Vector generalized linear model

In statistics, the class of vector generalized linear models was proposed to
enlarge the scope of models catered for by generalized linear models.
In particular, VGLMs allow for response variables outside the classical exponential family
and for more than one parameter. Each parameter can be transformed by a link function.
The VGLM framework is also large enough to naturally accommodate multiple responses; these are
several independent responses each coming from a particular statistical distribution with
possibly different parameter values.
Vector generalized linear models are described in detail in Yee.
The central algorithm adopted is the iteratively reweighted least squares method,
for maximum likelihood estimation of usually all the model parameters. In particular,
Fisher scoring is implemented by such, which, for most models,
uses the first and expected second derivatives of the log-likelihood function.

Motivation

GLMs essentially cover one-parameter models from the classical exponential family,
and include 3 of the most important statistical regression models:
the linear model, Poisson regression for counts, and logistic regression
for binary responses.
However, the exponential family is far too limiting for regular data analysis.
For example, for counts, zero-inflation, zero-truncation and overdispersion are regularly
encountered, and the makeshift adaptations made to the binomial and
Poisson models in the form of quasi-binomial and
quasi-Poisson can be argued as being ad hoc and unsatisfactory.
But the VGLM framework readily handles models such as
zero-inflated Poisson regression,
zero-altered Poisson regression,
positive-Poisson regression, and
negative binomial regression.
As another example, for the linear model,
the variance of a normal distribution is relegated
as a scale parameter and it is treated
often as a nuisance parameter.
But the VGLM framework allows the variance to be modelled using covariates.
As a whole, one can loosely think of VGLMs as GLMs that handle many models
outside the classical exponential family and are not restricted to estimating
a single mean.
During estimation,
rather than using weighted least squares
during IRLS, one uses generalized least squares to handle the
correlation between the M linear predictors.

Data and notation

We suppose that the response or outcome or the dependent variable,, are assumed to be generated from a particular distribution. Most distributions are univariate, so that, and an example of is the bivariate normal distribution.
Sometimes we write our data as
for. Each of the n observations are considered to be
independent.
Then.
The are known positive prior weights, and often.
The explanatory or independent variables are written,
or when i is needed, as.
Usually there is an intercept, in which case
or.
Actually, the VGLM framework allows for S responses, each of dimension.
In the above S = 1. Hence the dimension of is more generally. One handles S responses by code such
as vglm for S = 3.
To simplify things, most of this article has S = 1.

Model components

The VGLM usually consists of four elements:

Linear predictors

Each linear predictor is a quantity which incorporates
information about the independent variables into the model.
The symbol
denotes a linear predictor and a subscript j is used to denote the jth one.
It relates the jth parameter to the explanatory variables, and
is expressed as linear combinations
of unknown parameters
i.e., of regression coefficients.
The jth parameter,, of the distribution depends on the
independent variables, through
Let be the vector of
all the linear predictors..
Thus all the covariates comprising potentially affect all the parameters through the linear predictors. Later, we will allow the linear predictors to be generalized to additive predictors, which is the sum of smooth functions of each and each function is estimated from the data.

Link functions

Each link function provides the relationship between a linear predictor and a
parameter of the distribution.
There are many commonly used link functions, and their choice can be somewhat arbitrary. It makes sense to try to match the domain of the link function to
the range of the distribution's parameter value.
Notice above that the allows a different link function for each parameter.
They have similar properties as with generalized linear models, for example,
common link functions include the logit link for parameters in,
and the log link for positive parameters. The VGAM package has function identitylink for parameters that can assume both positive and negative values.

Constraint matrices

More generally, the VGLM framework allows for any linear constraints between the regression coefficients of each linear predictors. For example, we may want to set some to be equal to 0, or constraint some of them to be equal. We have
where the are the constraint matrices.
Each constraint matrix is known and prespecified, and has M rows, and between 1 and M columns. The elements of constraint matrices are finite-valued, and often they are just 0 or 1.
For example, the value 0 effectively omits that element while a 1 includes it.
It is common for some models to have a parallelism assumption, which means that
for, and
for some models, for too.
The special case when for
all is known as trivial constraints; all the
regression coefficients are estimated and are unrelated.
And is known as an intercept-only parameter
if the jth row of all the are equal to for, i.e., equals an intercept only. Intercept-only parameters are thus modelled as simply as possible, as a scalar.
The unknown parameters,,
are typically estimated by the method of maximum likelihood.
All the regression coefficients may be put into a matrix as follows:

The xij facility

With even more generally, one can allow the value of a variable
to have a different value for each.
For example, if each linear predictor is for a different time point then
one might have a time-varying covariate.
For example,
in discrete choice models, one has
conditional logit models,
nested logit models,
generalized logit models,
and the like, to distinguish between certain variants and
fit a multinomial logit model to, e.g., transport choices.
A variable such as cost differs depending on the choice, for example,
taxi is more expensive than bus, which is more expensive than walking.
The xij facility of VGAM allows one to
generalize
to.
The most general formula is
Here the is an optional offset; which translates
to be a matrix in practice.
The VGAM package has an xij argument that allows
the successive elements of the diagonal matrix to be inputted.

Software

Yee describes an R package
implementation in the
called VGAM.
Currently this software fits approximately 150 models/distributions.
The central modelling functions are vglm and vgam.
The family argument is assigned a VGAM family function,
e.g., family = negbinomial for negative binomial regression,
family = poissonff for Poisson regression,
family = propodds for the proportional odd model or
cumulative logit model for ordinal categorical regression.

Fitting

Maximum likelihood

We are maximizing a log-likelihood
where the are positive and known prior weights.
The maximum likelihood estimates can be found
using an iteratively reweighted least squares algorithm using
Fisher's scoring method, with updates of the form:
where is
the Fisher information matrix at iteration a.
It is also called the expected information matrix, or EIM.

VLM

For the computation, the model matrix constructed
from the RHS of the formula in vglm
and the constraint matrices are combined to form a big model matrix.
The IRLS is applied to this big X. This matrix is known as the VLM
matrix, since the vector linear model is the underlying least squares
problem being solved. A VLM is a weighted multivariate regression where the
variance-covariance matrix for each row of the response matrix is not
necessarily the same, and is known.
.
In particular, the VLM minimizes the weighted sum of squares
This quantity is minimized at each IRLS iteration.
The working responses are
where the are known as working weights or working weight matrices. They are symmetric and positive-definite. Using the EIM helps ensure that they are all positive-definite over much of the parameter space. In contrast, using Newton–Raphson would mean the observed information matrices would be used, and these tend to be positive-definite in a smaller subset of the parameter space.
Computationally, the Cholesky decomposition is used to invert the working weight matrices and to convert the overall generalized least squares problem into an ordinary least squares problem.

Examples

Generalized linear models

Of course, all generalized linear models are a special cases of VGLMs.
But we often estimate all parameters by full maximum likelihood estimation rather
than using the method of moments for the scale parameter.

Ordered categorical response

If the response variable is an ordinal measurement with M + 1 levels, then one may fit a model function of the form:
for
Different links g lead to proportional odds models or ordered probit models,
e.g., the VGAM family function cumulative assigns a probit link to the cumulative
probabilities, therefore this model is also called the cumulative probit model.
In general they are called cumulative link models.
For categorical and multinomial distributions, the fitted values are an -vector of probabilities, with the property that all probabilities add up to 1. Each probability indicates the likelihood of occurrence of one of the M + 1 possible values.

Unordered categorical response

If the response variable is a nominal measurement,
or the data do not satisfy the assumptions of an ordered model, then one may fit a model of the following form:
for The above link is sometimes called the multilogit link,
and the model is called the multinomial logit model.
It is common to choose the first or the last level of the response as the
reference or baseline group; the above uses the last level.
The VGAM family function multinomial fits the above model,
and it has an argument called refLevel that can be assigned
the level used for as the reference group.

Count data

Classical GLM theory performs Poisson regression for count data.
The link is typically the logarithm, which is known as the canonical link.
The variance function is proportional to the mean:
where the dispersion parameter is typically fixed at exactly one. When it is not, the resulting quasi-likelihood model is often described as Poisson with overdispersion, or quasi-Poisson; then is commonly estimated by the method-of-moments and as such,
confidence intervals for are difficult to obtain.
In contrast, VGLMs offer a much richer set of models to handle overdispersion with respect to the Poisson, e.g., the negative binomial distribution and several variants thereof. Another count regression model is the generalized Poisson distribution. Other possible models are the zeta distribution and the Zipf distribution.

Extensions

Reduced-rank vector generalized linear models

RR-VGLMs are VGLMs where a subset of
the B matrix is of a lower rank.
Without loss of generality, suppose that is a partition of the covariate vector. Then the part of the B matrix corresponding to is of the form where and
are thin matrices, e.g., vectors if the rank R = 1. RR-VGLMs potentially offer several advantages when applied to certain
models and data sets. Firstly, if M and p are large then the number of regression coefficients
that are estimated by VGLMs is large . Then RR-VGLMs can reduce the number of estimated regression coefficients enormously if R is low, e.g., R = 1
or R = 2. An example of a model where this is particularly useful is the RR-multinomial logit model, also known as the stereotype model.
Secondly,
is an R-vector of latent variables, and often these can be usefully interpreted.
If R = 1 then we can write
so that the latent variable comprises loadings on the explanatory variables.
It may be seen that RR-VGLMs take optimal linear combinations of the
and then a VGLM is fitted to the explanatory variables. Thirdly, a biplot can be produced if R = 2, and this allows the model to be visualized.
It can be shown that RR-VGLMs are simply VGLMs where the constraint matrices for
the variables in are unknown and to be estimated.
It then transpires that for
such variables.
RR-VGLMs can be estimated by an alternating algorithm which fixes
and estimates and then fixes and estimates, etc.
In practice, some uniqueness constraints are needed for
and/or. In VGAM, the rrvglm function uses corner constraints by default, which means that the top R rows of is set to. RR-VGLMs were proposed in 2003.

Two to one

A special case of RR-VGLMs is when R = 1 and M = 2. This is dimension reduction from 2 parameters to 1 parameter. Then it can be shown that
where elements and are estimated. Equivalently,
This formula provides a coupling of and. It induces a relationship between two parameters of a model that can be useful, e.g., for modelling a mean-variance relationship. Sometimes there is some choice of link functions, therefore it offers a little flexibility when coupling the two parameters, e.g., a logit, probit, cauchit or cloglog link for parameters in the unit interval. The above formula is particularly useful for the negative binomial distribution, so that the RR-NB has variance function
This has been called the NB-P variant by some authors. The and are estimated, and it is also possible to obtain approximate confidence intervals for them too.
Incidentally, several other useful NB variants can also be fitted, with the help of selecting the right combination of constraint matrices. For example, NB − 1, NB − 2, NB − H; see Yee and Table 11.3 of Yee.

RCIMs

The subclass of row-column interaction models
has also been proposed; these are a special type of RR-VGLM.
RCIMs apply only to a matrix Y response and there are
no explicit explanatory variables.
Instead, indicator variables for each row and column are explicitly set up, and an order-R
interaction of the form is allowed.
Special cases of this type of model include the Goodman RC association model
and the quasi-variances methodology as implemented by the qvcalc R package.
RCIMs can be defined as a RR-VGLM applied to Y with
For the Goodman RC association model, we have so that
if R = 0 then it is a Poisson regression fitted to a matrix of counts with row effects and column effects; this has a similar idea to a no-interaction two-way ANOVA model.
Another example of a RCIM is if is the identity link and the parameter is the median and the model corresponds to an asymmetric Laplace distribution; then a no-interaction RCIM is similar to a technique called median polish.
In VGAM, rcim and grc functions fit the above models.
And also Yee and Hadi
show that RCIMs can be used to fit unconstrained quadratic ordination
models to species data; this is an example of indirect gradient analysis in
ordination.

Vector generalized additive models

Vector generalized additive models are a major
extension to VGLMs in which the linear predictor is not restricted to be
linear in the covariates but is the
sum of smoothing functions applied to the :
where
These are M ''additive predictors.
Each smooth function is estimated from the data.
Thus VGLMs are model-driven while VGAMs are data-driven.
Currently, only smoothing splines are implemented in the VGAM package.
For M'' > 1 they are actually vector splines, which estimate the component functions
in simultaneously.
Of course, one could use regression splines with VGLMs.
The motivation behind VGAMs is similar to
that of
Hastie and Tibshirani
and
Wood.
VGAMs were proposed in 1996
Currently, work is being done to estimate VGAMs using P-splines
of Eilers and Marx
This allows for several advantages over using smoothing splines and vector backfitting, such as the
ability to perform automatic smoothing parameter selection easier.

Quadratic reduced-rank vector generalized linear models

These add on a quadratic in the latent variable to the RR-VGLM class.
The result is a bell-shaped curve can be fitted to each response, as
a function of the latent variable.
For R = 2, one has bell-shaped surfaces as a function of the 2
latent variables---somewhat similar to a
bivariate normal distribution.
Particular applications of QRR-VGLMs can be found in ecology,
in a field of multivariate analysis called ordination.
As a specific rank-1 example of a QRR-VGLM,
consider Poisson data with S species.
The model for Species s is the Poisson regression
for. The right-most parameterization which uses the symbols has particular ecological meaning, because they relate to the species abundance, optimum and tolerance respectively. For example, the tolerance is a measure of niche width, and a large value means that that species can live in a wide range of environments. In the above equation, one would need in order
to obtain a bell-shaped curve.
QRR-VGLMs fit Gaussian ordination models by maximum likelihood estimation, and
they are an example of direct gradient analysis.
The cqo function in the VGAM package currently
calls optim to search for the optimal
, and given that, it is easy to calculate
the site scores and fit a suitable generalized linear model to that.
The function is named after the acronym CQO, which stands for
constrained quadratic ordination: the constrained is for direct
gradient analysis and the quadratic is for the
quadratic form in the latent variables
on the scale.
Unfortunately QRR-VGLMs are sensitive to outliers in both the response
and explanatory variables, as well as being computationally expensive, and
may give a local solution rather than a global solution.
QRR-VGLMs were proposed in 2004.