optical {optical}

Optimal item calibration

Description

Calibrate items following a 2PL, 3PL, mixture of 2PL and 3PL model, or 2PL with common discrimination for all items (Rasch-type).

Usage

optical(
  ip,
  oc = "D",
  uncert = FALSE,
  ipop,
  imf = c(0.005, 0.01, 0.02, 0.05, 0.1, 0.2, 0.45),
  maxiter = rep(300, 6),
  eps = rep(0.002, 6),
  nnn = c(0, 50, 50, 200, 200, 200),
  nsp = c(0.001, 1e-04, 1e-04, 1e-05, 1e-05, 1e-05),
  sss = 0.001,
  falpha = 1.08,
  sdr = TRUE,
  ig = 3,
  ex = 0,
  integ = TRUE,
  show_progress = 1
)

Arguments

ip

matrix with item parameters for all items (number of rows determines number of items; number of column is 2 (2PL or Rasch-type with NA from second item in first column) or 3 (3PL or mixed 2/3-PL with NA for 2PL-items in third column).

oc

optimality criterion: "D" (D-optimality, default), "I" (I-optimality with standard normal weight function), "A" (A-optimality).

uncert

if false (default), abilities are assumed to be known; if true, handling of uncertainties of Bjermo et al. (2021) is used.

ipop

matrix with item parameters for operational items (used if uncert=TRUE, only).

imf

the vector of step-lengths; default c(0.005, 0.01, 0.02, 0.05, 0.1, 0.2, 0.45).

maxiter

maximal number of iterations in each inner loop, the length of this vector defines the number of outer loops.

eps

convergence criterion (maximum violation of eq.th.), vector with value for each iteration in the outer loop, but the same number for all iterations is recommended.

nnn

number of new nodes added at each position in the adaptive grid, vector with value for each iteration in the outer loop (nnn [1] not used).

nsp

node spacing between new nodes, vector with value for each iteration in the outer loop (nsp [1] is the spacing between nodes of the starting grid).

sss

step size stopping criterion.

falpha

factor alpha for adjusting the step size vector (should be > 1).

sdr

stop if design repeated (flag TRUE/FALSE).

ig

inner grid between -ig and ig.

ex

intervals of size < ex will be removed (consolidate); if ex=0, no consolidation will be done.

integ

if true (default), integrate() is used for computation of partial information matrices; if false, Riemann rule is used.

show_progress

if 1 (default), no output will be printed for each iteration. If 2, the + symbols will be printed on a line for each Iteration If 3, some output of the function will be printed.

Value

Result of this function is a list with following instances:

dd

directional derivatives of optimal solution.

xi

optimal solution.

t

final grid of ability values which was used.

viomax

largest violation of eq.th. from final solution (if < eps, alg. has converged, otherwise not).

h1

interval boundaries for optimal solution.

ht

Refined table of interval boundaries for optimal design with calibrated items and their corresponding probabilities

mooiter

monitoring iterations; information about each iteration to produce convergence plots.

time

running time of algorithm in minutes.

oc

optimality criterion ("D", "I", "A", "L").

L

L-matrix (not for D-optimality).

Author(s)

Mahmood Ul Hassan ([e-mail]); Frank Miller ([e-mail])

References

Ul Hassan and Miller (2021). An exchange algorithm for optimal calibration of items in computerized achievement tests.Computational Statistics and Data Analysis, 157: 107177.

Ul Hassan and Miller (2019). Optimal item calibration for computerized achievement tests. Psychometrika, 84, 1101-1128.

Bjermo, Fackle-Fornius, and Miller (2021). Optimizing Calibration Designs with Uncertainty in Abilities. Manuscript.

See Also

drawdesign, convergenceplot, efficiency

Examples

# 2PL-models for two items; parameters (a, b)=(1.6, -1) and (1.6, 1), respectively

ip <- cbind(c(1.6, 1.6),c(-1, 1))

yyy <- optical(ip)

# Table of interval boundaries for D-optimal design with items and
# probabilities (expected proportion of examinees in this interval)
yyy$ht



# 1PL-models with common discrimination parameter for two items
# (model assumption is that both have same discrimination);
# parameters (a, b)=(1.6, -1) and (1.6, 1), respectively;
# NA for discrimination means that item has same parameter as preceeding item
ip <- cbind(c(1.6, NA), c(-1, 1))

yyy <- optical(ip)

# Table of interval boundaries for D-optimal design with items and
# probabilities (expected proportion of examinees in this interval)
yyy$ht


# 3PL-models for three items; parameters (a, b, c)=(1, 2, 2.5),
# (-1.5, 0.5, 2) and (0.2, 0.1, 0.05), respectively.
ip <- cbind(c(1, 2, 2.5),c(-1.5, 0.5, 2),c(0.2, 0.1, 0.05))

yyy <- optical(ip)

# Table of interval boundaries for D-optimal design with items and
# probabilities (expected proportion of examinees in this interval)
yyy$ht