To install the stable version of mcglm, use
devtools::install_git(). For more information, visit mcglm/README.
library(devtools)
install_git("wbonat/mcglm")
library(mcglm)
packageVersion("mcglm")
The mcglm package implements the multivariate covariance generalized
linear models (McGLMs) proposed by Bonat and J$\o$rgensen (2016).
The core fit function mcglm is employed for fitting a set of models.
In this introductory vignette we restrict ourselves to model
independent data, although a simple model for longitudinal data analysis
in the Gaussian case is also presented. We present models to deal with
continuous, binomial/bounded and count univariate response variables.
We explore the specification of different link, variance and covariance
functions.
Consider a simple regression model, for univariate and independent Gaussian data: \[Y \sim N(X \beta, \tau_0 Z_0).\]
# Loading extra packages
require(mcglm)
require(Matrix)
require(mvtnorm)
require(tweedie)
# Setting the seed
set.seed(2503)
# Fixed component
x1 <- seq(-1,1, l = 100)
X <- model.matrix(~ x1)
mu1 <- mcglm::mc_link_function(beta = c(1,0.8), X = X, offset = NULL,
link = "identity")
# Random component
y1 <- rnorm(100, mu1$mu, sd = 0.5)
# Data structure
data <- data.frame("y1" = y1, "x1" = x1)
# Matrix linear predictor
Z0 <- mc_id(data)
# Fit
fit1.id <- mcglm(linear_pred = c(y1 ~ x1),
matrix_pred = list(Z0),
data = data)
## Automatic initial values selected.
The mcglm package offers the following set of S3-methods for
model summarize.
print(methods(class = "mcglm"))
## [1] anova coef confint fitted plot print
## [7] residuals summary vcov
## see '?methods' for accessing help and source code
The traditional summary for a fitted model can be obtained by
summary(fit1.id)
## Call: y1 ~ x1
##
## Link function: identity
## Variance function: constant
## Covariance function: identity
## Regression:
## Estimates Std.error Z value Pr(>|z|)
## (Intercept) 1.0029066 0.05383753 18.62839 1.891184e-77
## x1 0.9488496 0.09232145 10.27767 8.884812e-25
##
## Dispersion:
## Estimates Std.error Z value Pr(>|z|)
## 1 0.2898479 0.04137255 7.005803 2.455725e-12
##
## Algorithm: chaser
## Correction: TRUE
## Number iterations: 2
The function summary.mcglm was designed to be similar to the native
summary functions for the classes lm and glm. Extra features were
included to describe the dispersion structure. The mean formula call,
along with the the link, variance and covariance functions are
presented. The parameter estimates are presented in two blocks,
the first presents the regression estimates while the second presents
the dispersion estimates. In both cases the parameter estimates are
summarized by point estimates, standard errors, Z-values and p-values associated
with the Wald test whose null hypothesis is defined as \(\beta = 0\) and
\(\tau = 0\) for the mean and dispersion structures, respectively.
Finally, the selected algorithm, if the correction term is employed or
not and the number of iterations is printed.
The same linear regresion model can be fitted by using a different covariance link function, for example the inverse covariance link function.
# Fit using inverse covariance link function
fit1.inv <- mcglm(linear_pred = c(y1 ~ x1),
matrix_pred = list(Z0),
covariance = "inverse", data = data)
## Automatic initial values selected.
Furthermore, we can use a less conventional covariance link function, as the exponential-matrix.
# Fit using expm covariance link function
fit1.expm <- mcglm(linear_pred = c(y1 ~ x1),
matrix_pred = list(Z0),
covariance = "expm", data = data)
## Automatic initial values selected.
The function mcglm returns an object of mcglm class, for which we can
use the method coef to extract the parameters estimates.
# Comparing estimates using different covariance link functions
cbind(coef(fit1.id)$Estimates,
coef(fit1.inv)$Estimates,
coef(fit1.expm)$Estimates)
## [,1] [,2] [,3]
## [1,] 1.0029066 1.0029066 1.0029066
## [2,] 0.9488496 0.9488496 0.9488496
## [3,] 0.2898479 3.4500851 -1.2383989
# Applying the inverse transformation
c(coef(fit1.id)$Estimates[3],
1/coef(fit1.inv)$Estimates[3],
exp(coef(fit1.expm)$Estimates[3]))
## [1] 0.2898479 0.2898479 0.2898479
Consider an extension of the linear regression models to deal with heteroscedasticity:
\[ Y \sim N(X \beta, \tau_0 Z_0 + \tau_1 Z_1),\]
where \(Z_0\) is a identity matrix and \(Z_1\) is a diagonal matrix whose
elements are given by the values of a known covariate. Such a model,
can be fitted easily using the mcglm package.
# Mean model
set.seed(1811)
x1 <- seq(-1,1, l = 100)
X <- model.matrix(~ x1)
mu1 <- mcglm::mc_link_function(beta = c(1,0.8), X = X, offset = NULL,
link = "identity")
z1 <- rnorm(100, mean = 0, sd = 0.25)
data <- data.frame("id" = 1, "x1" = x1, "z1" = z1)
# Matrix linear predictor
Z <- mc_dglm(~ z1, id = 'id', data = data)
# Covariance model
Sigma <- mcglm::mc_matrix_linear_predictor(tau = c(0.2, 0.15), Z = Z)
# Simulating the response variable
y1 <- rnorm(100, mu1$mu, sd = sqrt(diag(Sigma)))
data$y <- y1
# Fitting
fit2.id <- mcglm(linear_pred = c(y1 ~ x1), matrix_pred = list(Z), data = data)
## Automatic initial values selected.
We can also extend the linear regression model to deal with longitudinal data analysis. The code below presents an example of such a model.
# Mean model
x1 <- seq(-1,1, l = 100)
X <- model.matrix(~ x1)
mu1 <- mcglm::mc_link_function(beta = c(1,0.8), X = X, offset = NULL,
link = "identity")
# Data structure
data <- data.frame("id" = as.factor(rep(1:10, each = 10)), "x1" = x1)
# Covariance model
Z0 <- mc_id(data)
Z1 <- mc_mixed(~ 0 + id, data = data)
Sigma <- mcglm::mc_matrix_linear_predictor(tau = c(0.2, 0.15), Z = c(Z0,Z1))
# Simulating the Response variable
y1 <- as.numeric(rmvnorm(1, mean = mu1$mu, sigma = as.matrix(Sigma)))
data <- data.frame("y1" = y1, "x1" = x1)
# Fit
fit3.id <- mcglm(linear_pred = c(y1 ~ x1),
matrix_pred = list("resp1" = c(Z0,Z1)), data = data)
## Automatic initial values selected.
The model summary
summary(fit3.id)
## Call: y1 ~ x1
##
## Link function: identity
## Variance function: constant
## Covariance function: identity
## Regression:
## Estimates Std.error Z value Pr(>|z|)
## (Intercept) 0.8547952 0.05768440 14.818482 1.112705e-49
## x1 0.6100739 0.09849275 6.194099 5.861920e-10
##
## Dispersion:
## Estimates Std.error Z value Pr(>|z|)
## 1 0.1775319 0.02291873 7.746151 9.471961e-15
## 2 0.0155217 0.01511449 1.026942 3.044476e-01
##
## Algorithm: chaser
## Correction: TRUE
## Number iterations: 5
Note that, the dispersion structure now has two parameters. In that case, the parameter \(\tau_1\) represents the longitudinal structure for which we are assuming a compound symmetry model. This model is an equivalent to a random intercept model in the context of Linear Mixed Models (LMMs).
The mcglm package offers a rich set of models to deal with binomial
and bounded response variables. The logit, probit, cauchit,
cloglog, and loglog link functions along with the extended binomial
variance function combined with the linear covariance structure,
provide a flexible class of models for handling binomial and
bounded response variables. The extended binomial variance function is
given by \(\mu^p (1- \mu)^q\) where the two extra power parameters
offer more flexibility to model the relationship between mean and
variance. Consider the following simulated dataset.
# Mean model
x1 <- seq(-1,1, l = 500)
X <- model.matrix(~ x1)
mu1 <- mcglm::mc_link_function(beta = c(1,0.8), X = X, offset = NULL, link = "logit")
# Data structure
data <- data.frame("x1" = x1)
# Covariance model
Z0 <- mc_id(data)
# Simulating the response variable
set.seed(123)
data$y <- rbinom(500, prob = mu1$mu, size = 10)/10
The most traditional regression model to deal with binomial data is the
logistic regression model that can be fitted using the mcglm package
using the following code:
# Fit
fit4.logit <- mcglm(linear_pred = c(y ~ x1),
matrix_pred = list(Z0),
link = "logit", variance = "binomialP",
power_fixed = TRUE,
Ntrial = list(rep(10,500)), data = data)
## Automatic initial values selected.
It is important to highlight that the response variable collumn should
be between \(0\) and \(1\) and in the case of more than one trial the
argument Ntrial should be used for fitting the model.
The argument link specifies the link function whereas the argument
variance specifies the variance function in that case binomialP.
The variance function binomialP represents a simplification of the
extended binomial variance function given by \(\mu^p (1- \mu)^p\).
Note that, in this example the argument 'power_fixed = TRUE' specifies
that the power parameter \(p\) will not be estimated, but fixed at the
initial value \(p = 1\) corresponding to the orthodox binomial variance
function.
We can easily fit the model using a different link function, for example
the cauchit.
fit4.cauchit <- mcglm(linear_pred = c(y ~ x1),
matrix_pred = list(Z0),
link = "cauchit", variance = "binomialP",
Ntrial = list(rep(10,250)), data = data)
## Automatic initial values selected.
We can also estimate the extra power parameter \(p\).
fit4.logitP <- mcglm(linear_pred = c(y ~ x1),
matrix_pred = list(Z0),
link = "logit", variance = "binomialP",
power_fixed = FALSE,
Ntrial = list(rep(10,500)), data = data)
## Automatic initial values selected.
Furthermore, we can estimate the two extra power parameters involved in the extended binomial variance function.
fit4.logitPQ <- mcglm(linear_pred = c(y ~ x1),
matrix_pred = list(Z0),
link = "logit", variance = "binomialPQ",
power_fixed = FALSE,
Ntrial = list(rep(10,500)),
control_algorithm = list(tuning = 0.5, max_iter = 100),
data = data)
## Automatic initial values selected.
The estimation of the extra power parameters involved in the
extended binomial variance function is challenging mainly for small data
sets. Note that, in this simulated example, we have to control the
step-length of the chaser algorithm to avoid unrealistic values for
the parameters involved in the dispersion structure. To do that, we used
the extra argument control_algorithm that should be a named list.
For a detailed description of the arguments that can be passed to the
control_algorithm function see ?fit_mcglm.
The analysis of count data in the mcglm package relies on the
structure of the Poisson-Tweedie distribution. Such a distribution is
characterized by the following dispersion function:
\[ \nu(\mu, p) = \mu + \tau_0 \mu^p. \] The power parameter is an index that identify different distributions, examples include the Hermite (\(p = 0\)), Neyman-Type A (\(p = 1\)) and the negative binomial (\(p = 2\)).
The orthodox Poisson model can be fitted using the mcglm package using
the Tweedie variance function \(\nu(\mu, p ) = \mu^p\) where the power
parameter \(p\) is fixed at \(1\). For example,
# Mean model
x1 <- seq(-2,2, l = 200)
X <- model.matrix(~ x1)
mu <- mcglm::mc_link_function(beta = c(1,0.8), X = X, offset = NULL, link = "log")
# Data structure
data <- data.frame("x1" = x1)
# Covariance model
Z0 <- mc_id(data)
# Data structure
data$y <- rpois(200, lambda = mu$mu)
# Fit
fit.poisson <- mcglm(linear_pred = c(y ~ x1),
matrix_pred = list(Z0),
link = "log", variance = "tweedie",
power_fixed = TRUE, data = data)
## Automatic initial values selected.
Another very useful model for count data is the negative binomial.
# Simulating negative binomial models
set.seed(1811)
x <- rtweedie(200, mu = mu$mu, power = 2, phi = 0.5)
y <- rpois(200, lambda = x)
data <- data.frame("y1" = y, "x1" = x1)
fit.pt <- mcglm(linear_pred = c(y ~ x1), matrix_pred = list(Z0),
link = "log", variance = "poisson_tweedie",
power_fixed = FALSE, data = data)
## Automatic initial values selected.
summary(fit.pt)
## Call: y ~ x1
##
## Link function: log
## Variance function: poisson_tweedie
## Covariance function: identity
## Regression:
## Estimates Std.error Z value Pr(>|z|)
## (Intercept) 0.8551973 0.07095214 12.05316 1.866611e-33
## x1 0.9285076 0.06336780 14.65267 1.295191e-48
##
## Power:
## Estimates Std.error Z value Pr(>|z|)
## 1 2.233986 0.3291973 6.78616 1.151575e-11
##
## Dispersion:
## Estimates Std.error Z value Pr(>|z|)
## 1 0.2827984 0.1906259 1.483525 0.1379349
##
## Algorithm: chaser
## Correction: TRUE
## Number iterations: 9
Note that, we estimate the power parameter rather than fix it at \(p = 2\).