pop_iita {DAKS} | R Documentation |
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.
pop_iita(imp, ce, lg, items, dataset = NULL, A = NULL, v)
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). |
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 Sargin and Uenlue (2009), and in
the papers by Sargin and Uenlue (2009) and
Uenlue and Sargin (2008).
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 & 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.
If the arguments imp
, ce
, lg
, items
,
dataset
, and v
are of required types, pop_iita
returns a named list consisting of the following four 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). |
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.
Anatol Sargin anatol.sargin@math.uni-augsburg.de, Ali Uenlue ali.uenlue@math.uni-augsburg.de
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. Manuscript under revision. http://www.math.uni-augsburg.de/~uenlueal/
Sargin, A. and Uenlue, A. (2009) DAKS: An R package for data analysis in knowledge space theory. Manuscript submitted for publication. http://www.math.uni-augsburg.de/~uenlueal/
Uenlue, A. and Sargin, A. (2008) Maximum likelihood methodology for diff fit measures for quasi orders. Manuscript submitted for publication. http://www.math.uni-augsburg.de/~uenlueal/
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.
x <- simu(7, 10000, ce = 0.01, lg = 0.01, delta = 0.12) y <- iita(x$dataset, v = 2) z <- pop_iita(x$implications, 0.01, 0.01, 7, x$dataset, v = 2) ## similar sample and population diff values are obtained (y$diff) / (10000^2) z