iita: Inductive Item Tree Analysis
In DAKS: Data Analysis and Knowledge Spaces

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

iita can be used to perform one of the three inductive item tree analysis algorithms (original, corrected, and minimized corrected) selectively.

1	iita(dataset, v)

`dataset`	a required data frame or matrix consisting of binary, 1 or 0, numeric data.
`v`	a required numeric giving the inductive item tree analysis algorithm to be performed; `v = 1` (minimized corrected), `v = 2` (corrected), and `v = 3` (original).

The three inductive item tree analysis algorithms are exploratory methods for extracting quasi orders (surmise relations) from data. In each algorithm, competing binary relations are generated (in the same way for all three versions), and a fit measure (differing from version to version) is computed for every relation of the selection set in order to find the quasi order that fits the data best. In all three algorithms, the idea is to estimate the numbers of counterexamples for each quasi order, and to find, over all competing quasi orders, the minimum value for the discrepancy between the observed and expected numbers of counterexamples. The three data analysis methods differ in their choices of estimates for the expected numbers of counterexamples. (For an item pair (i, j), the number of subjects solving item j but failing to solve item i, is the corresponding number of counterexamples. Their response patterns contradict the interpretation of (i, j) as 'mastering item j implies mastering item i.') The algorithms are described in the paper about the DAKS package by Uenlue and Sargin (2010), and in the paper by Sargin and Uenlue (2009).

iita calls ind_gen for constructing the set of competing quasi orders according to the inductive generation procedure. Subject to the selected version to be performed, iita computes the discrepancies between observed and expected numbers of counterexamples under each relation, and finds a quasi order with the minimum discrepancy (diff) value.

A set of implications, an object of the class set, consists of 2-tuples (i, j) of the class tuple, where a 2-tuple (i, j) is interpreted as 'mastering item j implies mastering item i.'

The data must contain only ones and zeros, which encode solving or failing to solve an item, respectively.

If the arguments dataset and v are of required types, iita returns a named list consisting of the following five components:

`diff`	a vector giving the diff values corresponding to the (inductively generated) competing quasi orders.
`implications`	an object of class `set` representing the solution quasi order (with smallest diff value) under the selected algorithm.
`error.rate`	a value giving the estimated error rate corresponding to the best fitting quasi order.
`selection.set.index`	a numeric giving the index of the solution quasi order in the selection set.
`v`	the version used; `v = 1` (minimized corrected), `v = 2` (corrected), and `v = 3` (original).

The function iita can be used to perform one of the three inductive item tree analysis procedures selectively. Whereas for the functions orig_iita, corr_iita, mini_iita selection sets of competing quasi orders have to be passed via an argument manually, iita automatically generates a selection set from the data using the inductive generation procedure implemented in ind_gen.

The latter approach using iita is common so far, in knowledge space theory, where the inductive data analysis methods have been utilized for exploratory derivations of surmise relations from data. The functions orig_iita, corr_iita, mini_iita, on the other hand, can be used to select among surmise relations for instance obtained from querying experts or from competing psychological theories.

Anatol Sargin, Ali Uenlue

Sargin, A. and Uenlue, A. (2009) Inductive item tree analysis: Corrections, improvements, and comparisons. Mathematical Social Sciences, 58, 376–392.

Schrepp, M. (1999) On the empirical construction of implications between bi-valued test items. Mathematical Social Sciences, 38, 361–375.

Schrepp, M. (2003) A method for the analysis of hierarchical dependencies between items of a questionnaire. Methods of Psychological Research, 19, 43–79.

Uenlue, A. and Sargin, A. (2010) DAKS: An R package for data analysis methods in knowledge space theory. Journal of Statistical Software, 37(2), 1–31. URL http://www.jstatsoft.org/v37/i02/.

orig_iita for original inductive item tree analysis; corr_iita for corrected inductive item tree analysis; mini_iita for minimized corrected inductive item tree analysis; ind_gen for inductive generation procedure; pop_variance for population asymptotic variances of diff coefficients; variance for estimated asymptotic variances of diff coefficients; z_test for one- and two-sample Z-tests; pop_iita for population inductive item tree analysis. See also DAKS-package for general information about this package.

1 2	iita(pisa, v = 1) iita(pisa, v = 3)

Loading required package: relations
Loading required package: sets

 	 Inductive Item Tree Analysis

Algorithm: minimized corrected IITA

quasi order: {(1L, 2L), (1L, 3L), (1L, 4L), (1L, 5L), (2L, 3L), (2L, 4L), (2L, 5L),
 (3L, 4L), (3L, 5L)}

 	 Inductive Item Tree Analysis

Algorithm: original IITA

quasi order: {(1L, 4L), (1L, 5L), (2L, 4L), (2L, 5L)}