Why not multiple t-tests? i.e. Placebo vs Lucozade Placebo vs Gatorade Placebo vs Powerade Lucozade vs Gatorade Lucozade vs Powerade Gatorade vs Powerade We accept significance and reject the null hypothesis at P 0.05 (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%.
Post-hoc Tests A popular solution is the Tukey HSD (Honestly Significant Difference) test This uses the omnibus error term from the ANOVA to determine which means are significantly different T = (q) Error Variance n
q table for Tukeys HSD
Tukey Test Critique As you learnt last week, the omnibus error term is not reflective of all contrasts if sphericity is violated So Tukey tests commit many type I errors with even a slight degree of asphericity. Placebo Lucozade Gatorade Powerade
Solution for Aspherical Data There are alternatives to the Tukey HSD test which use specific error terms for each contrast –Fishers LSD (Least Significant Difference) –Sidak –Bonferroni –Many others… e.g. Newman-Kewls, Scheffe, Duncan, Dunnett, Gabriel, R-E-G-W, etc.
Bonferroni Correction Critique Correction of LSD values successfully controls for type I errors following a 1-way ANOVA However, factorial designs often involve a larger number of contrasts, many of which may not be relevant. Recovery Supp. 1 Recovery Supp. 2 See also Perneger (1998) BMJ 316: 1236
Solution for Factorial Designs An adjustment to the standard Bonferroni correction can be applied for factorial designs This Ryan-Holm-Bonferroni or stepwise method involves returning to the P values of interest from our LSD test These 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.
Summary Post-Hoc Tests A Tukey test may be appropriate when sphericity can be assumed Multiple t-tests with a Bonferroni correction are more appropriate for aspherical data Stepwise correction of standard Bonferroni procedures maintain power with factorial designs Best option is to keep your study simple: –Pre-planned contrast at a specific time point –Summary statistics (e.g. rate of change, area under curve) –Just make an informed based on the data available.
Atkinson, G. (2001) Analysis of repeated measurements in physical therapy research Physical Therapy in Sport 2: p Atkinson, 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 Further reading from this 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 Compulsory reading for next weeks lecture…