Fitting generalized linear models using the mcglm package

library(knitr)
opts_chunk$set(
    dev.args=list(family="Palatino"))
options(width=68)

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")
Abstract

The mcglm package implements the multivariate covariance generalized linear models (McGLMs) proposed by Bonat and J$\o$rgensen (2016). In this introductory vignette we employed the mcglm package for fitting a set of generalized linear models and compare our results with the ones obtained by ordinary R functions like lm and glm.


Example 1 - Count data

Consider the count data obtained in Dobson (1990).

## Dobson (1990) Page 93: Randomized Controlled Trial :
counts <- c(18,17,15,20,10,20,25,13,12)
outcome <- gl(3,1,9)
treatment <- gl(3,3)
print(d.AD <- data.frame(treatment, outcome, counts))

Ordinary analysis using quasi-Poisson model.

fit.glm <- glm(counts ~ outcome + treatment, family = quasipoisson)

The orthodox quasi-Poisson model is obtained by specifying the variance function as tweedie and fix the power parameter at $1$. Since, we are dealing with independent data, the matrix linear predictor is composed of a diagonal matrix.

require(mcglm)
require(Matrix)
# Matrix linear predictor
Z0 <- mc_id(d.AD)
fit.qglm <- mcglm(linear_pred = c(counts ~ outcome + treatment),
                  matrix_pred = list("resp1" = Z0),
                  link = "log", variance = "tweedie", data = d.AD,
                  control_algorithm = list(verbose = FALSE,
                                           method = "chaser",
                                           tuning = 0.8))

Comparing regression parameter estimates and standard errors.

cbind("GLM" = coef(fit.glm),
      "McGLM" = coef(fit.qglm, type = "beta")$Estimates)

cbind("GLM" = sqrt(diag(vcov(fit.glm))), 
      "McGLM" = coef(fit.qglm, type = "beta", std.error = TRUE)$Std.error)

Example 2 - Continuous data with offset

Consider the example from Venables & Ripley (2002, p.189). The response variable is continuous for which we can assume the Gaussian distribution. In this example, we exemplify the use of the offset argument.

# Loading the data set
utils::data(anorexia, package = "MASS")

# GLM fit
anorex.1 <- glm(Postwt ~ Prewt + Treat + offset(Prewt),
               family = gaussian, data = anorexia)

# McGLM fit
Z0 <- mc_id(anorexia)
fit.anorexia <- mcglm(linear_pred = c(Postwt ~ Prewt + Treat),
                      matrix_pred = list(Z0),
                      offset = list(anorexia$Prewt),
                      power_fixed = TRUE, data = anorexia,
                      control_algorithm = list("correct" = TRUE))

Comparing parameter estimates and standard errors.

# Estimates
cbind("McGLM" = round(coef(fit.anorexia, type = "beta")$Estimates,5),
      "GLM" = round(coef(anorex.1),5))

# Standard errors
cbind("McGLM" = sqrt(diag(vcov(fit.anorexia))),
      "GLM" = c(sqrt(diag(vcov(anorex.1))),NA))

Example 3 - Continuous positive data

Consider the following data set from McCullagh & Nelder (1989, pp.300-2). It is an example of Gamma regression model.

clotting <- data.frame(
  u = c(5,10,15,20,30,40,60,80,100),
  lot1 = c(118,58,42,35,27,25,21,19,18),
  lot2 = c(69,35,26,21,18,16,13,12,12))

Analysis using generalized linear models glm function in R.

fit.lot1 <- glm(lot1 ~ log(u), data = clotting,
                family = Gamma(link = "inverse"))
fit.lot2 <- glm(lot2 ~ log(u), data = clotting,
                family = Gamma(link = "inverse"))

The code below exemplify how to use the control_initial argument for fixing the power parameter at $p = 2$.

list_initial = list()
list_initial$regression <- list(coef(fit.lot1))
list_initial$power <- list(c(2))
list_initial$tau <- list(summary(fit.lot1)$dispersion)
list_initial$rho = 0

The control_initial argument should be a named list with initial values for all parameters involved in the model. Note that, in this example we used the parameter estimates from the glm fit as initial values for the regression and dispersion parameters. The power parameter was fixed at $p = 2$, since we want to fit Gamma regression models. In that case, we have only one response variable, but the initial value for correlation parameter $\rho$ should be specified.

Z0 <- mc_id(clotting)
fit.lot1.mcglm <- mcglm(linear_pred = c(lot1 ~ log(u)),
                        matrix_pred = list(Z0),
                        link = "inverse", variance = "tweedie",
                        data = clotting,
                        control_initial = list_initial)

Comparing parameter estimates and standard errors.

# Estimates
cbind("mcglm" = round(coef(fit.lot1.mcglm, type = "beta")$Estimates,5),
      "glm" = round(coef(fit.lot1),5))
# Standard errors
cbind("mcglm" = sqrt(diag(vcov(fit.lot1.mcglm))),
      "glm" = c(sqrt(diag(vcov(fit.lot1))),NA))

Initial values for the response variable lot2

list_initial$regression <- list("resp1" = coef(fit.lot2))
list_initial$tau <- list("resp1" = c(var(1/clotting$lot2)))

Note that, since the list_initial object already have all components required, we just modify the entries regression and tau.

fit.lot2.mcglm <- mcglm(linear_pred = c(lot2 ~ log(u)),
                        matrix_pred = list(Z0),
                        link = "inverse", variance = "tweedie",
                        data = clotting,
                        control_initial = list_initial)

Comparing parameter estimates and standard errors.

# Estimates
cbind("mcglm" = round(coef(fit.lot2.mcglm, type = "beta")$Estimates,5),
      "glm" = round(coef(fit.lot2),5))
# Standard errors
cbind("mcglm" = sqrt(diag(vcov(fit.lot2.mcglm))),
      "glm" = c(sqrt(diag(vcov(fit.lot2))),NA))

The main contribution of the mcglmpackage is that it easily fits multivariate regression models. For example, for the clotting data a bivariate Gamma model is a suitable choice.

# Initial values
list_initial = list()
list_initial$regression <- list(coef(fit.lot1), coef(fit.lot2))
list_initial$power <- list(c(2),c(2))
list_initial$tau <- list(c(0.00149), c(0.001276))
list_initial$rho = 0.80

# Matrix linear predictor
Z0 <- mc_id(clotting)

# Fit bivariate Gamma model
fit.joint.mcglm <- mcglm(linear_pred = c(lot1 ~ log(u), lot2 ~ log(u)),
                         matrix_pred = list(Z0, Z0),
                         link = c("inverse", "inverse"),
                         variance = c("tweedie", "tweedie"),
                         data = clotting,
                         control_initial = list_initial,
                         control_algorithm = list("correct" = TRUE,
                                                 "method" = "chaser",
                                                 "tuning" = 0.1,
                                                 "max_iter" = 1000))
summary(fit.joint.mcglm)

We also can easily change the link function. The code below fit a bivariate Gamma model using the log link function.

# Initial values
list_initial = list()
list_initial$regression <- list(c(log(mean(clotting$lot1)),0),
                                c(log(mean(clotting$lot2)),0))
list_initial$power <- list(c(2), c(2))
list_initial$tau <- list(c(0.023), c(0.024))
list_initial$rho = 0

# Fit bivariate Gamma model
fit.joint.log <- mcglm(linear_pred = c(lot1 ~ log(u), lot2 ~ log(u)),
                       matrix_pred = list(Z0,Z0),
                       link = c("log", "log"),
                       variance = c("tweedie", "tweedie"),
                       data = clotting,
                       control_initial = list_initial)
summary(fit.joint.log)

Example 4 - Binomial data

Consider the example menarche from the MASS R package.

require(MASS)
data(menarche)
data <- data.frame("resp" = menarche$Menarche/menarche$Total,
                   "Ntrial" = menarche$Total,
                   "Age" = menarche$Age)

Logistic regression model.

glm.out = glm(cbind(Menarche, Total-Menarche) ~ Age,
              family=binomial(logit), data=menarche)

The same fitted by mcglm function.

# Matrix linear predictor
Z0 <- mc_id(data)
fit.logit <- mcglm(linear_pred = c(resp ~ Age),
                   matrix_pred = list(Z0),
                   link = "logit", variance = "binomialP",
                   Ntrial = list(data$Ntrial), data = data)

Comparing parameter estimates and standard errors.

# Estimates
cbind("GLM" = coef(glm.out),
      "McGLM" = coef(fit.logit, type = "beta")$Estimates)
# Standard error
cbind("GLM" = c(sqrt(diag(vcov(glm.out))),NA),
      "McGLM" =  sqrt(diag(vcov(fit.logit))))

We can estimate a more flexible model using the extended binomial variance function.

fit.logit.power <- mcglm(linear_pred = c(resp ~ Age),
                         matrix_pred = list(Z0),
                         link = "logit", variance = "binomialP",
                         Ntrial = list(data$Ntrial),
                         power_fixed = FALSE, data = data)
summary(fit.logit.power)


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mcglm documentation built on Sept. 16, 2022, 1:06 a.m.