View source: R/classification.R
Beta.tp.fit | R Documentation |
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.
Beta.tp.fit(
x,
min,
max,
etl = NULL,
reliability = NULL,
true.model = "4P",
failsafe = FALSE,
l = 0,
u = 1,
output = "parameters"
)
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 |
reliability |
Optional specification of the test-score reliability coefficient. If specified, overrides the input of the |
true.model |
The type of Beta distribution which is to be fit to the moments of the true-score distribution. Options are |
failsafe |
Logical. Whether to revert to a fail-safe two-parameter solution should the four-parameter solution contain invalid parameter estimates. |
l |
If |
u |
If |
output |
Option to specify true-score distribution moments as output if the value of the output argument does not equal |
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.
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
# 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)
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