nrm-methods: Nominal Response Model Probabilities
In plink: IRT Separate Calibration Linking Methods

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

This function computes the probability of responding in a specific category for one or more items for a given set of theta values using the nominal response model or multidimensional nominal response model.

nrm(x, cat, theta, dimensions = 1, items, information = FALSE, angle, ...)

## S4 method for signature 'matrix', 'numeric'
nrm(x, cat, theta, dimensions, items, information, angle, ...)

## S4 method for signature 'data.frame', 'numeric'
nrm(x, cat, theta, dimensions, items, information, angle, ...)

## S4 method for signature 'list', 'numeric'
nrm(x, cat, theta, dimensions, items, information, angle, ...)

## S4 method for signature 'irt.pars', 'ANY'
nrm(x, cat, theta, dimensions, items, information, angle, ...)

## S4 method for signature 'sep.pars', 'ANY'
nrm(x, cat, theta, dimensions, items, information, angle, ...)

`x`	Object containing item parameters. See below for more details.
`cat`	vector identifying the number of response categories for each item
`theta`	vector, matrix, or list of theta values for which probabilities will be computed. If `theta` is not specified, an equal interval range of values from -4 to 4 is used with an increment of 0.5. See details below for more information.
`dimensions`	number of modeled dimensions
`items`	numeric vector identifying the items for which probabilities should be computed
`information`	logical value. If `TRUE` compute item information. In the multidimensional case, information will be computed in the directions specified by `angle` or default angles of 0 - 90 in increments of 10 degrees.
`angle`	vector or matrix of angles between the dimension 1 axis and the corresponding axes for each of the other dimensions for which information will be computed. When there are more than two dimensions and `angle` is a vector, the same set of angles will be used relative to each of the corresponding axes.
`...`	further arguments passed to or from other methods

theta can be specified as a vector, matrix, or list. For the unidimensional case, theta should be a vector. If a matrix or list of values is supplied, they will be converted to a single vector of theta values. For the multidimensional case, if a vector of values is supplied it will be assumed that this same set of values should be used for each dimension. Probabilities will be computed for each combination of theta values. Similarly, if a list is supplied, probabilities will be computed for each combination of theta values. In instances where probabilities are desired for specific combinations of theta values, a j x m matrix should be specified for j ability points and m dimensions where the columns are ordered from dimension 1 to m.

Returns an object of class irt.prob

x = "matrix", cat = "numeric"

This method allows one to specify an n x (m x 2k) matrix for n items, m dimensions, and k equal to the maximum number of response categories across items. The first (m x k) columns are for category slope parameters and the remaining columns are for the category difficulty parameters. For any items with fewer categories than the maximum, the remaining cells in each block of (m x k) columns should be NA.

Unidimensional Specification:: Say we have one four category item and one five category item, the first four columns of the four response item would include the slope parameters. The fifth column for this item would be NA. The next four columns would include the category difficulty values, and the last column would be NA.
Multidimensional Specification:: In the multidimensional case, the columns for the slope and difficulty parameters should be grouped first by dimension and then by category. Using the same example for the two items with two dimensions there will be 20 columns. The first four columns for the four category item would include the slope parameters associated with the first dimension for each of the four categories respectively. Columns 9-10 would be NA. Columns 11-14 would include the category difficulties associated with the first dimension and columns 19-20 would be NA.

x = "data.frame", cat = "numeric"

See the method for x = "matrix"

x = "list", cat = "numeric"

This method is for a list with two elements. The first element is an n x (m x k) matrix of category slope values for n items, m dimensions, and k equal to the maximum number of response categories across items. The second list element is an n x (m x k) matrix of category difficulty parameters. For either element, for items with fewer categories than the maximum, the remaining cells in the rows should be NA (see the examples for method x = "matrix" for specification details).

x = "irt.pars", cat = "ANY"

This method can be used to compute probabilities for the nrm items in an object of class "irt.pars". If x contains dichotomous items or items associated with another polytomous model, a warning will be displayed stating that probabilities will be computed for the nrm items only. If x contains parameters for multiple groups, a list of "irt.prob" objects will be returned.

x = "sep.pars", cat = "ANY"

This method can be used to compute probabilities for the mcm items in an object of class sep.pars. If x contains dichotomous items or items associated with another polytomous model, a warning will be displayed stating that probabilities will be computed for the nrm items only.

Jonathan P. Weeks weeksjp@gmail.com

Bock, R.D. (1972) Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika, 37(1), 29-51.

Bock, R.D. (1996) The nominal categories model. In W.J. van der Linden & Hambleton, R. K. (Eds.) Handbook of Modern Item Response Theory. New York: Springer-Verlag

Bolt, D. M. & Johnson, T. J. (in press) Applications of a MIRT model to self-report measures: Addressing score bias and DIF due to individual differences in response style. Applied Psychological Measurement.

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

Takane, Y., & De Leeuw, J. (1987) On the relationship between item response theory and factor analysis of discretized variables. Psychometrika, 52(3), 393-408.

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/

mixed: compute probabilities for mixed-format items
plot: plot item characteristic/category curves
irt.prob, irt.pars, sep.pars: classes

###### Unidimensional Example ######
## Item parameters from Bock (1972, p. 46,47)
a <- matrix(c(.905, .522, -.469, -.959, NA, 
  .828, .375, -.357, -.079, -.817), 2,5,byrow=TRUE)
c <- matrix(c(.126, -.206, -.257, .336, NA, 
  .565, .865, -1.186, -1.199, .993), 2,5,byrow=TRUE)
pars <- cbind(a,c)
x <- nrm(pars, c(4,5))
plot(x,auto.key=list(space="right"))

###### Multidimensional Example ######
# From Bolt & Johnson (in press)
pars <- matrix(c(-1.28, -1.029, -0.537, 0.015, 0.519, 0.969, 1.343,
1.473, -0.585, -0.561, -0.445, -0.741, -0.584, 1.444,
0.29, 0.01, 0.04, 0.34, 0, -0.04, -0.63), 1,21)
x <- nrm(pars, cat=7, dimensions=2)
# Plot separated surfaces
plot(x,separate=TRUE,drape=TRUE)