| Title: | Filtered Monotonic Polynomial IRT Models |
|---|---|
| Description: | Estimates Filtered Monotonic Polynomial IRT Models as described by Liang and Browne (2015) <DOI:10.3102/1076998614556816>. |
| Authors: | Niels G. Waller <[email protected]> |
| Maintainer: | Niels G. Waller <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 1.4 |
| Built: | 2025-02-10 02:57:15 UTC |
| Source: | https://github.com/cran/FMP |
Compute eap trait estimates for items fit by filtered monotonic polynomial IRT models.
eap(data, bParams, NQuad = 21, priorVar = 2, mintheta = -4, maxtheta = 4)eap(data, bParams, NQuad = 21, priorVar = 2, mintheta = -4, maxtheta = 4)
data |
N(subjects)-by-p(items) matrix of 0/1 item response data. |
bParams |
A p-by-9 matrix of FMP or FUP item parameters and model designations. Columns 1 - 8 hold the (possibly zero valued) polynomial coefficients; column 9 holds the value of |
NQuad |
Number of quadrature points used to calculate the eap estimates. |
priorVar |
Variance of the normal prior for the eap estimates. The prior mean equals 0. |
mintheta, maxtheta
|
NQuad quadrature points will be evenly spaced between |
eap trait estimates. |
Niels Waller
## this example demonstrates how to calculate ## eap trait estimates for a scale composed of items ## that have been fit to FMP models of different ## degree NSubjects <- 2000 ## Assume that ## items 1 - 5 fit a k=0 model, ## items 6 - 10 fit a k=1 model, and ## items 11 - 15 fit a k=2 model. itmParameters <- matrix(c( # b0 b1 b2 b3 b4 b5, b6, b7, k -1.05, 1.63, 0.00, 0.00, 0.00, 0, 0, 0, 0, #1 -1.97, 1.75, 0.00, 0.00, 0.00, 0, 0, 0, 0, #2 -1.77, 1.82, 0.00, 0.00, 0.00, 0, 0, 0, 0, #3 -4.76, 2.67, 0.00, 0.00, 0.00, 0, 0, 0, 0, #4 -2.15, 1.93, 0.00, 0.00, 0.00, 0, 0, 0, 0, #5 -1.25, 1.17, -0.25, 0.12, 0.00, 0, 0, 0, 1, #6 1.65, 0.01, 0.02, 0.03, 0.00, 0, 0, 0, 1, #7 -2.99, 1.64, 0.17, 0.03, 0.00, 0, 0, 0, 1, #8 -3.22, 2.40, -0.12, 0.10, 0.00, 0, 0, 0, 1, #9 -0.75, 1.09, -0.39, 0.31, 0.00, 0, 0, 0, 1, #10 -1.21, 9.07, 1.20,-0.01,-0.01, 0.01, 0, 0, 2, #11 -1.92, 1.55, -0.17, 0.50,-0.01, 0.01, 0, 0, 2, #12 -1.76, 1.29, -0.13, 1.60,-0.01, 0.01, 0, 0, 2, #13 -2.32, 1.40, 0.55, 0.05,-0.01, 0.01, 0, 0, 2, #14 -1.24, 2.48, -0.65, 0.60,-0.01, 0.01, 0, 0, 2),#15 15, 9, byrow=TRUE) # generate data using the above item parameters ex1.data<-genFMPData(NSubj = NSubjects, bParams = itmParameters, seed = 345)$data ## calculate eap estimates for mixed models thetaEAP<-eap(data = ex1.data, bParams = itmParameters, NQuad = 25, priorVar = 2, mintheta = -4, maxtheta = 4) ## compare eap estimates with initial theta surrogates if(FALSE){ #set to TRUE to see plot thetaInit <- svdNorm(ex1.data) plot(thetaInit,thetaEAP, xlim = c(-3.5,3.5), ylim = c(-3.5,3.5), xlab = "Initial theta surrogates", ylab = "EAP trait estimates (Mixed models)") }## this example demonstrates how to calculate ## eap trait estimates for a scale composed of items ## that have been fit to FMP models of different ## degree NSubjects <- 2000 ## Assume that ## items 1 - 5 fit a k=0 model, ## items 6 - 10 fit a k=1 model, and ## items 11 - 15 fit a k=2 model. itmParameters <- matrix(c( # b0 b1 b2 b3 b4 b5, b6, b7, k -1.05, 1.63, 0.00, 0.00, 0.00, 0, 0, 0, 0, #1 -1.97, 1.75, 0.00, 0.00, 0.00, 0, 0, 0, 0, #2 -1.77, 1.82, 0.00, 0.00, 0.00, 0, 0, 0, 0, #3 -4.76, 2.67, 0.00, 0.00, 0.00, 0, 0, 0, 0, #4 -2.15, 1.93, 0.00, 0.00, 0.00, 0, 0, 0, 0, #5 -1.25, 1.17, -0.25, 0.12, 0.00, 0, 0, 0, 1, #6 1.65, 0.01, 0.02, 0.03, 0.00, 0, 0, 0, 1, #7 -2.99, 1.64, 0.17, 0.03, 0.00, 0, 0, 0, 1, #8 -3.22, 2.40, -0.12, 0.10, 0.00, 0, 0, 0, 1, #9 -0.75, 1.09, -0.39, 0.31, 0.00, 0, 0, 0, 1, #10 -1.21, 9.07, 1.20,-0.01,-0.01, 0.01, 0, 0, 2, #11 -1.92, 1.55, -0.17, 0.50,-0.01, 0.01, 0, 0, 2, #12 -1.76, 1.29, -0.13, 1.60,-0.01, 0.01, 0, 0, 2, #13 -2.32, 1.40, 0.55, 0.05,-0.01, 0.01, 0, 0, 2, #14 -1.24, 2.48, -0.65, 0.60,-0.01, 0.01, 0, 0, 2),#15 15, 9, byrow=TRUE) # generate data using the above item parameters ex1.data<-genFMPData(NSubj = NSubjects, bParams = itmParameters, seed = 345)$data ## calculate eap estimates for mixed models thetaEAP<-eap(data = ex1.data, bParams = itmParameters, NQuad = 25, priorVar = 2, mintheta = -4, maxtheta = 4) ## compare eap estimates with initial theta surrogates if(FALSE){ #set to TRUE to see plot thetaInit <- svdNorm(ex1.data) plot(thetaInit,thetaEAP, xlim = c(-3.5,3.5), ylim = c(-3.5,3.5), xlab = "Initial theta surrogates", ylab = "EAP trait estimates (Mixed models)") }
Utility function to compute empirical response functions.
erf(theta, data, whichItem, min = -3, max = 3, Ncuts = 12)erf(theta, data, whichItem, min = -3, max = 3, Ncuts = 12)
theta |
Vector of estimated latent trait scores. |
data |
A matrix of binary item responses. |
whichItem |
Data for an erf will be generated for whichItem. |
min |
Default = -3. Minimum value of theta. |
max |
Default = 3. Maximum value of theta. |
Ncuts |
Number of score groups for erf. |
probs |
A vector (of length Ncuts) of bin response probabilities for the empirical response function. |
centers |
A vector of bin centers. |
Ni |
Bin sample sizes. |
se.p |
Standard errors of the estimated bin response probabilities. |
Niels Waller
NSubj <- 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) theta <- rnorm(NSubj) data<-genFMPData(NSubj = NSubj, bParam = b, theta = theta, seed = 345)$data erfItem1 <- erf(theta, data, whichItem = 1, min = -3, max = 3, Ncuts = 12) plot( erfItem1$centers, erfItem1$probs, type="b", main="Empirical Response Function", xlab = expression(theta), ylab="Probability", cex.lab=1.5)NSubj <- 2000 #generate sample k=1 FMP data b <- matrix(c( #b0 b1 b2 b3 b4 b5 b6 b7 k 1.675, 1.974, -0.068, 0.053, 0, 0, 0, 0, 1, 1.550, 1.805, -0.230, 0.032, 0, 0, 0, 0, 1, 1.282, 1.063, -0.103, 0.003, 0, 0, 0, 0, 1, 0.704, 1.376, -0.107, 0.040, 0, 0, 0, 0, 1, 1.417, 1.413, 0.021, 0.000, 0, 0, 0, 0, 1, -0.008, 1.349, -0.195, 0.144, 0, 0, 0, 0, 1, 0.512, 1.538, -0.089, 0.082, 0, 0, 0, 0, 1, 0.122, 0.601, -0.082, 0.119, 0, 0, 0, 0, 1, 1.801, 1.211, 0.015, 0.000, 0, 0, 0, 0, 1, -0.207, 1.191, 0.066, 0.033, 0, 0, 0, 0, 1, -0.215, 1.291, -0.087, 0.029, 0, 0, 0, 0, 1, 0.259, 0.875, 0.177, 0.072, 0, 0, 0, 0, 1, -0.423, 0.942, 0.064, 0.094, 0, 0, 0, 0, 1, 0.113, 0.795, 0.124, 0.110, 0, 0, 0, 0, 1, 1.030, 1.525, 0.200, 0.076, 0, 0, 0, 0, 1, 0.140, 1.209, 0.082, 0.148, 0, 0, 0, 0, 1, 0.429, 1.480, -0.008, 0.061, 0, 0, 0, 0, 1, 0.089, 0.785, -0.065, 0.018, 0, 0, 0, 0, 1, -0.516, 1.013, 0.016, 0.023, 0, 0, 0, 0, 1, 0.143, 1.315, -0.011, 0.136, 0, 0, 0, 0, 1, 0.347, 0.733, -0.121, 0.041, 0, 0, 0, 0, 1, -0.074, 0.869, 0.013, 0.026, 0, 0, 0, 0, 1, 0.630, 1.484, -0.001, 0.000, 0, 0, 0, 0, 1), nrow=23, ncol=9, byrow=TRUE) theta <- rnorm(NSubj) data<-genFMPData(NSubj = NSubj, bParam = b, theta = theta, seed = 345)$data erfItem1 <- erf(theta, data, whichItem = 1, min = -3, max = 3, Ncuts = 12) plot( erfItem1$centers, erfItem1$probs, type="b", main="Empirical Response Function", xlab = expression(theta), ylab="Probability", cex.lab=1.5)
Estimate the coefficients of a filtered monotonic polynomial IRT model.
FMP(data, thetaInit, item, startvals, k, eps = 1e-06)FMP(data, thetaInit, item, startvals, k, eps = 1e-06)
data |
N(subjects)-by-p(items) matrix of 0/1 item response data. |
thetaInit |
Initial theta ( |
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. |
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, , of degree 2k + 1.
When k = 0,
(the slope intercept form of the 2PL). When k = 1, 2k + 1 equals 3 resulting in
. 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).
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. |
Niels Waller
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.
## 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$convergence } # print output print(bmat) ## End(Not run)## Not run: ## In this example we will generate 2000 item response vectors ## for a k = 1 order filtered polynomial model and then recover ## the estimated item parameters with the FMP function. k <- 1 # order of polynomial NSubjects <- 2000 ## generate a sample of 2000 item response vectors ## for a k = 1 FMP model using the following ## coefficients b <- matrix(c( #b0 b1 b2 b3 b4 b5 b6 b7 k 1.675, 1.974, -0.068, 0.053, 0, 0, 0, 0, 1, 1.550, 1.805, -0.230, 0.032, 0, 0, 0, 0, 1, 1.282, 1.063, -0.103, 0.003, 0, 0, 0, 0, 1, 0.704, 1.376, -0.107, 0.040, 0, 0, 0, 0, 1, 1.417, 1.413, 0.021, 0.000, 0, 0, 0, 0, 1, -0.008, 1.349, -0.195, 0.144, 0, 0, 0, 0, 1, 0.512, 1.538, -0.089, 0.082, 0, 0, 0, 0, 1, 0.122, 0.601, -0.082, 0.119, 0, 0, 0, 0, 1, 1.801, 1.211, 0.015, 0.000, 0, 0, 0, 0, 1, -0.207, 1.191, 0.066, 0.033, 0, 0, 0, 0, 1, -0.215, 1.291, -0.087, 0.029, 0, 0, 0, 0, 1, 0.259, 0.875, 0.177, 0.072, 0, 0, 0, 0, 1, -0.423, 0.942, 0.064, 0.094, 0, 0, 0, 0, 1, 0.113, 0.795, 0.124, 0.110, 0, 0, 0, 0, 1, 1.030, 1.525, 0.200, 0.076, 0, 0, 0, 0, 1, 0.140, 1.209, 0.082, 0.148, 0, 0, 0, 0, 1, 0.429, 1.480, -0.008, 0.061, 0, 0, 0, 0, 1, 0.089, 0.785, -0.065, 0.018, 0, 0, 0, 0, 1, -0.516, 1.013, 0.016, 0.023, 0, 0, 0, 0, 1, 0.143, 1.315, -0.011, 0.136, 0, 0, 0, 0, 1, 0.347, 0.733, -0.121, 0.041, 0, 0, 0, 0, 1, -0.074, 0.869, 0.013, 0.026, 0, 0, 0, 0, 1, 0.630, 1.484, -0.001, 0.000, 0, 0, 0, 0, 1), nrow=23, ncol=9, byrow=TRUE) ex1.data<-genFMPData(NSubj = NSubjects, bParams = b, seed = 345)$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)
Utility function for checking whether candidate FMP coefficients yield a monotonically increasing polynomial.
FMPMonotonicityCheck(b, lower = -20, upper = 20)FMPMonotonicityCheck(b, lower = -20, upper = 20)
b |
A vector of 8 polynomial coefficients ( |
lower, upper
|
Theta bounds for monotonicity check. |
minDeriv |
Minimum value of the derivative for the polynomial. |
Niels Waller
## A set of candidate coefficients for an FMP model. ## These coefficients fail the test and thus ## should not be used with genFMPdata to generate ## item response data that are consistent with an ## FMP model. b <- c(1.21, 1.87, -1.02, 0.18, 0.18, 0, 0, 0) FMPMonotonicityCheck(b)## A set of candidate coefficients for an FMP model. ## These coefficients fail the test and thus ## should not be used with genFMPdata to generate ## item response data that are consistent with an ## FMP model. b <- c(1.21, 1.87, -1.02, 0.18, 0.18, 0, 0, 0) FMPMonotonicityCheck(b)
Estimate the coefficients of a filtered unconstrained polynomial IRT model.
FUP(data, thetaInit, item, startvals, k)FUP(data, thetaInit, item, startvals, k)
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). |
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. . |
Niels Waller
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.
## 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$convergence } # print results print(bmat) ## End(Not run)## Not run: NSubjects <- 2000 ## generate sample k=1 FMP data b <- matrix(c( #b0 b1 b2 b3 b4 b5 b6 b7 k 1.675, 1.974, -0.068, 0.053, 0, 0, 0, 0, 1, 1.550, 1.805, -0.230, 0.032, 0, 0, 0, 0, 1, 1.282, 1.063, -0.103, 0.003, 0, 0, 0, 0, 1, 0.704, 1.376, -0.107, 0.040, 0, 0, 0, 0, 1, 1.417, 1.413, 0.021, 0.000, 0, 0, 0, 0, 1, -0.008, 1.349, -0.195, 0.144, 0, 0, 0, 0, 1, 0.512, 1.538, -0.089, 0.082, 0, 0, 0, 0, 1, 0.122, 0.601, -0.082, 0.119, 0, 0, 0, 0, 1, 1.801, 1.211, 0.015, 0.000, 0, 0, 0, 0, 1, -0.207, 1.191, 0.066, 0.033, 0, 0, 0, 0, 1, -0.215, 1.291, -0.087, 0.029, 0, 0, 0, 0, 1, 0.259, 0.875, 0.177, 0.072, 0, 0, 0, 0, 1, -0.423, 0.942, 0.064, 0.094, 0, 0, 0, 0, 1, 0.113, 0.795, 0.124, 0.110, 0, 0, 0, 0, 1, 1.030, 1.525, 0.200, 0.076, 0, 0, 0, 0, 1, 0.140, 1.209, 0.082, 0.148, 0, 0, 0, 0, 1, 0.429, 1.480, -0.008, 0.061, 0, 0, 0, 0, 1, 0.089, 0.785, -0.065, 0.018, 0, 0, 0, 0, 1, -0.516, 1.013, 0.016, 0.023, 0, 0, 0, 0, 1, 0.143, 1.315, -0.011, 0.136, 0, 0, 0, 0, 1, 0.347, 0.733, -0.121, 0.041, 0, 0, 0, 0, 1, -0.074, 0.869, 0.013, 0.026, 0, 0, 0, 0, 1, 0.630, 1.484, -0.001, 0.000, 0, 0, 0, 0, 1), nrow=23, ncol=9, byrow=TRUE) # generate data using the above item parameters ex1.data<-genFMPData(NSubj = NSubjects, bParams = b, seed = 345)$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)
Generate item response data for or 1, 2, 3 or 4-parameter IRT Models.
gen4PMData(NSubj, abcdParams, D = 1.702, seed = NULL, theta = NULL, thetaMN = 0, thetaVar = 1)gen4PMData(NSubj, abcdParams, D = 1.702, seed = NULL, theta = NULL, thetaMN = 0, thetaVar = 1)
NSubj |
the desired number of subject response vectors. |
abcdParams |
a p(items)-by-4 matrix of IRT item parameters: a = discrimination, b = difficulty, c = lower asymptote, and d = upper asymptote. |
D |
Scaling constant to place the IRF on the normal ogive or logistic metric. Default = 1.702 (normal ogive metric) |
seed |
Optional seed for the random number generator. |
theta |
Optional vector of latent trait scores. If theta = NULL (the default value) then gen4PMData will simulate theta from a normal distribution. |
thetaMN |
Mean of simulated theta distribution. Default = 0. |
thetaVar |
Variance of simulated theta distribution. Default = 1 |
data |
N(subject)-by-p(items) matrix of item response data. |
theta |
Latent trait scores. |
seed |
Value of the random number seed. |
Niels Waller
## Generate simulated 4PM data for 2,000 subjects # 4PM Item parameters from MMPI-A CYN scale Params<-matrix(c(1.41, -0.79, .01, .98, #1 1.19, -0.81, .02, .96, #2 0.79, -1.11, .05, .94, #3 0.94, -0.53, .02, .93, #4 0.90, -1.02, .04, .95, #5 1.00, -0.21, .02, .84, #6 1.05, -0.27, .02, .97, #7 0.90, -0.75, .04, .73, #8 0.80, -1.42, .06, .98, #9 0.71, 0.13, .05, .94, #10 1.01, -0.14, .02, .81, #11 0.63, 0.18, .18, .97, #12 0.68, 0.18, .02, .87, #13 0.60, -0.14, .09, .96, #14 0.85, -0.71, .04, .99, #15 0.83, -0.07, .05, .97, #16 0.86, -0.36, .03, .95, #17 0.66, -0.64, .04, .77, #18 0.60, 0.52, .04, .94, #19 0.90, -0.06, .02, .96, #20 0.62, -0.47, .05, .86, #21 0.57, 0.13, .06, .93, #22 0.77, -0.43, .04, .97),23,4, byrow=TRUE) data <- gen4PMData(NSubj=2000, abcdParams = Params, D = 1.702, seed = 123, thetaMN = 0, thetaVar = 1)$data cat("\nClassical item difficulties for simulated data") print( round( apply(data,2,mean),2) )## Generate simulated 4PM data for 2,000 subjects # 4PM Item parameters from MMPI-A CYN scale Params<-matrix(c(1.41, -0.79, .01, .98, #1 1.19, -0.81, .02, .96, #2 0.79, -1.11, .05, .94, #3 0.94, -0.53, .02, .93, #4 0.90, -1.02, .04, .95, #5 1.00, -0.21, .02, .84, #6 1.05, -0.27, .02, .97, #7 0.90, -0.75, .04, .73, #8 0.80, -1.42, .06, .98, #9 0.71, 0.13, .05, .94, #10 1.01, -0.14, .02, .81, #11 0.63, 0.18, .18, .97, #12 0.68, 0.18, .02, .87, #13 0.60, -0.14, .09, .96, #14 0.85, -0.71, .04, .99, #15 0.83, -0.07, .05, .97, #16 0.86, -0.36, .03, .95, #17 0.66, -0.64, .04, .77, #18 0.60, 0.52, .04, .94, #19 0.90, -0.06, .02, .96, #20 0.62, -0.47, .05, .86, #21 0.57, 0.13, .06, .93, #22 0.77, -0.43, .04, .97),23,4, byrow=TRUE) data <- gen4PMData(NSubj=2000, abcdParams = Params, D = 1.702, seed = 123, thetaMN = 0, thetaVar = 1)$data cat("\nClassical item difficulties for simulated data") print( round( apply(data,2,mean),2) )
Generate item response data for the filtered polynomial IRT model.
genFMPData(NSubj, bParams, theta = NULL, thetaMN = 0, thetaVar = 1, seed)genFMPData(NSubj, bParams, theta = NULL, thetaMN = 0, thetaVar = 1, seed)
NSubj |
the desired number of subject response vectors. |
bParams |
a p(items)-by-9 matrix of polynomial coefficients and model designations. Columns 1 - 8 hold the polynomial coefficients; column 9 holds the value of |
theta |
A user-supplied vector of latent trait scores. Default theta = NULL. |
thetaMN |
If theta = NULL genFMPdata will simulate random normal deviates from a population with mean thetaMN and variance thetaVar. |
thetaVar |
If theta = NULL genFMPData will simulate random normal deviates from a population with mean thetaMN and variance thetaVar. |
seed |
initial seed for the random number generator. |
theta |
theta values used for data generation |
data |
N(subject)-by-p(items) matrix of item response data. |
seed |
Value of the random number seed. |
Niels Waller
# The following code illustrates data generation for # an FMP of order 3 (i.e., 2k+1) # data will be generated for 2000 examinees NSubjects <- 2000 ## Example item paramters, k=1 FMP 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 paramters data<-genFMPData(NSubj = NSubjects, bParams=b, seed=345)$data# The following code illustrates data generation for # an FMP of order 3 (i.e., 2k+1) # data will be generated for 2000 examinees NSubjects <- 2000 ## Example item paramters, k=1 FMP b <- matrix(c( #b0 b1 b2 b3 b4 b5 b6 b7 k 1.675, 1.974, -0.068, 0.053, 0, 0, 0, 0, 1, 1.550, 1.805, -0.230, 0.032, 0, 0, 0, 0, 1, 1.282, 1.063, -0.103, 0.003, 0, 0, 0, 0, 1, 0.704, 1.376, -0.107, 0.040, 0, 0, 0, 0, 1, 1.417, 1.413, 0.021, 0.000, 0, 0, 0, 0, 1, -0.008, 1.349, -0.195, 0.144, 0, 0, 0, 0, 1, 0.512, 1.538, -0.089, 0.082, 0, 0, 0, 0, 1, 0.122, 0.601, -0.082, 0.119, 0, 0, 0, 0, 1, 1.801, 1.211, 0.015, 0.000, 0, 0, 0, 0, 1, -0.207, 1.191, 0.066, 0.033, 0, 0, 0, 0, 1, -0.215, 1.291, -0.087, 0.029, 0, 0, 0, 0, 1, 0.259, 0.875, 0.177, 0.072, 0, 0, 0, 0, 1, -0.423, 0.942, 0.064, 0.094, 0, 0, 0, 0, 1, 0.113, 0.795, 0.124, 0.110, 0, 0, 0, 0, 1, 1.030, 1.525, 0.200, 0.076, 0, 0, 0, 0, 1, 0.140, 1.209, 0.082, 0.148, 0, 0, 0, 0, 1, 0.429, 1.480, -0.008, 0.061, 0, 0, 0, 0, 1, 0.089, 0.785, -0.065, 0.018, 0, 0, 0, 0, 1, -0.516, 1.013, 0.016, 0.023, 0, 0, 0, 0, 1, 0.143, 1.315, -0.011, 0.136, 0, 0, 0, 0, 1, 0.347, 0.733, -0.121, 0.041, 0, 0, 0, 0, 1, -0.074, 0.869, 0.013, 0.026, 0, 0, 0, 0, 1, 0.630, 1.484, -0.001, 0.000, 0, 0, 0, 0, 1), nrow=23, ncol=9, byrow=TRUE) # generate data using the above item paramters data<-genFMPData(NSubj = NSubjects, bParams=b, seed=345)$data
Plot model-implied (and possibly empirical) item response function for polynomial IRT models.
irf(data, bParams, item, plotERF = TRUE, thetaEAP = NULL, minCut = -3, maxCut = 3, NCuts = 9)irf(data, bParams, item, plotERF = TRUE, thetaEAP = NULL, minCut = -3, maxCut = 3, NCuts = 9)
data |
N(subjects)-by-p(items) matrix of 0/1 item response data. |
bParams |
p(items)-by-9 matrix. The first 8 columns of the matrix should contain the FMP or FUP polynomial coefficients for the p items. The 9th column contains the value of k for each item (where the item specific order of the polynomial is 2k+1). |
item |
The IRF for |
plotERF |
A logical that determines whether to plot discrete values of the empirical response function. |
thetaEAP |
If |
minCut, maxCut
|
If |
NCuts |
Desired number of bins for the empirical response function. |
Niels Waller
NSubjects <- 2000 NItems <- 15 itmParameters <- matrix(c( # b0 b1 b2 b3 b4 b5, b6, b7, k -1.05, 1.63, 0.00, 0.00, 0.00, 0, 0, 0, 0, #1 -1.97, 1.75, 0.00, 0.00, 0.00, 0, 0, 0, 0, #2 -1.77, 1.82, 0.00, 0.00, 0.00, 0, 0, 0, 0, #3 -4.76, 2.67, 0.00, 0.00, 0.00, 0, 0, 0, 0, #4 -2.15, 1.93, 0.00, 0.00, 0.00, 0, 0, 0, 0, #5 -1.25, 1.17, -0.25, 0.12, 0.00, 0, 0, 0, 1, #6 1.65, 0.01, 0.02, 0.03, 0.00, 0, 0, 0, 1, #7 -2.99, 1.64, 0.17, 0.03, 0.00, 0, 0, 0, 1, #8 -3.22, 2.40, -0.12, 0.10, 0.00, 0, 0, 0, 1, #9 -0.75, 1.09, -0.39, 0.31, 0.00, 0, 0, 0, 1, #10 -1.21, 9.07, 1.20,-0.01,-0.01, 0.01, 0, 0, 2, #11 -1.92, 1.55, -0.17, 0.50,-0.01, 0.01, 0, 0, 2, #12 -1.76, 1.29, -0.13, 1.60,-0.01, 0.01, 0, 0, 2, #13 -2.32, 1.40, 0.55, 0.05,-0.01, 0.01, 0, 0, 2, #14 -1.24, 2.48, -0.65, 0.60,-0.01, 0.01, 0, 0, 2),#15 15, 9, byrow=TRUE) ex1.data<-genFMPData(NSubj = NSubjects, bParams = itmParameters, seed = 345)$data ## compute initial theta surrogates thetaInit <- svdNorm(ex1.data) ## For convenience we assume that the item parameter ## estimates equal their population values. In practice, ## item parameters would be estimated at this step. itmEstimates <- itmParameters ## calculate eap estimates for mixed models thetaEAP <- eap(data = ex1.data, bParams = itmEstimates, NQuad = 21, priorVar = 2, mintheta = -4, maxtheta = 4) ## plot irf and erf for item 1 irf(data = ex1.data, bParams = itmEstimates, item = 1, plotERF = TRUE, thetaEAP) ## plot irf and erf for item 12 irf(data = ex1.data, bParams = itmEstimates, item = 12, plotERF = TRUE, thetaEAP)NSubjects <- 2000 NItems <- 15 itmParameters <- matrix(c( # b0 b1 b2 b3 b4 b5, b6, b7, k -1.05, 1.63, 0.00, 0.00, 0.00, 0, 0, 0, 0, #1 -1.97, 1.75, 0.00, 0.00, 0.00, 0, 0, 0, 0, #2 -1.77, 1.82, 0.00, 0.00, 0.00, 0, 0, 0, 0, #3 -4.76, 2.67, 0.00, 0.00, 0.00, 0, 0, 0, 0, #4 -2.15, 1.93, 0.00, 0.00, 0.00, 0, 0, 0, 0, #5 -1.25, 1.17, -0.25, 0.12, 0.00, 0, 0, 0, 1, #6 1.65, 0.01, 0.02, 0.03, 0.00, 0, 0, 0, 1, #7 -2.99, 1.64, 0.17, 0.03, 0.00, 0, 0, 0, 1, #8 -3.22, 2.40, -0.12, 0.10, 0.00, 0, 0, 0, 1, #9 -0.75, 1.09, -0.39, 0.31, 0.00, 0, 0, 0, 1, #10 -1.21, 9.07, 1.20,-0.01,-0.01, 0.01, 0, 0, 2, #11 -1.92, 1.55, -0.17, 0.50,-0.01, 0.01, 0, 0, 2, #12 -1.76, 1.29, -0.13, 1.60,-0.01, 0.01, 0, 0, 2, #13 -2.32, 1.40, 0.55, 0.05,-0.01, 0.01, 0, 0, 2, #14 -1.24, 2.48, -0.65, 0.60,-0.01, 0.01, 0, 0, 2),#15 15, 9, byrow=TRUE) ex1.data<-genFMPData(NSubj = NSubjects, bParams = itmParameters, seed = 345)$data ## compute initial theta surrogates thetaInit <- svdNorm(ex1.data) ## For convenience we assume that the item parameter ## estimates equal their population values. In practice, ## item parameters would be estimated at this step. itmEstimates <- itmParameters ## calculate eap estimates for mixed models thetaEAP <- eap(data = ex1.data, bParams = itmEstimates, NQuad = 21, priorVar = 2, mintheta = -4, maxtheta = 4) ## plot irf and erf for item 1 irf(data = ex1.data, bParams = itmEstimates, item = 1, plotERF = TRUE, thetaEAP) ## plot irf and erf for item 12 irf(data = ex1.data, bParams = itmEstimates, item = 12, plotERF = TRUE, thetaEAP)
Plot an empirical response function using rest scores.
restScore(data, item, NCuts)restScore(data, item, NCuts)
data |
N(subjects)-by-p(items) matrix of 0/1 item response data. |
item |
Generate a rest score plot for item |
NCuts |
Divide the rest scores into |
A restscore plot with 95% confidence interval bars for the conditional probability estimates.
item |
The item number. |
bins |
A vector of bin limits and bin sample sizes. |
binProb |
A vector of bin conditional probabilities. |
Niels Waller
NSubj <- 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) data<-genFMPData(NSubj = NSubj, bParam = b, seed = 345)$data ## generate a rest score plot for item 12. ## the grey horizontal lines in the plot ## respresent pseudo asymptotes that ## are significantly different from the ## (0,1) boundaries restScore(data, item = 12, NCuts = 9)NSubj <- 2000 #generate sample k=1 FMP data b <- matrix(c( #b0 b1 b2 b3 b4 b5 b6 b7 k 1.675, 1.974, -0.068, 0.053, 0, 0, 0, 0, 1, 1.550, 1.805, -0.230, 0.032, 0, 0, 0, 0, 1, 1.282, 1.063, -0.103, 0.003, 0, 0, 0, 0, 1, 0.704, 1.376, -0.107, 0.040, 0, 0, 0, 0, 1, 1.417, 1.413, 0.021, 0.000, 0, 0, 0, 0, 1, -0.008, 1.349, -0.195, 0.144, 0, 0, 0, 0, 1, 0.512, 1.538, -0.089, 0.082, 0, 0, 0, 0, 1, 0.122, 0.601, -0.082, 0.119, 0, 0, 0, 0, 1, 1.801, 1.211, 0.015, 0.000, 0, 0, 0, 0, 1, -0.207, 1.191, 0.066, 0.033, 0, 0, 0, 0, 1, -0.215, 1.291, -0.087, 0.029, 0, 0, 0, 0, 1, 0.259, 0.875, 0.177, 0.072, 0, 0, 0, 0, 1, -0.423, 0.942, 0.064, 0.094, 0, 0, 0, 0, 1, 0.113, 0.795, 0.124, 0.110, 0, 0, 0, 0, 1, 1.030, 1.525, 0.200, 0.076, 0, 0, 0, 0, 1, 0.140, 1.209, 0.082, 0.148, 0, 0, 0, 0, 1, 0.429, 1.480, -0.008, 0.061, 0, 0, 0, 0, 1, 0.089, 0.785, -0.065, 0.018, 0, 0, 0, 0, 1, -0.516, 1.013, 0.016, 0.023, 0, 0, 0, 0, 1, 0.143, 1.315, -0.011, 0.136, 0, 0, 0, 0, 1, 0.347, 0.733, -0.121, 0.041, 0, 0, 0, 0, 1, -0.074, 0.869, 0.013, 0.026, 0, 0, 0, 0, 1, 0.630, 1.484, -0.001, 0.000, 0, 0, 0, 0, 1), nrow=23, ncol=9, byrow=TRUE) data<-genFMPData(NSubj = NSubj, bParam = b, seed = 345)$data ## generate a rest score plot for item 12. ## the grey horizontal lines in the plot ## respresent pseudo asymptotes that ## are significantly different from the ## (0,1) boundaries restScore(data, item = 12, NCuts = 9)
Compute theta surrogates by calculating the normalized left singular vector of a (mean-centered) data matrix.
svdNorm(data)svdNorm(data)
data |
N(subjects)-by-p(items) matrix of 0/1 item response data. |
the normalized left singular vector of the mean centered data matrix. |
|
Niels Waller
NSubj <- 2000 ## example item parameters for sample data: k=1 FMP 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 paramters data<-genFMPData(NSubj=NSubj, bParam=b, seed=345)$data # compute (initial) surrogate theta values from # the normed left singular vector of the centered # data matrix thetaInit<-svdNorm(data)NSubj <- 2000 ## example item parameters for sample data: k=1 FMP b <- matrix(c( #b0 b1 b2 b3 b4 b5 b6 b7 k 1.675, 1.974, -0.068, 0.053, 0, 0, 0, 0, 1, 1.550, 1.805, -0.230, 0.032, 0, 0, 0, 0, 1, 1.282, 1.063, -0.103, 0.003, 0, 0, 0, 0, 1, 0.704, 1.376, -0.107, 0.040, 0, 0, 0, 0, 1, 1.417, 1.413, 0.021, 0.000, 0, 0, 0, 0, 1, -0.008, 1.349, -0.195, 0.144, 0, 0, 0, 0, 1, 0.512, 1.538, -0.089, 0.082, 0, 0, 0, 0, 1, 0.122, 0.601, -0.082, 0.119, 0, 0, 0, 0, 1, 1.801, 1.211, 0.015, 0.000, 0, 0, 0, 0, 1, -0.207, 1.191, 0.066, 0.033, 0, 0, 0, 0, 1, -0.215, 1.291, -0.087, 0.029, 0, 0, 0, 0, 1, 0.259, 0.875, 0.177, 0.072, 0, 0, 0, 0, 1, -0.423, 0.942, 0.064, 0.094, 0, 0, 0, 0, 1, 0.113, 0.795, 0.124, 0.110, 0, 0, 0, 0, 1, 1.030, 1.525, 0.200, 0.076, 0, 0, 0, 0, 1, 0.140, 1.209, 0.082, 0.148, 0, 0, 0, 0, 1, 0.429, 1.480, -0.008, 0.061, 0, 0, 0, 0, 1, 0.089, 0.785, -0.065, 0.018, 0, 0, 0, 0, 1, -0.516, 1.013, 0.016, 0.023, 0, 0, 0, 0, 1, 0.143, 1.315, -0.011, 0.136, 0, 0, 0, 0, 1, 0.347, 0.733, -0.121, 0.041, 0, 0, 0, 0, 1, -0.074, 0.869, 0.013, 0.026, 0, 0, 0, 0, 1, 0.630, 1.484, -0.001, 0.000, 0, 0, 0, 0, 1), nrow=23, ncol=9, byrow=TRUE) # generate data using the above item paramters data<-genFMPData(NSubj=NSubj, bParam=b, seed=345)$data # compute (initial) surrogate theta values from # the normed left singular vector of the centered # data matrix thetaInit<-svdNorm(data)