tpm  R Documentation 
Fit Birnbaum's three parameter model under the Item Response Theory approach.
tpm(data, type = c("latent.trait", "rasch"), constraint = NULL, max.guessing = 1, IRT.param = TRUE, start.val = NULL, na.action = NULL, control = list())
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 threecolumn numeric matrix specifying fixedvalue constraints. The first column represents the item (i.e., 1 denotes the first item, 2 the second, etc.); the second column denotes the type of parameter to fix for the item specified in the first column (i.e., 1 denotes the guessing parameters, 2 the easiness parameters, and 3 the discrimination parameters); the third column specifies the value at which the corresponding parameter should be fixed. See Examples for more info. 
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 p rows and
3 columns, with p denoting the number of items. If 
na.action 
the 
control 
a list of control values with elements,

Birnbaum's three parameter model is usually employed to handle the phenomenon of nonrandom guessing in the case of difficult items.
The model is defined as follows
π_i = c_i + (1  c_i) / [1 + exp{(beta_{1i} + beta_{2i} z)}],
where
π_i denotes the conditional probability of responding correctly to the ith 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.,
π_i = c_i + (1  c_i) / [1 + exp{beta_{2i} (z  beta_{1i}^*)}].
The fit of the model is based on approximate marginal Maximum Likelihood, using the GaussHermite quadrature rule for the approximation of the required integrals.
An object of class tpm
with components,
coefficients 
a matrix with the parameter values at convergence. These are always the estimates of
beta_i, beta parameters, even if 
log.Lik 
the loglikelihood 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 GaussHermite 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. 
The three parameter model is known to have numerical problems like nonconvergence 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.
Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl
Baker, F. and Kim, SH. (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: AddisonWesley.
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
coef.tpm
,
fitted.tpm
,
summary.tpm
,
anova.tpm
,
plot.tpm
,
vcov.tpm
,
item.fit
,
person.fit
,
margins
,
factor.scores
# 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))
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