mleEst {catIrt} | R Documentation |
mleEst
, wleEst
, bmeEst
, and eapEst
estimate
ability in IRT models. mleEst
is Maximum Likelihood Information, wleEst
is
Weighted Likelihood Information (see Details), bmeEst
is
Bayesian-Modal Estimation, and eapEst
is Expected-A-Posterior Estimation.
mleEst( resp, params, range = c(-6, 6), mod = c("brm", "grm"), ... ) wleEst( resp, params, range = c(-6, 6), mod = c("brm", "grm"), ... ) bmeEst( resp, params, range = c(-6, 6), mod = c("brm", "grm"), ddist = dnorm, ... ) eapEst( resp, params, range = c(-6, 6), mod = c("brm", "grm"), ddist = dnorm, quad = 33, ... )
resp |
numeric: either a N \times J matrix (where N indicates the
number of simulees and J indicates the number of items), a
N length vector (if there is only one item) or a J length
vector (if there is only one simulee). For the binary response model
("brm"), |
params |
numeric: a vector or matrix of item parameters. If specified as a matrix, the rows must index the items, and the columns must designate the item parameters. |
range |
numeric: a two-element numeric vector indicating the minimum and maximum over
which to optimize a likelihood function ( |
mod |
character: a character string indicating the IRT model. Current support
is for the 3-parameter binary response model ("brm"),
and Samejima's graded response model ("grm").
See |
ddist |
function: a function that calculates prior densities for Bayesian
estimation. For instance, if you wish to specify a normal prior, |
quad |
numeric: a scalar indicating the number of quadrature points when
using |
... |
arguments passed to |
These functions return estimated "ability" for the binary response model ("brm") and the graded response model ("grm"). The only difference between the functions is how they estimate ability.
The function mleEst
searches for a maximum of the log-likelihood with respect to
each individual θ_i and uses [T(θ)]^{-1/2} as the corresponding standard
error of measurement (SEM), where T(θ) is the observed test information function
at θ, as described in FI
.
The function bmeEst
searches for the maximum of the log-likelihood after a
log-prior is added, which effectively maximizes the posterior distribution for each
individual θ_i. The SEM of the bmeEst
estimator uses the well known
relationship (Keller, 2000, p. 10)
V[θ | u_i]^{-1} = T(θ) - d l[p(θ)]/dθ^2
where V[θ | u_i] is the variance of θ after
taking into consideration the prior distribution and p(θ) is the prior distribution
of θ. The function bmeEst
estimates the second derivative of the prior
distribution uses the hessian
function in the numDeriv
package.
The function wleEst
searches for the root of a modified score function (i.e.
the first derivative of the log-likelihood with something added to it). The modification
corrects for bias in fixed length tests, and estimation using this modification results in
what is called Weighted Maximum Likelihood (or alternatively, the Warm estimator) (see Warm,
1989). So rather than maximizing the likelihood, wleEst
finds a root of:
d l(θ)/dθ + H(θ)/[2I(θ)]
where l(θ) is the log-likelihood of θ given a set of responses and item parameters, I(θ) is expected test information to this point, and H(θ) is a correction constant defined as:
H(θ) = ∑_j[p_{ij}'p_{ij}''/[p_{ij}(1 - p_{ij})]
for the binary response model, where p_{ij}' is the first derivative
of p_{ij} with respect to θ, p_{ij}'' is
the second derivative of p_{ij} with respect to θ, and p_{ij} is the
probability of response, as indicated in the help page for simIrt
, and
H(θ) = ∑_j∑_k[P_{ijk}'P_{ijk}''/P_{ijk}]
for the graded response model, where P_{ijk}' is the first derivative
of P_{ijk} with respect to θ, P_{ijk}'' is
the second derivative of P_{ijk}, and P_{ijk} is the probability of responding
in category k as indicated in the help page for simIrt
. The SEM of the wleEst
estimator uses an approximation based on Warm (1989, p. 449):
V(θ) \approx (T(θ) + H(θ)/[2I(θ)])/T^2(θ).
The function eapEst
finds the mean and standard deviation of the posterior distribution
given the log-likelihood, a prior distribution (with specified parameters), and the number of
quadrature points using the standard Bayesian identity with summations in place of integrations
(see Bock and Mislevy, 1982). Rather than using the adaptive, quadrature based integrate
,
eapEst
uses the flexible integrate.xy
function in the sfsmisc
package.
As long as the prior distribution is reasonable (such that the joint distribution is relatively smooth),
this method should work.
mleEst
, wleEst
, bmeEst
, and eapEst
return a list of the
following elements:
theta |
an N-length vector of ability values, one for each simulee. |
info |
an N-length vector of observed test information, one for each simulee.
Test information is the sum of item information across items. See |
sem |
an N-length vector of observed standard error of measurement (or posterior standard
deviation) for each simulee. See |
For the binary response model ("brm"), it makes no sense to estimate ability with a non-mixed response pattern (all 0s or all 1s). The user might want to include enough items in the model to allow for reasonable estimation.
Weighted likelihood estimation (wleEst
) uses uniroot
to find the root
of the modified score function, so that the end points of range must evaluate
to opposite signs (or zero). Rarely, the end points of range will evaluate
to the same sign, so that uniroot
will error. In these cases, uniroot will
extend the interval until the end points of the (modified) range are opposite signs.
Steven W. Nydick nydic001@umn.edu
Bock, R. D., & Mislevy, R. J. (1982). Adaptive EAP estimation of ability in a microcomputer environment. Applied Psychological Measurement, 6, 431 – 444.
Embretson, S. E., & Reise, S. P. (2000). Item Response Theory for Psychologists. Mahway, NJ: Lawrence Erlbaum Associates.
Keller (2000). Ability estimation procedures in computerized adaptive testing (Technical Report). New York, NY: American Institute of Certified Public Accountants.
Warm, T. A. (1989). Weighted likelihood estimation of ability in item response theory. Psychometrika, 54, 427 – 450.
van dr Linden, W. J. & Pashley, P. J. (2010). Item selection and ability estimation in adaptive testing. In W. J. van der Linden & C. A. W. Glas (Eds.), Elements of Adaptive Testing. New York, NY: Springer.
catIrt
, eap
,
hessian
, mlebme
,
simIrt
, uniroot
## Not run: ######################### # Binary Response Model # ######################### set.seed(888) # generating random theta: theta <- rnorm(201) # generating an item bank under a 2-parameter binary response model: b.params <- cbind(a = runif(100, .5, 1.5), b = rnorm(100, 0, 2), c = 0) # simulating responses using specified theta: b.resp <- simIrt(theta = theta, params = b.params, mod = "brm")$resp # estimating theta using all four methods: est.mle1 <- mleEst(resp = b.resp, params = b.params, mod = "brm")$theta est.wle1 <- wleEst(resp = b.resp, params = b.params, mod = "brm")$theta est.bme1 <- bmeEst(resp = b.resp, params = b.params, mod = "brm", ddist = dnorm, mean = 0, sd = 1)$theta est.eap1 <- eapEst(resp = b.resp, params = b.params, mod = "brm", ddist = dnorm, mean = 0, sd = 1, quad = 33)$theta # eap takes a while! # all of the methods are highly correlated: cor(cbind(theta = theta, mle = est.mle1, wle = est.wle1, bme = est.bme1, eap = est.eap1)) # you can force eap to be positive: est.eap2 <- eapEst(resp = b.resp, params = b.params, range = c(0, 6), mod = "brm", ddist = dunif, min = 0, max = 6)$theta est.eap2 # if you only have a single response, MLE will give junk! mleEst(resp = 0, params = c(1, 0, .2), mod = "brm")$theta # the others will give you answers that are not really determined by the response: wleEst(resp = 0, params = c(1, 0, .2), mod = "brm")$theta bmeEst(resp = 0, params = c(1, 0, .2), mod = "brm")$theta eapEst(resp = 0, params = c(1, 0, .2), mod = "brm")$theta ######################### # Graded Response Model # ######################### set.seed(999) # generating random theta theta <- rnorm(400) # generating an item bank under a graded response model: g.params <- cbind(a = runif(100, .5, 1.5), b1 = rnorm(100), b2 = rnorm(100), b3 = rnorm(100), b4 = rnorm(100)) # simulating responses using random theta: g.mod <- simIrt(params = g.params, theta = theta, mod = "grm") # pulling out the responses and the parameters: g.params2 <- g.mod$params[ , -1] # now the parameters are sorted g.resp2 <- g.mod$resp # estimating theta using all four methods: est.mle3 <- mleEst(resp = g.resp2, params = g.params2, mod = "grm")$theta est.wle3 <- wleEst(resp = g.resp2, params = g.params2, mod = "grm")$theta est.bme3 <- bmeEst(resp = g.resp2, params = g.params2, mod = "grm", ddist = dnorm, mean = 0, sd = 1)$theta est.eap3 <- eapEst(resp = g.resp2, params = g.params2, mod = "grm", ddist = dnorm, mean = 0, sd = 1, quad = 33)$theta # and the correlations are still pretty high: cor(cbind(theta = theta, mle = est.mle3, wle = est.wle3, bme = est.bme3, eap = est.eap3)) # note that the graded response model is just a generalization of the brm: cor(est.mle1, mleEst(resp = b.resp + 1, params = b.params[ , -3], mod = "grm")$theta) cor(est.wle1, wleEst(resp = b.resp + 1, params = b.params[ , -3], mod = "grm")$theta) cor(est.bme1, bmeEst(resp = b.resp + 1, params = b.params[ , -3], mod = "grm")$theta) cor(est.eap1, eapEst(resp = b.resp + 1, params = b.params[ , -3], mod = "grm")$theta) ## End(Not run)