Description Usage Arguments Details Value Note Author(s) References See Also Examples
pop_variance
computes the population (exact) asymptotic
variances of the maximum likelihood estimators diff, assuming
a multinomial probability distribution on the set of all response
patterns.
1 | pop_variance(pop_matrix, imp, error_pop, v)
|
pop_matrix |
a required matrix of all possible response
patterns and their corresponding population occurrence
probabilities, for instance obtained from a call to
|
imp |
a required object of class |
error_pop |
a required numeric giving the gamma
rate to be used for computing diff, for instance obtained
from a call to |
v |
a required numeric giving the inductive item tree analysis
algorithm to be performed, in population quantities; |
Subject to the selected version to be performed, pop_variance
computes the population asymptotic variance of the maximum
likelihood estimator diff, which here is formulated for the
relation specified in imp
and for the gamma
rate in error_pop
. This population variance is obtained
using the delta method, which requires calculating the Jacobian
matrix of the diff coefficient and the inverse of the
expected Fisher information matrix for the multinomial distribution
with cell probabilities as specified in pop_matrix
.
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.'
If the arguments pop_matrix
, imp
, error_pop
,
and v
are of required types, pop_variance
returns a
numeric giving the population asymptotic variance of the maximum
likelihood estimator diff (formulated for the relation in
imp
and the gamma rate in error_pop
).
The current version of the package DAKS does not support computing population asymptotic variances for the original inductive item tree analysis algorithm; population asymptotic variances can be calculated only for the corrected and minimized corrected algorithms.
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.
Estimated asymptotic variances of the maximum likelihood estimators
diff are implemented in the function variance
.
Anatol Sargin, Ali Uenlue
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/.
variance
for estimated asymptotic variances of
diff coefficients; pop_iita
for population
inductive item tree analysis; 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.
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Loading required package: relations
Loading required package: sets
[1] 8.493509e-06
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