Fitting Generalized Linear Models for Multivariate Abundance Data

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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

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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.

show.coef, show.fitted, show.residuals, show.warning

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.

iter

the number of iterations of IWLS used.

data

a data frame storing the input data.

stderr.coefficients

the estimated standard error of each coefficient.

phi

the inverse of theta

tol

the tolerance used in estimations.

maxiter,maxiter2,family,theta.method,cor.type,formula

arguments supplied in the manyglm call.

aic

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.

See Also

anova.manyglm, summary.manyglm, residuals.manyglm, plot.manyglm

Examples

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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.

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