linking {flexmet}

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

bmat1

FMP item parameters on an anchor test.

bmat2

FMP item parameters to be rescaled.

maxncat

Maximum number of response categories (the first maxncat - 1 columns of bmat1 and bmat2 are intercepts)

cvec1

Vector of lower asymptote parameters for the anchor test.

cvec2

Vector of lower asymptote parameters corresponding to the rescaled item parameters.

dvec1

Vector of upper asymptote parameters for the anchor test.

dvec2

Vector of upper asymptote parameters corresponding to the rescaled item parameters.

k_theta

Complexity of the latent trait transformation (k_theta = 0 is linear, k_theta > 0 is nonlinear).

int

Matrix with two columns, used for numerical integration. Column 1 is a grid of theta values, column 2 are normalized densities associated with the column 1 values.

...

Additional arguments passed to optim.

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

par

(Greek-letter) parameters estimated by optim.

value

Value of the minimized criterion function.

counts

Number of function counts in optim.

convergence

Convergence criterion given by optim.

message

Message given by optim.

tvec

Vector of theta transformation coefficients (t = t0,....,t(2k_\theta+1))

bmat

Transformed bmat2 item parameters.

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 Exploring alternate latent trait metrics with the filtered monotonic polynomial IRT model (Unpublished dissertation).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)