fmp {flexmet}

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

dat

Vector of item responses for N (# subjects) examinees. Binary data should be coded 0/1, and polytomous data should be coded 0, 1, 2, etc.

k

Vector of item complexities for each item, see details. If k < ncol(dat), k's will be recycled.

tsur

Vector of N (# subjects) surrogate theta values.

start_vals

Start values, For fmp_1, a vector of length 2k+2 in the following order:

If k = 0: (xi_1, ..., x_C_i - 1, omega)

If k = 1: (xi_1, ..., x_C_i - 1, omega, alpha1, tau1)

If k = 2: (xi_1, ..., x_C_i - 1, omega, alpha1, tau1, alpha2, tau2)

and so forth. For fmp, add start values for item 1, followed by those for item 2, and so forth. For further help, first fit the model without start values, then inspect the outputted parmat data frame.

method

Optimization method passed to optim.

priors

List of prior information used to estimate the item parameters. The list should have up to 4 elements named xi, omega, alpha, tau. Each list should be a vector of length 3: the name of the prior distribution ("norm" or "none"), the first parameter of the prior distribution, and the second parameter of the prior distribution. Currently, "norm" and 'none" are the only available prior distributions.

em

If "mirt", use the mirt (Chalmers, 2012) package to estimate item parameters. If TRUE, random-effects estimation is used via the EM algorithm. If FALSE, fixed effects estimation is used with theta surrogates.

eps

Covergence tolerance for the EM algorithm. The EM algorithm is said to converge is the maximum absolute difference between parameter estimates for successive iterations is less than eps. Ignored if em = FALSE.

n_quad

Number of quadrature points for EM integration. Ignored if em = FALSE

max_em

Maximum number of EM iterations (for em = TRUE only).

...

Additional arguments passed to optim (if em != "mirt") or mirt (if em == "mirt").

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

bmat

Matrix of estimated b-matrix parameters, each row corresponds to an item, and contains b0, b1, ...b(max(k)).

parmat

Data frame of parameter estimation information, including the Greek-letter parameterization, starting value, and parameter estimate.

k

Vector of item complexities chosen for each item.

log_lik

Model log likelihood.

mod

If em == "mirt", the mirt object. Otherwise, optimization information, including output from optim.

AIC

Model AIC.

BIC

Model BIC.

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")