Model class specification #

The glmmrBase package defines the Model class, which is built on the R6 class system. R6 classes are object orientated, which means that when you create a new object of an R6 class, it has a set of methods and objects that “belong” to it. In the case of the Model class, it generates and stores all the relevant matrices for the GLMM internally. One can then call functions that belong the object to access or use the stored data. We discuss the

The arguments to create a new Model object are:

• formula: A formula specifying the GLMM, see Formula specification. Can include non-linear functions of the fixed effects.
• covariance: Optional. Generally, a vector of the parameter values otherwise it can be a list with a formula, parameter values, and data.
• mean: Optional. Generally, a vector of the parameter values otherwise it can be a list with a formula, parameter values, and data.
• var_par: Optional. The (starting) value of the scale parameter.
• data: A data frame with the data for the model. This can be omitted if data are provided separately to the mean and covariance arguments via their lists.
• family: An R family.
• offset: An optional vector of offset values
• trials: For binomially distributed data, the number of trials can be specified here.
• weights: An optional vector of regression weights. For a Gaussian model, these weights divide the individual level variance. For non-Gaussian models they weight the log-likelihood. We discuss each of these below. Examples are provided here.

Formula #

The formula for the model. The formula can either be a single formula including both fixed and random effects, or each component can be specified separately in the argument for the mean and covariance. Separate specifications may be desirable if one wants to quick change the mean or covariance separately in an existing Model object. See Formula specification for more details.

Family and link function #

The package currently supports the following families and link functions, which are specified using the standard R families in the stats package. Link functions are specified in the same way, for example binomial(link="logit").

The Beta family is provided by the package function Beta(), which generates a barebones list specifying the family and link. We use a mean-variance parameterisation of the Beta family. The likelihood is:

$$ f(y_i | \mu_i, \phi) = \frac{y_i^{\mu_i\phi - 1}(1-y_i)^{(1-\mu_i)\phi - 1}}{B(\mu_i\phi, (1-\mu_i)\phi)} $$

where $B()$ is the Beta function, and we use logit link

$$ \log\left( \frac{\mu_i}{1-\mu_i} \right) = \mathbf{x}_i\beta + \mathbf{z}_i \mathbf{u} $$

We similarly use a mean-variance parameterisation for the Gamma regression function:

$$ f(y_i | \mu_i, \nu) = \frac{1}{\Gamma(\nu)}\left( \frac{\nu y_i}{\mu_i} \right)^\nu \exp \left( -\frac{\nu y_i}{\mu_i} \right) \frac{1}{y} $$

where we also provide logit, inverse, and identity link functions for the specification of $\mu_i$.

Data #

Data are required to build the matrices and perform the calculations. When we want to analyse a study design prior to data being collected we will have to create our own covariate data for this purpose. We provide several useful functions to create data.frames with different structures to support generating dummy data, see Generating data.

Parameter values #

Parameter values are required to generate the matrices of the GLMM or act as starting values for model fitting algorithms. If they are not specified then random parameter values are used (Uniform(-2,2) for fixed effect parameters, and Uniform(0,1) for covariance paramters). Parameter values are given separately to the mean and covariance arguments when specifying a new model object; values are passed as a vector in the order they appear in the formula. For example, for the mean function we might specify (where we include the formula for clarity):

mean = list(formula = ~x1 + x2,
            parameters = c(-1,1,2))

to get the formula (or starting values) $-1 + x_1 + 2x_2$. Similarly, for a covariance function:

covariance = list(formula = ~ (1|gr(cl)*ar0(t)),
                  paramters = c(0.05, 0.8))