectiveness of matching with an F test (distinct from the main F test of differences between columns). If the P value for matching is large (say larger than 0.05), you should question whether it made sense to use a repeated measures test. Ideally, your choice of whether to use a repeated measures test should be based not only on this one P value, but also on the experimental design and the results you have seen in other similar experiments.
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Are the subjects independent?
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The results of repeated measures ANOVA only make sense when the subjects are independent. Prism cannot test this assumption. You must think about the experimental design. For example, the errors are not independent if you have six rows of data, but these were obtained from three animals, with duplicate measurements in each animal. In this case, some factor may affect the measurements from one animal. Since this factor would affect data in two (but not all) rows, the rows (subjects) are not independent. See The need for independent samples.
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Is the random variability distributed according to a Gaussian distribution?
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Repeated measures ANOVA assumes that each measurement is the sum of an overall mean, a treatment effect (the average difference between subjects given a particular treatment and the overall mean), an individual effect (the average difference between measurements made in a certain subject and the overall mean) and a random component. Furthermore, it assumes that the random component follows a Gaussian distribution and that the standard deviation does not vary between individuals (rows) or treatments (columns). While this assumption is not too important with large samples, it can be important with small sample sizes. Prism does not test for violations of this assumption. |
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Is there only one factor?
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One-way ANOVA compares three or more groups defined by one factor. For example, you might compare a control group, with a drug treatment group and a group treated with drug plus antagonist. Or you might compare a control group with five different drug treatments.
Some experiments involve more than one factor. For example, you might compare three different drugs in men and women. There are two factors in that experiment: drug treatment and gender. Similarly, there are two factors if you wish to compare the effect of drug treatment at several time points. These data need to be analyzed by two-way ANOVA, also called two factor ANOVA.
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Is the factor "fixed" rather than "random"?
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Prism performs Type I ANOVA, also known as fixed-effect ANOVA. This tests for differences among the means of the particular groups you have collected data from. Type II ANOVA, also known as random-effect ANOVA, assumes that you have randomly selected groups from an infinite (or at least large) number of possible groups, and that you want to reach conclusions about differences among ALL the groups, even the ones you didn't include in this experiment. Type II random-effects ANOVA is rarely used, and Prism does not perform it. |
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