added Week 2 materials

Week 2 -- mixed-effects/Lab 2/Introduction to GLMMs.R

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+
+setwd( "C:/Users/James.Thorson/Desktop/Project_git/2018_FSH556/Week 2 -- mixed-effects/Lab 2" )
+Use_REML = TRUE
+
+devtools::install_github("kaskr/TMB_contrib_R/TMBhelper")
+
+############
+# Generalized linear mixed model
+############
+library(lme4)
+
+###### Simulate data
+# Parameters
+Nsite = 10
+Nobs_per_site = 10
+Site_logMean = log(10)
+Site_logSd = 1
+
+# Bookkeeping
+s_i = rep( 1:Nsite, each=Nobs_per_site)
+
+# Simulation
+z_s = rnorm( Nsite, mean=0, sd=Site_logSd )
+Mean_s = exp( Site_logMean + z_s )
+y_i = rpois( Nsite*Nobs_per_site, lambda=Mean_s[s_i] )
+
+# Plot data
+library(lattice)
+histogram( ~ y_i | factor(s_i), breaks=seq( min(y_i), max(y_i), length=10), type="density", panel=function(x,...){ panel.histogram(x, ...); panel.mathdensity(dmath=dnorm, col="black", args = list(mean=mean(x),sd=sd(x))) } )      #
+
+###### Fit using R
+# No site level (Not recommended)
+GLM = glm( y_i ~ 1, family="poisson" )
+print( summary(GLM) )
+
+# Using fixed effects (Not recommended)
+GLM = glm( y_i ~ 0 + factor(s_i), family="poisson" )
+print( summary(GLM) )
+
+# Using mixed effects (Recommended) -- doesn't appear to use REML
+library(lme4)
+GLMM = glmer( y_i ~ 1 + (1 | factor(s_i)), family="poisson" )
+print( summary(GLMM) )
+
+####################
+# Fit using TMB
+####################
+library(TMB)
+
+# Compile model
+Version = "glmm"
+compile( paste0(Version,".cpp") )
+
+# Build inputs
+Data = list( "n_y"=length(y_i), "n_s"=length(unique(s_i)), "s_i"=s_i-1, "y_i"=y_i)
+Parameters = list( "beta0"=-10, "log_sdz"=2, "z_s"=rep(0,Data$n_s) )
+Random = c("z_s")
+if( Use_REML==TRUE ) Random = union( Random, "beta0")
+
+# Build object
+dyn.load( dynlib(Version) )
+Obj = MakeADFun(data=Data, parameters=Parameters, random=Random)  #
+
+# Prove that function and gradient calls work
+Obj$fn( Obj$par )
+Obj$gr( Obj$par )
+
+# Optimize
+Opt = TMBhelper::Optimize( obj=Obj, newtonsteps=1 )
+
+# Get reporting and SEs
+Report = Obj$report()
+ParHat = as.list( Opt$SD, "Estimate" )
+
+######################
+# Compare estimates
+######################
+
+# Global mean
+c( "Lme4"=fixef(GLMM), "TMB"=ParHat$beta0 )
+
+# Random effects
+cbind( "True"=z_s, "Lme4"=ranef(GLMM)[['factor(s_i)']], "TMB"=ParHat$z_s )
+
+

Week 2 -- mixed-effects/Lab 2/glmm.cpp

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+#include <TMB.hpp>
+template<class Type>
+Type objective_function<Type>::operator() ()
+{
+  // Data
+  DATA_INTEGER( n_y );
+  DATA_INTEGER( n_s );
+  DATA_IVECTOR( s_i );
+  DATA_VECTOR( y_i );
+
+  // Parameters
+  PARAMETER( beta0 );
+  PARAMETER( log_sdz );
+  PARAMETER_VECTOR( z_s );
+
+  // Objective funcction
+  Type jnll = 0;
+
+  // Probability of data conditional on fixed and random effect values
+  vector<Type> ypred_i(n_y);
+  for( int i=0; i<n_y; i++){
+    ypred_i(i) = exp( beta0 + z_s(s_i(i)) );
+    jnll -= dpois( y_i(i), ypred_i(i), true );
+  }
+
+  // Probability of random coefficients
+  for( int s=0; s<n_s; s++){
+    jnll -= dnorm( z_s(s), Type(0.0), exp(log_sdz), true );
+  }
+
+  // Reporting
+  Type sdz = exp(log_sdz);
+
+  REPORT( sdz );
+
+  ADREPORT( sdz );
+
+  return jnll;
+}

Week 2 -- mixed-effects/Lecture 2/linear_mixed_model.R

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+
+setwd( "C:/Users/James.Thorson/Desktop/Project_git/2018_FSH556/Week 2 -- mixed-effects/Lecture 2" )
+Use_REML = FALSE
+
+devtools::install_github("kaskr/TMB_contrib_R/TMBhelper")
+
+######################
+# Simulate data
+######################
+
+# Simulate predictors
+group_i = rep( 1:10, each=10)
+z_g = rnorm( length(unique(group_i)), mean=0, sd=1)
+beta0 = 0
+
+# Simulate response
+y_i = z_g[group_i] + beta0 + rnorm( length(group_i), mean=0, sd=1)
+
+######################
+# Run in R
+######################
+
+library(lme4)
+Lme = lmer( y_i ~ 1|factor(group_i), REML=Use_REML)
+
+######################
+# Run in TMB
+######################
+
+library(TMB)
+
+# Compile model
+Version = "linear_mixed_model"
+compile( paste0(Version,".cpp") )
+
+# Build inputs
+Data = list( "n_groups"=length(unique(group_i)), "g_i"=group_i-1, "y_i"=y_i)
+Parameters = list( "beta0"=-10, "log_SD0"=2, "log_SDZ"=2, "z_g"=rep(0,Data$n_groups) )
+Random = c("z_g")
+if( Use_REML==TRUE ) Random = union( Random, "beta0")
+
+# Build object
+dyn.load( dynlib("linear_mixed_model") )
+Obj = MakeADFun(data=Data, parameters=Parameters, random=Random)  #
+
+# Prove that function and gradient calls work
+Obj$fn( Obj$par )
+Obj$gr( Obj$par )
+
+# Optimize
+Opt = TMBhelper::Optimize( obj=Obj, newtonsteps=1 )
+
+# Get reporting and SEs
+Report = Obj$report()
+ParHat = as.list( Opt$SD, "Estimate" )
+
+######################
+# Shrinkage estimator
+######################
+
+# jim's attempt at replicating this from first principles
+Mu = mean(y_i)
+  Mu_s = tapply( y_i, INDEX=group_i, FUN=mean)
+Sigma = sd( Mu_s )
+  Sigma_s = sd( y_i - Mu_s[group_i] )
+Weights_hat = c( 1/Sigma^2, length(y_i)/length(unique(group_i))/Sigma_s^2 )
+  Weights_hat = Weights_hat / sum(Weights_hat)
+
+# Predictions
+Mu_s_hat = ( Mu*Weights_hat[1] + Mu_s*Weights_hat[2] )
+
+######################
+# Compare estimates
+######################
+
+# Global mean
+c( fixef(Lme), ParHat$beta0, Mu )
+
+# Random effects
+cbind( "True"=z_g, "Lme4"=ranef(Lme)[['factor(group_i)']], "TMB"=ParHat$z_g, "Shrinkage_estimator"=Mu_s-Mu )
+
+# Variances
+summary(Lme)
+unlist( Report[c("SDZ","SD0")] )
+

Week 2 -- mixed-effects/Lecture 2/linear_mixed_model.cpp

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+#include <TMB.hpp>
+template<class Type>
+Type objective_function<Type>::operator() ()
+{
+  // Data
+  DATA_INTEGER( n_groups );
+  DATA_IVECTOR( g_i );
+  DATA_VECTOR( y_i );
+
+  // Parameters
+  PARAMETER( beta0 );
+  PARAMETER( log_SD0 );
+  PARAMETER( log_SDZ );
+  PARAMETER_VECTOR( z_g );
+
+  // Objective funcction
+  Type jnll = 0;
+  int n_i = y_i.size();
+
+  // Probability of data conditional on fixed and random effect values
+  for( int i=0; i<n_i; i++){
+    jnll -= dnorm( y_i(i), beta0 + z_g(g_i(i)), exp(log_SD0), true );
+  }
+
+  // Probability of random coefficients
+  for( int g=0; g<n_groups; g++){
+    jnll -= dnorm( z_g(g), Type(0.0), exp(log_SDZ), true );
+  }
+
+  // Reporting
+  Type SDZ = exp(log_SDZ);
+  Type SD0 = exp(log_SD0);
+
+  REPORT( SDZ );
+  REPORT( SD0 );
+  ADREPORT( SDZ );
+  ADREPORT( SD0 );
+
+  return jnll;
+}