Model Terms

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. The definition of terms is pretty much like the R syntax, woth a few restrictions:

• Interactions are always defined with the column :, never with the star.
• Intercepts should always be defined with 1 or 0. For fixed effect, the intercept may be omitted but a warning is issued. Better explicitly declare it.

Model Coefficients

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 \(1\). There are a few rules to follows:

• Each term should have a coefficient.
• Coefficients for categorical variables are passed with the syntax [1,2,3]*x, where x has 4 levels, so it requires 3 contrast variables to estimate the effect.

For instance, if x is categorical with 3 levels, the syntax would be:

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

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 slopes whose variance is \(1.5\) across levels of clustervar. If the random slopes regards a categorical variable, the variance coefficients corresponding to the contrasts variable representing the variable should be cast with the brackets syntax:

y~1*1+[.5,1,1.5]*x+(1*1+[1.1,2,3.1]*x|clustervar)

indicating a random coefficient for the three contrasts variables representing the effect of x, with variances \(1.1\), \(2\), and \(3.1\) respectively.

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

For categorical variable, indicate the sign of the coefficient within the brackets:

y~1*1+[.5,-1,-1.5]*x+(1*1|clustervar)

Model Coefficients sign

Coefficients can be positive or negative. For continuous independent variables (terms) simply use the the negative sign in the formula, like in:

Multiple clustering variables can specified, for instance:

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

Symbolic coefficients

PAMLj allow passing also symbolic coefficients in the formula. Symbolic coefficients are labels that can be used to refer to the term/coefficient in additional syntax lines. Here an example:

y~1*1+a*.5*x+b*2*z+(1*1|school)

In this example, a and b would refer to the coefficient of x and z respectively.

Additional directives

The field Module Structure accepts additional directive over the specified model. At the moment there is only one directive that is recognized.

• test: alabel

This directive implies that the estimated sample size, Required Number of cluster levels or Required N per cluster are solved for the effect whose label is alabel. By default, in fact, PAMLj estimated the required sample size for the smallest fixed effect in the model. If one wishes to estimate the required sample size for a coefficient which is not the smallest, simply use a symbolic coefficient for the target term and use test: directive. In practice:

y~1*1+a*.5*x+b*2*z+(1*1|school)
test:b

computes the required sample size for 2*z, irrespective of the fact that .5*x will have a smaller effect.