guess: Adjust Estimates of Learning for Guessing

#  
# Simulating Data without Don't Know
#  
# Details:
#
# Inference should be at the item level
#
# Parameter Distribution and Ranges:
#	Ability 
#		rnorm, roughly -3,3 
#		we may want to draw a fresh respondent sample for each item
#	Difficulty 
#		uniform, item-level, covers most of the range of theta (-2.5,2.5)
#	Learning
#		Each item: A base chance that everyone who doesn't know the item gets (this varies between (.05 and .6))
#		By person: b*ability; b varies from 0 to 1.5 (so covers all scenarios)
#	Gamma 
#		uniform, covers broad range (.1, .6), orthogonal to item diff.
#		item level
#
# Creating data
# 	1. sample ability: random normal (just do it once for now)
# 	2. sample difficulty: random uniform
# 	3. Produce time 1 data using IRT
# 	4. sample alpha (learn): random uniform
# 	5. sample beta (learn): random uniform
# 	6. t2: Convert 0s to 1s as impact of learning, rbinom = alpha + beta*(ability at t1)
# 	7. sample gamma. random uniform.
#	8. create observed: convert 0s to 1s probabilistically using gamma
#
# ------------------------------------------------------------------------------------------------

# Load libraries
library(ltm)
library(goji)
library(devtools)

set.seed(76543)			# set seed for reproducibility
	
n 			<- 500 		# number of respondents
nitems		<- 1000		# number of items

theta	<- rnorm(n)												# standard normal abilities
diff 	<- seq(-3, 3, 	 length.out=nitems)						# difficulty
gamma  	<- seq(.1, .60,  length.out=nitems)[sample(1:nitems)]	# lucky guessing
alpha	<- seq(.1,.50,   length.out=nitems)[sample(1:nitems)] 	# base level of learning by item
beta	<- seq(0, .60,   length.out=nitems)[sample(1:nitems)]	# rich get richer process
	
# Creating data
wave1		<- rmvlogis(n, cbind(diff, 1), z.vals = theta)
wave2		<- wave1
		
# Setting the knowledge gains
for (i in 1:ncol(wave1)){
	probab 		<- ifelse(range(alpha[i] + beta[i]*theta)[2] >= 1 | range(alpha[i] + beta[i]*theta)[1] < 0 , zero1(alpha[i] + beta[i]*theta), alpha[i] + beta[i]*theta) 
	wave2[,i]	<- ifelse(wave1[,i]==0, rbinom(n, 1, probab), wave1[,i])
}
			
# Posthoc Introduction of guessing
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Combine data from diff. waves; create observed data
waves 	<- cbind(wave1, wave2)
wavem 	<- waves

# Add Guessing
# Gamma by item
gamma2   <- rep(gamma, 2)		

# Only the ignorant can guess
# Those who guess are lucky gamma_i of the time on the ith item
for (i in 1:ncol(waves)){
	wavem[,i] <- ifelse(!is.na(wavem[,i]) & wavem[,i]==0, rbinom(n, 1, gamma2[i]), wavem[,i])
}
		
# Check
mean(rowMeans(wave2 - wave1, na.rm=T))
mean(rowMeans(wavem[,(nitems+1):ncol(wavem)] - wavem[,1:nitems], na.rm=T))
	
# Create fake data dataset
rawdata 	<- waves
guessdata	<- wavem
		
alldat 									<- data.frame(rawdata, guessdata)
names(alldat)[1:nitems]					<- paste0("true.t1", 1:nitems)
names(alldat)[(nitems +1):(2*nitems)]	<- paste0("true.t2", 1:nitems)
names(alldat)[(2*nitems+1):(3*nitems)]	<- paste0("guess.t1", 1:nitems)
names(alldat)[(3*nitems +1):(4*nitems)]	<- paste0("guess.t2", 1:nitems)

# Output data
params <- cbind(diff, alpha, beta, gamma)

use_data(alldat, params)