Example10_MeasurementError.Rmd

---
title: "Measurement Error Models"
author: "coreysparks"
date: "November 5, 2014"
output: html_document
---

In this example, I will illustrate the use of the Berkson measurement error model on data from the BRFSS from Texas. 

Traditionally, we only assume error in our outcome variables: e.g.
$y_i ~ \alpha + \beta *x_i + e_i$

But often, both y and x have errors in their measurement. This leads to a joint model for both x and y. Two basic forms of this include the Berkson and classical error models.

The Berkson model represents the situation where the observed predictors are less variable than their actual underlying true covariates, such as if you have a covariate that is recorded in intervals (income quartiles), vs the true continuous predictor (income in dollars). The former is less variable that the latter. 

The classical model uses a latent variable specificaiton for the true underlying covariate. This is equivalent to stating that a variable *x* is an imperfect measure of the **true** covariate *z*.

We will use the BRFSS data for the state of Texas for our example, and use BMI as a continous outcome, and obesity status outcome (BMI >= 30) as a dichotomous outcome.
To examine the Berkson model, I will use a categorized measure of poverty at the county level.
To examine the classical error model, I will use the ACS estimates of the errors in the poverty rate estimates directly. 

First we load our data and recode some variables:
```{r}
library(rjags)
library(dplyr)
library(car)
library(lme4)
load("~/Google Drive/dem7903_App_Hier/data/brfss_11.Rdata")
nams<-names(brfss_11)
newnames<-gsub("_", "", nams)
names(brfss_11)<-tolower(newnames)
brfss_11$statefip<-sprintf("%02d", brfss_11$state )
brfss_11$cofip<-sprintf("%03d", brfss_11$cnty )
brfss_11$cofips<-paste(brfss_11$statefip, brfss_11$cofip, sep="")
brfss_11$obese<-ifelse(brfss_11$bmi5/100 >=30, 1,0)
brfss_11$black<-recode(brfss_11$racegr2, recodes="2=1; 9=NA; else=0", as.factor.result=F)
brfss_11$white<-recode(brfss_11$racegr2, recodes="1=1; 9=NA; else=0", as.factor.result=F)
brfss_11$other<-recode(brfss_11$racegr2, recodes="3:4=1; 9=NA; else=0", as.factor.result=F)
brfss_11$hispanic<-recode(brfss_11$racegr2, recodes="5=1; 9=NA; else=0", as.factor.result=F)
#education level
brfss_11$lths<-recode(brfss_11$educa, recodes="1:3=1;9=NA; else=0", as.factor.result=F)
brfss_11$coll<-recode(brfss_11$educa, recodes="5:6=1;9=NA; else=0", as.factor.result=F)
brfss_11$agez<-scale(brfss_11$age, center=T, scale=T)
brfss_11$bmiz<-scale(brfss_11$bmi5/100, center=T, scale=T)
brfss_11$lowinc<-recode(brfss_11$incomg, recodes = "1:3=1; 4:5=0; else=NA")


#Read in ACS poverty rates at the county level
#poverty is measured with error, and we know the error in this case
acsecon<-read.csv("~/Google Drive/dem7903_App_Hier/data/aff_download/ACS_10_5YR_DP03_with_ann.csv")
acsecon$povrate<-acsecon[, "HC03_VC156"]
acsecon$poverr<-acsecon[, "HC04_VC156"]

#the next way I do it is the classical way, where you have a predictor that is given in ranges, not actual values.
#So I make grouped poverty rates using quantiles
quantile(acsecon$povrate, p=c(0, .25, .5, .75, 1))
acsecon$pov_cut<-recode(acsecon$povrate, recodes = "0:7.4=1; 7.41:10.4=2; 10.41:14.1=3; 14.11:44.9=4")

acsecon$cofips<-substr(acsecon$GEO.id, 10,14)
acsecon<-acsecon[, c("cofips", "povrate", "poverr", "pov_cut")]
head(acsecon)
#and merge the data back to the brfss data
merged<-merge(x=brfss_11, y=acsecon, by.x="cofips", by.y="cofips", all.x=T)


```

Next, I subset the data to select the observations from Texas.
```{r}
tx<-subset(merged, subset = state=="48"& is.na(bmiz)==F& is.na(black)==F& is.na(lths)==F& is.na(lowinc)==F)
nwncos<-table(tx$cofips)
nwncos #Number of people within counties
tx$conum<-rep(1:length(unique(tx$cofips)), nwncos[nwncos!=0])
length(unique(tx$conum)) #Number of counties
```



## Model specifications

The first model will ignore the errors in the predictor.
```{r}
model1<-"
model{

  for (i in 1:n){
      bmi[i]~dnorm(mu[i], tau)
      mu[i]<-b[1]*black[i]+b[2]*hisp[i]+b[3]*other[i]+b[4]*lths[i]+b[5]*coll[i]+u[cofips[i]]
  }
for (j in 1:ncos)
  {
    u[j]~dnorm(muu[j], tau_u)
    muu[j]<-u0+gam*(pov[j]-mean(pov[]))
  }
#priors
u0~dnorm(0, .01)
for(j in 1:5) { b[j]~dnorm(0, .01)}
gam~dnorm(0, .01)
tau<-pow(sd, -2)
sd~dunif(0,100)

tau_u<-pow(sd_u, -2)
sd_u~dunif(0,100)

}

"

#get initial values from a linear model
b<-as.numeric(coef(lm(bmiz~black+hispanic+other+lths+coll+scale(povrate, center=T), tx)))
b

#make the data
dat<-list(bmi=as.numeric(tx$bmiz), obese=tx$obese, black=tx$black, hisp=tx$hispanic, other=tx$other, lths=tx$lths, coll=tx$coll, age=as.numeric(tx$agez), lowinc=tx$lowinc, pov=as.numeric(tapply(tx$povrate, tx$cofips, mean)), n=length(tx$bmiz),cofips=tx$conum, ncos=length(unique(tx$cofips)))

init.rng1<-list(".RNG.seed" = 1234, ".RNG.name" = "base::Mersenne-Twister", u0=b[1], b=b[2:6], gam=b[7])
init.rng2<-list(".RNG.seed" = 5678, ".RNG.name" = "base::Mersenne-Twister", u0=b[1], b=b[2:6], gam=b[7])

#start the model
mod1<-jags.model(file=textConnection(model1), data=dat,inits =list(init.rng1, init.rng2) , n.chains=2)

#next, we update the model, this is the "burn in" period
update(mod1, 20000)

#sample samples from each chain, thinning every 25th iteration
samps<-coda.samples(mod1, variable.names=c("u0", "gam", "b", "sd", "sd_u"), n.iter=2000, n.thin=25)
effectiveSize(samps)
gelman.diag(samps, multivariate = F)

#Numerical summary of each parameter, here I also include the 90% credible interval:
summary(samps, quantiles =  c(.025, .975))

dic.samples(mod1, n.iter = 1000, type = "pD")

```


## Berkson Model
Here, I have error in the poverty rate because the rate has been measured in an interval fashion, instead of a continuous fashion. One thing you may encounter in this type of situation is where you have a calibration estimate for the intervaled data, such as a regression of the true value on the intervaled value, so you know the parameters from such an equation. I estimate these calibration values below using a linear model of the intervaled estimate on the true estimate.


```{r}
model2<-"
model{

  for (i in 1:n){
      bmi[i]~dnorm(mu[i], tau)
      mu[i]<-b[1]*black[i]+b[2]*hisp[i]+b[3]*other[i]+b[4]*lths[i]+b[5]*coll[i]+u[cofips[i]]
      
  }
for (j in 1:ncos)
  {
     u[j]~dnorm(muu[j], tau_u)
    muu[j]<-u0+gam*(z[j])
    z[j]~dnorm(muz[j], tauz)
    muz[j]<-alphaz+bz*povcut[j]
  }
#priors
for(j in 1:5) { b[j]~dnorm(0, .01)}
gam~dnorm(0,.01)
u0~dnorm(0, .0001)

tau<-pow(sd, -2)
sd~dunif(0,100)

tau_u<-pow(sd_u, -2)
sd_u~dunif(0,100)

}
"
#get calibration values for error model. 
summary(lm(povrate~pov_cut, tx))

#Make the data
dat<-list(bmi=as.numeric(tx$bmiz), obese=tx$obese, black=tx$black, hisp=tx$hispanic, other=tx$other, lths=tx$lths, coll=tx$coll, age=as.numeric(tx$agez), lowinc=tx$lowinc, povcut=tapply(tx$pov_cut, tx$cofips, median ), n=length(tx$bmiz),cofips=tx$conum, ncos=length(unique(tx$cofips)), alphaz=3.367, bz=3.094, tauz=1/(.9128^2))

#get initial values

b<-as.numeric(fixef(lmer(bmiz~black+hispanic+other+lths+coll+pov_cut+(1|cofips), tx)))
b
VarCorr(lmer(bmiz~black+hispanic+other+lths+coll+pov_cut+(1|cofips), tx))

#initialize
init.rng1<-list( b=b[2:6], u0=b[1], gam=b[7], sd=1.001549, sd_u=.046201)
init.rng2<-list( b=b[2:6], u0=b[1], gam=b[7], sd=1.001549, sd_u=.046201)

#Start the model
mod2<-jags.model(file=textConnection(model2), data=dat,inits =list(init.rng1, init.rng2) , n.chains=2)

#next, we update the model, this is the "burn in" period
update(mod2, 30000)

#sample samples from each chain, thinning every 25th iteration
samps2<-coda.samples(mod2, variable.names=c("u0", "b","gam", "sd", "sd_u", "z", "sd", "sd_u"), n.iter=2000, n.thin=25)
effectiveSize(samps2)
gelman.diag(samps2, multivariate = F)

#Numerical summary of each parameter, here I also include the 90% credible interval:
summary(samps2, quantiles =  c(.025, .975))

dic.samples(mod2, n.iter = 1000, type = "pD")

```




## classical Model
Here the poverty rate is treated as a latent variable with known error (from the ACS margin of error). Alternatively, you could put a prior on the latent variable variance. 

```{r}
model3<-"
model{

  for (i in 1:n){
      bmi[i]~dnorm(mu[i], tau)
      mu[i]<-b[1]*black[i]+b[2]*hisp[i]+b[3]*other[i]+b[4]*lths[i]+b[5]*coll[i]+u[cofips[i]]
          
  }

for (j in 1:ncos)
  {
    u[j]~dnorm(muu[j], tau_u)
    muu[j]<-u0+gam*(povtrue[j]-mean(povtrue[]))
    povrate[j]~dnorm(povtrue[j],taupov[j] )
    povtrue[j]~dnorm(0, taup)T(0,100)
    taupov[j]<-pow(poverr[j],-2)
  }
#priors
u0~dnorm(0, .01)
for(j in 1:5) { b[j]~dnorm(0, .01)}
gam~dnorm(0, .01)

tau<-pow(sd, -2)
sd~dunif(0,100)

tau_u<-pow(sd_u, -2)
sd_u~dunif(0,100)

taup~dgamma(.01, .01)
}
"

dat<-list(bmi=as.numeric(tx$bmiz), obese=tx$obese, black=tx$black, hisp=tx$hispanic, other=tx$other, lths=tx$lths, coll=tx$coll, age=as.numeric(tx$agez), lowinc=tx$lowinc, povrate=as.numeric(tapply(tx$povrate, tx$cofips, mean)), poverr=tapply(tx$poverr/1.645, tx$cofips, mean), n=length(tx$bmiz),cofips=tx$conum, ncos=length(unique(tx$cofips)))

#get initial values

b<-as.numeric(fixef(lmer(bmiz~black+hispanic+other+lths+coll+scale(povrate, center=T)+(1|cofips), tx)))
b
VarCorr(lmer(bmiz~black+hispanic+other+lths+coll+scale(povrate, center=T)+(1|cofips), tx))


init.rng1<-list( u0=b[1], b=b[2:6], gam=b[7])
init.rng2<-list( u0=b[1], b=b[2:6], gam=b[7])
mod3<-jags.model(file=textConnection(model3), data=dat,inits =list(init.rng1, init.rng2) , n.chains=2)

#next, we update the model, this is the "burn in" period
update(mod3, 20000)

#sample 1000 samples from each chain, thinning every 25th iteration
samps3<-coda.samples(mod3, variable.names=c("u0", "b","gam", "sd", "sd_u", "povtrue", "povrate"), n.iter=2000, n.thin=25)
effectiveSize(samps3)
gelman.diag(samps3, multivariate = F)

#Numerical summary of each parameter:
summary(samps3, quantiles =  c(.025, .975))
dat$povrate
dat$poverr

dic.samples(mod3, n.iter = 1000, type = "pD")

```


In this case, the first model, without any measurement error fits best, followed by the berkson model, then the classical model, using the DIC.