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**Locating Variance: Post-Hoc Tests**

Developing Study Skills and Research Methods (HL20107) Dr James Betts

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**Lecture Outline: Influence of multiple comparisons on P**

Tukey’s HSD test Bonferroni Corrections Ryan-Holm-Bonferroni Adjustments.

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Placebo Placebo Lucozade Lucozade Gatorade Gatorade Powerade Powerade

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Placebo Placebo Lucozade Lucozade Gatorade Gatorade Powerade Powerade

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**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 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%.

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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

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q table for Tukey’s HSD

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Tukey Test Critique As you learnt last week, the omnibus error term is not reflective of all contrasts if sphericity is violated Placebo Lucozade So Tukey tests commit many type I errors with even a slight degree of asphericity. Gatorade Powerade

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**Solution for Aspherical Data**

There are alternatives to the Tukey HSD test which use specific error terms for each contrast Fisher’s LSD (Least Significant Difference) Sidak Bonferroni Many others… e.g. Newman-Kewls, Scheffe, Duncan, Dunnett, Gabriel, R-E-G-W, etc.

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Trial 2 Trial 1 Trial 4 Fisher’s LSD Bonferroni Trial 3

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**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

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**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.

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**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.

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**Further reading from this lecture…**

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

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**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

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