plink-methods: Separate Calibration
In plink: IRT Separate Calibration Linking Methods

Description Usage Arguments Details Value Methods Note Author(s) References Examples

This function conducts separate calibration of unidimensional or multidimensional IRT single-format or mixed-format item parameters for multiple groups. The unidimensional methods include the Mean/Mean, Mean/Sigma, Haebara, and Stocking-Lord methods. The multidimensional methods include a least squares approach (Li & Lissitz, 2000; Min, 2003; Reckase & Martineau, 2004) and extensions of the Haebara and Stocking-Lord method using a single dilation parameter (Li & Lissitz, 2000), multiple dilation parameters (Min, 2003), or an oblique Procrustes approach (Oshima, Davey, & Lee, 2000; Reckase, 2009). The methods allow for symmetric and non-symmetric optimization and chain-linked rescaling of item and ability parameters.

plink(x, common, rescale, ability, method, weights.t, weights.f, 
  startvals, exclude, score = 1, base.grp = 1, symmetric = FALSE, 
  rescale.com = TRUE, grp.names = NULL, dilation = "oblique", 
  md.center = FALSE, dim.order = NULL, ...)

## S4 method for signature 'list', 'matrix'
plink(x, common, rescale, ability, method, weights.t, weights.f, 
  startvals, exclude, score, base.grp, symmetric, rescale.com, 
  grp.names, dilation, md.center, dim.order, ...)

## S4 method for signature 'list', 'data.frame'
plink(x, common, rescale, ability, method, weights.t, weights.f, 
  startvals, exclude, score, base.grp, symmetric, rescale.com, 
  grp.names, dilation, md.center, dim.order, ...)

## S4 method for signature 'list', 'list'
plink(x, common, rescale, ability, method, weights.t, weights.f, 
  startvals, exclude, score, base.grp, symmetric, rescale.com, 
  grp.names, dilation, md.center, dim.order, ...)

## S4 method for signature 'irt.pars', 'ANY'
plink(x, common, rescale, ability, method, weights.t, weights.f, 
  startvals, exclude, score, base.grp, symmetric, rescale.com, 
  grp.names, dilation, md.center, dim.order, ...)

`x`	an object of class `irt.pars` with multiple groups or a list of `irt.pars` and/or `sep.pars` objects.
`common`	matrix or list of common items. See below for more details.
`rescale`	if missing (default), the parameters in `x` will not be transformed to the base scale. To transform the parameters use `"MM","MS","HB","SL","LS"` for the Mean/Mean, Mean/Sigma, Haebara, Stocking-Lord, and least squares linking constants respectively.
`ability`	list of theta values with length equal to the number of groups. If supplied, these values will be transformed to the base scale using the constants identified in `rescale` or the Haebara constants if `rescale` is missing.
`method`	character vector identifying the linking methods to use. Values can include `"MM","MS","HB","SL","LS"` for the Mean/Mean, Mean/Sigma, Haebara, Stocking-Lord, and least squares linking constants respectively, or if missing, constants will be estimated for all methods.
`weights.t`	list containing information about the theta values and weights on the To scale for use with the characteristic curve methods. See below for more details.
`weights.f`	list containing information about the theta values and weights on the From scale for use with the characteristic curve methods. This argument will be ignored if `symmetric`=FALSE. See below for more details.
`startvals`	vector or list of slope and intercept starting values for the characteristic curve methods or character value(s) of "MM", "MS", or "LS" to identify that values from the Mean/Mean, Mean/Sigma, or least squares method should be used for the starting values. See below for more details.
`exclude`	character vector or list identifying common items that should be excluded when estimating the linking constants. See below for more details.
`score`	if `score` = 1, score responses for the Stocking-Lord method with zero for the lowest category and k-1 for the highest, k, category for each item. If `score` = 2, score responses with one for the lowest category and k for the highest, k, category for each item. A vector or list of scoring weights for each response category can be supplied, but this is only recommended for advanced users.
`base.grp`	integer identifying the group for the base scale
`symmetric`	if `TRUE` use symmetric minimization for the characteristic curve methods. See Kim and Lee (2006) for more information
`rescale.com`	if `TRUE` rescale the common item parameters using the estimated linking constants; otherwise, insert the non-transformed common item parameters into the set of unique transformed item parameters
`grp.names`	character vector of group names
`dilation`	character value identifying whether an oblique Procrustes approach (Oshima, Davey, & Lee, 2000; Reckase 2009) `"oblique"`, an orthogonal Procrustes approach with a fixed dilation parameter (Li & Lissitz, 2000) `"orth.fd"`, or an orthogonal Procrustes approach with variable dilations (Min, 2003) `"orth.vd"` should be used to estimate the linking constants for the multidimensional methods. Both orthogonal approaches can be constrained to exclude reflections by using `"orth.fdc"` or `"orth.vdc"`.
`md.center`	if `TRUE` mean center the slope matrices prior to estimating the rotation matrix and dilation constants
`dim.order`	matrix for identifying the ordering of factors across groups. See below for details.
`...`	further arguments passed to or from other methods. See below for details.

If x contains only two elements, common should be a matrix. If x contains more than two elements, common should be a list. In any of the common matrices the first column identifies the common items for the first group of two adjacent list elements in x. The second column in common identifies the corresponding set of common items from the next list element in x. For example, if x contains only two list elements, a single set of common items links them together. If item 4 in group one (row 4 in slot pars) is the same as item 6 in group two, the first row of common would be (4,6).

startvals can be a vector or list of starting values for the slope(s) and intercept(s). For unidimensional methods, when there are only two groups, this argument should be a vector of length of two with the first value for the slope and the second value for the intercept or a character value equal to "MM" or"MS". When there are more than two groups a vector of starting values or a character value can be supplied (the same numeric values, if a vector is supplied, will be used for all pairs, or the corresponding MM/MS values for each pair of tests will be used) or a list of vectors/character values can be supplied with the number of list elements equal to the number of groups minus one. For the multidimensional methods, the same general structure applies (a vector or character value for a single group or a vector, character value or list for multiple groups); however, the length of the vector will vary depending on the dilation approach used. If dilation is "obliquw", the first m*m values in startvals, for m dimensions, should correspond to the values in the transformation matrix (starting with the value in the upper-left corner, then the next value in the column, ..., then the first value in the next column, etc.). The remaining m values should be for the translation vector. If dilation is "orth.fd", the first value will be the slope parameter and the remaining m values will be for the translation vector. If dilation is "orth.vd", the first m values are the slopes for each dimension and the remaining m values are for the translation vector.

weights.t can be a list or a list of lists. The purpose of this object is to specify the theta values on the To scale to integrate over in the characteristic curve methods as well as any weights associated with the theta values. See Kim and Lee (2006) or Kolen and Brennan (2004) for more information of these weights. The function as.weight can be used to facilitate the creation of this object. If weights.t is missing, the default–in the unidimensional case–is to use equally spaced theta values ranging from -4 to 4 with an increment of 0.05 and theta weights equal to one for all theta values. In the multidimensional case, the default is to use 1000 randomly sampled values from a multivariate normal distribution with correlations equal to 0.6 for all dimensions. The theta weights are equal to the normal distribution weights at these points.

To better understand the elements of weights.t, let us assume for a moment that x has parameters for only two groups and that we are using non-symmetric linking. In this instance, weights.t would be a single list with length two. The first element should be a vector of theta values corresponding to points on the To scale. The second list element should be a vector of weights corresponding the theta values. If x contains multiple groups, a single weights.t object can be supplied, and the same set of thetas and weights will be used for all adjacent groups. However, a separate list of theta values and theta weights for each adjacent group in x can be supplied.

The specification of weights.f is the same as that for weights.t, although the theta values and weights for this object correspond to theta values on the From scale. This argument is only used when symmetric=TRUE. If weights.f is missing and symmetric=TRUE, the same theta values and weights used for weights.t will be used for weights.f.

In general, all of the common items identified in x or common will be used in estimating linking constants; however, there are instances where there is a need to exclude certain common items (e.g., NRM or MCM items or items exhibiting parameter drift). Instead of creating a new matrix or list of matrices for common, the exclude argument can be used. exclude can be specified as a character vector or a list. In the former case, a vector of model names (i.e., "drm", "grm", "gpcm", "nrm", "mcm") would be supplied, indicating that any common item on any test associated with the given model(s) would be excluded from the set of items used to estimate the linking constants. If the argument is specified as a list, exclude should have as many elements as groups in x. Each list element can include model names and/or item numbers corresponding to the common items that should be excluded for the given group. If no items need to be excluded for a given group, the list element should be NULL or NA. For example, say we have two groups and we would like to exclude the NRM items and item 23 from the first group, we would specify exclude as exclude <- list(c("nrm",23),NA). Notice that the item number corresponding item 23 in group 2 does not need to be specified.

The argument dim.order is a k x r matrix for k groups and r unique dimensions across groups. This object identifies the common dimensions across groups. The elements in the matrix should correspond to the dimension (i.e., the column in the matrix of slope parameters) for a given group. For example, say there are four unique dimensions across two groups, each group only measures three dimensions, and there are only two common dimensions. We might specify a matrix as follows dim.order <- matrix(c(1:3,NA,NA,1:3),2,4). In words, this means that dimensions 2 and 3 in the first group correspond to dimensions 1 and 2 in the second group respectively. If no dim.order is specified, it is assumed that all of the dimensions are common, or in instances with different numbers of factors, that the first m dimensions for each group are common and the remaining r-m dimensions for a given group are unique.

For the characteristic curve methods, response probabilities are computed using the functions drm, grm, gpcm, nrm, and mcm for the associated models respectively. Various arguments from these functions can be passed to plink. Specifically, the argument incorrect can be passed to drm and catprob can be passed to grm. In the functions drm, grm, and gpcm there is an argument D for the value of a scaling constant. In plink, a single argument D can be passed that will be applied to all applicable models, or arguments D.drm, D.grm, and D.gpcm can be specified for each model respectively. If an argument is specified for D and, say D.drm, the values for D.grm and D.gpcm (if applicable) will be set equal to D. If only D.drm is specified, the values for D.grm and D.gpcm (if applicable) will be set to 1.

Returns an object of class link. The labels for the linking constants are specified in the following manner "group1/group2", meaning the group1 parameters were transformed to the group2 test. The base group is indicated by an asterisk.

x = "list", common = "matrix": This method is used when x contains only two list elements. If either of the list elements is of class irt.pars, they can include multiple groups. common is the matrix of common items between the two groups in x. See details for more information on common.
x = "list", common = "data.frame": See the method for signature x="list", common="matrix".
x = "list", common = "list": This method is used when x includes two or more list elements. When x has length two, common (although a single matrix) should be a list with length one. If x has more than two list elements common identifies the common items between adjacent list elements. If objects of class irt.pars are included with multiple groups, common should identify the common items between the first or last group in the irt.pars object, depending on its location in x, and the adjacent list element(s) in x. For example, if x has three elements: an irt.pars object with one group, an irt.pars object with four groups, and a sep.pars object, common will be a list with length two. The first element in common is a matrix identifying the common items between the items in the first irt.pars object and the first group in the second irt.pars object. The second element in common should identify the common items between the fourth group in the second irt.pars object and the items in the sep.pars object.
x = "irt.pars", common = "ANY": This method is intended for an irt.pars object with multiple groups.

The translation vector for the multidimensional Stocking-Lord method may converge to odd values. The least squares method has been shown to produce more accurate constants than the characteristic curve methods (in less time); however, if use of a characteristic curve approach is desired, it is recommended that the Haebara method be used and/or the relative tolerance for the optimization be lowered to 0.001 using the argument control=list(rel.tol=0.001).

Jonathan P. Weeks weeksjp@gmail.com

Haebara, T. (1980). Equating logistic ability scales by a weighted least squares method. Japanese Psychological Research, 22(3), 144-149.

Kim, S. & Lee, W.-C. (2006). An Extension of Four IRT Linking Methods for Mixed-Format Tests. Journal of Educational Measurement, 43(1), 53-76.

Kolen, M. J. & Brennan, R. L. (2004) Test Equating, Scaling, and Linking (2nd ed.). New York: Springer

Li, Y. H., & Lissitz, R. W. (2000). An evaluation of the accuracy of multidimensional IRT linking. Applied Psychological Measurement, 24(2), 115-138.

Loyd, B. H. & Hoover, H. D. (1980). Vertical Equating Using the Rasch Model. Journal of Educational Measurement, 17(3), 179-193.

Marco, G. L. (1977). Item Characteristic Curve Solutions to Three Intractable Testing Problems. Journal of Educational Measurement, 14(2), 139-160.

Min, K. -S. (2007). Evaluation of linking methods for multidimensional IRT calibrations. Asia Pacific Education Review, 8(1), 41-55.

Oshima, T. C., Davey, T., & Lee, K. (2000). Multidimensional linking: Four practical approaches. Journal of Educational Measurement, 37(4), 357-373.

Reckase, M. D. (2009). Multidimensional item response theory New York, Springer

Reckase, M. D., & Martineau, J. A. (2004). The vertical scaling of science achievement tests. Research report for the Center for Education and National Research Council.

Stocking, M. L. & Lord, F. M. (1983). Developing a common metric in item response theory. Applied Psychological Measurement, 7(2), 201-210.

Weeks, J. P. (2010) plink: An R package for linking mixed-format tests using IRT-based methods. Journal of Statistical Software, 35(12), 1–33. URL http://www.jstatsoft.org/v35/i12/

###### Unidimensional Examples ######
# Create irt.pars object with two groups (all dichotomous items),
# rescale the item parameters using the Mean/Sigma linking constants,
# and exclude item 27 from the common item set
pm <- as.poly.mod(36)
x <- as.irt.pars(KB04$pars, KB04$common, cat=list(rep(2,36),rep(2,36)), 
  poly.mod=list(pm,pm))
out <- plink(x, rescale="MS", base.grp=2, D=1.7, exclude=list(27,NA))
summary(out, descrip=TRUE)
pars.out <- link.pars(out)


# Create object with six groups (all dichotomous items)
pars <- TK07$pars
common <- TK07$common
cat <- list(rep(2,26),rep(2,34),rep(2,37),rep(2,40),rep(2,41),rep(2,43))
pm1 <- as.poly.mod(26)
pm2 <- as.poly.mod(34)
pm3 <- as.poly.mod(37)
pm4 <- as.poly.mod(40)
pm5 <- as.poly.mod(41)
pm6 <- as.poly.mod(43)
pm <- list(pm1, pm2, pm3, pm4, pm5, pm6)
x <- as.irt.pars(pars, common, cat, pm, 
  grp.names=paste("grade",3:8,sep=""))
out <- plink(x)
summary(out)
constants <- link.con(out) # Extract linking constants

# Create an irt.pars object and a sep.pars object for two groups of
# nominal response model items. Compare non-symmetric and symmetric 
# minimization. Note: This example may take a minute or two to run
## Not run: 
pm <- as.poly.mod(60, "nrm", 1:60)
pars1 <- as.irt.pars(act.nrm$yr97, cat=rep(5,60), poly.mod=pm)
pars2 <- sep.pars(act.nrm$yr98, cat=rep(5,60), poly.mod=pm)
out <- plink(list(pars1, pars2), matrix(1:60,60,2))
out1 <- plink(list(pars1, pars2), matrix(1:60,60,2), symmetric=TRUE)
summary(out, descrip=TRUE)
summary(out1, descrip=TRUE)

## End(Not run)

# Compute linking constants for two groups with multiple-choice model
# item parameters. Rescale theta values and item parameters using
# the Haebara linking constants
# Note: This example may take a minute or two to run
## Not run: 
theta <- rnorm(100) # In practice, estimated theta values would be used
pm <- as.poly.mod(60, "mcm", 1:60)
x <- as.irt.pars(act.mcm, common=matrix(1:60,60,2), cat=list(rep(6,60),
  rep(6,60)), poly.mod=list(pm,pm))
out <- plink(x, ability=list(theta,theta), rescale="HB")
pars.out <- link.pars(out)
ability.out <- link.ability(out)
summary(out, descrip=TRUE)

## End(Not run)

# Compute linking constants for two groups using mixed-format items and 
# a mixed placement of common items. Compare calibrations with the
# inclusion or exclusion of NRM items. This example uses the dgn dataset.
pm1 <- as.poly.mod(55,c("drm","gpcm","nrm"),dgn$items$group1)
pm2 <- as.poly.mod(55,c("drm","gpcm","nrm"),dgn$items$group2)
x <- as.irt.pars(dgn$pars,dgn$common,dgn$cat,list(pm1,pm2))
# Run with the NRM common items included
out <- plink(x) 
# Run with the NRM common items excluded
out1 <- plink(x,exclude="nrm") 
summary(out)
summary(out1)


# Compute linking constants for six groups using mixed-format items and 
# a mixed placement of common items. This example uses the reading dataset.
# See the information on the dataset for an interpretation of the output.
pm1 <- as.poly.mod(41, c("drm", "gpcm"), reading$items[[1]])
pm2 <- as.poly.mod(70, c("drm", "gpcm"), reading$items[[2]])
pm3 <- as.poly.mod(70, c("drm", "gpcm"), reading$items[[3]])
pm4 <- as.poly.mod(70, c("drm", "gpcm"), reading$items[[4]])
pm5 <- as.poly.mod(72, c("drm", "gpcm"), reading$items[[5]])
pm6 <- as.poly.mod(71, c("drm", "gpcm"), reading$items[[6]])
pm <- list(pm1, pm2, pm3, pm4, pm5, pm6)
x <- as.irt.pars(reading$pars, reading$common, reading$cat, pm, base.grp=4)
out <- plink(x)
summary(out)


###### Multidimensional Examples ######
# Reckase Chapter 9
pm <- as.poly.mod(80, "drm", list(1:80))
x <- as.irt.pars(reckase9$pars, reckase9$common, 
  list(rep(2,80),rep(2,80)), list(pm,pm), dimensions=2)
# Compute constants using the least squares method and 
# the orthongal rotation with variable dilation. 
# Rescale the item parameters using the LS method
out <- plink(x, dilation="orth.vd", rescale="LS")
summary(out, descrip=TRUE)
# Extract the rescaled item parameters
pars.out <- link.pars(out)

# Compute constants using an oblique Procrustes method
# Display output and descriptives
out <- plink(x, dilation="oblique")
summary(out, descrip=TRUE)

# Compute constants using and orthogonal rotation with
# a fixed dilation parameter 
# Rescale the item parameters and ability estimates 
# using the "HB" and "LS" methods.
# Display the optimization iterations
ability <- matrix(rnorm(40),20,2)
out <- plink(x, method=c("HB","LS"), dilation="orth.fd", 
   rescale="HB", ability=list(ability,ability),
   control=list(trace=1,rel.tol=0.001))
summary(out)
# Extract rescaled ability estimates
ability.out <- out$ability


# Compute linking constants for two groups using mixed-format items 
# modeled with the M3PL and MGPCM. Only compute constants using the 
# orth.vd dilation.
pm <- as.poly.mod(60,c("drm","gpcm"), list(c(1:60)[md.mixed$cat==2],
  c(1:60)[md.mixed$cat>2]))
x <- as.irt.pars(md.mixed$pars, matrix(1:60,60,2), 
  list(md.mixed$cat, md.mixed$cat), 
  list(pm, pm), dimensions=4, grp.names=c("Form.X","Form.Y"))
out <- plink(x,dilation="orth.vd")
summary(out, descrip=TRUE)


# Illustration of construct shift and how to specify common dimensions
pm <- as.poly.mod(80, "drm", list(1:80))
pars <- cbind(round(runif(80),2),reckase9$pars[[1]])
x <- as.irt.pars(list(pars,reckase9$pars[[2]]), reckase9$common, 
list(rep(2,80),rep(2,80)), list(pm,pm), dimensions=c(3,2))
dim.order <- matrix(c(1,2,3,NA,1,2),2,3,byrow=TRUE)
out <- plink(x, dilation="oblique", dim.order=dim.order, rescale="LS")
summary(out)
pars.out <- link.pars(out)