est_item {irtQ}

Fixed ability parameter calibration

Description

This function performs the fixed ability parameter calibration (FAPC), often called Method A, which is the maximum likelihood estimation of item parameters given the ability estimates (Baker & Kim, 2004; Ban, Hanson, Wang, Yi, & Harris, 2001; Stocking, 1988). Also, this could be considered as a special type of the joint maximum likelihood estimation where only one cycle of parameter estimation is implemented given the ability estimates (Birnbaum, 1968). FAPC is one of potentially useful online item calibration methods for computerized adaptive testing (CAT) to put the parameter estimates of pretest items on the same scale of operational item parameter estimates and recalibrate the operational items to evaluate the parameter drifts of the operational items (Chen & Wang, 2016; Stocking, 1988).

Usage

est_item(
  x = NULL,
  data,
  score,
  D = 1,
  model = NULL,
  cats = NULL,
  item.id = NULL,
  fix.a.1pl = FALSE,
  fix.a.gpcm = FALSE,
  fix.g = FALSE,
  a.val.1pl = 1,
  a.val.gpcm = 1,
  g.val = 0.2,
  use.aprior = FALSE,
  use.bprior = FALSE,
  use.gprior = TRUE,
  aprior = list(dist = "lnorm", params = c(0, 0.5)),
  bprior = list(dist = "norm", params = c(0, 1)),
  gprior = list(dist = "beta", params = c(5, 17)),
  missing = NA,
  use.startval = FALSE,
  control = list(eval.max = 500, iter.max = 500),
  verbose = TRUE
)

Arguments

x

A data frame containing the item metadata. This metadata is necessary to obtain the information of each item (i.e., number of score categories and IRT model) to be calibrated. You can easily create an empty item metadata using the function shape_df. When use.startval = TRUE, the item parameters specified in the item metadata are used as the starting values for the item parameter estimation. If x = NULL, the arguments of model and cats must be specified. See irtfit, info or simdat for more details about the item metadata. Default is NULL.

data

A matrix containing examinees' response data for the items in the argument x. A row and column indicate the examinees and items, respectively.

score

A vector of examinees' ability estimates. Length of the vector must be the same as the number of rows in the response data set.

D

A scaling factor in IRT models to make the logistic function as close as possible to the normal ogive function (if set to 1.7). Default is 1.

model

A vector of character strings indicating what IRT model is used to calibrate each item. Available IRT models are "1PLM", "2PLM", "3PLM", and "DRM" for dichotomous items, and "GRM" and "GPCM" for polytomous items. "GRM" and "GPCM" represent the graded response model and (generalized) partial credit model, respectively. Note that "DRM" is considered as "3PLM" in this function. If a single character of the IRT model is specified, that model will be recycled across all items. The provided information in the model argument is used only when x = NULL. Default is NULL.

cats

A numeric vector specifying the number of score categories for each item. For example, a dichotomous item has two score categories. If a single numeric value is specified, that value will be recycled across all items. If cats = NULL and all specified models in the model argument are the dichotomous models (i.e., 1PLM, 2PLM, 3PLM, or DRM), it assumes that all items have two score categories. The provided information in the cats argument is used only when x = NULL. Default is NULL.

item.id

A character vector of item IDs. If NULL, the item IDs are generated automatically. Default is NULL.

fix.a.1pl

A logical value. If TRUE, the slope parameters of the 1PLM items are fixed to a specific value specified in the argument a.val.1pl. Otherwise, the slope parameters of all 1PLM items are constrained to be equal and estimated. Default is FALSE.

fix.a.gpcm

A logical value. If TRUE, the GPCM items are calibrated with the partial credit model and the slope parameters of the GPCM items are fixed to a specific value specified in the argument a.val.gpcm. Otherwise, the slope parameter of each GPCM item is estimated. Default is FALSE.

fix.g

A logical value. If TRUE, the guessing parameters of the 3PLM items are fixed to a specific value specified in the argument g.val. Otherwise, the guessing parameter of each 3PLM item is estimated. Default is FALSE.

a.val.1pl

A numeric value. This value is used to fixed the slope parameters of the 1PLM items.

a.val.gpcm

A numeric value. This value is used to fixed the slope parameters of the GPCM items.

g.val

A numeric value. This value is used to fixed the guessing parameters of the 3PLM items.

use.aprior

A logical value. If TRUE, a prior distribution for the slope parameters is used for the parameter calibration across all items. Default is FALSE.

use.bprior

A logical value. If TRUE, a prior distribution for the difficulty (or threshold) parameters is used for the parameter calibration across all items. Default is FALSE.

use.gprior

A logical value. If TRUE, a prior distribution for the guessing parameters is used for the parameter calibration across all 3PLM items. Default is TRUE.

aprior

A list containing the information of the prior distribution for item slope parameters. Three probability distributions of Beta, Log-normal, and Normal distributions are available. In the list, a character string of the distribution name must be specified in the first internal argument and a vector of two numeric values for the two parameters of the distribution must be specified in the second internal argument. Specifically, when Beta distribution is used, "beta" should be specified in the first argument. When Log-normal distribution is used, "lnorm" should be specified in the first argument. When Normal distribution is used, "norm" should be specified in the first argument. In terms of the two parameters of the three distributions, see dbeta(), dlnorm(), and dnorm() in the stats package for more details.

bprior

A list containing the information of the prior distribution for item difficulty (or threshold) parameters. Three probability distributions of Beta, Log-normal, and Normal distributions are available. In the list, a character string of the distribution name must be specified in the first internal argument and a vector of two numeric values for the two parameters of the distribution must be specified in the second internal argument. Specifically, when Beta distribution is used, "beta" should be specified in the first argument. When Log-normal distribution is used, "lnorm" should be specified in the first argument. When Normal distribution is used, "norm" should be specified in the first argument. In terms of the two parameters of the three distributions, see dbeta(), dlnorm(), and dnorm() in the stats package for more details.

gprior

A list containing the information of the prior distribution for item guessing parameters. Three probability distributions of Beta, Log-normal, and Normal distributions are available. In the list, a character string of the distribution name must be specified in the first internal argument and a vector of two numeric values for the two parameters of the distribution must be specified in the second internal argument. Specifically, when Beta distribution is used, "beta" should be specified in the first argument. When Log-normal distribution is used, "lnorm" should be specified in the first argument. When Normal distribution is used, "norm" should be specified in the first argument. In terms of the two parameters of the three distributions, see dbeta(), dlnorm(), and dnorm() in the stats package for more details.

missing

A value indicating missing values in the response data set. Default is NA.

use.startval

A logical value. If TRUE, the item parameters provided in the item metadata (i.e., the argument x) are used as the starting values for the item parameter estimation. Otherwise, internal starting values of this function are used. Default is FALSE.

control

A list of control parameters to be passed to the optimization function of nlminb() in the stats package. The control parameters set the conditions of the item parameter estimation process such as the maximum number of iterations. See nlminb() in the stats package for details.

verbose

A logical value. If FALSE, all progress messages are suppressed. Default is TRUE.

Details

In most cases, the function est_item will return successfully converged item parameter estimates using the default internal starting values. However, if there is a convergence problem in the calibration, one possible solution is using different starting values. When the item parameter values are specified in the item metadata (i.e., the argument x), those values can be used as the starting values for the item parameter calibration by setting use.startval = TRUE.

Value

This function returns an object of class est_item. Within this object, several internal objects are contained such as:

estimates

A data frame containing both the item parameter estimates and the corresponding standard errors of estimates.

par.est

A data frame containing the item parameter estimates.

se.est

A data frame containing the standard errors of the item parameter estimates. Note that the standard errors are estimated using observed information functions.

pos.par

A data frame containing the position number of item parameters being estimated. The position information is useful when interpreting the variance-covariance matrix of item parameter estimates.

covariance

A matrix of variance-covariance matrix of item parameter estimates.

loglikelihood

A sum of the log-likelihood values of the complete data set across all estimated items.

data

A data frame of the examinees' response data set.

score

A vector of the examinees' ability values used as the fixed effects.

scale.D

A scaling factor in IRT models.

convergence

A string indicating the convergence status of the item parameter estimation.

nitem

A total number of items included in the response data.

deleted.item

The items which have no item response data. Those items are excluded from the item parameter estimation.

npar.est

A total number of the estimated parameters.

n.response

An integer vector indicating the number of item responses for each item used to estimate the item parameters.

TotalTime

Time (in seconds) spent for total compuatation.

The internal objects can be easily extracted using the function getirt.

Author(s)

Hwanggyu Lim [e-mail]

References

Baker, F. B., & Kim, S. H. (2004). Item response theory: Parameter estimation techniques. CRC Press.

Ban, J. C., Hanson, B. A., Wang, T., Yi, Q., & Harris, D., J. (2001) A comparative study of on-line pretest item calibration/scaling methods in computerized adaptive testing. Journal of Educational Measurement, 38(3), 191-212.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In F. M. Lord & M. R. Novick (Eds.), Statistical theories of mental test scores (pp. 397-479). Reading, MA: Addison-Wesley.

Chen, P., & Wang, C. (2016). A new online calibration method for multidimensional computerized adaptive testing. Psychometrika, 81(3), 674-701.

Stocking, M. L. (1988). Scale drift in on-line calibration (Research Rep. 88-28). Princeton, NJ: ETS.

See Also

irtfit, info, simdat, shape_df, sx2_fit, traceline.est_item, getirt

Examples

## import the "-prm.txt" output file from flexMIRT
flex_sam <- system.file("extdata", "flexmirt_sample-prm.txt", package = "irtQ")

# select the item metadata
x <- bring.flexmirt(file=flex_sam, "par")$Group1$full_df

# modify the item metadata so that some items follow 1PLM, 2PLM and GPCM
x[c(1:3, 5), 3] <- "1PLM"
x[c(1:3, 5), 4] <- 1
x[c(1:3, 5), 6] <- 0
x[c(4, 8:12), 3] <- "2PLM"
x[c(4, 8:12), 6] <- 0
x[54:55, 3] <- "GPCM"

# generate examinees' abilities from N(0, 1)
set.seed(23)
score <- rnorm(500, mean=0, sd=1)

# simulate the response data
data <- simdat(x=x, theta=score, D=1)


# 1) item parameter estimation: constrain the slope parameters of the 1PLM to be equal
(mod1 <- est_item(x, data, score, D=1, fix.a.1pl=FALSE, use.gprior=TRUE,
                  gprior=list(dist="beta", params=c(5, 17)), use.startval=FALSE))
summary(mod1)

# extract the item parameter estimates
getirt(mod1, what="par.est")

# 2) item parameter estimation: fix the slope parameters of the 1PLM to 1
(mod2 <- est_item(x, data, score, D=1, fix.a.1pl=TRUE, a.val.1pl=1, use.gprior=TRUE,
                  gprior=list(dist="beta", params=c(5, 17)), use.startval=FALSE))
summary(mod2)

# extract the standard error estimates
getirt(mod2, what="se.est")

# 3) item parameter estimation: fix the guessing parameters of the 3PLM to 0.2
(mod3 <- est_item(x, data, score, D=1, fix.a.1pl=TRUE, fix.g=TRUE, a.val.1pl=1, g.val=.2,
                  use.startval=FALSE))
summary(mod3)

# extract both item parameter and standard error estimates
getirt(mod2, what="estimates")