Find the Greek-Letter Parameterization corresponding to a b Vector of Item Parameters

Description

Convert the b vector of item parameters (polynomial coefficients) to the corresponding Greek-letter parameterization (used to ensure monotonicitiy).

Usage

b2greek(bvec, ncat = 2, eps = 1e-08)

Arguments

Details

See greek2b for more information about the b (polynomial coefficient) and Greek-letter parameterizations of the FMP model.

Value

A vector of item parameters in the Greek-letter parameterization.

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. doi: 10.3102/1076998614556816

See Also

greek2b

Examples

(bvec <- greek2b(xi = 0, omega = 1, alpha = c(.1, .1), tau = c(-2, -2)))
## 0.00000000  2.71828183 -0.54365637  0.29961860 -0.03950623  0.01148330

(b2greek(bvec))
##  0.0  1.0  0.1 -2.0  0.1 -2.0

Estimate FMP Item Parameters

Description

Estimate FMP item parameters for a single item using user-specified theta values (fixed-effects) using fmp_1, or estimate FMP item parameters for multiple items using fixed-effects or random-effects with fmp.

Usage

fmp_1(
  dat,
  k,
  tsur,
  start_vals = NULL,
  method = "CG",
  priors = list(xi = c("none", NaN, NaN), omega = c("none", NaN, NaN), alpha =
    c("none", NaN, NaN), tau = c("none", NaN, NaN)),
  ...
)

fmp(
  dat,
  k,
  start_vals = NULL,
  em = TRUE,
  eps = 1e-04,
  n_quad = 49,
  method = "CG",
  max_em = 500,
  priors = list(xi = c("none", NaN, NaN), omega = c("none", NaN, NaN), alpha =
    c("none", NaN, NaN), tau = c("none", NaN, NaN)),
  ...
)

Arguments

Details

The FMP item response function for a single item i with responses in categories c = 0, ..., C_i - 1 is specified using the composite function,

P(X_i = c | \theta) = exp(\sum_{v=0}^c(b_0i_{v} + m_i(\theta))) / (\sum_{u=0}^{C_i - 1} exp(\sum_{v=0}^u(b_{0i_{v}} + m_i(\theta))))

where m(\theta) is an unbounded and monotonically increasing polynomial function of the latent trait \theta, excluding the intercept (s).

The item complexity parameter k controls the degree of the polynomial:

m(\theta)=b_1\theta+b_2\theta^{2}+...+b_{2k+1} \theta^{2k+1},

where 2k+1 equals the order of the polynomial, k is a nonnegative integer, and

b=(b1,...,b(2k+1))'

are item parameters that define the location and shape of the IRF. The vector b is called the b-vector parameterization of the FMP Model. When k=0, the FMP IRF equals either the slope-threshold parameterization of the two-parameter item response model (if maxncat = 2) or Muraki's (1992) generalized partial credit model (if maxncat > 2).

For m(\theta) to be a monotonic function, the FMP IRF can also be expressed as a function of the vector

\gamma = (\xi, \omega, \alpha_1, \tau_1, \alpha_2, \tau_2, \cdots \alpha_k,\tau_k)'.

The \gamma vector is called the Greek-letter parameterization of the FMP model. See Falk & Cai (2016a), Feuerstahler (2016), or Liang & Browne (2015) for details about the relationship between the b-vector and Greek-letter parameterizations.

Value

References

Chalmers, R. P. (2012). mirt: A multidimensional item response theory package for the R environment. Journal of Statistical Software, 48, 1–29. doi: 10.18637/jss.v048.i06

Elphinstone, C. D. (1983). A target distribution model for nonparametric density estimation. Communication in Statistics–Theory and Methods, 12, 161–198. doi: 10.1080/03610928308828450

Elphinstone, C. D. (1985). A method of distribution and density estimation (Unpublished dissertation). University of South Africa, Pretoria, South Africa.

Falk, C. F., & Cai, L. (2016a). Maximum marginal likelihood estimation of a monotonic polynomial generalized partial credit model with applications to multiple group analysis. Psychometrika, 81, 434–460. doi: 10.1007/s11336-014-9428-7

Falk, C. F., & Cai, L. (2016b). Semiparametric item response functions in the context of guessing. Journal of Educational Measurement, 53, 229–247. doi: 10.1111/jedm.12111

Feuerstahler, L. M. (2016). Exploring alternate latent trait metrics with the filtered monotonic polynomial IRT model (Unpublished dissertation). University of Minnesota, Minneapolis, MN. http://hdl.handle.net/11299/182267

Feuerstahler, L. M. (2019). Metric Transformations and the Filtered Monotonic Polynomial Item Response Model. Psychometrika, 84, 105–123. doi: 10.1007/s11336-018-9642-9

Liang, L. (2007). A semi-parametric approach to estimating item response functions (Unpublished dissertation). The Ohio State University, Columbus, OH. Retrieved from https://etd.ohiolink.edu/

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. doi: 10.3102/1076998614556816

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16, 159–176. doi: 10.1177/014662169201600206

Examples

set.seed(2345)
bmat <- sim_bmat(n_items = 5, k = 2, ncat = 4)$bmat

theta <- rnorm(50)
dat <- sim_data(bmat = bmat, theta = theta, maxncat = 4)

## fixed-effects estimation for item 1

tsur <- get_surrogates(dat)

# k = 0
fmp0_it_1 <- fmp_1(dat = dat[, 1], k = 0, tsur = tsur)

# k = 1
fmp1_it_1 <- fmp_1(dat = dat[, 1], k = 1, tsur = tsur)

## fixed-effects estimation for all items

fmp0_fixed <- fmp(dat = dat, k = 0, em = FALSE)

## random-effects estimation


fmp0_random <- fmp(dat = dat, k = 0, em = TRUE)


## random-effects estimation using mirt's estimation engine

fmp0_mirt <- fmp(dat = dat, k = 0, em = "mirt")

Find Theta Surrogates

Description

Compute surrogate theta values as the set of normalized first principal component scores.

Usage

get_surrogates(dat)

Arguments

Details

Compute surrogate theta values as the normalized first principal component scores.

Value

Vector of surrogate theta values.

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. doi: 10.3102/1076998614556816

Examples

set.seed(2342)
bmat <- sim_bmat(n_items = 5, k = 2)$bmat

theta <- rnorm(50)
dat <- sim_data(bmat = bmat, theta = theta)

tsur <- get_surrogates(dat)

Find the b Vector from a Greek-Letter Parameterization of Item Parameters.

Description

Convert the Greek-letter parameterization of item parameters (used to ensure monotonicitiy) to the b-vector parameterization (polynomial coefficients).

Usage

greek2b(xi, omega, alpha = NULL, tau = NULL)

Arguments

Details

For

m(\theta) = b_{0} + b_{1}\theta + b_{2}\theta^2 + \cdots + b_{2k+1}\theta^{2k+1}

to be a monotonic function, a necessary and sufficient condition is that its first derivative,

p(\theta) = a_{0} + a_{1}\theta + ... + a_{2k}\theta^{2k},

is nonnegative at all theta. Here, let

b_{0} = \xi

be the constant of integration and

b_{s} = a_{s-1}/s

for s = 1, 2, ..., 2k+1. Notice that p(\theta) is a polynomial function of degree 2k. A nonnegative polynomial of an even degree can be re-expressed as the product of k quadratic functions.

If k \geq 1:

p(\theta) = \exp{\omega} \Pi_{s=1}^{k}[1 - 2\alpha_{s}\theta + (\alpha_{s}^2+ \exp(\tau_{s}))\theta^2]

If k = 0:

p(\theta) = 0.

Value

A vector of item parameters in the b parameterization.

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. doi: 10.3102/1076998614556816

See Also

b2greek

Examples

(bvec <- greek2b(xi = 0, omega = 1, alpha = .1, tau = -1))
## 0.0000000  2.7182818 -0.2718282  0.3423943

(b2greek(bvec))
##  0.0  1.0  0.1 -1.0

FMP Item Information Function

Description

Find FMP item information for user-supplied item and person parameters.

Usage

iif_fmp(theta, bmat, maxncat = 2, cvec = NULL, dvec = NULL)

Arguments

Value

Matrix of item information.

Examples

# plot the IIF for a dichotomous item with k = 2

set.seed(2342)
bmat <- sim_bmat(n_items = 1, k = 2)$bmat

theta <- seq(-3, 3, by = .01)

information <- iif_fmp(theta = theta, bmat = bmat)

plot(theta, information, type = 'l')

Numerical Integration Matrix

Description

Create a matrix for numerical integration.

Usage

int_mat(
  distr = dnorm,
  args = list(mean = 0, sd = 1),
  lb = -4,
  ub = 4,
  npts = 10000
)

Arguments

Value

Matrix of two columns. Column 1 is a sequence of x-coordinates, and column 2 is a sequence of y-coordinates from a normalized distribution.

See Also

rimse th_est_ml th_est_eap sl_link hb_link

@importFrom stats dnorm

Polynomial Functions

Description

Evaluate a forward or inverse (monotonic) polynomial function.

Usage

inv_poly(x, coefs, lb = -1000, ub = 1000)

fw_poly(y, coefs)

Arguments

Details

x = t_0 + t_1y + t_2y^2 + ...

Then, for coefs = (t_0, t_1, t_2, ...)^\prime, this function finds the corresponding y value (inv_poly) or x value (fw_poly).

FMP Item Response Function

Description

Find FMP item response probabilities for user-supplied item and person parameters.

Usage

irf_fmp(theta, bmat, maxncat = 2, returncat = NA, cvec = NULL, dvec = NULL)

Arguments

Value

Matrix of item response probabilities.

Examples

# plot the IRF for an item with 4 response categories and k = 2

set.seed(2342)
bmat <- sim_bmat(n_items = 1, ncat = 4, k = 2)$bmat

theta <- seq(-3, 3, by = .01)

probability <- irf_fmp(theta = theta, bmat = bmat,
                       maxncat = 4, returncat = 0:3)

plot(theta, probability[, , 1], type = 'l', ylab = "probability")
points(theta, probability[, , 2], type = 'l')
points(theta, probability[, , 3], type = 'l')
points(theta, probability[, , 4], type = 'l')

Linear and Nonlinear Item Parameter Linking

Description

Link two sets of FMP item parameters using linear or nonlinear transformations of the latent trait.

Usage

sl_link(
  bmat1,
  bmat2,
  maxncat = 2,
  cvec1 = NULL,
  cvec2 = NULL,
  dvec1 = NULL,
  dvec2 = NULL,
  k_theta,
  int = int_mat(),
  ...
)

hb_link(
  bmat1,
  bmat2,
  maxncat = 2,
  cvec1 = NULL,
  cvec2 = NULL,
  dvec1 = NULL,
  dvec2 = NULL,
  k_theta,
  int = int_mat(),
  ...
)

Arguments

Details

The goal of item parameter linking is to find a metric transformation such that the fitted parameters for one test can be transformed to the same metric as those for the other test. In the Haebara approach, the overall sum of squared differences between the original and transformed individual item response functions is minimized. In the Stocking-Lord approach, the sum of squared differences between the original and transformed test response functions is minimized. See Feuerstahler (2016, 2019) for details on linking with the FMP model.

Value

References

Feuerstahler, L. M. (2016). Exploring alternate latent trait metrics with the filtered monotonic polynomial IRT model (Unpublished dissertation). University of Minnesota, Minneapolis, MN. http://hdl.handle.net/11299/182267

Feuerstahler, L. M. (2019). Metric Transformations and the Filtered Monotonic Polynomial Item Response Model. Psychometrika, 84, 105–123. doi: 10.1007/s11336-018-9642-9

Haebara, T. (1980). Equating logistic ability scales by a weighted least squares method. Japanese Psychological Research, 22, 144–149. doi: 10.4992/psycholres1954.22.144

Stocking, M. L., & Lord, F. M. (1983). Developing a common metric in item response theory. Applied Psychological Measurement, 7, 201–210. doi: 10.1002/j.2333-8504.1982.tb01311.x

Examples

set.seed(2342)
bmat <- sim_bmat(n_items = 10, k = 2)$bmat

theta1 <- rnorm(100)
theta2 <- rnorm(100, mean = -1)

dat1 <- sim_data(bmat = bmat, theta = theta1)
dat2 <- sim_data(bmat = bmat, theta = theta2)

# estimate each model with fixed-effects and k = 0
fmp0_1 <- fmp(dat = dat1, k = 0, em = FALSE)
fmp0_2 <- fmp(dat = dat2, k = 0, em = FALSE)

# Stocking-Lord linking


sl_res <- sl_link(bmat1 = fmp0_1$bmat[1:5, ],
                  bmat2 = fmp0_2$bmat[1:5, ],
                  k_theta = 0)


hb_res <- hb_link(bmat1 = fmp0_1$bmat[1:5, ],
                  bmat2 = fmp0_2$bmat[1:5, ],
                  k_theta = 0)

Root Integrated Mean Squared Difference Between FMP IRFs

Description

Compute the root integrated mean squared error (RIMSE) between two FMP IRFs.

Usage

rimse(
  bvec1,
  bvec2,
  ncat = 2,
  c1 = NULL,
  d1 = NULL,
  c2 = NULL,
  d2 = NULL,
  int = int_mat()
)

Arguments

Value

Root integrated mean squared difference between two IRFs (dichotomous items) or expected item scores (polytomous items).

References

Ramsay, J. O. (1991). Kernel smoothing approaches to nonparametric item characteristic curve estimation. Psychometrika, 56, 611–630. doi: 10.1007/BF02294494

Examples

set.seed(2342)
bmat <- sim_bmat(n_items = 2, k = 2, ncat = c(2, 5))$bmat

theta <- rnorm(500)
dat <- sim_data(bmat = bmat, theta = theta, maxncat = 5)

# k = 0
fmp0a <- fmp_1(dat = dat[, 1], k = 0, tsur = theta)
fmp0b <- fmp_1(dat = dat[, 2], k = 0, tsur = theta)


# k = 1
fmp1a <- fmp_1(dat = dat[, 1], k = 1, tsur = theta)
fmp1b <- fmp_1(dat = dat[, 2], k = 1, tsur = theta)


## compare estimated curves to the data-generating curve
rimse(fmp0a$bmat, bmat[1, -c(2:4)])
rimse(fmp0b$bmat, bmat[2, ], ncat = 5)


rimse(fmp1a$bmat, bmat[1, -c(2:4)])
rimse(fmp1b$bmat, bmat[2, ], ncat = 5)

Randomly Generate FMP Parameters

Description

Generate monotonic polynomial coefficients for user-specified item complexities and prior distributions.

Usage

sim_bmat(
  n_items,
  k,
  ncat = 2,
  xi_dist = list(runif, min = -1, max = 1),
  omega_dist = list(runif, min = -1, max = 1),
  alpha_dist = list(runif, min = -1, max = 0.5),
  tau_dist = list(runif, min = -3, max = 0)
)

Arguments

Details

Randomly generate FMP item parameters for a given k value.

Value

Examples

## generate FMP item parameters for 5 dichotomous items all with k = 2
set.seed(2342)
pars <- sim_bmat(n_items = 5, k = 2)
pars$bmat

## generate FMP item parameters for 5 items with varying k values and 
##  varying numbers of response categories
set.seed(2432)
pars <- sim_bmat(n_items = 5, k = c(1, 2, 0, 0, 2), ncat = c(2, 3, 4, 5, 2))
pars$bmat

Simulate FMP Data

Description

Simulate data according to user-specified FMP item parameters and latent trait parameters.

Usage

sim_data(bmat, theta, maxncat = 2, cvec = NULL, dvec = NULL)

Arguments

Value

Matrix of randomly generated binary item responses.

Examples

## generate 5-category item responses for normally distributed theta
##   and 5 items with k = 2

set.seed(2342)
bmat <- sim_bmat(n_items = 5, k = 2, ncat = 5)$bmat

theta <- rnorm(50)
dat <- sim_data(bmat = bmat, theta = theta, maxncat = 5)

Latent Trait Estimation

Description

Compute latent trait estimates using either maximum likelihood (ML) or expected a posteriori (EAP) trait estimation.

Usage

th_est_ml(dat, bmat, maxncat = 2, cvec = NULL, dvec = NULL, lb = -4, ub = 4)

th_est_eap(
  dat,
  bmat,
  maxncat = 2,
  cvec = NULL,
  dvec = NULL,
  int = int_mat(npts = 33)
)

Arguments

Value

Matrix with two columns: est and either sem or psd

Examples

set.seed(3453)
bmat <- sim_bmat(n_items = 20, k = 0)$bmat

theta <- rnorm(10)
dat <- sim_data(bmat = bmat, theta = theta)

## mle estimates
mles <- th_est_ml(dat = dat, bmat = bmat)

## eap estimates
eaps <- th_est_eap(dat = dat, bmat = bmat)

cor(mles[,1], eaps[,1])
# 0.9967317

Transform FMP Item Parameters

Description

Given FMP item parameters for a single item and the polynomial coefficients defining a latent trait transformation, find the transformed FMP item parameters.

Usage

transform_b(bvec, tvec, ncat = 2)

inv_transform_b(bstarvec, tvec, ncat = 2)

Arguments

Details

Equivalent item response models can be written

P(\theta) = b_0 + b_1\theta + b_2\theta^2 + \cdots + b_{2k+1}\theta^{2k+1}

and

P(\theta^\star) = b^\star_0 + b^\star_1\theta^\star + b^\star_2\theta^{\star2}+\cdots + b^\star_{2k^\star+1}\theta^{2k^\star+1}

where

\theta = t_0 + t_1\theta^\star + t_2\theta^{\star 2} + \cdots + t_{2k_\theta+1}\theta^{\star2k_\theta+1}

When using inv_transform_b, be aware that multiple tvec/bstarvec pairings will lead to the same bvec. Users are advised not to use the inv_transform_b function unless bstarvec has first been calculated by a call to transform_b.

Value

Vector of transformed FMP item parameters.

Examples

## example parameters from Table 7 of Reise & Waller (2003)
## goal: transform IRT model to sum score metric

a <- c(0.57, 0.68, 0.76, 0.72, 0.69, 0.57, 0.53, 0.64,
       0.45, 1.01, 1.05, 0.50, 0.58, 0.58, 0.60, 0.59,
       1.03, 0.52, 0.59, 0.99, 0.95, 0.39, 0.50)
b <- c(0.87, 1.02, 0.87, 0.81, 0.75, -0.22, 0.14, 0.56,
       1.69, 0.37, 0.68, 0.56, 1.70, 1.20, 1.04, 1.69,
       0.76, 1.51, 1.89, 1.77, 0.39, 0.08, 2.02)

## convert from difficulties and discriminations to FMP parameters

b1 <- 1.702 * a
b0 <- - 1.702 * a * b
bmat <- cbind(b0, b1)

## theta transformation vector (k_theta = 3)
##  see vignette for details about how to find tvec

tvec <- c(-3.80789e+00, 2.14164e+00, -6.47773e-01, 1.17182e-01,
          -1.20807e-02, 7.02295e-04, -2.13809e-05, 2.65177e-07)

## transform bmat
bstarmat <- t(apply(bmat, 1, transform_b, tvec = tvec))

## inspect transformed parameters
signif(head(bstarmat), 2)

## plot test response function
##  should be a straight line if transformation worked

curve(rowSums(irf_fmp(x, bmat = bstarmat)), xlim = c(0, 23),
      ylim = c(0, 23), xlab = expression(paste(theta,"*")),
      ylab = "Expected Sum Score")
abline(0, 1, col = 2)