keywords Mixed models, hierarchical linear model, multilevel model, ANOVA, crossing, nesting

GAMLj version ≥ 1.0.3 Draft version, mistakes may be around

# Intro

In multilevel designs, there are at least two layers or levels in the sample: the within-cluster level and the between-cluster level. This structure is simply declared in the models by specifying a cluster variable and at least one random coefficient that varies across clusters. In many applications of the mixed model, however, there are more than two levels. In a classical educational program, one can have pupils (within layer), classes (second layer), and schools (third layer). An experiment may involve several participants exposed to multiple stimuli, creating two clustering variables. Cluster variable groups may be nested or cross-classified (referred to as crossed for simplicity). Whether the cluster variable groups are nested or crossed makes a substantial difference in the model structure and the parameters that one can estimate. Therefore, it is important to clarify how different data structures can be included in the mixed model. Here, we explore the possibilities in GAMLj

In terms of logic of defining the mixed model GAMLj follows Bates et al. (2015) R package implementation, so further details can be found in lme4 documentation.

# Cross-classified clusters

Two or more clustering variables are said to be crossed when each level (group) of one cluster variable is repeated for all levels (groups) of the other clustering variable. An example can be an experiment in which each participant is exposed to a series of stimuli. All participants see all stimuli. An example can be found Mixed Models: Subjects by Stimuli random effects, using the subjects_by_stimuli dataset. There we have 50 participants measured in two conditions using 30 stimuli. The first cells of a contingency table subj(in columns) by stimulus is shown here.

What is crucial here is that both subj and stimulus clustering variables are a the same level (or layer). This means that each measurement (each row) belongs to one combination of subj and stimulus, that is is measured for one particular participant and one particular stimulus. The classifications are orthogonal, meaning that they are independent one each other. In the experimental terminology , they create a balanced 50 X 30 design.

To understand the structure of the model estimable in these designs, we can focus on the random intercepts (the same applies to slopes, but the latter depend on the specific design at hand). What random intercepts can we estimate in this kind of design? The random intercepts varying across participants, each intercept corresponding to the average score of the participant averaged across the 30 stimuli. The random intercepts varying across stimuli, each intercept corresponding to the average score associated with a stimulus, averaged across the 50 participants. Thus, the number of intercepts across participant will be N, where N is the number of participants, and the number of intercepts across stimuli would be K, where K is the number of stimuli.

If within each combination of subj and stimulus there are several measurements, one can also estimate the intercepts vary across subj by stimulus cells. If subj and stimulus were factors and not clusters, one could say that the random intercepts can represent the variance of the main effect of subj, the main effect of stimulus and (if there are enough data), their interaction. For this variance, the number of intercepts are N X K.

In terms of coding the variable, one simply needs to assign one unique value to each participants and one unique value to each stimulus.

Then one specifies the random structure as for any other mixed model:

The results will show the correct variances in the tables

# Structure by data vs by formula

What if I want to insert also a random intercept that varies across the cells representing the combinations of subj and stimulus (notice that in this dataset it does not make much sense because there are only two scores in each subj by stimulus combination, but we show this anyway as an example).

There are two ways in GAMLj to obtain the model structure we desire. Going by data or going by formula: The two methods give exactly the same results, so which way to go depends on which way is more convenient from a practical point of view.

## By data

Since we want to estimate an intercept for each combination of subj and stimulus, we need to create a variable in the dataset that represents those combinations: We can do this by simply using the compute command in Data tab.

and specify the value of the new variable as a combination of the values of subj and stimulus

Then we can add subj_stimuli as a clustering variable of our model, and ask for the random intercepts across it.

The results will now show an additional variance, which we know will capture the variability due to the combinations of the clustering variables.

(notice that in this example the variance is zero, so the random intercepts across subj_stimulus can be removed, but the general idea is working)

## By formula

Some user may find this method tiring. For those, the module can create the combination variable automatically. When more than one cluster variables are defined, the option Crossing by formula option appears. By selecting it, we ask the module to list all possible crossing between clustering variables, so we can select the one we need

As expected, the results are identical to the results obtained with the by data set up.

# Nested clusters

One clustering variable is said to be nested within another clustering variable if its levels (groups) are distinct within each level of the parent variable. A typical example are classes nested within schools: Class 1 of school 1 is clearly a different class, with different pupils, as compared with Class 1 of school 2. In our experimental example, stimuli are nested within participants if each participant is exposed to a different set of stimuli as compared with the other participants.

An example can be found in the subjects_on_stimuli dataset.

Here each participant gets their own set of stimuli, so the stimulus 1 of participant 1 is different from stimulus 1 of participant 2. This is clear in the contingency table (showing only 3 participants in columns and 3 stimuli in rows), in which every stimulus appears only in combination with one participant. There are 20 participants, 600 stimuli, 30 per participant, and each stimulus is measured over 10 trials, for a total of 6000 observations.

To understand the structure of the model estimable in these designs, we can focus on the random intercepts (the same applies to slopes, but the latter depends on the specific design at hand). What random intercepts can we estimate in this kind of designs? The random intercepts varying across participants, each intercepts representing the average score of the participant, averaged across the stimuli, and the random intercepts of each particular stimuli by participant combination. Thus, the number of intercepts across participant will be N=20, where N is the number of participants, and the number of intercepts across stimuli would be N=20 X K=30, where K is the number of stimuli, thus 600.

# Structure by data vs by formula

The way to structure the correct model in terms of nesting clustering variables depends on the way the clustering variables are coded in the dataset.

## By data

If one has coded each nested level (group) with a different code, one simply needs to add the parent and the nested clustering variable, and ask for their random intercepts (or any other coefficients we wish to estimate as random). In the dataset of the example, we do have a variable named uni_stimulus (unique stimulus) which uses a different code for each combination of stimulus and participant.

(showing only 3 participant and 3 stimuli)

Being coded as such, the model will recognize each stimulus as a different cluster group, and thus estimate the correct number of intercepts, and thus the correct variances. We simply list the clustering variables and their random coefficients.

Please notice the numerosity of the clusters. There are 6000 observations, 600 stimuli and 20 participants.

## By Formula

Sometimes the nested variable is not coded with a different value for each unique level. A dataset of schools data, may have classes coded as 1, 2, 3 within each school. In our example data, for instance, there is a variable named within_stimulus that codes each stimulus as s1, s2 etc within each participant, so the code s1 refers to different stimuli for different people. Very often, data are coded like this because the s1 represents the first stimulus of each participant, s2 the second and so forth, even if they are actually different objects.

If we use a nested clustering variable as such, the model will be mis-specified.

Indeed, the model will estimate 30 intercepts across stimuli, pooling together the scores of every stimulus coded with the same value (s1,s2 etc), as if they were the same stimulus. But they are not the same, so the model is wrong.

As we did for cross-classified data, one can create a new variable to explicitly identify different stimuli (nested groups) across participant (parent groups). So using a variable like uni_stimulus as we did in the by data approach. However, GAMLj offers the option to do that automatically. We can select Nesting by formula and select participant/within_stimulus. This notation, which is the R notation for nested random coefficients , will automatically identifies as different stimuli with the same value in within_stimulus (the nested clustering variable) but with different value in participant (the parent clustering variable).

The numerosity is now correct and the results are the same as the ones obtained with the by data approach.

# Differences in notation

Defining the structure of the clustering groups by data assures control over the model being estimated and correct results. The rule is simple: each clustering variable or variable representing combination of clustering variable should have its own random intercept (or possibly random slopes).

When variables are not coded in order to convey the correct structure of the data, one can use the Nesting by formula and Nesting by formula. Crossing by formula creates a clustering variable that combines the parent and the nested values into an unique code, and then estimates the random coefficients for the parent variable and for the combination of parent and nested. Crossing by formula creates a clustering variable that combines the levels (groups) of two or more clustering variables and estimates the coefficients across this new variable levels.

In practice, if uni_stimulus uniquely identifies the levels of a nested variable, within_stimulus` differentiates stimuli only with each participant, the following three commands gives exactly the same results.