betafunctions (version 1.8.1)

HB.beta.tp.fit: Estimate Beta True-Score Distribution Based on Observed-Score Raw-Moments and Lord's k.

Description

Estimator for the Beta true-score distribution shape-parameters from the observed-score distribution and Lord's k. Returns a list with entries representing the lower- and upper shape parameters (l and u), and the shape parameters (alpha and beta) of the four-parameters beta distribution, as well as Lord's k and the test length.

Usage

HB.beta.tp.fit(x, N, k, true.model = "4P", failsafe = FALSE, l = 0, u = 1)

Value

A list with the parameter values of a four-parameter Beta distribution. "l" is the lower location-parameter, "u" the upper location-parameter, "alpha" the first shape-parameter, and "beta" the second shape-parameter. Also includes Lord's k and the test length.

Arguments

x

Vector of observed-scores.

N

The test length.

k

Lord's k (see documentation for the Lords.k() function).

true.model

The type of Beta distribution which is to be fit to the moments of the true-score distribution. Options are "4P" and "2P", where "4P" refers to the four-parameter (with the same mean, variance, skewness, and kurtosis), and "2P" the two-parameter solution where both location-parameters are specified (with the same mean and variance).

failsafe

Logical. Whether to revert to a fail-safe two-parameter solution should the four-parameter solution contain invalid parameter estimates.

l

If failsafe = TRUE or true.model = "2P": The lower-bound of the Beta distribution. Default is 0 (i.e., the lower-bound of the Standard, two-parameter Beta distribution).

u

If failsafe = TRUE or true.model = "2P": The upper-bound of the Beta distribution. Default is 1 (i.e., the upper-bound of the Standard, two-parameter Beta distribution).

References

Hanson, B. A. (1991). Method of Moments Estimates for the Four-Parameter Beta Compound Binomial Model and the Calculation of Classification Consistency Indexes. American College Testing Research Report Series. Retrieved from https://files.eric.ed.gov/fulltext/ED344945.pdf

Lord, F. M. (1965). A strong true-score theory, with applications. Psychometrika. 30(3). pp. 239--270. doi: 10.1007/BF02289490

Examples

# Generate some fictional data. Say 1000 individuals take a 100-item test
# where all items are equally difficult, and the true-score distribution
# is a four-parameter Beta distribution with location parameters l = 0.25,
# u = 0.75, alpha = 5, and beta = 3, and the error distribution is a
# compound Binomial with Lord's k = 2:
set.seed(12)
testdata <- rcBinom(1000, 100, 2, rBeta.4P(1000, 0.25, 0.75, 5, 3))

# To estimate the four-parameter Beta distribution parameters from this
# sample of observations:
HB.beta.tp.fit(testdata, 100, 2)