offset+ X beta + Z b, where
X is the design matrix of fixed effects and Z is a design matrix of random effects.
The function also handles a linear predictor (with only fixed effects) for the residual variance.HLfit(formula, data, family = gaussian(), rand.family = gaussian(),
resid.model = ~1, resid.formula, REMLformula = NULL,
verbose = c(warn = TRUE, trace = FALSE, summary = FALSE),
HLmethod = "HL(1,1)", control.HLfit = list(),
control.glm = list(), init.HLfit = list(), ranFix = list(),
etaFix = list(), prior.weights = NULL, processed = NULL)
## see 'rand.family' argument for inverse.Gammafamily object describing the distribution of the response variable.
Possible values include the gaussian, poisson, binomial, Gamma, negbin and
COMPoisson families. Possible combinations of family and link are those allowed by each of these families
(see family for the first four, and specific documentation pages for the last two).
family object describing the distribution of the random effect, or a list of
family objects for different random effects (see Examples). Possible options are
gaussian(), Gamma(log), Gamma(identity) (see Details), Beta(logit), inverse.Gamma(-1/mu), and inverse.Gamma(log).
For discussion of these alternatives see Lee and Nelder 2001 or Lee et al. 2006, p. 178-.
Here the family gives the distribution of a random effect \(u\)
and the link gives v as function of \(u\) (see Details).
If there are several random effects and only one family is given, this family holds for all random effects.
phi of the residual error. A log link is assumed by default;
or a list, with at most two possible elements if its formula involves only fixed effects:
Gamma(identity) can be tried but may fail because only the log link ensures that the fitted \(\phi\) is positive. and additional possible elements (all named as fitme arguments) if its formula involves random effects: see phiHGLM.
resid.model.
formula that allows the estimation of dispersion parameters, and
computation of restricted likelihood (p_bv) under a model different from the predictor formula. For example, if only random effects are included in REMLformula, an ML fit is performed and p_bv equals
the marginal likelihood (or its approximation), p_v. This ML fit can be performed more simply by setting
HLmethod="ML" and leaving REMLformula at its default NULL value.
trace controls various diagnostic (possibly messy) messages about the iterations.
summary controls whether a summary of the fit is called by HLfit.
warn is for programming purposes and best ignored.
"REML", "ML", "EQL-" and "EQL+" for all models;
"PQL" (="REPQL") and "PQL/L" for GLMMs only; and further values
for those curious to experiment (see Details). The default is REML (standard REML for LMMs,
an extended definition for other models). REML can be viewed as a fom of conditional inference, and non-standard conditionings can be called as “REML” with a non-standard REMLformula. See Details for further information.
resid.family allows one to change the link for modeling of residual variance \(\phi\), which is "log" by default. The family is always Gamma, so the non-default possible values of resid.family are Gamma(identity) or Gamma(inverse). Only the default value ensures that the fitted \(\phi\) is positive.
Controls for the fitting algorithms should be ignored in routine use. They are
conv.threshold,
a convergence threshold for the iterative algorithm, which controls whether
linear predictor terms (fixed effects and inferred random effects), and dispersion parameter estimates have converged. Defaults to 1e-05;
break_conv_logL, a boolean specifying whether the iterative algorithm should terminate when log-likelihood appears to have converged (roughly, when its relative variation over on iteration is lower than 1e-8). Default is FALSE (convergence is then assessed on the parameter estimates rather than on log-likelihood).
iter.mean.dispFix, the number of iterations of the iterative algorithm for coefficients of the linear predictor,
if no dispersion parameters are estimated by the iterative algorithm. Defaults to 200;
iter.mean.dispVar, the number of iterations of the iterative algorithm for coefficients of the linear predictor,
if some dispersion parameter(s) is estimated by the iterative algorithm. Defaults to 50;
max.iter, the number of iterations of the iterative algorithm for joint estimation of dispersion parameters and
of coefficients of the linear predictor. Defaults to 200. This is typically much more than necessary,
unless there is little information to separately estimate \(\lambda\) and \(\phi\) parameters.
glm.control; e.g. control.glm=list(maxit=100). See glm.control for further details.
fixef for fixed effect estimates (beta),
v_h for random effects vector v in the linear predictor,
lambda for the parameter determining the variance of random effects \(u\) as drawn from the rand.family distribution
phi for the residual variance.
However, this argument can be ignored in routine use.
lambda, and also phi for gaussian and Gamma HGLMs.
Inhibits the estimation of these parameters.
beta. etaFix$beta should be a vector with names matching (a subset of) coefficient names of a fit without fixed values. It provides a convenient interface for fixing (some of) the fixed-effects coefficients (\(\beta\)). In contrast to an offset specification, it affects the REML correction for estimation of dispersion parameters, which depends only on which \(\beta\) coefficients are estimated. However, for non-standard use, REML can still be performed as if all \(\beta\) coefficients were estimated, by adding attribute keepInREML=TRUE to etaFix$beta (see Examples). These different behaviours will be overridden whenever a non-null REMLformula is provided.
glm. This fits the data to a model with residual variance phi/prior.weights, so that increasing the weights by a constant factor f will yield (Intercept) estimates of phi also increased by f (this effect cannot be generally achieved if a non-trivial resid.formula with log link is used). This is not necessarily the way prior weights are interpreted in widely used packages, but this is consistent with what glm.
corrHLfit).
HLfit, which is a list with many elements, not all of which are documented.A few extractor functions are available (see extractors),
and should be used as far as possible as they should be backward-compatible from version 1.4 onwards, while the structure of the return object may still evolve. The following information will be useful for extracting further elements of the object.
Elements include descriptors of the fit:
fitted function);fixef function);ranef(*,type="uncorrelated");HLfit, corrHLfit or fitme;p_v and the (approximate) restricted likelihood p_bv (the latter two available through the logLik function). See the extractor function get_any_IC for information criteria (“AIC”) and effective degrees of freedom;vcov;Information about the input is contained in output elements named as HLfit or corrHLfit arguments (data,family,resid.family,ranFix,prior.weights), with the following notable exceptions or modifications:
formula (possibly reformatted) and several attributes;predictor, for the residual variance;rand.family input;Further miscellaneous diagnostics and descriptors of model structure:
get_ZALMatrix can be used to reconstruct a single “ZL” matrix for all terms.Finally, the object includes programming tools: call, spaMM.version, fit_time and envir.
I. Fitting methods: Many approximations for likelihood have been defined to fit mixed models (e.g. Noh and Lee (2007) for some overview), and this function implements several of them, and some additional ones. In particular, PQL as originally defined by Breslow and Clayton (1993) uses REML to estimate dispersion parameters, but this function allows one to use an ML variant of PQL. Moreover, it allows some non-standard specification of the model formula that determines the conditional distribution used in REML.
EQL stands for the EQL method of Lee and Nelder (2001). The '+' version includes the d v/ d tau correction described p. 997 of that paper, and the '-' version ignores it. PQL can be seen as the version of EQL- for GLMMs. It estimates fixed effects by maximizing h-likelihood and dispersion parameters by an approximation of REML, i.e. by maximization of an approximation of restricted likelihood. PQL/L is PQL without the leverage corrections that define REML estimation of random-effect parameters. Thus, it estimates dispersion parameters by an approximation of marginal likelihood.
HLmethod also accepts values of the form "HL(<...>)", "ML(<...>)" and "RE(<...>)", e.g. HLmethod="RE(1,1)", which allow a more direct specification of the approximations used.
HL and RE are equivalent (both imply an REML correction).
The first '1' means that a first order Laplace approximation to the likelihood is used to estimate fixed effects
(a '0' would instead mean that the h likelihood is used as the objective function).
The second '1' means that a first order Laplace approximation to the likelihood or restricted likelihood
is used to estimate dispersion parameters, this approximation including the dv/d tau term specifically discussed by Lee & Nelder 2001, p. 997 (a '0' would instead mean that these terms are ignored).
It is possible to enforce the EQL approximation for estimation of dispersion parameter (i.e., Lee and Nelder's (2001) method) by adding a third index with value 0. "EQL+" is thus "HL(0,1,0)", while "EQL-" is "HL(0,0,0)". "PQL" is EQL- for GLMMs. "REML" is "HL(1,1)". "ML" is "ML(1,1)".
Some of these distinctions make sense for GLMs, and glm methods use approximations, which make a difference for Gamma GLMs. This means in particular that, (as stated in logLik) the logLik of a Gamma GLM fit by glm differs from the exact likelihood. Further, the dispersion estimate returned by summary.glm differs from the one implied by logLik, because summary.glm uses Pearson residuals instead of deviance residuals, and no HLmethod tries to reproduce this behaviour. logLik gives the approximation returned by an "ML(0,0,0)" fit. The dispersion estimate returned by an "HL(.,.,0)" fit matches what can be computed from residual deviance and residual degrees of freedom of a glm fit, but this is not the estimate displayed by summary.glm. With a log link, the fixed effect estimates are unaffected by these distinctions.
II. Random effects are constructed in several steps. first, a vector u of independent and identically distributed (iid) random effects is drawn from some distribution;
second, a transformation v=f(u) is applied to each element (this defines v which elements are still iid); third, correlated random effects are obtained as Lv
where L is the “square root” of a correlation matrix (this may be meaningful only for Gaussian random effects). Coefficients in a random-coefficient model correspond to Lv.
Finally, a matrix Z (or sometimes ZA, see Predictor) allows to specify how the correlated random effects
affect the response values. In particular, Z is the identity matrix if there is a single observation (response) for each location, but otherwise
its elements \(z_{ji}\) are 1 for the \(j\)th observation in the \(i\)th location.
The design matrix for v is then of the form ZL.
The specification of the random effects u and v handles the following cases:
Gaussian with zero mean, unit variance, and identity link; Beta-distributed, where \(u ~ B(1/(2\lambda),1/(2\lambda))\) with mean=1/2, and var\(=\lambda/[4(1+\lambda)]\); and with logit link v=logit(u);
Gamma-distributed random effects, where \(u ~ \)Gamma(shape=1+1/\(\lambda\),scale=1/\(\lambda\)): see Gamma for allowed links and further details; and Inverse-Gamma-distributed random effects, where \(u ~ \)inverse-Gamma(shape=1+1/\(\lambda\),rate=1/\(\lambda\)): see inverse.Gamma for allowed links and further details.
III. The standard errors reported may sometimes be misleading. For each set of parameters among \(\beta\), \(\lambda\), and \(\phi\) parameters these are computed assuming that the other parameters are known without error. This is why they are labelled Cond. SE (conditional standard error). This is most uninformative in the unusual case where \(\lambda\) and \(\phi\) are not separately estimable parameters. Further, the SEs for \(\lambda\) and \(\phi\) are rough approximations as discussed in particular by Smyth et al. (2001; \(V_1\) method).
Lee, Y., Nelder, J. A. (2001) Hierarchical generalised linear models: A synthesis of generalised linear models, random-effect models and structured dispersions. Biometrika 88, 987-1006.
Lee, Y., Nelder, J. A. and Pawitan, Y. (2006). Generalized linear models with random effects: unified analysis via h-likelihood. Chapman & Hall: London.
Noh, M., and Lee, Y. (2007). REML estimation for binary data in GLMMs, J. Multivariate Anal. 98, 896-915.
Smyth GK, Huele AF, Verbyla AP (2001). Exact and approximate REML for heteroscedastic regression. Statistical Modelling 1, 161-175.
HLCor for estimation with given spatial correlation parameters;
corrHLfit for joint estimation with spatial correlation parameters;
fitme as an alternative to all these functions.data(wafers)
## Gamma GLMM with log link
<!-- % example also in main page... -->
HLfit(y ~X1+X2+X1*X3+X2*X3+I(X2^2)+(1|batch),family=Gamma(log),
resid.model = ~ X3+I(X3^2) ,data=wafers)
<!-- %- : tested in update.Rd -->
## Gamma - inverseGamma HGLM with log link
HLfit(y ~X1+X2+X1*X3+X2*X3+I(X2^2)+(1|batch),family=Gamma(log),
HLmethod="HL(1,1)",rand.family=inverse.Gamma(log),
resid.model = ~ X3+I(X3^2) ,data=wafers)
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