Calculation of the Elementary Symmetric Functions and Their Derivatives

Calculation of elementary_symmetric_functions (ESFs), their first and, in the case of dichotomous items, second derivatives with sum or difference algorithm for the Rasch, rating scale and partial credit model.

elementary_symmetric_functions(par, order = 0L, log = TRUE,
  diff = FALSE, engine = NULL)

numeric vector or a list. Either a vector of item difficulty parameters of dichotomous items (Rasch model) or a list of item-category parameters of polytomous items (rating scale and partial credit model).


integer between 0 and 2, specifying up to which derivative the ESFs should be calculated. Please note, second order derivatives are currently only possible for dichtomous items in an R implementation engine == "R".


logical. Are the parameters given in par on log scale? Primarily used for internal recursive calls of elementary_symmetric_functions.


logical. Should the first and second derivatives (if requested) of the ESFs calculated with sum (FALSE) or difference algorithm (TRUE).


character, either "C" or "R". If the former, a C implementation is used to calculcate the ESFs and their derivatives, otherwise ("R") pure R code is used.


Depending on the type of par, the elementary symmetric functions for dichotomous (par is a numeric vector) or polytomous items (par is a list) are calculated.

For dichotomous items, the summation and difference algorithm published in Liou (1994) is used. For calculating the second order derivatives, the equations proposed by Jansens (1984) are employed.

For polytomous items, the summation and difference algorithm published by Fischer and Pococny (1994) is used (see also Fischer and Pococny, 1995).


elementary_symmetric_function returns a list of length 1 + order.

If order = 0, then the first (and only) element is a numeric vector with the ESFs of order 0 to the maximum score possible with the given parameters.

If order = 1, the second element of the list contains a matrix, with the rows corresponding to the possible scores and the columns corresponding to the derivatives with respect to the i-th parameter of par.

For dichotomous items and order = 2, the third element of the list contains an array with the second derivatives with respect to every possible combination of two parameters given in par. The rows of the individual matrices still correspond to the possibles scores (orders) starting from zero.


Liou M (1994). More on the Computation of Higher-Order Derivatives of the Elementary Symmetric Functions in the Rasch Model. Applied Psychological Measurement, 18, 53--62.

Jansen PGW (1984). Computing the Second-Order Derivatives of the Symmetric Functions in the Rasch Model. Kwantitatieve Methoden, 13, 131--147.

Fischer GH, and Ponocny I (1994). An Extension of the Partial Credit Model with an Application to the Measurement of Change. Psychometrika, 59(2), 177--192.

Fischer GH, and Ponocny I (1995). “Extended Rating Scale and Partial Credit Models for Assessing Change.” In Fischer GH, and Molenaar IW (eds.). Rasch Models: Foundations, Recent Developments, and Applications.

  • elementary_symmetric_functions
 ## zero and first order derivatives of 100 dichotomous items
 di <- rnorm(100)
 system.time(esfC <- elementary_symmetric_functions(di, order = 1))
 ## again with R implementation
 system.time(esfR <- elementary_symmetric_functions(di, order = 1,
 engine = "R"))

 ## are the results equal?
 all.equal(esfC, esfR)
# }
 ## calculate zero and first order elementary symmetric functions
 ## for 10 polytomous items with three categories each.
 pi <- split(rnorm(20), rep(1:10, each = 2))
 x <- elementary_symmetric_functions(pi)

 ## use difference algorithm instead and compare results
 y <- elementary_symmetric_functions(pi, diff = TRUE)
 all.equal(x, y)
# }
Documentation reproduced from package psychotools, version 0.5-1, License: GPL-2 | GPL-3

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