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DATA ANALYSIS USINGHIERARCHICAL GENERALIZED
LINEAR MODELS WITH R
Youngjo Lee, Lars Rönnegård & Maengseok Noh
What did John Nelder achieve in the 1970’s?
Gaussian
Poisson Binomial
Gamma
𝐿𝐿 = �𝑖𝑖=1
𝑛𝑛12𝜋𝜋𝜎𝜎
𝑒𝑒12𝜎𝜎2 𝑦𝑦𝑖𝑖−𝜇𝜇 2
𝜇𝜇 = 𝑋𝑋𝑋𝑋
𝐿𝐿 = �𝑖𝑖=1
𝑛𝑛𝜇𝜇𝑦𝑦𝑖𝑖𝑒𝑒−𝜇𝜇𝑦𝑦𝑖𝑖!
log(𝜇𝜇) = 𝑋𝑋𝑋𝑋
𝐿𝐿 = �𝑖𝑖=1
𝑛𝑛𝑛𝑛𝑦𝑦𝑖𝑖 𝜇𝜇𝑦𝑦𝑖𝑖(1 − 𝜇𝜇)𝑛𝑛−𝑦𝑦𝑖𝑖
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙(𝜇𝜇) = 𝑋𝑋𝑋𝑋
𝐿𝐿 = �𝑖𝑖=1
𝑛𝑛1
Γ(𝑘𝑘) 𝜃𝜃𝑘𝑘𝑦𝑦𝑖𝑖𝑘𝑘−1𝑒𝑒
−𝑦𝑦𝑖𝑖𝜃𝜃
k𝜃𝜃 ≡ 𝜇𝜇; 1/𝜇𝜇 = 𝑋𝑋𝑋𝑋
Gaussian
Poisson Binomial
Gamma
𝐿𝐿 = �𝑖𝑖=1
𝑛𝑛12𝜋𝜋𝜎𝜎
𝑒𝑒12𝜎𝜎2 𝑦𝑦𝑖𝑖−𝜇𝜇 2
𝜇𝜇 = 𝑋𝑋𝑋𝑋
𝐿𝐿 = �𝑖𝑖=1
𝑛𝑛𝜇𝜇𝑦𝑦𝑖𝑖𝑒𝑒−𝜇𝜇𝑦𝑦𝑖𝑖!
log(𝜇𝜇) = 𝑋𝑋𝑋𝑋
𝐿𝐿 = �𝑖𝑖=1
𝑛𝑛𝑛𝑛𝑦𝑦𝑖𝑖 𝜇𝜇𝑦𝑦𝑖𝑖(1 − 𝜇𝜇)𝑛𝑛−𝑦𝑦𝑖𝑖
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙(𝜇𝜇) = 𝑋𝑋𝑋𝑋
𝐿𝐿 = �𝑖𝑖=1
𝑛𝑛1
Γ(𝑘𝑘) 𝜃𝜃𝑘𝑘𝑦𝑦𝑖𝑖𝑘𝑘−1𝑒𝑒
−𝑦𝑦𝑖𝑖𝜃𝜃
k𝜃𝜃 ≡ 𝜇𝜇; 1/𝜇𝜇 = 𝑋𝑋𝑋𝑋
GLM
Gaussian
Poisson Binomial
Gamma
𝐿𝐿 = �𝑖𝑖=1
𝑛𝑛12𝜋𝜋𝜎𝜎
𝑒𝑒12𝜎𝜎2 𝑦𝑦𝑖𝑖−𝜇𝜇 2
𝜇𝜇 = 𝑋𝑋𝑋𝑋
𝐿𝐿 = �𝑖𝑖=1
𝑛𝑛𝜇𝜇𝑦𝑦𝑖𝑖𝑒𝑒−𝜇𝜇𝑦𝑦𝑖𝑖!
log(𝜇𝜇) = 𝑋𝑋𝑋𝑋
𝐿𝐿 = �𝑖𝑖=1
𝑛𝑛𝑛𝑛𝑦𝑦𝑖𝑖 𝜇𝜇𝑦𝑦𝑖𝑖(1 − 𝜇𝜇)𝑛𝑛−𝑦𝑦𝑖𝑖
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙(𝜇𝜇) = 𝑋𝑋𝑋𝑋
𝐿𝐿 = �𝑖𝑖=1
𝑛𝑛1
Γ(𝑘𝑘) 𝜃𝜃𝑘𝑘𝑦𝑦𝑖𝑖𝑘𝑘−1𝑒𝑒
−𝑦𝑦𝑖𝑖𝜃𝜃
k𝜃𝜃 ≡ 𝜇𝜇; 1/𝜇𝜇 = 𝑋𝑋𝑋𝑋
GLMCommon estimation algorithm using iterativeregression - Fast and easy to implement- Linear regression model checking tools!!!
Lee Y. & Nelder J. A. (1996) ”Hierarchicalgeneralized linear models” JRSS B 619-678
GLM approach for fitting• Linear mixed models• Generalized linear mixed models (Laplace approximation)
• Mixed models with non-Gaussian random effects• Above models + dispersion modelling
Coming out July 2017
Linear model
Generalized linear model (GLM)
Joint GLM Generalized linear model
including dispersion model with fixed effects
Linear mixed model (LMM)
Generalized linear mixedmodel (GLMM)
Generalized linear model including Gaussian random effects
Hierarchical GLM (HGLM)Generalized linear model including Gaussian
and/or non-Gaussian random effects. Dispersion can be modelled using fixed
effects.
Linear model
Generalized linear model (GLM)
Joint GLM Generalized linear model
including dispersion model with fixed effects
Linear mixed model (LMM)
Generalized linear mixedmodel (GLMM)
Generalized linear model including Gaussian random effects
Double HGLM (DHGLM)HGLM including dispersion model with both fixed and random effects
Frailty HGLMHGLMs for survival analysis including
competing risk models
Structural Equation Models (SEM)
Hierarchical GLM (HGLM)Generalized linear model including Gaussian
and/or non-Gaussian random effects. Dispersion can be modelled using fixed
effects.
HGLMs with correlated random effects
Including spatial, temporal correlations, splines, GAM.
Factor analysis
Why use HGLMs?
• Fast deterministic algorithms for complex models
• All parts of the model are checkable• Predictions of unobservables
Crack growth data
Hudak et al. (1978) presented data from an experiment where crack lengths are measured on a compact tension steel test.
• There are 21 metallic specimens with the crack lengths recorded every 104 cycles.
• y = increment of crack length• Covariate for the mean part of the model: crack length
previously recorded (crack0)• Covariate for the dispersion part of the model: number of cycles
(cycle)
## GLM ##res_glm <- glm(y ~ crack0, family= Gamma(link=log), data=data_crack_growth)
## GLM ##res_glm <- glm(y ~ crack0, family= Gamma(link=log), data=data_crack_growth)
library(hglm)## GLMM ##res_glmm1 <- hglm2(y ~ crack0 + (1|specimen), family = Gamma(link=log) , data = data_crack_growth )
## GLM ##res_glm <- glm(y ~ crack0, family= Gamma(link=log), data=data_crack_growth)
library(hglm)## GLMM ##res_glmm1 <- hglm2(y ~ crack0 + (1|specimen), family = Gamma(link=log) , data = data_crack_growth )
library(dhglm)## HGLM I ##model_mu <- DHGLMMODELING(Model="mean", Link="log",
LinPred = y ~ crack0 + (1|specimen), RandDist = "inverse-gamma")model_phi <- DHGLMMODELING(Model="dispersion")res_hglm1 <- dhglmfit(RespDist="gamma", DataMain=data_crack_growth,
MeanModel=model_mu, DispersionModel=model_phi)
## GLM ##res_glm <- glm(y ~ crack0, family= Gamma(link=log), data=data_crack_growth)
library(hglm)## GLMM ##res_glmm1 <- hglm2(y ~ crack0 + (1|specimen), family = Gamma(link=log) , data = data_crack_growth )
library(dhglm)## HGLM I ##model_mu <- DHGLMMODELING(Model="mean", Link="log",
LinPred = y ~ crack0 + (1|specimen), RandDist = "inverse-gamma")model_phi <- DHGLMMODELING(Model="dispersion")res_hglm1 <- dhglmfit(RespDist="gamma", DataMain=data_crack_growth,
MeanModel=model_mu, DispersionModel=model_phi)## HGLM II ##model_mu <- DHGLMMODELING(Model="mean", Link="log",
LinPred = y ~ crack0 + (1|specimen), RandDist="inverse-gamma")model_phi <- DHGLMMODELING(Model = "dispersion", Link = "log",
LinPred = phi ~ cycle)res_hglm2 <- dhglmfit(RespDist = "gamma", DataMain = data_crack_growth,
MeanModel = model_mu, DispersionModel = model_phi)
V(y)= 𝜙𝜙𝜇𝜇2log(𝜙𝜙)=Xb
## GLM ##res_glm <- glm(y ~ crack0, family= Gamma(link=log), data=data_crack_growth)
library(hglm)## GLMM ##res_glmm1 <- hglm2(y ~ crack0 + (1|specimen), family = Gamma(link=log) , data = data_crack_growth )
library(dhglm)## HGLM I ##model_mu <- DHGLMMODELING(Model="mean", Link="log",
LinPred = y ~ crack0 + (1|specimen), RandDist = "inverse-gamma")model_phi <- DHGLMMODELING(Model="dispersion")res_hglm1 <- dhglmfit(RespDist="gamma", DataMain=data_crack_growth,
MeanModel=model_mu, DispersionModel=model_phi)## HGLM II ##model_mu <- DHGLMMODELING(Model="mean", Link="log",
LinPred = y ~ crack0 + (1|specimen), RandDist="inverse-gamma")model_phi <- DHGLMMODELING(Model = "dispersion", Link = "log",
LinPred = phi ~ cycle)res_hglm2 <- dhglmfit(RespDist = "gamma", DataMain = data_crack_growth,
MeanModel = model_mu, DispersionModel = model_phi)## DHGLM I ##model_mu <- DHGLMMODELING(Model="mean", Link="log",
LinPred = y ~ crack0 + (1|specimen), RandDist="inverse-gamma")model_phi <- DHGLMMODELING(Model="dispersion", Link="log",
LinPred = phi ~ cycle + (1|specimen), RandDist="gaussian")res_dhglm1 <- dhglmfit(RespDist = "gamma", DataMain = data_crack_growth,
MeanModel = model_mu, DispersionModel = model_phi)
hglm1
hglm2
dhglm1
Use HGLMs!
• Fast deterministic algorithms for complex models
• All parts of the model are checkable• Predictions of unobservables
DATA ANALYSIS USINGHIERARCHICAL GENERALIZED
LINEAR MODELS WITH R
Lee, Rönnegård, Noh
www.larsronnegard.se
