Statistics
Chapter 15
Repeated measures designs

Introduction to repeated-measures designs

Repeated measures: when the same entities participate in all conditions of an experiment or provide data at multiple time points.

Repeated measures and the linear model

Repeated measures can also be considered as a variation of the general linear model.

For example.

Y_gi = b_0i +b_1iX_gi +Ɛ_gi

b_0i = b₀ + u_0i

b_1i = b₁ + u_1i

Y_gi for outcome g within person i from the specific predictor X_gi with the error Ɛ_gi

g is the level of treatment condition
i for the individuals

u_0i for the deviation of the individual’s intercept from the group-level intercept

The ANOVA approach to repeated-measures designs

The way that people typically handle repeated measures in IBM SPSS is to use a repeated-measures ANOVA approach.

The assumption of sphericity

The assumption that permits us to use a simpler model to analyse repeated-measures data is sphericity.

Sphericity: assuming that the relationship between scores in pairs of treatment conditions is similar.

It is a form of compound symmetry: holds true when both the variances across conditions are equal and the covariances between pairs of conditions are equal.
We assume that the variation within conditions is similar and that no two conditions are any more dependent than any other two.
Sphericity is a more general, less restrictive form of compound symmetry and refers to the equality of variances of the differences between treatment levels.

For example:

variance_A-B = variance_A-C = variance_B-C

Assessing the severity of departures from sphericity

Mauchly’s test: assesses the hypothesis that the variances of the differences between conditions are equal.
If the test is statistically significant, it implies that there are significant differences between the variances of differences and, therefore, sphericity is not met.
If it is not significant, the implication is that the variances of differences are roughly equal and sphericity is met.
It depends upon sample size.

What’s the effect of violating the assumption of sphericity?

A lack of sphericity creates a loss of power and an F-statistic that doesn’t have the distribution that it’s supposed to have.
It also causes some complications for post hoc tests.

What do you do if you violate sphericity?

Adjust the degrees of freedom of any F-statistic affected.
The greater the violation of sphericity, the smaller the degrees of freedom become.

Or fit the multilevel model or use a MANOVA

The F-statistic for repeated-measures designs

In a repeated-measures design the effect of the experiment is shown up in the within-participant variance.
We look for the experimental effect (the model sum of squares) within the individual rather than within the group.
The types of variances are the same as in independent designs: the total sum of squares, a model sum of squares, and residual sum of squares.

In repeated-measures designs the model and residual of squares are both part of the within-participant variance.

The total sum of squares, SS_T

SS_T = s²_Grand(N-1)

The grand variance is the variance of all scores when we ignore the group to which they belong. We treat data as one big group

Df = N-1

Within-participant sum of squares SS_W

A within-participant variance component, which represents individual differences within participants.

SS_W = s²_entitiy1(n₁-1) + s²_entitiy2(n₂-1) + s²_entitiy3(n₃-1) + … + s²_entitiyn(n_n-1)

Looking at the variation in each individual’s score and then adding these variances for all the entities in the study. The ns represents the number of scores within the person.

The df = n-1

to get the total degrees of freedom we add the df’s for all participants.

The model sum of squares SS_M

How much variance is explained by our manipulation and how much is not.

SS_M = Σ^k_g-1n_g(mean_g-mean_grand)²

df = k -1

Residual sum of squares SS_R

how much of the variation cannot be explained by the model.
The amount of variation caused by extraneous factors outside experimental control.

SS_R = SS_W-SS_M

df_R= df_W-df_M

The mean squares MS_M

the mean squares represents the average variation explained by the model

MS_M = SS_M/df_M

MS_R = SS_R/df_R

The F-statistic

The F-statistic is the ratio of variation explained by the model and the variation explained by unsystematic factors.

F = MS_M/MS_R

The experimental effect on performance to the effect of unmeasured factors.
If the values is greater than 1, it indicates that the experimental manipulation had some effect above and beyond the effect of unmeasured factors.

The between-participant sum of squares

SS_T = SS_B-SS_W

SS_B = SS_T-SS_W

SS_B is the individual differences between cases.

Assumptions in repeated-measures designs

All sources of potential bias in chapter 6 apply.

Using the ANOVA approach, the assumption of independence is replaced by assumptions about the relationships between differences scores (sphericity)

In the multilevel approach, sphericity isn’t required.

If assumptions are not met, there is a robust variant of one-way repeated-measures ANOVA.

One-way repeated-measures designs

• one-way repeated-measures designs compares several means, when those means come form the same entities.
• 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-Subject Effects shows the main F-statistic. Other things being equal, always read the row labelled Greenhouse-Geisser. 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 called sig to discover if you comparisons are significant.

Effect sizes for one-way repeated-measures designs

The best measure of the overall effect size is omega squared ω².

ω² = ((k-1)/nk) (MS_M – MS_R) / (MS_R + ((MS_B – MS_R)/k) +((k-1)/nk) (MS_M – MS_R))

MS_B = SS_B/df_B = SS_B/(N-1)

Effect size for the contrasts

r = square root ( (F(1,df_R))/ (F(1,df_R) +df_R) )

Reporting one-way repeated-measures designs

F, the correct degrees of freedom, p and effect size.

Factorial repeated-measures designs

• Two-way repeated-measures designs compare means when there are two 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 Test of Within-Subject Effects show the F-statistic 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. 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 Test of Within-Subjects Contrasts. If the values in the column labelled Sig are less than 0.05, the contrast is significant.

Effect sizes for factorial repeated-measures designs

Effect sizes are more useful when they describe a focused effect, so calculate effect sizes for you contrasts in factorial designs.

Reporting the results from factorial repeated-measures designs

We have multiple effects to report and we need to report corrected degrees of freedom for each, and these effects might have different degrees of freedom.

F, degrees of freedom, p and effect size.