utilities: Utility Functions

In irtProb: Utilities and Probability Distributions Related to Multidimensional Person Item Response Models

Description

Some functions to help in the generation of binary data or to interpret m4pl models results.

rmultinomial is used to draw n values from a x vector of values according to a multinomial probability distribution. rmultinomial is different from the stats::rmultinom function in that it return only the value of the selected draw and not a binary vector corresponding tho the position of the draw

propCorrect computes the expected proportion of correct responses to a test according to the item and person parameters of a m4pl models.

Usage

 rmultinomial(x, n = 100, prob = rep(1, length(x))/length(x))

 propCorrect(theta, S, C, D, s, b, c, d)
 

Arguments

x	numeric: vector of values to draw from.
n	numeric: number of draws.
prob	numeric: probability assigned to each value of the x vector. Default to equiprobability.
theta	numeric; vector of person proficiency (θ) levels scaled on a normal z score.
S	numeric: positive vector of personal fluctuation parameters (σ).
C	numeric: positive vector of personal pseudo-guessing parameters (χ, a probability between 0 and 1).
D	numeric: positive vector of personal inattention parameters (δ, a probability between 0 and 1).
s	numeric: positive vector of item fluctuation parameters.
b	numeric: vector of item difficulty parameters.
c	numeric: positive vector of item pseudo-guessing parameters (a probability between 0 and 1).
d	numeric: positive vector of item inattention parameters (a probability between 0 and 1).

Value

propCorrect

numeric:

return n draws from a multinimial distribution.

rmultinomial

numeric:

return the expected proportion of correcte responses to a test.

Author(s)

Gilles Raiche, Universite du Quebec a Montreal (UQAM),

Departement d'education et pedagogie

[e-mail], http://www.er.uqam.ca/nobel/r17165/

See Also

Multinom

Examples

## Not run: 
## Comparison of the results from the multinomial and multinom functions
 x <- c(1,4,9)
 # Values draws
 rmultinomial(x=x, n=10)
 # Binary vectors draws
 rmultinom(n=10, size = 1, prob=rep(1,length(x))/length(x))

## Computation of the expected proportion of correct responses varying values
 # of theta (-3 to 3) and of pseudo-guessing (C = 0.0 to 0.6) person parameters
 nItems  <- 40
 a       <- rep(1.702,nItems); b <- seq(-3,3,length=nItems)
 c       <- rep(0,nItems); d <- rep(1,nItems)
 theta   <- seq(-3.0, 3.0, by=1.0)
 C       <- seq( 0.0, 1.0, by=0.1)
 D       <- S <- 0
 
 results <- matrix(NA, ncol=length(C), nrow=length(theta))
 colnames(results) <- C; rownames(results) <- theta
 for (i in (1:length(theta))) {
  results[i,] <- propCorrect(theta = theta[i], S = 0, C = C, D = 0,
                             s = 1/a, b = b, c = c, d = d)
  }
  round(results, 2)

## Computation of the expected proportion of correct responses varying values
 # of theta (-3 to 3) and of pseudo-guessing (C = 0.0 to 0.6) person parameters
 # if we choose the correct modelisation itegrating the C pseudo-guessing papameter
 # and if we choose according to a model selection by LL criteria
 nItems <- 40
 a      <- rep(1.702,nItems); b <- seq(-3,3,length=nItems)
 c      <- rep(0,nItems); d <- rep(1,nItems)
 nSubjects <- 300
 theta     <- rmultinomial(c(-1), nSubjects)
 S         <- rmultinomial(c(0), nSubjects)
 C         <- rmultinomial(seq(0,0.9,by=0.1), nSubjects)
 D         <- rmultinomial(c(0), nSubjects)
 set.seed(seed = 100)
 X         <- ggrm4pl(n=nItems, rep=1,
                      theta=theta, S=S, C=C, D=D,
                      s=1/a, b=b,c=c,d=d)
 # Results for each subjects for each models
 essai     <- m4plModelShow(X, b=b, s=1/a, c=c, d=d, m=0, prior="uniform")
 total     <- rowSums(X)
 pourcent  <- total/nItems * 100
 pCorrect  <- numeric(dim(essai)[1])
 for ( i in (1:dim(essai)[1]))
  pCorrect[i] <- propCorrect(essai$T[i],0,0,0,s=1/a,b=b,c=c,d=d)
 resultLL  <- summary(essai, report="add", criteria="LL")
 resultLL  <- data.frame(resultLL, theta=theta, TS=S, TC=C, errorT=resultLL$T - theta,
                         total=total, pourcent=pourcent, tpcorrect=pCorrect)
 # If the only theta model is badly choosen
 results <- resultLL[which(resultLL$MODEL == "T" ),]
 byStats <- "TC"; ofStats <- "tpcorrect"
 MeansByThetaT <- cbind(
  aggregate(results[ofStats], by=list(Theta=factor(results[,byStats]) ),
            mean, na.rm=TRUE),
  aggregate(results[ofStats], by=list(Theta=factor(results[,byStats])), sd, na.rm=TRUE),
  aggregate(results["SeT"], by=list(Theta=factor(results[,byStats])), mean, na.rm=TRUE),
  aggregate(results[ofStats], by=list(theta=factor(results[,byStats])), length)
  )[,-c(3,5,7)]
  names(MeansByThetaT) <- c("C", "pCorrect", "seE", "SeT", "n")
  MeansByThetaT[,-c(1,4,5)] <- round(MeansByThetaT[,-c(1,4,5)], 2)
  MeansByThetaT[,-c(4,5)]
 # Only for the TC model
 results <- resultLL[which(resultLL$MODEL == "TC" ),]
 byStats <- "TC"; ofStats <- "tpcorrect"
 MeansByThetaC <- cbind(
  aggregate(results[ofStats], by=list(Theta=factor(results[,byStats]) ),
            mean, na.rm=TRUE),
  aggregate(results[ofStats], by=list(Theta=factor(results[,byStats])), sd, na.rm=TRUE),
  aggregate(results["SeT"], by=list(Theta=factor(results[,byStats])), mean, na.rm=TRUE),
  aggregate(results[ofStats], by=list(theta=factor(results[,byStats])), length)
  )[,-c(3,5,7)]
  names(MeansByThetaC) <- c("C", "pCorrect", "seE", "SeT", "n")
  MeansByThetaC[,-c(1,4,5)] <- round(MeansByThetaC[,-c(1,4,5)], 2)
  MeansByThetaC[,-c(4,5)]
 # For the model choosen according to the LL criteria
 results <- resultLL[which(resultLL$critLL == TRUE),]
 byStats <- "TC"; ofStats <- "tpcorrect"
 MeansByThetaLL <- cbind(
  aggregate(results[ofStats], by=list(Theta=factor(results[,byStats]) ),
            mean, na.rm=TRUE),
  aggregate(results[ofStats], by=list(Theta=factor(results[,byStats])), sd, na.rm=TRUE),
  aggregate(results["SeT"], by=list(Theta=factor(results[,byStats])), mean, na.rm=TRUE),
  aggregate(results[ofStats], by=list(theta=factor(results[,byStats])), length)
  )[,-c(3,5,7)]
  names(MeansByThetaLL) <- c("C", "pCorrect", "seE", "SeT", "n")
  MeansByThetaLL[,-c(1,4,5)] <- round(MeansByThetaLL[,-c(1,4,5)], 2)
  MeansByThetaLL[,-c(4,5)]
  # Grapical comparison of the estimation of the 
  # by means of the 3 preceeding models
  plot(MeansByThetaT$pCorrect   ~ levels(MeansByThetaT$C),  type="l", lty=1,
       xlab="Pseudo-Guessing", ylab="% of Correct Responses")
  lines(MeansByThetaC$pCorrect  ~ levels(MeansByThetaC$C),  type="l", lty=2)
  lines(MeansByThetaLL$pCorrect ~ levels(MeansByThetaLL$C), type="l", lty=3)
  text(x=0.60, y=0.80, "Without correction", cex=0.8)
  text(x=0.50, y=0.38, "Without Knowledge of the Correct Model", cex=0.8)
  text(x=0.65, y=0.50, "With Knowledge of the Correct Model", cex=0.8)
 
## End(Not run)
 

irtProb documentation built on May 2, 2019, 1:30 p.m.