One-way repeated-measures ANOVA design

Assume we have a repeated-measures design with four time points (see an example in Mixed vs RM-ANOVA), and one measurement per participant (identified by the variable ID) at each time point. To define this design, we would input:

y~1*1+1*time+(1*1|ID)

Then we specify that time is a categorical variable with 4 levels.

and we set the cluster N per cluster to 4, indicating that one participant (the cluster unit) has 4 observations.

The resulting data look like this:

             y time      ID
1   0.64342217    1       1
2   0.15874249    2       1
3  -0.70961322    3       1
4  -1.46269242    4       1
5  -0.80773575    1       2
6   0.38518147    2       2
7  -1.95162686    3       2
8   2.98249154    4       2
9  -0.04821753    1       3
10 -1.61386528    2       3
11 -0.68877198    3       3
12  1.33885528    4       3

We can see that time is repeated within ID, with one measurement for each level of time.

Assume now we have a similar design, but within each time condition the participant is measured 5 times (5 trials). We still have the same model:

y~1*1+1*time+(1*1|ID)

and the same variable definition (time with 4 levels), but now N per cluster is 20, which is 4 levels of time × 5 trials.

The resulting data look like this (first participant):

              y time      ID
1    0.38563267    1       1
2    1.24833377    1       1
3   -0.76549380    1       1
4   -0.94633446    1       1
5    0.72747486    1       1
6   -0.58658441    2       1
7   -1.01631241    2       1
8   -2.66731584    2       1
9    0.62401507    2       1
10   0.15654240    2       1
11  -0.26135861    3       1
12   0.11923442    3       1
13   0.66327673    3       1
14  -0.63811091    3       1
15   0.31462810    3       1
16   0.27796168    4       1
17   0.32836307    4       1
18  -0.34867872    4       1
19  -0.20470434    4       1
20  -0.71480958    4       1

Cross-classified clusters

A typical example of cross-classified clustering is an experimental design in which \(P\) participants (identified by the variable PID) are tested with \(S\) stimuli (variable SID). Every participant is exposed to each stimulus, and each stimulus is seen by all participants. We start with the simpler example in which there is only one measurement for each combination of participant and stimulus (a participant sees a stimulus only once).

The mixed model is:

y~1*1+1*x+(1*1|PID)+(1*1|SID)

Let’s say we want a sample of 10 participants and 5 stimuli. In total we have 50 observations, in which each participant has N per cluster equal to 1, and each stimulus has N per cluster equal to 1 as well. It may seem counterintuitive, but the participant is actually repeated only once, but their measurement is spread across 5 stimuli. Thus, we need to set:

The resulting data (first participant) will look like this:

        y       PID SID
1   0.741008569   1   1
2  -1.001295926   1   2
3   0.728456584   1   3
4   0.649785351   1   4
5  -1.373459856   1   5
6  -2.073459856   1   6

Now, what about a categorical variable (i.e., an experimental factor named cond)? If the variable is between-subjects, we simply declare it as above. For a 4-level cond, we have (first two participants):

             y    cond PID SID
1   0.21678164    1   1   1
2  -1.89497678    2   1   2
3   0.68536365    3   1   3
4  -0.50112451    4   1   4
5  -0.03173882    1   1   5
6   0.10935699    2   1   6
7  -0.03479272    3   2   1
8   0.79953612    4   2   2
9   1.16768781    1   2   3
10  0.01980561    2   2   4
11 -0.67931988    3   2   5
12  0.12628890    4   2   6

(Recall that for between-subjects factors, their structure does not affect the results.)

If the variable is within-subjects, we must ensure that the number of observations per participant (here 6) is a multiple of its levels. With 6 observations, cond can have 2 levels, 3 levels, or 6 levels; otherwise it cannot be repeated within participant. A 2-level factor would look like this:

             y cond PID SID
1   0.49915152    1   1   1
2   0.74412274    2   1   2
3  -1.13346819    1   1   3
4   0.86378973    2   1   4
5  -0.13518382    1   1   5
6   1.16554628    2   1   6

A more complex design arises when the repeated-measures factor cond is repeated both within participants and within stimuli. Suppose each participant sees each stimulus in both conditions of factor cond. This implies that each stimulus has two observations, so we change its N per cluster:

and we obtain the desired design:

              y cond PID SID
1    1.61187894    1   1   1
2    0.41588941    2   1   1
3   -0.88306925    1   1   2
4   -0.42600560    2   1   2
5   -0.71458546    1   1   3
6   -1.00448085    2   1   3
7   -0.98879046    1   1   4
8    0.91321802    2   1   4
9    0.81331414    1   1   5
10   0.54411584    2   1   5
11  -0.15936292    1   1   6
12  -1.03945842    2   1   6

With the same logic, we can simulate designs with several trials within each combination of participant and stimulus. For instance, with the following setup:

each participant sees all stimuli 9 times.

Nested cluster classification

Let us consider one of the classic multilevel design: pupils within classes within schools. In this design we have two clustering variables: class and school. In multilevel terminology, pupils are level 0, classes are level 1 with pupils nested within classes, and schools are level 2 with classes nested within schools. To be practical, let us assume we want a sample with 10 schools, each with 6 classes, and each class (on average) with 12 pupils.

First, we set the model:

y~1*1+1*x+(1*1|school/class)

Then we specify that we have 10 schools, each with 6 classes, and each class has 12 observations (pupils).

The generated data look similar to the cross-classified data:

row   y             x           school  class
1    0.456717064 -1.017673137      1     1
2   -1.495551991  0.694440645      1     1
3   -0.480543438  0.194002872      1     1
4    0.293003240 -0.116523555      1     1
5   -1.702547420 -1.203825265      1     1
6    1.014035495  0.461345298      1     1
7   -0.415440850 -0.585482460      1     1
8   -1.524924282  0.544323536      1     1
9   -0.106578314 -0.196693360      1     1
10  -0.035543818 -2.741920747      1     1
11   0.229491601  0.170131727      1     1
12  -1.909614050 -1.159812029      1     1
13   1.190966855 -0.356247480      1     2
14  -2.028748115  0.489991721      1     2
15  -1.079830560  1.315918615      1     2
16   0.099924682 -0.330729467      1     2
17  -0.075449795 -2.424778270      1     2
....

However, the estimated model is different. If one looks at the Effect size indices table, one finds that the clusters are now school and school:class, which is exactly how the model specification expands.

(1 | school) + (1 | school:class)