Estimate the coefficients of a filtered unconstrained polynomial IRT model

Description

Estimate the coefficients of a filtered unconstrained polynomial IRT model.

Usage

1
 FUP(data, thetaInit, item, startvals, k)

Arguments

data

N(subjects)-by-p(items) matrix of 0/1 item response data.

thetaInit

Initial theta surrogates (e.g., calculated by svdNorm).

item

item number for coefficient estimation.

startvals

start values for function minimization.

k

order of monotonic polynomial = 2k+1 (see Liang & Browne, 2015).

Value

b

Vector of polynomial coefficients.

FHAT

Function value at convergence.

counts

Number of function evaluations during minimization (see optim documentation for further details).

AIC

Pseudo scaled Akaike Information Criterion (AIC). Candidate models that produce the smallest AIC suggest the optimal number of parameters given the sample size. Scaling is accomplished by dividing the non-scaled AIC by sample size.

BIC

Pseudo scaled Bayesian Information Criterion (BIC). Candidate models that produce the smallest BIC suggest the optimal number of parameters given the sample size. Scaling is accomplished by dividing the non-scaled BIC by sample size.

convergence

Convergence = 0 indicates that the optimization algorithm converged; convergence=1 indicates that the optimization failed to converge.

.

Author(s)

Niels Waller

References

Liang, L. & Browne, M. W. (2015). A quasi-parametric method for fitting flexible item response functions. Journal of Educational and Behavioral Statistics, 40, 5–34.

Examples

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## Not run: 
NSubjects <- 2000


## generate sample k=1 FMP data
b <- matrix(c(
    #b0    b1     b2    b3      b4   b5 b6 b7  k
  1.675, 1.974, -0.068, 0.053,  0,  0,  0,  0, 1,
  1.550, 1.805, -0.230, 0.032,  0,  0,  0,  0, 1,
  1.282, 1.063, -0.103, 0.003,  0,  0,  0,  0, 1,
  0.704, 1.376, -0.107, 0.040,  0,  0,  0,  0, 1,
  1.417, 1.413,  0.021, 0.000,  0,  0,  0,  0, 1,
 -0.008, 1.349, -0.195, 0.144,  0,  0,  0,  0, 1,
  0.512, 1.538, -0.089, 0.082,  0,  0,  0,  0, 1,
  0.122, 0.601, -0.082, 0.119,  0,  0,  0,  0, 1,
  1.801, 1.211,  0.015, 0.000,  0,  0,  0,  0, 1,
 -0.207, 1.191,  0.066, 0.033,  0,  0,  0,  0, 1,
 -0.215, 1.291, -0.087, 0.029,  0,  0,  0,  0, 1,
  0.259, 0.875,  0.177, 0.072,  0,  0,  0,  0, 1,
 -0.423, 0.942,  0.064, 0.094,  0,  0,  0,  0, 1,
  0.113, 0.795,  0.124, 0.110,  0,  0,  0,  0, 1,
  1.030, 1.525,  0.200, 0.076,  0,  0,  0,  0, 1,
  0.140, 1.209,  0.082, 0.148,  0,  0,  0,  0, 1,
  0.429, 1.480, -0.008, 0.061,  0,  0,  0,  0, 1,
  0.089, 0.785, -0.065, 0.018,  0,  0,  0,  0, 1,
 -0.516, 1.013,  0.016, 0.023,  0,  0,  0,  0, 1,
  0.143, 1.315, -0.011, 0.136,  0,  0,  0,  0, 1,
  0.347, 0.733, -0.121, 0.041,  0,  0,  0,  0, 1,
 -0.074, 0.869,  0.013, 0.026,  0,  0,  0,  0, 1,
  0.630, 1.484, -0.001, 0.000,  0,  0,  0,  0, 1), 
  nrow=23, ncol=9, byrow=TRUE)  
 
# generate data using the above item parameters 
ex1.data<-genFMPData(NSubj = NSubjects, bParams = b, seed = 345)$data

NItems <- ncol(ex1.data)

# compute (initial) surrogate theta values from 
# the normed left singular vector of the centered 
# data matrix
thetaInit <- svdNorm(ex1.data)

# Choose model
k <- 1  # order of polynomial = 2k+1

# Initialize matrices to hold output
if(k == 0) {
  startVals <- c(1.5, 1.5)
  bmat <- matrix(0,NItems,6)
  colnames(bmat) <- c(paste("b", 0:1, sep = ""),"FHAT", "AIC", "BIC", "convergence") 
}

if(k == 1) {
  startVals <- c(1.5, 1.5, .10, .10)
  bmat <- matrix(0,NItems,8)
  colnames(bmat) <- c(paste("b", 0:3, sep = ""),"FHAT", "AIC", "BIC", "convergence") 
}

if(k == 2) {
  startVals <- c(1.5, 1.5, .10, .10, .10, .10)
  bmat <- matrix(0,NItems,10)
  colnames(bmat) <- c(paste("b", 0:5, sep = ""),"FHAT", "AIC", "BIC", "convergence") 
}

if(k == 3) {
  startVals <- c(1.5, 1.5, .10, .10, .10, .10, .10, .10)
  bmat <- matrix(0,NItems,12)
  colnames(bmat) <- c(paste("b", 0:7, sep = ""),"FHAT", "AIC", "BIC", "convergence") 
}   


# estimate item parameters and fit statistics
for(i in 1:NItems){
  out<-FUP(data = ex1.data,thetaInit = thetaInit, item = i, startvals = startVals, k = k)
  Nb <- length(out$b)
  bmat[i,1:Nb] <- out$b
  bmat[i,Nb+1] <- out$FHAT
  bmat[i,Nb+2] <- out$AIC
  bmat[i,Nb+3] <- out$BIC
  bmat[i,Nb+4] <- out$convergence
}

# print results
print(bmat)

## End(Not run)