tpm {ltm} | R Documentation |
Birnbaum's Three Parameter Model
Description
Fit Birnbaum's three parameter model under the Item Response Theory approach.
Usage
tpm(data, type = c("latent.trait", "rasch"), constraint = NULL,
max.guessing = 1, IRT.param = TRUE, start.val = NULL,
na.action = NULL, control = list())
Arguments
data |
a |
type |
a character string indicating the type of model to fit. Available options are ‘rasch’ that assumes equal discrimination parameter among items, and ‘latent.trait’ (default) that assumes a different discrimination parameter per item. |
constraint |
a three-column numeric matrix specifying fixed-value constraints. The first column represents
the item (i.e., |
max.guessing |
a scalar between 0 and 1 denoting the upper bound for the guessing parameters. |
IRT.param |
logical; if |
start.val |
the character string "random" or a numeric matrix supplying starting values with |
na.action |
the |
control |
a list of control values with elements,
|
Details
Birnbaum's three parameter model is usually employed to handle the phenomenon of non-random guessing in the case of difficult items.
The model is defined as follows
\pi_i = c_i + (1 - c_i) \frac{\exp(\beta_{1i} + \beta_{2i} z)}{1 +
\exp(\beta_{1i} + \beta_{2i} z)},
where
\pi_i
denotes the conditional probability of responding correctly to the i
th item given z
,
c_i
denotes the guessing parameter, \beta_{1i}
is the easiness parameter,
\beta_{2i}
is the discrimination parameter, and z
denotes the
latent ability. In case type = "rasch"
, \beta_{2i}
is assumed equal for all items.
If IRT.param = TRUE
, then the parameters estimates are reported under the usual IRT parameterization,
i.e.,
\pi_i = c_i + (1 - c_i) \frac{\exp[\beta_{2i} (z - \beta_{1i}^*)]}{1 +
\exp[\beta_{2i} (z - \beta_{1i}^*)]}.
The fit of the model is based on approximate marginal Maximum Likelihood, using the Gauss-Hermite quadrature rule for the approximation of the required integrals.
Value
An object of class tpm
with components,
coefficients |
a matrix with the parameter values at convergence. These are always the estimates of
|
log.Lik |
the log-likelihood value at convergence. |
convergence |
the convergence identifier returned by |
hessian |
the approximate Hessian matrix at convergence obtained using a central difference approximation. |
counts |
the number of function and gradient evaluations used by the optimization algorithm. |
patterns |
a list with two components: (i) |
GH |
a list with two components used in the Gauss-Hermite rule: (i) |
max.sc |
the maximum absolute value of the score vector at convergence. |
type |
the value of the |
constraint |
the value of the |
max.guessing |
the value of the |
IRT.param |
the value of the |
X |
a copy of the response data matrix. |
control |
the values used in the |
na.action |
the value of the |
call |
the matched call. |
Warning
The three parameter model is known to have numerical problems like non-convergence or convergence on the boundary,
especially for the guessing parameters. These problems usually result in a zero estimate for some guessing
parameters and/or in a non positive definite Hessian matrix or in a high absolute value for the score vector
(returned by the summary
method) at convergence. In case of estimates on the boundary, the constraint
argument can be used to set the guessing parameter(s) for the problematic item(s) to zero. In addition,
tpm()
has a number of control parameters that can be tuned in order to obtain successful convergence;
the most important of these are the starting values, the parameter scaling vector and the optimizer.
Author(s)
Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl
References
Baker, F. and Kim, S-H. (2004) Item Response Theory, 2nd ed. New York: Marcel Dekker.
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In F. M. Lord and M. R. Novick (Eds.), Statistical Theories of Mental Test Scores, 397–479. Reading, MA: Addison-Wesley.
Rizopoulos, D. (2006) ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17(5), 1–25. URL doi: 10.18637/jss.v017.i05
See Also
coef.tpm
,
fitted.tpm
,
summary.tpm
,
anova.tpm
,
plot.tpm
,
vcov.tpm
,
item.fit
,
person.fit
,
margins
,
factor.scores
Examples
# the three parameter model
tpm(LSAT)
# use 'nlminb' as optimizer
tpm(LSAT, control = list(optimizer = "nlminb"))
# the three parameter model with equal
# discrimination parameter across items
# fix the guessing parameter for the third item to zero
tpm(LSAT, type = "rasch", constraint = cbind(3, 1, 0))
# the three parameter model for the Abortion data
fit <- tpm(Abortion)
fit
# the guessing parameter estimates for items 1, 3, and 4 seem to be on
# the boundary; update the fit by fixing them to zero
update(fit, constraint = cbind(c(1, 3, 4), 1, 0))