Linear Mixed Models

Linear Mixed Models allows users to compute achievable power and sample size parameters for testing coefficients in a random coefficients linear model.
Because of the complexity and flexibility of mixed models, this sub-module includes several features that differ from other sub-modules of PAMLj.
Particular attention should therefore be paid to the specific options available and their implications. Furthermore, power analysis of mixed models is computed using complex simulations, that often require very long estimation times. So, some patience is required to perform a correct and complete analysis.

Simulations

Mixed models can be considered as linear models applied to multi-level designs, where observations are sampled at different hierarchical levels. In a standard multi-level design, clusters (groups of observations) are randomly sampled, and within each cluster, a number of observations is sampled. Here, \(k\) indicates the number of clusters, and \(n\) indicates the number of observations within each cluster.

In PAMLj, grouping variables are referred to as clustering variables, the number of clusters (or groups) as # Clusters levels, and the number of observations per cluster as N per cluster.

Power parameters are estimated by Monte Carlo simulations. A sample with the required characteristics — such as independent variables, variable types, and clustering variables — is generated from a data-generating model (DGM) defined by the user. The DGM reproduces the structure and coefficients specified in the Model Syntax field of the interface.
For each simulation run, \(R\) models are estimated, and power (the proportion of significant results) is computed.

When the aim is to determine the required sample size (Aim: N), the sample characteristics (\(k\) or \(n\)) are varied until the desired power is reached. When the aim is to assess achievable power, the \(R\) simulations are run to estimate power given the \(k\) and \(n\) specified by the user.

User interface (UI)

As opposed to other modules and command in jamovi, the power analysis for mixed models does not update the results whenever an option is changed in the UI. This is because power analysis simulation algorithm may take very long time to achieve an estimation, thus blocking the user work when setting several options. Thus, to run the analysis the user should always click the ▶ run button.

The first requirement of the analysis, however, is to specify a model and its coefficients. This can be done in Model Syntax field.

Model Syntax

Power analysis for mixed model requires to input a mixed model with all the expected coefficients. PAMLj employs a custom syntax based on R package lme4 (Bates et al. 2015) standard formulas, modified to easily pass coefficients.

First, a model in the R package lme4 (Bates et al. 2015) is to be defined. For instance, a simple random intercept-only model may look like this

y~1+x+(1|clustervar)

Recall that 1 indicates the intercept: this model has a fixed intercept, a fixed effect of x and a random intercept across a variable named clustervar.

Second, one needs to input the coefficients of each term in the model using the syntax value*x. Thus, the syntax:

y~1*1+.5*x+(1*1|clustervar)

This means that we expect the fixed intercept to be \(1\), the fix coefficient associated with \(x\) to be equal to \(.5\) and we expect the variance of the random intercept to be \(1\). The residual variance of the model can be set in the UI in the field Residual Variance, default to 1.

If a model has also a random coefficient of the independent variable (random slopes), it should be added to the model with its expected coefficient. For instance:

y~1*1+.5*x+(1*1+1.5*x|clustervar)

indicates that \(x\) has a random slope whose variance is \(1.5\) across levels of clustervar. Random coefficients are expected to be independent in the population, but their correlation is estimated in the simulation model (if not explicitly denied, such as in model like y~1*1+.5*x+(1*1|clustervar)+(0*0+1.5*x|clustervar)).

Multiple clustering variables can specified, for instance:

y~1*1+.5*x+(1*1+1.5*x|class)+(1*1|school)

Notice, however, that the power sample size parameters (\(k\) or \(n\)) are solved for the first cluster specified, the other clusters parameters should be set by the user in the UI (see Model Structure).

Once the model is set up, one should decide which is the aim of the analysis.

Aim: N (Required sample size)

In mixed models, the sample size depends on the number of clusters (\(k\) ) of the clustering variable or the number of cases (\(n\) observations) within each cluster. So, the user should decide what the algorithm should find, either the required Number of cluster levels or the required Cases within cluster.

Variables

PAMLj extracts from the input model the name of the independent variables included in the model. By default, they are considered as continuous variables.

y~1*1+[.5,1,-.6]*x+(1*1|clustervar)

Additional material