ltm (version 0.8-9)

tpm: 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 data.frame (that will be converted to a numeric matrix using data.matrix()) or a numeric matrix of manifest variables.

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., $1$ denotes the first item, $2$ the second, etc.); the second column denotes the type of parameter to fix for the item specif

max.guessing

a scalar between 0 and 1 denoting the upper bound for the guessing parameters.

IRT.param

logical; if TRUE then the coefficients' estimates are reported under the usual IRT parameterization. See Details for more info.

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 NULL starting values are automatically computed. If "random", random starti

na.action

the na.action to be used on data. In case of missing data, if na.action = NULL the model uses the available cases, i.e., it takes into account the observed part of sample units with missing values

control

a list of control values with elements, [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Value

• 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 IRT.param = TRUE.
• log.Lik
  the log-likelihood value at convergence.
• convergence
  the convergence identifier returned by optim().
• 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) X: a numeric matrix that contains the observed response patterns, and (ii) obs: a numeric vector that contains the observed frequencies for each observed response pattern.
• GH
  a list with two components used in the Gauss-Hermite rule: (i) Z: a numeric matrix that contains the abscissas, and (ii) GHw: a numeric vector that contains the corresponding weights.
• max.sc
  the maximum absolute value of the score vector at convergence.
• type
  the value of the type argument.
• constraint
  the value of the constraint argument.
• max.guessing
  the value of the max.guessing argument.
• IRT.param
  the value of the IRT.param argument.
• X
  a copy of the response data matrix.
• control
  the values used in the control argument.
• na.action
  the value of the na.action argument.
• 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.

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.

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 http://www.jstatsoft.org/v17/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), max.guessing = 1)


# 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))