betafunctions (version 1.9.0)

Beta.tp.fit: Estimate Beta true-score distribution based on observed-score raw-moments and the effective test length.

Description

Estimator for the Beta true-score distribution shape-parameters from the observed-score distribution and Livingston and Lewis' effective test length. 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, and the effective test length.

Usage

Beta.tp.fit(
  x,
  min,
  max,
  etl = NULL,
  reliability = NULL,
  true.model = "4P",
  failsafe = FALSE,
  l = 0,
  u = 1,
  output = "parameters"
)

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, "beta" the second shape-parameter, and "etl" the effective test length.

Arguments

x

Vector of observed-scores.

min

The minimum possible score to attain on the test.

max

The maximum possible score to attain on the test.

etl

The value of Livingston and Lewis' effective test length. See ?ETL(). Not necessary to specify if reliability is supplied to the reliability argument.

reliability

Optional specification of the test-score reliability coefficient. If specified, overrides the input of the etl argument.

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).

output

Option to specify true-score distribution moments as output if the value of the output argument does not equal "parameters".

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

Rogosa, D. & Finkelman, M. (2004). How Accurate Are the STAR Scores for Individual Students? An Interpretive Guide. Retrieved from http://statweb.stanford.edu/~rag/accguide/guide04.pdf

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:
set.seed(12)
testdata <- rbinom(1000, 100, rBeta.4P(1000, 0.25, 0.75, 5, 3))

# Since this test contains items which are all equally difficult, the true
# effective test length (etl) is the actual test length. I.e., etl = 100.
# To estimate the four-parameter Beta distribution parameters underlying
# the draws from the binomial distribution:
Beta.tp.fit(testdata, 0, 100, 100)

# Imagine a case where the fitting procedure produces an impermissible
# estimate (e.g., l < 0 or u > 1).
set.seed(1234)
testdata <- rbinom(1000, 50, rBeta.4P(1000, 0.25, 0.75, 5, 3))
Beta.tp.fit(testdata, 0, 50, 50)

# This example produced an l-value estimate less than 0. One way of
# dealing with such an occurrence is to revert to a two-parameter
# model, specifying the l and u parameters and estimating the
# alpha and beta parameters necessary to produce a Beta distribution
# with the same mean and variance as the estimated true-score distribution.

# Suppose you have good theoretical reasons to fix the l parameter at a
# value of 0.25 (e.g., the test is composed of multiple-choice questions
# with four response-options, resulting in a 25% chance of guessing the
# correct answer). The l-parameter could be specified to this theoretically
# justified value, and the u-parameter could be specified to be equal to the
# estimate above (u = 0.7256552) as such:
Beta.tp.fit(testdata, 0, 50, 50, true.model = "2P", l = 0.25, u = 0.7256552)