vignettes/GLMExamples.Rmd
GLMExamples.RmdTo 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). 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.
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 12Ordinary 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.
## 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 -1.751522e-16
## treatment3 1.421085e-15 -6.040927e-17cbind("GLM" = sqrt(diag(vcov(fit.glm))), "McGLM" = coef(fit.qglm, type = "beta", std.error = TRUE)$Std.error)
## GLM McGLM
## (Intercept) 0.1943517 0.1943517
## outcome2 0.2299154 0.2299154
## outcome3 0.2191931 0.2191931
## treatment2 0.2274467 0.2274467
## treatment3 0.2274467 0.2274467Consider the example from Venables & Ripley (2002, p.189). The response variable is continuous for which we can assume the Gaussian distribution. In this example, we exemplify the use of the offset argument.
# Loading the data set utils::data(anorexia, package = "MASS") # GLM fit anorex.1 <- glm(Postwt ~ Prewt + Treat + offset(Prewt), family = gaussian, data = anorexia) # McGLM fit Z0 <- mc_id(anorexia) fit.anorexia <- mcglm(linear_pred = c(Postwt ~ Prewt + Treat), matrix_pred = list(Z0), offset = list(anorexia$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 NAConsider 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")$Estimates,5), "glm" = round(coef(fit.lot1),5))
## mcglm glm
## (Intercept) -0.01655 -0.01655
## log(u) 0.01534 0.01534# Standard errors cbind("mcglm" = sqrt(diag(vcov(fit.lot1.mcglm))), "glm" = c(sqrt(diag(vcov(fit.lot1))),NA))
## mcglm glm
## (Intercept) 0.0009275501 0.0009275466
## log(u) 0.0004149601 0.0004149596
## 0.0005340030 NAInitial values for the response variable lot2
list_initial$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")$Estimates,5), "glm" = round(coef(fit.lot2),5))
## mcglm glm
## (Intercept) -0.02391 -0.02391
## log(u) 0.02360 0.02360# Standard errors cbind("mcglm" = sqrt(diag(vcov(fit.lot2.mcglm))), "glm" = c(sqrt(diag(vcov(fit.lot2))),NA))
## mcglm glm
## (Intercept) 0.0013215825 0.0013264571
## log(u) 0.0005746644 0.0005767841
## 0.0002649003 NAThe main contribution of the mcglmpackage is that it easily fits multivariate regression models. For example, for the clotting data a bivariate Gamma model is a suitable choice.
# Initial values list_initial = list() list_initial$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$rho = 0.80 # Matrix linear predictor Z0 <- mc_id(clotting) # Fit bivariate Gamma model fit.joint.mcglm <- mcglm(linear_pred = c(lot1 ~ log(u), lot2 ~ log(u)), matrix_pred = list(Z0, Z0), link = c("inverse", "inverse"), variance = c("tweedie", "tweedie"), data = clotting, control_initial = list_initial, control_algorithm = list("correct" = TRUE, "method" = "chaser", "tuning" = 0.1, "max_iter" = 1000)) summary(fit.joint.mcglm)
## Call: lot1 ~ log(u)
##
## Link function: inverse
## Variance function: tweedie
## Covariance function: identity
## Regression:
## Estimates Std.error Z value Pr(>|z|)
## (Intercept) -0.01655946 0.0009226782 -17.94716 5.050529e-72
## log(u) 0.01534500 0.0004128619 37.16739 2.296479e-302
##
## Dispersion:
## Estimates Std.error Z value Pr(>|z|)
## 1 0.002422623 0.0005447457 4.447255 8.697457e-06
##
## Call: lot2 ~ log(u)
##
## Link function: inverse
## Variance function: tweedie
## Covariance function: identity
## Regression:
## Estimates Std.error Z value Pr(>|z|)
## (Intercept) -0.02385457 0.0013224002 -18.03884 9.654067e-73
## log(u) 0.02357948 0.0005747444 41.02603 0.000000e+00
##
## Dispersion:
## Estimates Std.error Z value Pr(>|z|)
## 1 0.001800727 0.0002773475 6.492676 8.432517e-11
##
## Correlation matrix:
## Parameters Estimates Std.error Z value Pr(>|z|)
## 1 rho12 0.7262051 NaN NaN NaN
##
## Algorithm: chaser
## Correction: TRUE
## Number iterations: 47We also can easily change the link function. The code below fit a bivariate Gamma model using the log link function.
# Initial values list_initial = list() list_initial$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$rho = 0 # Fit bivariate Gamma model fit.joint.log <- mcglm(linear_pred = c(lot1 ~ log(u), lot2 ~ log(u)), matrix_pred = list(Z0,Z0), link = c("log", "log"), variance = c("tweedie", "tweedie"), data = clotting, control_initial = list_initial) summary(fit.joint.log)
## Call: lot1 ~ log(u)
##
## Link function: log
## Variance function: tweedie
## Covariance function: identity
## Regression:
## Estimates Std.error Z value Pr(>|z|)
## (Intercept) 5.5032302 0.19030106 28.91855 6.979582e-184
## log(u) -0.6019177 0.05530784 -10.88304 1.388474e-27
##
## Dispersion:
## Estimates Std.error Z value Pr(>|z|)
## 1 0.02435442 0.004609548 5.283472 1.267584e-07
##
## Call: lot2 ~ log(u)
##
## Link function: log
## Variance function: tweedie
## Covariance function: identity
## Regression:
## Estimates Std.error Z value Pr(>|z|)
## (Intercept) 4.9187575 0.18554086 26.51037 7.359390e-155
## log(u) -0.5674356 0.05392437 -10.52280 6.782394e-26
##
## Dispersion:
## Estimates Std.error Z value Pr(>|z|)
## 1 0.02315125 0.00491656 4.708832 2.491407e-06
##
## Correlation matrix:
## Parameters Estimates Std.error Z value Pr(>|z|)
## 1 rho12 0.9807089 NaN NaN NaN
##
## Algorithm: chaser
## Correction: TRUE
## Number iterations: 11Consider the example menarche from the MASS R package.
require(MASS) data(menarche) data <- data.frame("resp" = menarche$Menarche/menarche$Total, "Ntrial" = menarche$Total, "Age" = menarche$Age)
Logistic regression model.
The same fitted by mcglm function.
# Matrix linear predictor Z0 <- mc_id(data) fit.logit <- mcglm(linear_pred = c(resp ~ Age), matrix_pred = list(Z0), link = "logit", variance = "binomialP", Ntrial = list(data$Ntrial), data = data)
## Automatic initial values selected.Comparing parameter estimates and standard errors.
## 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 0.09310035We 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