betafunctions (version 1.4.0)

LL.CA: An Implementation of the Livingston and Lewis (1995) Approach to Estimate Classification Consistency and Accuracy based on Observed Test Scores and Test Reliability.

Description

An implementation of what has been come to be known as the "Livingston and Lewis approach" to classification consistency and accuracy, which by employing a compound beta-binomial distribution assumes that true-scores conform to the four-parameter beta distribution, and errors of measurement to the binomial distribution. Under these assumptions, the expected classification consistency and accuracy of tests can be estimated from observed outcomes and test reliability.

Usage

LL.CA(
  x = NULL,
  reliability,
  cut,
  min = 0,
  max = 1,
  true.model = "4P",
  error.model = "binomial",
  truecut = NULL,
  output = c("accuracy", "consistency"),
  failsafe = FALSE,
  l = 0,
  u = 1,
  override = NULL
)

Arguments

x

A vector of observed scores for which a beta-distribution is to be fitted, or a list of pre-defined true-score distribution parameter values. If a list is provided, the list entries must be named after the parameters: l and u for the location parameters, and alpha and beta for the shape parameters.

reliability

The observed-score squared correlation (i.e., proportion of shared variance) with the true-score.

cut

The cutoff value for classifying observations into pass or fail categories.

min

The minimum value possible to attain on the test. Default is 0 (assuming x represent proportions).

max

The maximum value possible to attain on the test. Default is 1 (assuming x represent proportions).

true.model

The probability distribution to be fitted to the moments of the true-score distribution. Options are "4P" (default) and "2P", referring to four- and two-parameter Beta distributions. The "4P" method produces a four-parameter Beta distribution with the same first four moments (mean, variance, skewness, and kurtosis) as the estimated true-score distribution, while the "2P" method produces a two-parameter Beta distribution with the first two moments (mean and variance) as the estimated true-score distribution.

error.model

The probability distribution to be used for producing the sampling distributions at different points of the true-score scale. Options are binomial and beta. The binomial distribution is discrete, and is the distribution used originally by Livingston and Lewis. Use of the binomial distribution involves a rounding of the effective test length to the nearest integer value. The Beta distribution is continuous, and does not involve rounding of the effective test length.

truecut

Optional specification of a "true" cutoff. Useful for producing ROC curves (see documentation for the LL.ROC() function).

output

Character vector indicating which types of statistics (i.e, accuracy and/or consistency) are to be computed and included in the output. Permissible values are "accuracy" and "consistency".

failsafe

Logical value indicating whether to engage the automatic failsafe defaulting to the two-parameter Beta true-score distribution if the four-parameter fitting procedure produces impermissible parameter estimates. Default is FALSE (i.e., the function will not engage failsafe, and will likely produce an error if impermissible parameter estimates were produced.

l

If true.model = "2P" or failsafe = TRUE, the lower-bound location parameter to be used in the two-parameter fitting procedure. Default is 0 (i.e., the )

u

If true.model = "2P" or failsafe = TRUE, the upper-bound location parameter to be used in the two-parameter fitting procedure.

override

Inert artifact from betafunctions version 1.3.1 (replaced by the failsafe argument). Will be removed completely in a later update.

Value

A list containing the estimated parameters necessary for the approach (i.e., the effective test-length and the beta distribution parameters), the confusion matrix containing estimated proportions of true/false pass/fail categorizations for a test, diagnostic performance statistics, and / or a classification consistency matrix and indices. Accuracy output includes a confusion matrix and diagnostic performance indices, and consistency output includes a consistency matrix and consistency indices p (expected proportion of agreement between two independent test administrations), p_c (proportion of agreement on two independent administrations expected by chance alone), and Kappa (Cohen's Kappa).

References

Livingston, Samuel A. and Lewis, Charles. (1995). Estimating the Consistency and Accuracy of Classifications Based on Test Scores. Journal of Educational Measurement, 32(2).

Hanson, Bradley A. (1991). Method of Moments Estimates for the Four-Parameter Beta Compound Binomial Model and the Calculation of Classification Consistency Indexes. American College Testing.

Examples

# NOT RUN {
# Generate some fictional data. Say, 100 individuals take a test with a
# maximum score of 100 and a minimum score of 0.
set.seed(1234)
testdata <- rbinom(100, 100, rBeta.4P(100, 0.25, 0.75, 5, 3))
hist(testdata, xlim = c(0, 100))

# Suppose the cutoff value for attaining a pass is 50 items correct, and
# that the reliability of this test was estimated to 0.7. To estimate and
# retrieve the estimated parameters, confusion matrix, consistency and
# accuracy statistics using LL.CA():
LL.CA(x = testdata, reliability = .7, cut = 50, min = 0, max = 100)

# Suppose the true-score parameter estimation procedure arrived at
# impermissible parameter estimates (i.e., l < 0, u > 1, alpha < 0, or
# beta < 0). For example:
set.seed(9)
testdata <- rbinom(100, 25, rBeta.4P(100, 0.25, 1, 5, 3))
Beta.tp.fit(testdata, 0, 25, 25, failsafe = TRUE)

# Suppose further that you have good grounds for assuming that the lower-
# bound parameter is equal to 0.25 (e.g., the test consists of multiple-
# choice questions with four response options, leading to a 25% probability
# of guessing the correct answer per question), and good reason to believe
# that the upper-bound parameter is equal to 1 (i.e., there is no reason to
# believe that there are no members of the population who will attain a
# perfect score across all possible test-forms.) To set these lower and
# upper bounds for the fitting procedure in the LL.CA() function, set
# the argument true.model = "2p", and specify the location parameters
# l = 0.25 and u = 1:
LL.CA(testdata, 0.6287713, 12, 0, 25, true.model = "2p", l = 0.25, u = 1)

# Alternatively to supplying scores to which a true-score distribution is
# to be fit, a list with true-score distribution parameter values can be
# supplied manually along with the effective test length (see documentation
# for the ETL() function), foregoing the need for actual data. The list
# entries must be named. "l" is the lower-bound and "u" the upper-bound
# location parameters of the true-score distribution, "alpha" and "beta" for
# the shape parameters, and "etl" for the effective test-length..
trueparams <- list("l" = 0.25, "u" = 0.75, "alpha" = 5, "beta" = 3, "etl" = 50)
LL.CA(x = trueparams, cut = 50, min = 0, max = 100)
# }