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bpm19psqf7375_Lecture13.Rmd
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---
title: 'Lecture 13: Missing Data via Bayesian'
author: "Bayesian Psychometric Modeling"
output: pdf_document
---
```{r setup}
# Install/Load Packages ===============================================================================================
if (!require(R2jags)) install.packages("R2jags")
library(R2jags)
if (!require(mcmcplots)) install.packages("mcmcplots")
library(mcmcplots)
```
## Bayesian Methods for Missing Data
Look up the word "missing" from the JAGS user manual (v. 4.3; available at (http://people.stat.sc.edu/hansont/stat740/jags_user_manual.pdf)[http://people.stat.sc.edu/hansont/stat740/jags_user_manual.pdf]) and you will find only three cases: two regarding how missing data should be input into JAGS and one talking about missing values in initialized parameters. Despite this, Bayesian methods missing data are some of the more powerful techniques around.
Let's take an example. Say we create:
```{r DataSimReg}
set.seed(007)
nObs = 1000
nMiss = 1
# for x
probX = .5
# for y|x
beta0 = 10
beta1 = 10
sigma2e = 1
# generate x, then y
x = rbinom(n = nObs, size = 1, prob = probX)
y = beta0 + beta1*x + rnorm(n = nObs, mean = 0, sd = sqrt(sigma2e))
# make one value of X missing
x[sample(x = 1:nObs, size = nMiss, replace = FALSE)] = NA
x[1]= NA
# show plot
plot(x = x, y = y)
```
Bayesian missing data methods use Bayes Theorem to "impute" values of missing data using specs from a simultaneous analysis. For instance, we could model:
$$ Prob(x_p = 1) = \pi_x, $$
where $x_p \sim B(\pi_x)$, simultaneous to:
$$ y_p = \beta_0 + \beta_1x_p + e_p, $$
where $y_p|x_p \sim N(\beta_0+\beta_1x_p, \sigma^2_e)$.
Here, should we have a missing $x_p$, we can use Bayes' theorem to determine the distribution of the observation:
$$f(x_p|y_p) \propto f(y_p|x_p)f(x_p), $$
Where $f(y_p|x_p)$ is our regression model and $f(x_p)$ is the distribution of $x_p$, or Bernoulli.
We can even make this work in JAGS:
```{r regjags}
model01.function = function(){
for (i in 1:N){
x[i] ~ dbern(pi)
y[i] ~ dnorm(beta0+beta1*x[i], sigma2.inv)
}
pi ~ dbeta(1,1)
sigma2.inv ~ dgamma(4, 8)
sigma2 <- 1/sigma2.inv
beta0 ~ dnorm(0, .00001)
beta1 ~ dnorm(0, .00001)
}
# next, create data for JAGS to use:
model01.data = list(
N = length(x),
x = x,
y = y
)
model01.parameters = c("beta0", "beta1", "sigma2", "pi", "x[1]")
model01.seed = 23042019
model01.r2jags = jags.parallel(
data = model01.data,
parameters.to.save = model01.parameters,
model.file = model01.function,
n.chains = 4,
n.iter = 2000,
n.thin = 1,
n.burnin = 1000,
jags.seed = model01.seed
)
model01.r2jags
denplot(model01.r2jags, parms = "beta1")
model01.r2jags$BUGSoutput$sims.matrix[,6]
```
Comparing this with just modeling $x_p$, we see a difference:
```{r model2reg}
model02.function = function(){
for (i in 1:N){
x[i] ~ dbern(pi)
y[i] ~ dnorm(beta0+beta1*x[i], sigma2.inv)
}
pi ~ dbeta(1,1)
sigma2.inv ~ dgamma(4, 8)
sigma2 <- 1/sigma2.inv
beta0 ~ dnorm(0, .00001)
beta1 <- 0
}
model02.r2jags = jags.parallel(
data = model01.data,
parameters.to.save = c("beta0", "beta1", "sigma2", "pi", "x[1]"),
model.file = model02.function,
n.chains = 4,
n.iter = 2000,
n.thin = 1,
n.burnin = 1000,
jags.seed = model01.seed+1
)
model02.r2jags
```
The difference here is with the posterior distribution of $x_p$. In model 1 (the simultaneous regression and marginal $x_p$ model -- given by setting $\beta_1 = 0$) the posterior distribution for $x_p$ involved $y_p$. As you'll note in the plot above, knowing $y_p$ tells us everything about $x_p$ as there is no overlap in $y_p$ across values of $x_p$.
```{r DataSimReg2}
set.seed(007)
nObs = 1000
nMiss = 400
# for x
meanX = 50
sdX = 100
# for y|x
beta0 = 0
beta1 = 0
beta2 = -100
sigma2e = 1
# generate x, then y
x = rnorm(n = nObs, mean = meanX, sd = sdX)
y = beta0 + beta1*x + beta2*x^2 + rnorm(n = nObs, mean = 0, sd = sqrt(sigma2e))
# make one value of X missing
x[sample(x = 1:nObs, size = nMiss, replace = FALSE)] = NA
x[1]= NA
# show plot
plot(x = x, y = y)
model01.function = function(){
for (i in 1:N){
x[i] ~ dnorm(meanX, sigma2.invX)
y[i] ~ dnorm(beta0+beta1*x[i], sigma2.inv)
}
sigma2.invX ~ dgamma(1000, 2)
sigma2.inv ~ dgamma(4, 8)
sigma2 <- 1/sigma2.inv
sigma2X <- 1/sigma2.inv
beta0 ~ dnorm(0, .00001)
meanX ~ dnorm(0, .00001)
beta1 ~ dnorm(0, .00001)
}
# next, create data for JAGS to use:
model01.data = list(
N = length(x),
x = x,
y = y
)
model01.parameters = c("beta0", "beta1", "sigma2", "sigma2X", "meanX", "x", "y")
model01.seed = 23042019
model01.r2jags = jags.parallel(
data = model01.data,
parameters.to.save = model01.parameters,
model.file = model01.function,
n.chains = 4,
n.iter = 2000,
n.thin = 1,
n.burnin = 1000,
jags.seed = model01.seed
)
model01.r2jags
denplot(model01.r2jags, parms = "beta1")
model01.r2jags$BUGSoutput$sims.matrix[,6]
colnames(model01.r2jags$BUGSoutput$sims.matrix)
plot(x = model01.r2jags$BUGSoutput$sims.matrix[501, 7:1006],
y = model01.r2jags$BUGSoutput$sims.matrix[501, 1007:2006])
```
## Missing Item Response Example
We can show the same result in an IRT model when we make an observation missing as well. Let's revisit our conspiracy theories example:
```{r dataimport}
# read in data:
conspiracy = read.csv("conspiracies.csv")
```
To demonstrate how this works in a small-scale sense, let's make the first observation missing:
```{r delete1stobs}
conspiracy[4,1] = NA
```
Here, the frustrating part of JAGS kicks in: A matrix of data cannot have any missing values. So, everything has to be a vector. Note the extra syntax below (and also note there may be an easier way, but not one I could discover while prepping this lecture).
```{r missingIRT}
# marker item:
model03.function = function(){
# measurement model specification
for (person in 1:N){
# item 1 ---
# form cumulative probability item response functions
CProb1[person, 1] <- 1
for (cat in 2:5){
CProb1[person, cat] <- phi(a1*(theta[person]-b1[(cat-1)]))
}
# form probability response is equal to each category
for (cat in 1:4){
Prob1[person, cat] <- CProb1[person, cat] - CProb1[person, cat+1]
}
Prob1[person, 5] <- CProb1[person, 5]
item1[person] ~ dcat(Prob1[person, 1:5])
# item 2 ---
# form cumulative probability item response functions
CProb2[person, 1] <- 1
for (cat in 2:5){
CProb2[person, cat] <- phi(a2*(theta[person]-b2[(cat-1)]))
}
# form probability response is equal to each category
for (cat in 1:4){
Prob2[person, cat] <- CProb2[person, cat] - CProb2[person, cat+1]
}
Prob2[person, 5] <- CProb2[person, 5]
item2[person] ~ dcat(Prob2[person, 1:5])
# item 3 ---
# form cumulative probability item response functions
CProb3[person, 1] <- 1
for (cat in 2:5){
CProb3[person, cat] <- phi(a3*(theta[person]-b3[(cat-1)]))
}
# form probability response is equal to each category
for (cat in 1:4){
Prob3[person, cat] <- CProb3[person, cat] - CProb3[person, cat+1]
}
Prob3[person, 5] <- CProb3[person, 5]
item3[person] ~ dcat(Prob3[person, 1:5])
}
# prior distributions for the factor:
for (person in 1:N){
theta[person] ~ dnorm(0, 1)
}
# prior distributions for the measurement model mean/precision parameters
for (cat in 1:4){
b1.star[cat] ~ dnorm(b.mean.0, b.precision.0)
b2.star[cat] ~ dnorm(b.mean.0, b.precision.0)
b3.star[cat] ~ dnorm(b.mean.0, b.precision.0)
}
b1[1:4] <- sort(b1.star[1:4])
b2[1:4] <- sort(b2.star[1:4])
b3[1:4] <- sort(b3.star[1:4])
# loadings are set to be all positive
a1 ~ dnorm(a.mean.0, a.precision.0);T(0,)
a2 ~ dnorm(a.mean.0, a.precision.0);T(0,)
a3 ~ dnorm(a.mean.0, a.precision.0);T(0,)
}
nItems = 10
# specification of prior values for measurement model parameters:
# item intercepts
b.mean.0 = 0
b.variance.0 = 100
b.precision.0 = 1 / b.variance.0
# Factor loadings -- these are the discriminations
a.mean.0 = 0
a.variance.0 = 100
a.precision.0 = 1 / a.variance.0
# next, create data for JAGS to use:
model03.data = list(
N = nrow(conspiracy),
item1 = conspiracy[,1],
item2 = conspiracy[,2],
item3 = conspiracy[,3],
b.mean.0 = b.mean.0,
b.precision.0 = b.precision.0,
a.mean.0 = a.mean.0,
a.precision.0 = a.precision.0
)
model03.init = function(){
list("a1" = runif(1, 1, 2),
"a2" = runif(1, 1, 2),
"a3" = runif(1, 1, 2),
"b1.star" = c(1, 0, -1, -2),
"b2.star" = c(1, 0, -1, -2),
"b3.star" = c(1, 0, -1, -2))
}
model03.parameters = c("a1", "a2", "a3", "b1", "b2", "b3", "item1[4]", "theta[4]", "theta[6]")
model03.seed = 23042019+2
model03.r2jags = jags.parallel(
data = model03.data,
inits = model03.init,
parameters.to.save = model03.parameters,
model.file = model03.function,
n.chains = 4,
n.iter = 5000,
n.thin = 1,
n.burnin = 3000,
jags.seed = model03.seed
)
model03.r2jags
```
Here is the plot of the posterior distribution of the data (these are the imputed values). The observed value was ```1```.
```{r plothist}
hist(model03.r2jags$BUGSoutput$sims.matrix[,which(colnames(model03.r2jags$BUGSoutput$sims.matrix)=="item1[4]")])
conspiracy[1:10,1:3]
```
And look at how the posterior distribution of $\theta_p$ changes by removing that one observation. The dashed line is the posterior distribution from an observation with the same response pattern but complete data.
```{r posttheta}
par(mfrow = c(1,1))
plot(density(model03.r2jags$BUGSoutput$sims.matrix[,which(colnames(model03.r2jags$BUGSoutput$sims.matrix)=="theta[4]")]), main = "Theta Posterior")
lines(density(model03.r2jags$BUGSoutput$sims.matrix[,which(colnames(model03.r2jags$BUGSoutput$sims.matrix)=="theta[6]")]), lty=2)
legend(x = c(-4, -3), y = c(.5, .5), legend = c("Missing", "Complete"), lty=1:2)
```