ltparam and lparam refers the the parameterization used
by the software used to estimate item parameters. The R package ltm,
and the programs IRTPRO and flexMIRT use these parameterizations.
If ltparam is TRUE the latent trait parameterization is used.
Under this parameterization, the three-parameter logistic model is 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.
The two-parameter logistic model, the one-parameter logistic model
and the Rasch model present the same
formulation. The two-parameter logistic model can be obtained
by setting \(c_i\) equal to zero,
the one-parameter logistic model can be obtained
by setting \(c_i\) equal to zero and \(\beta_{2i}\)
costant across items,
while the Rasch model can be obtained by setting \(c_i\)
equal to zero and \(\beta_{2i}\) equal to 1.
If lparam is TRUE the guessing parameters are given under this
parameterization
$$c_i = \frac{\exp(c_i^*)}{1+\exp(c_i^*)}. $$
The modIRT function returns parameter estimates
under the usual IRT parameterization, that is,
$$\pi_i = c_i + (1 - c_i) \frac{\exp[D a_i (\theta - b_i)]}{1 +
\exp[D a_i (\theta - b_i)]},$$
where \(D a_i = \beta_{2i}\), \(b_i = -\beta_{1i}/\beta_{2i}\) and
\(\theta = z\).
If ltparam or lparam are TRUE, the covariance matrix
is calculated using the delta method.
If item parameters are already given under the usual IRT parameterization,
arguments ltparam and lparam should be set to FALSE.