Repeated Measures ANOVA

Why repeated measures?

- A repeated measures ANOVA uses a single sample with the same set of individuals measured in all of the different treatment conditions - Thus, one of the characteristics of a repeated-measures (aka within subjects) design is that it eliminates variance caused by individual differences - Individual differences are those participant characteristics that vary from one person to another and may influence the measurement that you obtain from each person E.g., age, gender etc.

The assumptions of sphericity

- Basically means that the correlation between treatment levels is the same - More specifically, it assumes that the variances of differences between conditions is equal - Estimated and adjusted for using: * Greenhouse-Geisser estimate * Huynh-Feldt estimate - Tested using Mauchly's test * P < .05, sphericity is violated * P > .05, sphericity is met

Final Structure of the F-Ratio for One-Way Repeated-Measures ANOVA

- Because individual differences can be eliminated from a repeated measures study, the structure of the final F-ratio is: 𝑭= 𝒗𝒂𝒓𝒊𝒂𝒏𝒄𝒆 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝒕𝒓𝒆𝒂𝒕𝒎𝒆𝒏𝒕𝒔_(𝒘𝒊𝒕𝒉𝒐𝒖𝒕 𝒊𝒏𝒅𝒊𝒗𝒊𝒅𝒖𝒂𝒍 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆𝒔)/ 𝒗𝒂𝒓𝒊𝒂𝒏𝒄𝒆 𝒆𝒙𝒑𝒆𝒄𝒕𝒆𝒅 𝒃𝒚 𝒄𝒉𝒂𝒏𝒄𝒆_(𝒘𝒊𝒕𝒉 𝒊𝒏𝒅𝒊𝒗𝒊𝒅𝒖𝒂𝒍 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆𝒔 𝒆𝒍𝒊𝒎𝒊𝒏𝒂𝒕𝒆𝒅) - = 𝑴𝑺_𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝒕𝒓𝒆𝒂𝒕𝒎𝒆𝒏𝒕𝒔/ 𝑴𝑺_𝒆𝒓𝒓𝒐𝒓__

post hoc tests

- Compare each mean against all others (t-tests). - In general terms they use a stricter criterion to accept an effect as significant. * Hence, control the family wise error rate. * Simplest example is the Bonferroni method

An example

- Field (2013): Effects of advertising on evaluations of different drink types. * IV 1 (Drink): Beer, Wine, Water * IV 2 (Imagery): Positive, negative, neutral * Dependent Variable (DV): Evaluation of product from -100 dislike very much to +100 like very much)

Correcting for sphericity

- G-G and H-F estimates of sphericity are used to adjust degrees of freedom - Example, with 3 and 21 degrees of freedom:

Which one do you use?

- G-G is conservative, - and H-F liberal - Won't make much of a difference which one you do use.

F-Ratio for One-Way Repeated-Measures ANOVA

- In terms of the F-ratio for a repeated measures design, the variance between treatments (the numerator) does not contain any individual differences. - A repeated measures design also allows you to eliminate individual differences from the denominator of the F-ratio In a repeated measures research study, individual differences are not random. - These individual difference can be measured and separated out from other sources of error This is because the same individuals are measured in every treatment condition

Recap two

- Independent measures ANOVA's (one way and two way) - Remember "One-way" just means that you have only one independent variable (IV) or factor - "Two-way" means that you have 2 independent variables or factors - IV can be either: * A between-subjects or independent measures factor(s) * I.e., each level of the IV has different participants (males and females) * A within-subjects or repeated measures factor * i.e., the same participates take part in each level of the IV

Recap from factor ANOVA

- More often that not, behaviour, affect and cognitions are influenced by more than one factor * i.e., children's level of resilience is likely to depend upon numerous factors: temperament, quality of attachment, number of adverse life events etc. - Therefore we often design studies which involve examining the impact of two or more factors (IV's) on a particular dependent variable (DV) * Different participants in all conditions. * Independent = 'different participants' - Several Independent Variables are known as a factorial design. - What are some of the benefits of factorial designs? - How many F-ratios does a two-way independent ANOVA have? Why?

Assumptions of repeated measures ANOVA

- No independence as use same group of participants. - Normality - Homoscedasticity - Sphericity (if more than 3 levels)

Summary: one way repeated measures design

- One-way repeated-measures designs compares several means, when those means come from the same entities - for example, if you measured people's statistical ability each month over a year-long course. * When you have three or more repeated-measures conditions there is an additional assumption: sphericity. * You can test for sphericity using Mauchly's test, but it is better to always adjust for the departure from sphericity in the data. * The table labelled Tests of Within-Subjects Effects shows the main F-statistic. § Other things being equal, always read the row labelled Greenhouse-Geisser (or Huynh-Feldt, but you'll have to read this chapter to find out the relative merits of the two procedures). § If the value in the column labelled Sig. is less than 0.05 then the means of the conditions are significantly different. * For contrasts and post hoc tests, again look to the columns labelled Sig. to discover if your comparisons are significant (i.e., the value is less than 0.05).

Aims

- Rationale of repeated measures ANOVA * One way and two way Benefits - Partitioning variance - Statistical Problems with Repeated Measures Designs * Sphericity * Overcoming these problems - Interpretation

Power

- Repeated measures designs are powerful because they remove individual differences - E.g., assume that you know: - Treatment effect = 10 units of variance - Individual differences = 1000 units of variance - Other error = 1 unit of variance

Problem with the variance approach

- Same participants in all conditions * Scores across conditions correlate * Violates assumption of independence - Assumption of sphericity * Crudely put: the correlation across conditions should be the same. * Adjust degrees of freedom

Problems with the variance approach

- Same participants in all conditions * Scores across conditions correlate * Violates assumption of independence - Assumption of sphericity * Crudely put: the correlation across conditions should be the same. * Adjust degrees of freedom

Benefits of Repeated Measures Designs

- Sensitivity * Unsystematic variance is reduced. * More sensitive to experimental effects. - Economy * Less participants are needed. * But, be careful of fatigue.

Dealing with individual differences

- The individual differences are automatically removed from the numerator because the design uses the same subjects in all treatments, but we must also remove them from the denominator - Remove individual differences from the denominator by measuring the variance within treatments and then subtracting the individual differences - The result is a measure of unsystematic error variance that does not include any individual differences

Estimates of sphericity

- Three measures: * Greenhouse-Geisser estimate, * Huynh-Feldt estimate, * Lower-bound estimate - Multiply df by these estimates to correct for the effect of sphericity * G-G is conservative, and H-F liberal * Corrects for the degree of non-sphericity, so routinely use these adjusted values

What is Two-Way Repeated Measures ANOVA?

- Two Independent Variables * Two-way = 2 IVs * Three-Way = 3 IVs - The same participants in all conditions. * Repeated Measures = 'same participants' * A.k.a. 'within-subjects'

Factorial repeated-measures designs

- Two-way repeated-measures designs compare means when there are two predictor/independent variables, and the same entities have been used in all conditions. * You can test the assumption of sphericity when you have three or more repeated-measures conditions with Mauchly's test, but a better approach is to routinely interpret F-statistics that have been corrected for the amount by which the data are not spherical. * The table labelled Tests of Within-Subjects Effects shows the F-statistics and their p-values. § In a two-way design you will have a main effect of each variable and the interaction between them. § For each effect, read the row labelled Greenhouse-Geisser (you can also look at Huynh-Feldt, but you'll have to read this chapter to find out the relative merits of the two procedures). § If the value in the column labelled Sig. is less than 0.05 then the effect is significant. * Break down the main effects and interactions using contrasts. § These contrasts appear in the table labelled Tests of Within-Subjects Contrasts. § If the values in the column labelled Sig. are less than 0.05 the contrast is significant.