5 Why not multiple t-tests? i.e.Placebo vs LucozadePlacebo vs GatoradePlacebo vs PoweradeLucozade vs GatoradeLucozade vs PoweradeGatorade vs PoweradeWe accept ‘significance’ and reject the null hypothesis at P0.05 (i.e. a 5% chance that we are wrong)Performing multiple tests therefore means that our overall chance of committing a type I error is >5%.
6 Post-hoc TestsA popular solution is the Tukey HSD (Honestly Significant Difference) testThis uses the omnibus error term from the ANOVA to determine which means are significantly differentT = (q)Error Variance√n
12 Tukey Test CritiqueAs you learnt last week, the omnibus error term is not reflective of all contrasts if sphericity is violatedPlaceboLucozadeSo Tukey tests commit many type I errors with even a slight degree of asphericity.GatoradePowerade
13 Solution for Aspherical Data There are alternatives to the Tukey HSD test which use specific error terms for each contrastFisher’s LSD (Least Significant Difference)SidakBonferroniMany others…e.g. Newman-Kewls, Scheffe, Duncan, Dunnett, Gabriel, R-E-G-W, etc.
17 Bonferroni Correction Critique Correction of LSD values successfully controls for type I errors following a 1-way ANOVAHowever, factorial designs often involve a larger number of contrasts, many of which may not be relevant.Recovery Supp. 1Recovery Supp. 2See also Perneger (1998) BMJ 316: 1236
18 Solution for Factorial Designs An adjustment to the standard Bonferroni correction can be applied for factorial designsThis ‘Ryan-Holm-Bonferroni’ or ‘stepwise’ method involves returning to the P values of interest from our LSD testThese P values are placed in numerical order and the most significant is Bonferroni corrected (i.e. P x m)However, all subsequent P values are multplied by m minus the number of contrasts already corrected.
19 Summary Post-Hoc Tests A Tukey test may be appropriate when sphericity can be assumedMultiple t-tests with a Bonferroni correction are more appropriate for aspherical dataStepwise correction of standard Bonferroni procedures maintain power with factorial designsBest option is to keep your study simple:Pre-planned contrast at a specific time pointSummary statistics (e.g. rate of change, area under curve)Just make an informed based on the data available.
20 Further reading from this lecture… Atkinson, G. (2001) Analysis of repeated measurements in physical therapy research Physical Therapy in Sport 2: pAtkinson, G. (2002) Analysis of repeated measurements in physical therapy research: multiple comparisons amongst level means and multi-factorial designs Physical Therapy in Sport 3: p
21 Compulsory reading for next week’s lecture… Batterham A. M. & Atkinson, G. (2005) How Big Does My Sample Need to Be? A primer on the Murky World of Sample Size Estimation Physical Therapy in Sport 6: p