# Estimated Coefficients and Covariance Matrix of IRT Models Creates an object of the class `modIRT` containing estimated coefficients and covariance matrices of IRT models. Rasch, one-parameter logistic, two-parameter logistic and three-parameter logistic models are included. ```r modIRT(est.mods = NULL, coef = NULL, var = NULL, names = NULL, ltparam = TRUE, lparam = TRUE, display = TRUE, digits = 2) ``` ## Arguments - `est.mods`: list of output objects from functions `rasch`, `ltm`, or `tpm` of the `ltm` package or from function `mirt` of the `mirt` package. - `coef`: necessary only if `est.mods` is `NULL`. List of matrices (one for each form) containing item parameter estimates. Guessing, difficulty and discrimination parameters should strictly be given in this order and they are contained in different columns of the matrix. The names of the rows of each matrix should be the names of the items. - `var`: necessary only if `est.mods` is `NULL`. list of matrices (one for each form) containing the covariance matrix of item parameter estimates. They should be given in the same order of coefficients. - `names`: character vector containing the names of the forms. This should have the same length of `est.mods`, `coef` and `var`. If NULL, the names of the forms are assigned by function `modIRT`. - `ltparam`: logical; if TRUE the latent trait parameterization is used for difficulty parameters and the `modIRT` function performs a transformation of item parameters to return them in the usual IRT parameterization. Set to FALSE to avoid transformations. Not necessary if `est.mods` is not `NULL`. See below for more details. - `lparam`: logical; if TRUE the logistic parameterization is used for guessing parameters and the `modIRT` function performs a transformation of item parameters to return them in the usual IRT parameterization. Set to FALSE to avoid transformations. Not necessary if `est.mods` is not `NULL`. See below for more details. - `display`: logical; if TRUE coefficients and standard errors are printed. - `digits`: integer indicating the number of decimal places to be used if `display` is `TRUE`. ## Details Since package version 2.5.0 this function takes as input `est.mods`. `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 $$ \pi_i = c_i + (1 - c_i) \frac{\exp(\beta_{1i} + \beta_{2i} z)}{1 +\exp(\beta_{1i} + \beta_{2i} z)},\pi_i = c_i + (1 - c_i) * {exp(\beta_{1i} + \beta_{2i} z)}/{1 + exp(\beta_{1i} + \beta_{2i} z)}, $$ 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 $$ c_i = \frac{\exp(c_i^*)}{1+\exp(c_i^*)}.c_i = exp(c_i^*)/{1+exp(c_i^*)}. $$ The `modIRT` function returns parameter estimates under the usual IRT parameterization, that is, $$ \pi_i = c_i + (1 - c_i) \frac{\exp[D a_i (\theta - b_i)]}{1 +\exp[D a_i (\theta - b_i)]},\pi_i = c_i + (1 - c_i) *exp{D a_i (\theta - b_i)} / [1 + exp{D a_i (\theta - b_i)}], $$ 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`. ## Returns An object of class `modIRT` consisting in a list with length equal to the number of forms containing lists with components - **coefficients**: item parameter estimates. - **var**: covariance matrix of item parameter estimates. - **itmp**: number of item parameters of the IRT model. This is 1 for the Rasch model, 2 for the one-parameter logistic model with constant discriminations, 2 for the two-parameter logistic model and 3 for the three-parameter logistic model. ## References Battauz, M. (2015). equateIRT: An R Package for IRT Test Equating. **Journal of Statistical Software**, 68 , 1--22. Bartholomew, D., Knott, M. and Moustaki, I. (2011) **Latent Variable Models and Factor Analysis: a Unified Approach**, 3rd ed. Wiley. Rizopoulos, D. (2006). ltm: an R package for latent variable modelling and item response theory analyses. **Journal of Statistical Software**, 17 , 1--25. ## Author(s) Michela Battauz ## See Also `direc`, `import.ltm` ## Examples ```r # the following code can be used to estimate the item parameters and then # extract the coefficients and their covariance matrices # skipping functions import.mirt or import.ltm ## Not run: library("mirt") data("data2pl") m1 <- mirt(data2pl[[1]], SE = TRUE) m2 <- mirt(data2pl[[2]], SE = TRUE) m3 <- mirt(data2pl[[3]], SE = TRUE) m4 <- mirt(data2pl[[4]], SE = TRUE) m5 <- mirt(data2pl[[5]], SE = TRUE) mlist<- list(m1,m2,m3,m4,m5) test <- paste("test", 1:5, sep = "") mod2pl <- modIRT(est.mods = mlist, names = test, display = FALSE) ## End(Not run) # =========================================================================== # the following code uses item parameter estimates previously obtained # three-parameter logistic model data(est3pl) # includes item parameter estimates test <- paste("test", 1:5, sep = "") mod3pl <- modIRT(coef = est3pl$coef, var = est3pl$var, names = test, display = FALSE) # two-parameter logistic model data(est2pl) # includes item parameter estimates test <- paste("test", 1:5, sep = "") mod2pl <- modIRT(coef = est2pl$coef, var = est2pl$var, names = test, display = FALSE) # Rasch model data(estrasch) # includes item parameter estimates test <- paste("test", 1:5, sep = "") modrasch <- modIRT(coef = estrasch$coef, var = estrasch$var, names = test, display = FALSE) # one-parameter logistic model imported from the R package ltm library(ltm) mod1pl <- rasch(LSAT) summary(mod1pl) est.mod1pl <- import.ltm(mod1pl) mod1pl.ltm <- modIRT(coef = list(est.mod1pl$coef), var = list(est.mod1pl$var), digits = 4) ```

modIRT function