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))
##   treatment outcome counts
## 1         1       1     18
## 2         1       2     17
## 3         1       3     15
## 4         2       1     20
## 5         2       2     10
## 6         2       3     20
## 7         3       1     25
## 8         3       2     13
## 9         3       3     12

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))
## Automatic initial values selected.

Comparing regression parameter estimates and standard errors.

cbind("GLM" = coef(fit.glm),
      "McGLM" = coef(fit.qglm, type = "beta")$Estimates)
##                       GLM         McGLM
## (Intercept)  3.044522e+00  3.044522e+00
## outcome2    -4.542553e-01 -4.542553e-01
## outcome3    -2.929871e-01 -2.929871e-01
## treatment2   1.337909e-15 -1.981259e-16
## treatment3   1.421085e-15 -6.040927e-17
cbind("GLM" = sqrt(diag(vcov(fit.glm))), 
      "McGLM" = coef(fit.qglm, type = "beta", std.error = TRUE)$Std.error)
##                   GLM     McGLM
## (Intercept) 0.1943517 0.1943517
## outcome2    0.2299154 0.2299154
## outcome3    0.2191931 0.2191931
## treatment2  0.2274467 0.2274467
## treatment3  0.2274467 0.2274467

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))
## Automatic initial values selected.

Comparing parameter estimates and standard errors.

# Estimates
cbind("McGLM" = round(coef(fit.anorexia, type = "beta")$Estimates,5),
      "GLM" = round(coef(anorex.1),5))
##                McGLM      GLM
## (Intercept) 49.77111 49.77111
## Prewt       -0.56554 -0.56554
## TreatCont   -4.09707 -4.09707
## TreatFT      4.56306  4.56306
# Standard errors
cbind("McGLM" = sqrt(diag(vcov(fit.anorexia))),
      "GLM" = c(sqrt(diag(vcov(anorex.1))),NA))
##                  McGLM        GLM
## (Intercept) 13.3909581 13.3909581
## Prewt        0.1611824  0.1611824
## TreatCont    1.8934926  1.8934926
## TreatFT      2.1333359  2.1333359
##              6.1832255         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))
##                mcglm      glm
## (Intercept) -0.01655 -0.01655
## log(u)       0.01534  0.01534
# Standard errors
cbind("mcglm" = sqrt(diag(vcov(fit.lot1.mcglm))),
      "glm" = c(sqrt(diag(vcov(fit.lot1))),NA))
##                    mcglm          glm
## (Intercept) 0.0009275501 0.0009275466
## log(u)      0.0004149601 0.0004149596
##             0.0005340030           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))
##                mcglm      glm
## (Intercept) -0.02391 -0.02391
## log(u)       0.02360  0.02360
# Standard errors
cbind("mcglm" = sqrt(diag(vcov(fit.lot2.mcglm))),
      "glm" = c(sqrt(diag(vcov(fit.lot2))),NA))
##                    mcglm          glm
## (Intercept) 0.0013215825 0.0013264571
## log(u)      0.0005746644 0.0005767841
##             0.0002649003           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)
## Call: lot1 ~ log(u)
## 
## Link function: inverse
## Variance function: tweedie
## Covariance function: identity
## Regression:
##               Estimates    Std.error   Z value      Pr(>|z|)
## (Intercept) -0.01655946 0.0009226782 -17.94716  5.050529e-72
## log(u)       0.01534500 0.0004128619  37.16739 2.296479e-302
## 
## Dispersion:
##     Estimates    Std.error  Z value     Pr(>|z|)
## 1 0.002422623 0.0005447457 4.447255 8.697457e-06
## 
## Call: lot2 ~ log(u)
## 
## Link function: inverse
## Variance function: tweedie
## Covariance function: identity
## Regression:
##               Estimates    Std.error   Z value     Pr(>|z|)
## (Intercept) -0.02385457 0.0013224002 -18.03884 9.654067e-73
## log(u)       0.02357948 0.0005747444  41.02603 0.000000e+00
## 
## Dispersion:
##     Estimates    Std.error  Z value     Pr(>|z|)
## 1 0.001800727 0.0002773475 6.492676 8.432517e-11
## 
## Correlation matrix:
##   Parameters Estimates Std.error Z value Pr(>|z|)
## 1      rho12 0.7262051       NaN     NaN      NaN
## 
## Algorithm: chaser
## Correction: TRUE
## Number iterations: 47

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)
## Call: lot1 ~ log(u)
## 
## Link function: log
## Variance function: tweedie
## Covariance function: identity
## Regression:
##              Estimates  Std.error   Z value      Pr(>|z|)
## (Intercept)  5.5032302 0.19030106  28.91855 6.979582e-184
## log(u)      -0.6019177 0.05530784 -10.88304  1.388474e-27
## 
## Dispersion:
##    Estimates   Std.error  Z value     Pr(>|z|)
## 1 0.02435442 0.004609548 5.283472 1.267584e-07
## 
## Call: lot2 ~ log(u)
## 
## Link function: log
## Variance function: tweedie
## Covariance function: identity
## Regression:
##              Estimates  Std.error   Z value      Pr(>|z|)
## (Intercept)  4.9187575 0.18554086  26.51037 7.359390e-155
## log(u)      -0.5674356 0.05392437 -10.52280  6.782394e-26
## 
## Dispersion:
##    Estimates  Std.error  Z value     Pr(>|z|)
## 1 0.02315125 0.00491656 4.708832 2.491407e-06
## 
## Correlation matrix:
##   Parameters Estimates Std.error Z value Pr(>|z|)
## 1      rho12 0.9807089       NaN     NaN      NaN
## 
## Algorithm: chaser
## Correction: TRUE
## Number iterations: 11

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)
## Automatic initial values selected.

Comparing parameter estimates and standard errors.

# Estimates
cbind("GLM" = coef(glm.out),
      "McGLM" = coef(fit.logit, type = "beta")$Estimates)
##                    GLM      McGLM
## (Intercept) -21.226395 -21.226395
## Age           1.631968   1.631968
# Standard error
cbind("GLM" = c(sqrt(diag(vcov(glm.out))),NA),
      "McGLM" =  sqrt(diag(vcov(fit.logit))))
##                    GLM      McGLM
## (Intercept) 0.77068466 0.75151473
## Age         0.05895308 0.05748669
##                     NA 0.09310035

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)
## Automatic initial values selected.
summary(fit.logit.power)
## Call: resp ~ Age
## 
## Link function: logit
## Variance function: binomialP
## Covariance function: identity
## Regression:
##              Estimates  Std.error   Z value      Pr(>|z|)
## (Intercept) -21.378210 0.75681833 -28.24748 1.528506e-175
## Age           1.643005 0.05784811  28.40205 1.907701e-177
## 
## Power:
##   Estimates Std.error  Z value     Pr(>|z|)
## 1  1.071814 0.0599217 17.88691 1.491508e-71
## 
## Dispersion:
##   Estimates Std.error  Z value     Pr(>|z|)
## 1  1.199726 0.2431969 4.933146 8.091577e-07
## 
## Algorithm: chaser
## Correction: TRUE
## Number iterations: 9