pop_variance function

Population Asymptotic Variance

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.

pop_variance(pop_matrix, imp, error_pop, v)

Arguments

• pop_matrix: a required matrix of all possible response patterns and their corresponding population occurrence probabilities, for instance obtained from a call to pop_iita.

• 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 pop_iita.

• error_pop: a required numeric giving the $gamma$

  rate to be used for computing diff, for instance obtained from a call to pop_iita.

• v: a required numeric giving the inductive item tree analysis algorithm to be performed, in population quantities; v = 1

  (minimized corrected) and v = 2 (corrected).

Details

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 $i$.'

Returns

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).

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 package for data analysis methods in knowledge space theory. Journal of Statistical Software, 37 (2), 1--31. URL http://www.jstatsoft.org/v37/i02/.

Author(s)

Anatol Sargin, Ali Uenlue

Note

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.

See Also

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.

Examples

## Not run:

x <- simu(5, 100, 0.05, 0.05, delta = 0.15)
y <- pop_iita(x$implications, 0.05, 0.05, 5, x$dataset, v = 2)
pop_variance(y$pop.matrix, 
             y$selection.set[[which(y$pop.diff == min(y$pop.diff))]],
             y$error.pop[which(y$pop.diff == min(y$pop.diff))], v = 2)
## End(Not run)