Prism 9.4 does not allow test for trend following one-way repeated measures ANOVA when sphericity is not assumed

With the release of Prism 9.4, the test for linear trend following one-way ANOVA with repeated measures is no longer available if you choose not to assume sphericity, but instead choose to use the Geisser-Greenhouse correction. In previous versions, Prism was not taking the Geisser-Greenhouse correction into account when calculating the test for linear trend, resulting in misleading results.

The ANOVA results are fine. The issue is only with the test for trend following repeated measures one-way ANOVA when you don't assume sphericity. In this case, the followup test doesn't really make sense unless you make an alternative assumption. As explained at the bottom of this page, we are planning a future version of Prism that will offer different choices of alternative assumptions (different covariance matrices).

Dialog options that are no longer available

Experimental design: Repeated measures one-way ANOVA, no assumption of sphericity (using the Geisser-Greenhouse Correction)

Disabled followup test: Test for linear trend between column mean and left-to-right column order

Fixed bug

While addressing the situation described above, a separate but related bug was discovered. In the case when no sphericity is assumed and a mixed-effects model is used (instead of GLM) to calculate the results, the Geisser-Greenhouse correction was mistakenly applied to the P value of the between columns (i.e. treatment) effect for the test for trend. As a result, the P value for this effect was incorrectly reported as being the same as the P value of the fixed effect reported on the Tabular Results tab. Note that - although the P value was incorrect for the test for trend - the degrees of freedom (df) for both values were reported correctly.

Note, although this bug was fixed, the test for trend will not be available when sphericity is not assumed starting in Prism 9.4, so there's no reason that you should encounter this issue unless using an older version of Prism.

What Prism Assumes for the Covariance Structure in Repeated Measures Designs

ANOVA estimates both the effect sizes and the error - or noise - in the experiment so that the effect sizes can be compared to something (think signal-to-noise ratio). The residual error is simply the difference between the observed value and the value predicted by the model. In the simplest form of ANOVA, this error is assumed to be the same throughout the experiment (regardless of which treatment or factor combination the value was observed in). Additionally, these errors are assumed to be independent of each other. However, in a repeated measures style experiment, multiple observations are obtained from the same subject. As a result, there are generally correlations in the resulting residual errors. Instead of assuming a constant error variance with no correlations, we need to specify or assume a covariance structure for the residuals in a repeated measures experiment.

The simplest covariance structure that accounts for correlated errors is compound symmetry. Here, the correlation of the errors between repeated measures (for example, different time points) for a single subject are assumed to be the same regardless of how "distant" the repeated measures are (for example, the error of the first and last measurements are assumed to have the same correlation as the error of the first and second measurement). With this structure, the variance at each time point is also assumed to be the same. This is the structure Prism 9 assumes when "Yes" is selected under "Assume sphericity". When you choose "No" under "Assume sphericity", the covariance structure allows for unique correlations for each pair of measurements from the same subject, and does not assume equal variation at each time point. In this scenario, the multiple comparisons tests are challenging because you can't use data from the entire experiment to estimate the error for a single comparison. Sometimes, a comparison can't be made, and - when it can - the power can be very poor. Other covariance structures, such as the autoregressive lag 1, assume correlations are highest for adjacent time points and systematically decrease as the distance between time points increases. This structure also assumes equal variation at each time point, but note that it is only appropriate for evenly spaced time intervals. Prism 9 does not offer this covariance structure, but we are considering offering it (and possibly others) in a future release. This would allow us to offer additional reasonable post tests after repeated measures ANOVA that we do not offer today.