The standard response families gaussian, binomial, poisson, and Gamma are handled, as well as negative binomial (see negbin), zero-truncated poisson and negative binomial, and Conway-Maxwell-Poisson response (see Tpoisson, Tnegbin and COMPoisson). A multi family look-alike is also available for multinomial response, with some constraints. The variance parameter of residual error is denoted \(\phi\) (phi): this is the residual variance for gaussian response, but for Gamma-distributed response, the residual variance is \(\phi\)\(\mu^2\) where \(\mu\) is expected response. A fixed-effects linear predictor for \(\phi\), modeling heteroscedasticity, can be considered (see Examples).
The package fits models including several nested or crossed random effects, including autocorrelated ones with the following correlation models: Matern, Cauchy, interpolated Markov Random Fields (IMRF), first-order autoregressive (AR1), conditional autoregressive as specified by an adjacency matrix, or any fixed correlation matrix (corrMatrix). GLMMs and HGLMs are fit via Laplace approximations for (1) the marginal likelihood with respect to random effects and (2) the restricted likelihood (as in REML), i.e. the likelihood of random effect parameters given the fixed effect estimates.
All handled models can be formulated in terms of a linear predictor of the standard form offset+ X\(\beta\) + Z b, where X is the design matrix of fixed effects, \(\beta\) (beta) is a vector of fixed-effect coefficients, Z is a design matrix of random effects (typically an incidence matrix), and b a vector of random effect values.
The structure of the random effects Zb can generally be described by the following steps. First, independent and identically distributed (iid) random effects u are drawn from one of the following distributions: gaussian, Beta-distributed, Gamma and inverse-Gamma distributed random effects, implemented as detailed in the HLfit documentation. The variance(s) of random effects (\(u\)) is (are) denoted \(\lambda\). Second, a transformation v\(=f\)(u) is applied (v elements are still iid). Third, correlated random effects are obtained as Mv, where the matrix M can describe spatial correlation between observed locations, block effects (or repeated observations in given locations), and correlations involving unobserved locations. See Details in covStruct for the general form of M as a matrix product ZAL. In most cases M is determined from the model formula, but it can also be input directly (e.g., to describe genetic correlations).
The syntax for formulas extends that used in the lme4 package. In particular, non-autocorrelated random effects are specified using the (1|<block>) syntax, and Gaussian random-coefficient terms by (<rhs>|<block>). Autocorrelated random effects are specified by adding some prefix to this syntax, <prefix>(1|.), e.g. Matern(1|long+lat). Since version 2.6.0, it is possible to fit some “autocorrelated random-coefficient” models by a syntax consistent with that of random-coefficient terms, <prefix>(<rhs>|.). For example, independent Mat<U+00E9>rn effects can be fitted for males and females by using the syntax Matern(male|.) + Matern(female|.), where male and female are TRUE/FALSE factors. A numeric variable z can also be considered, in which case the proper syntax is Matern(0+z|.), which represents an autocorrelated random-slope (only) term (or, equivalently, a direct specification of heteroscedasticy of the Mat<U+00E9>rn random effect). All these effects are achieved by direct control of the elements of the incidence matrix Z through the <rhs> term: for numeric z, such elements are multiplied by z values, and thus provide a variance of order O(z squared). By contrast, Matern(z|.) is not defined. It could mean that a correlation structure between random intercepts and random slopes is to be combined with a Mat<U+00E9>rn correlation structure, but no way of combining them is yet defined and implemented.
Since version 2.7.0, the syntax (z-1|.), for numeric z only, can also be used to fit some heteroscedastic non-Gaussian random effects. For example, a Gamma random-effect term (wei-1|block) specifies an heteroscedastic Gamma random effect \(u\) with constant mean 1 and variance wei^2 \(\lambda\), where \(\lambda\) is still the estimated variance parameter. See Details of negbin for a possible application. Here, this effect is not implemented through direct control of Z (multiplying the elements of an incidence matrix Z by wei), as this would have a different effect on the distribution of the random effect term. (z|.) is not defined. It could mean that a correlation structure between random intercepts and random slopes for Gamma-distributed random effects is considered, but such correlation structures are not well-specified by their correlation matrix.
The double-vertical syntax, ( rhs || lhs ), is interpreted as in lme4. Any such term is immediately converted to ( ( 1 | lhs ) + ( 0 + lhs | rhs ) ), and should be counted as two random effects for all purposes (e.g., for fixing the variances of the random effects).