R: Fitting Generalized Linear Models for Multivariate Abundance...
manyglm
R Documentation
Fitting Generalized Linear Models for Multivariate Abundance Data
Description
manyglm is used to fit generalized linear models to high-dimensional data, such as multivariate abundance data in ecology. This is the base model-fitting function - see plot.manyglm for assumption checking, and anova.manyglm or summary.manyglm for significance testing.
Usage
manyglm(formula, family="negative.binomial", K=1, data=NULL, subset=NULL,
na.action=options("na.action"), theta.method = "PHI", model = FALSE,
x = TRUE, y = TRUE, qr = TRUE, cor.type= "I", shrink.param=NULL,
tol=sqrt(.Machine$double.eps), maxiter=25, maxiter2=10,
show.coef=FALSE, show.fitted=FALSE, show.residuals=FALSE,
show.warning=FALSE, offset, ... )
Arguments
formula
an object of class "formula" (or one that
can be coerced to that class): a symbolic description of the
model to be fitted. The details of model specification are given
under Details.
family
a description of the distribution function to be used in the
in the model. The default is negative binomial regression (using a log
link, with unknown overdispersion parameter), the following family
functions are also accepted: binomial(), binomial(link="cloglog"),
poisson(), which can also be specified using character strings as
'binomial', 'cloglog' and 'poisson', respectively. In future we hope
to include other family functions as described in family.
K
number of trials in binomial regression. By default, K=1 for presence-absence data using logistic regression.
data
an optional data frame, list or environment (or object
coercible by as.data.frame to a data frame) containing
the variables in the model. If not found in data, the variables
are taken from environment(formula), typically the environment
from which glm is called.
subset
an optional vector specifying a subset of observations
to be used in the fitting process.
na.action
a function which indicates what should happen
when the data contain NAs. The default is set by
the na.action setting of options, and is
na.fail if that is unset. The ‘factory-fresh’
default is na.omit. Another possible value is
NULL, no action. Value na.exclude can be useful.
theta.method
the method used for the estimation of the overdisperson
parameter theta, such that the mean-variance relationship is V=m+m^2/theta for
the negative binomial family. Here offers three options
"PHI" = Maximum likelihood estimation with respect to phi (default)
"ML" = Maximum likelihood estimation with respect to theta, as in Lawless(1987),
"Chi2" = Moment estimation using chi-square dampening on the log scale, as
in Hilbe(2008).
model, x, y, qr
logicals. If TRUE the corresponding
components of the fit (the model frame, the model matrix, the model
matrix, the response, the QR decomposition of the model matrix) are
returned.
cor.type
the structure imposed on the estimated correlation
matrix under the fitted model. Can be "I"(default), "shrink", or "R".
See Details. This parameter is merely stored in manyglm, and
will be used as the default value for cor.type in subsequent
functions for inference.
shrink.param
shrinkage parameter to be used if cor.type="shrink". If a numerical
value is not supplied, it will be estimated from the data by cross
validation-penalised normal likelihood as in Warton (2008). The parameter
value is stored as an attribute of the manyglm object, and will be
used in subsequent functions for inference.
tol
the tolerance used in estimation.
maxiter
maximum allowed iterations in the weighted least square estimation of beta. The default value is 25.
maxiter2
maximum allowed iterations in the internal ML estimations of negative binomial regression. The default value is 10.
logical. Whether to show model coefficients, fitted values, standardized pearson residuals, or operation warnings.
offset
this can be used to specify an _a priori_ known component to be included in the linear predictor during fitting. This should be 'NULL' or a numeric vector of length equal to NROW (i.e. number of observations) or a matrix of NROW times p (i.e. number of species).
...
further arguments passed to or from other methods.
Details
manyglm is used to calculate the parameter estimates of generalised linear models fitted to each of many variables simultaneously as in Warton et. al. (2012) and Wang et.al.(2012). Models for manyglm are specified symbolically. For details on how to specify a formula see the details section of lm and formula.
Generalised linear models are designed for non-normal data for which a distribution can be specified that offers a reasonable model for data, as specified using the argument family. The manyglm function currently handles count and binary data, and accepts either a character argument or a family argument for common choices of family. For binary (presence/absence) data, family=binomial() can be used for logistic regression (logit link, "logistic regression"), or the complementary log-log link can be used family=binomial("cloglog"), arguably a better choice for presence-absence data. Poisson regression family=poisson() can be used for counts that are not "overdispersed" (that is, if the variance is not larger than the mean), although for multivariate abundance data it has been shown that the negative binomial distribution (family="negative.binomial") is usually a better choice (Warton 2005). In both cases, a log-link is used. If another link function or family is desired, this can be specified using the manyany function, which accepts regular family arguments.
In negative binomial regression, the overdispersion parameter (theta) is estimated separately for each variable from the data, as controlled by theta.method for negative binomial distributions. We iterate between updates of theta and generalised linear model updates for regression parameters, as many as maxiter2 times.
cor.type is the structure imposed on the estimated correlation
matrix under the fitted model. Possible values are: "I"(default) = independence is assumed (correlation matrix is the identity) "shrink" = sample correlation matrix is shrunk towards I to improve its stability. "R" = unstructured correlation matrix is used. (Only available when N>p.)
If cor.type=="shrink", a numerical value will be assigned to shrink.param either through the argument or by internal estimation. The working horse function for the internal estimation is ridgeParamEst, which is based on cross-validation (Warton 2008). The validation groups are chosen by random assignment, so some slight variation in the estimated values may be observed in repeat analyses. See ridgeParamEst for more details. The shrinkage parameter can be any value between 0 and 1 (0="I" and 1="R", values closer towards 0 indicate more shrinkage towards "I").
Value
manyglm returns an object inheriting from "manyglm",
"manylm" and "mglm".
The function summary (i.e. summary.manyglm) can be used to obtain or print a summary of the results and the function
anova (i.e. anova.manyglm) to produce an
analysis of variance table, although note that these functions use resampling so they can take a while to fit.
The generic accessor functions coefficients,
fitted.values and residuals can be used to
extract various useful features of the value returned by manyglm.
An object of class "manyglm" is a list containing at least the
following components:
coefficients
a named matrix of coefficients.
var.coefficients
the estimated variances of each coefficient.
fitted.values
the matrix of fitted mean values, obtained by transforming the linear predictors by the inverse of the link function.
linear.predictor
the linear fit on the scale of the linear predictor.
residuals
the matrix of working residuals, that is the Pearson residuals standardized by the leverage adjustment h obtained from the diagonal elements of the hat matrix H.
PIT.residuals
probability integral transform (PIT) residuals - for a model that fits well these should be approximately standard uniform values evenly scattered between 0 and 1. Their calculation involves some randomisation, so different fits will return slightly different values for PIT residuals.
sqrt.1_Hii
the matrix of scale terms used to standardize the Pearson reidusals.
var.estimator
the estimated variance of each observation, computed using the corresponding family function.
sqrt.weight
the matrix of square root of working weights, estimated for the corresponding family function.
theta
the estimated nuisance parameters accounting for overdispersion
two.loglike
two times the log likelihood.
deviance
up to a constant, minus twice the maximized log-likelihood. Where sensible, the constant is chosen so that a saturated model has deviance zero.
a vector returning Akaike's An Information Criterion for each response variable - minus twice the
maximized log-likelihood plus twice the number of coefficients.
AICsum
the sum of the AIC's over all variables.
shrink.param
the shrink parameter to be used in subsequent inference.
call
the matched call.
terms
the terms object used.
rank
the numeric rank of the fitted linear model.
xlevels
(where relevant) a record of the levels of the factors used in fitting.
df.residual
the residual degrees of freedom.
x
if the argument x is TRUE, this is the design matrix used.
y
if the argument y is TRUE, this is the response variables used.
model
if the argument model is TRUE, this is the model.frame.
qr
if the argument qr is TRUE, this is the QR decomposition of the design matrix.
show.coef,show.fitted,show.residuals
arguments supplied in the manyglm call concerning what it presented in output.
offset
the offset data used (where applicable).
Author(s)
Yi Wang, Ulrike Naumann and David Warton <David.Warton@unsw.edu.au>.
References
Lawless, J. F. (1987)
Negative binomial and mixed Poisson regression,
Canadian Journal of Statistics 15, 209-225.
Hilbe, J. M. (2008)
Negative Binomial Regression,
Cambridge University Press, Cambridge.
Warton D.I. (2005)
Many zeros does not mean zero inflation: comparing the
goodness of-fit of parametric models to multivariate abundance data,
Environmetrics 16(3), 275-289.
Warton D.I. (2008). Penalized normal likelihood and ridge regularization
of correlation and covariance matrices. Journal of the American
Statistical Association 103, 340-349.
Warton D.I. (2011). Regularized sandwich estimators for analysis of high dimensional data using generalized estimating equations. Biometrics, 67(1), 116-123.
Warton D. I., Wright S., and Wang, Y. (2012). Distance-based multivariate analyses confound location and dispersion effects. Methods in Ecology and Evolution, 3(1), 89-101.
Wang Y., Neuman U., Wright S. and Warton D. I. (2012). mvabund: an R package for model-based analysis of multivariate abundance data. Methods in Ecology and Evolution. online 21 Feb 2012.
data(spider)
spiddat <- mvabund(spider$abund)
X <- spider$x
#To fit a log-linear model assuming counts are poisson:
glm.spid <- manyglm(spiddat~X, family="poisson")
glm.spid
summary(glm.spid, resamp="residual")
#To fit a binomial regression model to presence/absence data:
pres.abs <- spiddat
pres.abs[pres.abs>0] = 1
X <- data.frame(spider$x) #turn into a data frame to refer to variables in formula
glm.spid.bin <- manyglm(pres.abs~soil.dry+bare.sand+moss, data=X, family="binomial")
glm.spid.bin
drop1(glm.spid.bin) #AICs for one-term deletions, suggests dropping bare.sand
glm2.spid.bin <- manyglm(pres.abs~soil.dry+moss, data=X, family="binomial")
drop1(glm2.spid.bin) #backward elimination suggests settling on this model.