pop_iita {DAKS}

Population Inductive Item Tree Analysis

Description

pop_iita can be used to perform one of the three inductive item tree analysis algorithms (original, corrected, and minimized corrected) in population quantities (in a known population) selectively.

Usage

pop_iita(imp, ce, lg, items, dataset = NULL, A = NULL, v)

Arguments

imp

a required object of class set representing the underlying set of implications (assumed to be a quasi order), for instance obtained from a call to iita.

ce

a required numeric giving the probability for a careless error.

lg

a required numeric giving the probability for a lucky guess.

items

a required numeric giving the number of items of the domain taken as basis for imp.

dataset

an optional data frame or matrix consisting of binary, 1 or 0, numeric data. The default dataset = NULL corresponds to no dataset being used.

A

an optional list of competing quasi orders (surmise relations).

v

a required numeric giving the inductive item tree analysis algorithm to be performed, in population quantities; v = 1 (minimized corrected), v = 2 (corrected), and v = 3 (original).

Details

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 details see iita. 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).

Compared to iita, the function pop_iita implements the three inductive item tree analysis algorithms in population, not sample, quantities. The argument imp must give a quasi order, and equipped with the error probabilities ce and lg, it is considered a special case of the basic local independence model (Doignon and Falmagne, 1999). The latter then is considered as the underlying population model. If dataset = NULL a set of competing quasi orders is constructed based on a population analog of the inductive generation procedure implemented in sample quantities in ind_gen. If a dataset is specified explicitly, that data are used to generate the set of competing quasi orders based on the sample version of the inductive generation procedure.

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 (in dataset) must contain only ones and zeros, which encode solving or failing to solve an item, respectively.

Value

If the arguments imp, ce, lg, items, dataset, A, and v are of required types, pop_iita returns a named list consisting of the following five components:

pop.diff

a vector giving the population diff values corresponding to the (inductively generated) competing quasi orders (subject to selected version that was performed).

pop.matrix

a matrix of all possible response patterns and their corresponding population occurrence probabilities.

error.pop

a vector of the population \gamma rates corresponding to the (inductively generated) competing quasi orders (subject to selected version that was performed).

selection.set

a list of the (inductively generated) competing quasi orders.

v

the version used; v = 1 (minimized corrected), v = 2 (corrected), and v = 3 (original).

Note

The single careless error ce and lucky guess lg probabilities are assumed to be constant over all items. The most general case that can be specified thus includes two error probabilities, which are the same for all items.

The sample diff coefficients of the three inductive item tree analysis algorithms can be transformed into maximum likelihood estimators, by division through the square of sample size. These transformed diff coefficients are considered in population quantities. The \gamma rates are the algorithms' specific estimates of the postulated response error probability.

Population and estimated asymptotic variances of the maximum likelihood estimators diff are implemented in the functions pop_variance and variance, respectively.

Author(s)

Anatol Sargin, Ali Uenlue

References

Doignon, J.-P. and Falmagne, J.-C. (1999) Knowledge Spaces. Berlin, Heidelberg, and New York: Springer-Verlag.

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

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/.

See Also

pop_variance for population asymptotic variances of diff coefficients; variance for estimated asymptotic variances of diff coefficients; simu for data simulation tool; ind_gen for (sample) inductive generation procedure; iita, the interface that provides the three (sample) inductive item tree analysis methods under one umbrella. See also DAKS-package for general information about this package.

Examples

x <- simu(3, 10000, ce = 0.05, lg = 0.05, delta = 0.12)
y <- iita(x$dataset, v = 2)
z <- pop_iita(x$implications, 0.05, 0.05, 3, x$dataset, v = 2)

## similar sample and population diff values are obtained
(y$diff) / (10000^2)
z