Estimate the coefficients of a filtered monotonic polynomial IRT model

Description

Estimate the coefficients of a filtered monotonic polynomial IRT model.

Usage

1
 FMP(data, thetaInit, item, startvals, k, eps = 1e-06)

Arguments

data

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

thetaInit

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

item

Item number for coefficient estimation.

startvals

Start values for function minimization. Start values are in the gamma metric (see Liang & Browne, 2015)

k

Order of monotonic polynomial = 2k+1 (see Liang & Browne, 2015). k can equal 0, 1, 2, or 3.

eps

Step size for gradient approximation, default = 1e-6. If a convergence failure occurs during function optimization reducing the value of eps will often produce a converged solution.

Details

As described by Liang and Browne (2015), the filtered polynomial model (FMP) is a quasi-parametric IRT model in which the IRF is a composition of a logistic function and a polynomial function, m(θ), of degree 2k + 1. When k = 0, m(θ) = b_0 + b_1 θ (the slope intercept form of the 2PL). When k = 1, 2k + 1 equals 3 resulting in m(θ) = b_0 + b_1 θ + b_2 θ^2 + b_3 θ^3. Acceptable values of k = 0,1,2,3. According to Liang and Browne, the "FMP IRF may be used to approximate any IRF with a continuous derivative arbitrarily closely by increasing the number of parameters in the monotonic polynomial" (2015, p. 2) The FMP model assumes that the IRF is monotonically increasing, bounded by 0 and 1, and everywhere differentiable with respect to theta (the latent trait).

Value

b

Vector of polynomial coefficients.

gamma

Polynomial coefficients in gamma metric (see Liang & Browne, 2015).

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: 
## In this example we will generate 2000 item response vectors 
## for a k = 1 order filtered polynomial model and then recover 
## the estimated item parameters with the FMP function.  

k <- 1  # order of polynomial

NSubjects <- 2000


## generate a sample of 2000 item response vectors 
## for a k = 1 FMP model using the following
## coefficients
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)  
  
ex1.data<-genFMPData(NSubj = NSubjects, bParams = b, seed = 345)$data

## number of items in the data matrix
NItems <- ncol(ex1.data)

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


## earlier we defined k = 1
  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 <- FMP(data = ex1.data, 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 output 
print(bmat)

## End(Not run)