variance {DAKS}

Estimated Asymptotic Variance

Description

variance computes estimated asymptotic variances of the maximum likelihood estimators diff from data, assuming a multinomial probability distribution on the set of all response patterns.

Usage

variance(dataset, imp, v)

Arguments

dataset

a required data frame or matrix consisting of binary, 1 or 0, numeric data.

imp

a required object of class set representing the set of implications (ought to be a quasi order) for which diff is computed, for instance obtained from a call to simu.

v

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

Details

Subject to the selected version to be performed, variance computes a consistent estimator for the population asymptotic variance of the maximum likelihood estimator diff, which here is formulated for the relation specified in imp and for the data in dataset. This estimated asymptotic 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 on the set of all response patterns. In the expression for the exact asymptotic variance, the true parameter vector of multinomial probabilities is estimated by its corresponding maximum likelihood estimate (vector of the relative frequencies of the response patterns).

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.

Value

If the arguments dataset, imp, and v are of required types, variance returns a numeric giving the estimated asymptotic variance of the maximum likelihood estimator diff (formulated for the relation in imp and the data in dataset).

Note

The current version of the package DAKS does not support computing estimated asymptotic variances for the original inductive item tree analysis algorithm; population asymptotic variances can be estimated only for the corrected and minimized corrected algorithms.

The two types of estimators for the population asymptotic variances of the diff coefficients obtained using the expected Fisher information matrix on the one hand, and the observed Fisher information matrix on the other, yield the same result, in the case of the multinomial distribution. Since computation based on expected Fisher information is faster, this is implemented in variance.

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 sample and population quantities.

Population (exact) asymptotic variances of the maximum likelihood estimators diff are implemented in the function pop_variance.

Author(s)

Anatol Sargin, Ali Uenlue

References

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; pop_iita for population inductive item tree analysis; iita, the interface that provides the three (sample) inductive item tree analysis methods under one umbrella; z_test for one- and two-sample Z-tests. See also DAKS-package for general information about this package.

Examples

x <- simu(5, 100, 0.05, 0.05, delta = 0.15)
variance(x$dataset, x$implications, v = 2)