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A full audit and refactor of every function, deprecation of functions without a theoretical basis, and new functionality. Arguments and output structures of existing functions are unchanged (new arguments are added at the end, with defaults). Outputs differ where the previous implementation was wrong, numerically inaccurate, or used an unfounded estimator; every such change is listed below. The major version reflects these changes in results.
The following functions now issue a deprecation warning and will
become defunct in the next major version (see
?"betafunctions-deprecated"):
MLA(), MLB(), MLM(): they
return the parameters of a Beta distribution with the same alpha + beta
whose mode equals an observed proportion. That is a re-parameterization,
not an estimate of a “most likely” true-score distribution.
dGammaBinom(), pGammaBinom(),
qGammaBinom(), rGammaBinom(),
dBeta.pGammaBinom(): interpolating the Binomial pmf with
Gamma functions does not give a density, and its renormalized version is
not derived from any model of item responses, so it has no basis as an
error model for non-integer effective test lengths.
gchoose() is retained.
dBeta.pBeta(): a Beta-distributed error with mean
equal to the true score and concentration n has no
grounding in the Livingston and Lewis or strong true-score models. Use
dBeta.pBinom().
MN.CA(), MN.CA.MC(), and
MN.ROC() implement the multinomial and compound multinomial
approach to classification accuracy and consistency of Lee (2007) and
Lee, Brennan, and Wan (2009), as in the MULT-CLASS program. They mirror
the LL. and HB. functions’ arguments and
output, so MC.out.tabular() works on
MN.CA.MC() output. Items can be grouped into fixed,
weighted item sets with any score points. Brennan and Lee’s (2006) bias
correction is optional (on by default), and examinee-level results are
available with conditional = TRUE. MN.CA.MC()
also reports the ordered over- and underclassification rates. The
implementation reproduces both example analyses distributed with
MULT-CLASS v3.0 to the five printed decimals.
LL.CA(), LL.CA.MC(),
HB.CA(), and HB.CA.MC() gain
conditional = FALSE. With TRUE, they return
accuracy and consistency conditional on each possible observed score,
together with each score’s expected proportion and the posterior
probability of each true category. Weighting the conditional values by
the expected proportions recovers the overall indices.
mdo() gains method = c("triad", "ml").
"ml" estimates the one-factor model by maximum likelihood
(via factanal(), rescaled to the covariance metric). It
returns a likelihood-ratio test of the one-factor model with
fit = TRUE, allows negative covariances (warning about
opposite-sign loadings), and warns on Heywood cases. Maximum likelihood
is now the default; method = "triad" reproduces the
estimates of earlier versions.
qBeta.4P(lower.tail = FALSE) (and therefore
qBetaMS(lower.tail = FALSE)) returned
1 - qbeta(p, alpha, beta), which is the upper-tail quantile
only when alpha == beta. It now returns
qbeta(p, alpha, beta, lower.tail = FALSE).
dBetaBinom(), pBetaBinom(),
rBetaBinom(), and betabinomialmoments()
integrated over [0, 1] even when the true-score distribution lives on a
narrower [l, u]. integrate() then missed or double-counted
mass, giving probabilities that could be negative or above 1 (errors
around 2e-4). Integration is now restricted to the support, and the
standard (l = 0, u = 1) case uses the closed-form Beta-Binomial
pmf.
pBetaBinom() and pcBinom() gave wrong
answers for non-integer q and for q > N.
Both now consistently return P[X < q] (lower tail) or P[X >= q]
(upper tail).
LL.CA() used round(N * cut) as the
observed cut-point for accuracy but floor(cut * N) for
consistency, so the two disagreed whenever N * cut had a
fractional part of at least 0.5. Neither matched the rule that a score
at or above the cut passes, which on the effective-test-length scale
means an integer score of at least ceiling(N * cut).
LL.CA(), LL.CA.MC() (which used
round()), and dBeta.pBinom() now all use
ceiling(), so results change whenever N * cut
is not an integer.
observedmoments(correct = TRUE) divided every
central moment by n - 1, which is a bias correction only
for the variance. It now returns the unbiased estimators of the second
to fourth central moments and the k-statistic based (adjusted
Fisher-Pearson) skewness and kurtosis. correct = FALSE
gives the plug-in moments. Because Beta.4p.fit(scores) uses
these defaults, four-parameter fits to raw scores change
slightly.
Model fit in LL.CA() and LL.CA.MC():
when bin widths were fractional (the usual case, since the default of
100 bins rarely divides the effective test length), observed scores were
binned with >=/< on the break-points
while expected frequencies were computed with
pbinom(break - 1). That put the two counts in bins offset
by one score, inflating the chi-square statistic. Both now use the same
binning rule.
Model fit in all LL. and HB. functions:
the p-value was always computed with ncol - 4 degrees of
freedom, even when the reported df was
ncol - 2 (two-parameter model, or fail-safe engaged).
Degrees of freedom are now the number of bins minus one minus the number
of moments matched (ncol - 5 for four-parameter fits,
ncol - 3 for two-parameter fits), following Lord (1965,
p. 244), who gives n - 4 df for an ungrouped four-parameter
fit. The p-value uses this df. A spurious
expected < number of bins condition that blocked
bin-merging when minbin exceeded the number of bins was
also removed. The documentation of the modelfit argument
now describes Lord’s grouping rule (merge only to reach an expected
frequency of 1.0; modelfit = 1 for HB.,
c(nbins, 1) for LL.). The default minimum of
10 is unchanged.
HB.CA() with cut = 0 counted a score of
0 as “below the cut”, which affected the first point of every
HB.ROC() curve. Non-integer cuts (from
HB.ROC() with grainsize != testlength)
silently evaluated the pmf at non-integer scores. Scores at or above
cut are now classified as above it.
LL.CA.MC() labelled categories with
cut / (max - min) * max, which is wrong when
min != 0 (for example, a cut of 150 on a 100–200 scale was
labelled “< 100”). Labels now show the cut-points as
supplied.
ccStats() normalized p but not
p_c by the total, so Kappa was wrong when frequencies
rather than proportions were supplied.
mdo(fit = TRUE): the GFI divided the mean squared
residual by the mean (not mean squared) covariance, so it depended on
the scale of the data. It now uses the standard ULS form
1 - sum(residual^2) / sum(S^2).
betamedian() returned the approximation
(alpha - 1/3) / (alpha + beta - 2/3), which is inaccurate
for small shape parameters. It now returns the exact median.
dGammaBinom() and gchoose() returned
NaN for size above about 170 (Gamma-function
overflow). These are now computed on the log scale where
needed.
dfac() set values of x below
r to zero. That is correct for integers but not for the
non-integer effective scores used when Beta.tp.fit()
estimates the true-score moments, where it made the estimator jump
discontinuously. It now computes the falling factorial for any real
x.
rcBinom() and rBetacBinom() handled
negative probabilities from Lord’s two-term approximation by adding the
absolute value of the smallest probability to every score, which spread
probability mass across the whole scale. Negative probabilities are now
set to zero before renormalizing.
qGammaBinom(p = 0 or 1, lower.tail = FALSE) returned
the wrong end-point. The bisection search, which could loop forever, is
replaced by uniroot().
rcBinom() and rBetacBinom() warned “Sum
of probabilities did not equal 1” on virtually every call (an exact
floating-point comparison, and a sum(weights != 1) typo).
rcBinom() with a vector p longer than
n returned length(p) draws instead of
n.
LL.CA()/LL.CA.MC() computed
classification-consistency statistics from the unnormalized matrix while
reporting the normalized one. Accuracy statistics in the
.MC functions likewise came from the unnormalized confusion
matrix. All statistics now come from the normalized matrices.
The fail-safe in the LL. and HB.
functions now also engages on non-finite or non-positive shape-parameter
estimates, as the documentation describes, instead of erroring.
dBetacBinom() is now exported and documented (its
examples were previously embedded in a parameter description).
betamode() warns when alpha <= 1 or
beta <= 1, where the formula does not give the mode.
LABMSU() warns “Not enough information.” like
UABMSL() does.
mdo() now gives an informative error with fewer than
three variables.
Performance: pBeta.4P() uses pbeta()
rather than two numerical integrals per value.
betamoments() uses closed-form moments.
HB.CA() and HB.CA.MC() compute
observed-category probabilities in closed form rather than one integral
per pair of scores. In testing, LL.CA(),
LL.CA.MC(), HB.CA(), and HB.ROC()
ran 3–12x faster, and HB.CA.MC() 50–200x faster (3 s to
0.015 s at 60 items).
Code quality: duplicated logic is now shared through internal
helpers in R/utils.R (fitting, model fit, ROC plotting,
category labels, accuracy/consistency summaries). Loops and
sapply() calls are vectorized.
&/| in scalar conditions are replaced by
&&/||. Dead code is removed, including
unreachable branches in LABMSU() and UABMSL(),
and HB.CA.MC() recomputing its full consistency matrix once
per category.
Added a testthat test suite. Added
stats and graphics to Imports,
and an .Rbuildignore.
Fixed a bug with fitting the two-parameter beta-distribution
model in the HB.CA(), HB.CA.MC(), and
HB.ROC() functions.
Significantly improved computation time for consistency in the
HB.CA() and HB.CA.MC() functions.
To reduce the verbosity of the output, the LL.and
HB. functions no longer include the positive and negative
likelihood ratios as part of their output.
The AMS() and BMS() functions have had
some arguments removed. Now only allows for specifying mean, variance
(or standard deviation), and location parameters.
The LABMSU() and UABMSL() functions
have had arguments removed (skewness and
kurtosis) due to arguments being removed from
AMS() and BMS().
The Beta.tp.fit() function is significantly
simplified and some arguments (alpha andbeta)
are removed.
The d/p/q/rBetaMS() functions now includes arguments
for specifying lower- and upper-bound location parameters. This allows
the mean to be specified outside the [0, 1] interval.
Changes to model fit argument in the LL. and
HB. functions.
The classification functions LL.CA(),
LL.CA.MC(), HB.CA, and HB.CA.MC()
functions now allows for specifying not to perform model-fit testing by
passing NULL to the modelfit
argument.
Fixed a bug that affected the binning of of observed values. Due to some strange behavior in R that resulted from scaling and re-scaling values, observations would sometimes be assigned to the wrong bins. This should now be fixed.
Added arguments to the mdlfit.gfx() function that
allows for manually controlling a number of plot aesthetics.
Added an implementation of the Hanson-Brennan approach to classification accuracy and consistency.
The HB.CA() and HB.MC.CA() functions
for binary and multiple classifications, respectively.
The HB.ROC() function for ROC analysis.
Added d/p/r functions for Lord’s two-term approximation to the
compound binomial distribution (xcBinom()).
Added d/p/r functions for Beta compound-Binomial distributions
where the compound Binomial is Lord’s two-term approximation
(xBetacBinom()).
Added the mdlfit.gfx() as a visual aid to gauge the
severity of model misfit for the LL and HB approaches.
Added d/p/r functions for Beta-Binomial distributions
(d/p/rBetaBinom(), no q function as of
yet).
Added the betabinomialmoments() function for
calculating raw-, central-, and standardized moments of Beta-Binomial
distributions.
Added the R.ETL() function for calculating the
model-implied reliability of test-scores given the “Effective Test
Length” of Livingston and Lewis (1995), and the mean, variance, and
minimum and maximum possible scores of the observed-score
distribution.
Added the mdo() function which allows for estimating
McDonald’s Omega reliability coefficient. Requires that there are no
negative item-covariances.
Added the MC.out.tabular() function which can be
used to organize the accuracy and consistency output from the
LL.CA.MC() function in tabular format. The
MC.out.tabular() function takes the output of the
LL.CA.MC() function as input.
Changes to the model-fit test procedure for the
LL.CA()and LL.CA.MC() functions. The initial
number of bins are now set to 100 and the minimum bin-size is set to 10.
The bins are now only grouped into the expected number of observations
rather than both expected and observed. The behaviour of this model-fit
test-procedure is still under scrutiny and should as of yet be
considered an experimental approximation procedure.
Added betamode() and betamedian()
functions that allow for calculating the mode and median (respectively)
of two- and four-parameter Beta distributions.
Added modelfit arguments to the LL.CA()
and LL.CA.MC() functions that allow for controlling the
maximum number of- and minimum size of the bins that are to be used for
the chi-square test of model fit. Tuning the maximum number and minimum
size of the bins should facilitate model-fit testing if the default
settings does not result in sufficient degrees of freedom to perform the
significance-test.
Fixed bug in the LL.CA.MC() function regarding the
assembly of the consistency matrix when more than two categories were
included which resulted in some entries having slightly too large
values. This change results in slight changes to the estimated
consistency-indices (now more precise).
The LL.CA() and LL.CA.MC() functions no
longer issue warnings if parameter estimates are out of bounds if the
true.model argument is specified to "2P" and
the failsafe argument is set to TRUE (the
default).
Major updates to the classification accuracy and consistency functionalities:
Added new LL.ROC() functionality, providing the
option to give the ROC-curve a classic, “staircase” look. This option is
made the default.
Added the LL.CA.MC() function extending the
Livingston and Lewis approach to using multiple cut-points. The output
of this function can grow quite verbose when operating with several
cut-points, as diagnostic performance and consistency indices are
estimated and reported for each group separately.
Removed the error.model argument of the
LL.CA() function. In essence, this means that one can no
longer specify a beta error model.
Added model-fit functionality to the LL.CA()
function. The model-fit is examined by comparing observed and
model-expected frequencies by binning frequencies into bins of at least
10 expected and observed frequencies. According to Lord, this model fit
procedure has N-bins - 4 degrees of freedom, meaning that model-fit
cannot be examined if a grouping of the observed-score distribution
reduces to fewer than 5 bins.
Added the gchoose() function generalizing the base-R
choose() function to work with non-integers and positive
integers by calculating the factorials of the Binomial coefficient by
drawing on the Gamma distribution.
Added new set of d/p/q/r functions for a new distribution: The “Gamma-Binomial” distribution. This distribution extends the Binomial distribution to all positive real numbers. Not defined for negative integers.
dGammaBinom(): Probability Density Distribution for
the Gamma-Binomial distribution.
pGammaBinom(): Cumulative Probability Density
Distribution for the Gamma-Binomial distribution.
qGammaBinom(): Quantile function for the
Gamma-Binomial distribution. Calls the pGammaBinom()
function and utilizes a bisecting search-algorithm to find the number of
“successful trials” corresponding to the quantile in question. This
algorithm is rather slow so the function might take longer to find the
appropriate quantile than what one might be used to.
rGammaBinom(): Random number generation for the
Gamma-Binomial distribution. Calls the qGammaBinom()
function. Since the qGammaBinom() function searches for the
appropriate values using a rather inefficient search-algorithm
(bisection), this random-number generation is somewhat slow.
Added the binomialmoments() function, which allows
for calculating the raw, central, and standardized moments of Binomial
distributions (for which the Beta distribution is the conjugate
prior).
Changes to the LL.ROC() function.
Changed the internal behavior of the function to estimate the true-score distribution only once. This should greatly improve the time required to produce the plots.
Added the locate argument where it is possible to
ask the function to locate the operational cut-point at which the values
of sensitivity or NPV are greater than or equal to some value, or
specificity or PPV are lesser than or equal to some value.
Added the maxAcc argument to locate the cut-point at
which the Accuracy statistic is maximized.
The raw-output print-out now contains the cut-point specific Accuracy, PPV, and NPV statistics as well.
Changes to the afac and dfac
functions.
"product" as part of the method
argument.Changes to the tsm function.
tsm function now calls the dfac
function with the direct-arithmetic method for calculating descending
factorials as default. In order to use the gamma function rather than
direct arithmetic, specify any value other than “product” as part of the
method argument.Correction to the LL.CA() function. The Binomial
error distribution evaluated up to and including the cut-point. The
intended behaviour was to evaluate up to but NOT including the
cut-point. Prior to this correction, it is expected that the
LL.CA() function will have underestimated accuracy
somewhat.
dBeta.pBinom()
function as well, which is contrary to the default behavior of base-R’s
pbinom() function.Fixed a bug with the true-score distribution fitting procedure in
the Beta.tp.fit() that could occur for low integer
values.
Minor changes to the documentation for various functions.
Added possibility of calculating descending (falling) and
ascending (rising) factorials by means of the dfac() and
afac()functions.
Added tsm() argument for calculating raw moments of
the true-score distribution under the Livingston and Lewis
approach.
Added confmat() function for organizing supplied
values of true and false positives and negatives into a confusion
matrix.
Changes to the LL.ROC() function. Now allows for
specifying the lower- and upper-bound parameters of the true-score
distribution should “2P” be specified.
A Shiny application providing a GUI for the Livingston and Lewis approach functionality of the package has been developed
The “3P” functionality of Beta.tp.fit(),
LL.CA() and LL.ROC has been removed, as it did
not perform satisfactorily.
The true.model argument of the
Beta.tp.fit() function now includes a "3P"
option, allowing for the specification of one location-parameter (l or
u) and estimating the remaining location-parameter and the
shape-parameters (alpha and beta) so as to make the resulting
distribution have the same skewness and kurtosis as the estimated
true-score distribution.
The AMS() and BMS() functions now issue
warnings if there was not enough information supplied to calculate the
target parameter.
The ETL() function: For the sake of argument
consistency, the l and u arguments are not
renamed min and max, respectively.
Added a number of new functions for working with Beta distributions.
The new LABMSU() function allows for finding the
lower-bound parameter for the four-parameter Beta distribution by
supplying the shape-parameters and moments of the resulting
distribution, and (optionally) the upper-bound location
parameter.
The new UABMSU() function allows for finding the
upper-bound parameter for the four-parameter Beta distribution by
supplying the shape-parameters and moments of the resulting
distribution, and (optionally) the lower-bound location
parameter.
Added additional functionality to some existing functions, allowing for specifying the lower- and upper-bounds, and/or moments of the resulting distribution.
The AMS() function now includes l and
u arguments, finding the Alpha shape parameter necessary to
produce a Beta distribution with target moments and specified lower and
upper bounds of the resulting distribution.
The BMS() function now includes l and
u arguments, finding the Beta shape parameter necessary to
produce a Beta distribution with target moments and specified lower and
upper bounds of the resulting distribution.
The Beta.2p.fit() function, essentially a
wrapper-function for AMS() and BMS(), now
includes l and u arguments for specifying the
bounds for the resulting distribution.
Functions relating to estimating classification accuracy and consistency:
Beta.tp.fit(), LL.CA(), and
LL.ROC():
Added true.model argument allowing for greater
control over the true-score estimation procedure in.
The true.model allows for specifying whether to fit
four- or two-parameter Beta distribution to the estimated moments of the
true-score distribution.
Currently, the options for the argument are “4P” and “2P”. Further options might be added in the future.
The Beta.tp.fit() function:
Now allows for specifying the lower- and/or upper bounds of the two-parameter solutions.
Now allows for specifying that the fitted distribution should
estimate two parameters rather than the default of four. Estimates then
the parameters necessary to produce a distribution with the same mean
and variance as the estimated true-score distribution, given specified
moments, shape- and location parameters (default is l = 0
and u = 1).
Added the option of specifying the true-score distribution
moments (sans a functional form) as output rather than the estimated
parameters of the Beta true-score distribution in the new
output argument. The default is to retrieve the estimated
parameters of the Beta true-score distribution.
When parameters is specified in the
output argument, the effective test length is included as
part of the output. As such, the default output of the
Beta.tp.fit() function is now complete for the purposes of
being used as input for the LL.CA() function.
The LL.CA() function:
The override and failsafe
arguments:
The override argument is rendered inert, to be
removed in a later version.
The failsafe argument is introduced to replace the
override argument. It is essentially an inversion of the
override argument.
The above change is to streamline and make consistent arguments structures across functions.
General code cleaning:
Changed some argument names so as to have the names be consistent across functions. Argument locations in the function calls remain unchanged.
Function arguments previously named lt now named
lower.tail.
Function arguments previously named a or
b now named alpha and beta
(respectively).
Function arguments previously named var now named
variance.
Added the possibility of specifying a Beta error model for the
LL.ROC() function by way of the true.model
argument.
Additional correction of typographical errors in documentation.
Fixed some typographical errors in documentation.
Minor modification to LL.CA() where the function now
won’t automatically terminate if true-score distribution fitting
procedure produces NA or NaN estimates.
True-score distribution fitting now performed using the new
Beta.tp.fit() function.
The grainsize argument is now fully removed from
LL.CA().
Added additional checks and diagnostics in LL.CA(),
issuing warnings for aberrant events.
If a list of parameter values are supplied in place of a
score-vector, it is now necessary to supply an etl
(effective test length) parameter as well. See documentation for the
ETL() function for more information.
Fixed an error in the fitting procedure of
Beta.4P.fit() which occurred during positive skew.
Added the Beta.tp.fit() function for estimating
four-parameter Beta distribution parameters for an underlying true-score
distribution, assuming that the observations are generated from a
Beta-Binomial model.
Fixed the LL.ROC() function which stopped working
after the previous update.
grainsize argument.Added functionality for calculating consistency statistics by way
of the new ccStats() function.
Substantial developments concerning primarily the
LL.CA() function, adding functionality and improving
performance.
Added calculation of consistency statistics by calling the new
ccStats() function.
Added output argument indicating which statistics to
calculate. Default is to compute both accuracy and consistency.
Calculation of distribution-based output now utilizes the
integrate() function.
grainsize argument inert, as
it was used for the previous method.Added possibility of supplying list of custom parameter values for the four-parameter Beta true-score distribution, forgoing the need to estimate one.
Added override argument providing the possibility of
overriding the default fail-safe reverting to a two-parameter Beta
true-score distribution, should the fitting procedure produce
impermissible parameter estimates.
Fixed small typographical errors in documentation.
Added the cba() function for computing Cronbach’s
alpha.
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.