# Direct Equating Coefficients Calculates direct equating coefficients and standard errors using IRT methods. ```r direc(mods, which, mod1, mod2, method = "mean-mean", suff1 = ".1", suff2 = ".2", D = 1, quadrature = TRUE, nq = 30, items.select = NULL) ``` ## Arguments - `mods`: an object of the class `modIRT` containing item parameter coefficients and their covariance matrix of the forms to be equated. - `which`: which forms to equate. Can be specified by name or number. - `mod1`: deprecated; please use mods instead. An object of the class `modIRT` containing item parameter coefficients and their covariance matrix of the first form. - `mod2`: deprecated; please use mods instead. An object of the class `modIRT` containing item parameter coefficients and their covariance matrix of the second form. - `method`: the equating method to be used. This should be one of "`mean-mean`", "`mean-gmean`", "`mean-sigma`", "`Haebara`" or "`Stocking-Lord`". - `suff1`: suffix to identify the first form to be equated. - `suff2`: suffix to identify the second form to be equated. - `D`: constant D of the IRT model used to estimate item parameters. See below for more details. - `quadrature`: logical; if TRUE the Gauss-Hermite quadrature is used to approximate the integral in the function that is minimized in the Haebara and Stocking-Lord methods. If FALSE the integral is replaced with a sum over 40 equally spaced values ranging from -4 to 4 with an increment of 0.05 and weights equal to one for all values. - `nq`: number of quadrature points used for the Gauss-Hermite quadrature if `quadrature` is TRUE. - `items.select`: optional character vector including the names of the items to use for equating. ## Details Equating coefficients perform the conversion from the scale of the first form to the scale of the second form. In the three-parameter logistic model the probability of a positive response on item $i$ 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 $a_i$ is the item discrimination parameter, $b_i$ is the item difficulty parameter, $c_i$ is the item guessing parameter and $\theta$ is the latent ability. The constant $D$ can be specified using argument `D` of the `direc` function. 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 $a_i$ costant across items, while the Rasch model can be obtained by setting $c_i$ equal to zero and $a_i$ equal to 1. The type of IRT model does not need to be specified as it is obtained from arguments `mod1` and `mod2`. ## Returns An object of class `eqc` with components - **tab1**: item parameters of the first form. - **tab2**: item parameters of the second form. - **tab**: Data frame containing item names (`Item`), item parameters of the first form (e.g. `test1`), item parameters of the second form (e.g. `test2`), and item parameters of the first form converted in the scale of the second form (e.g. `test1.as.test2`). - **var12**: covariance matrix of item parameters of the first and the second form (only items used for equating). - **varFull**: list of covariance matrices of the item parameters (one matrix for each form) - **partial**: partial derivatives of equating coefficients A and B with respect to item parameters. - **A**: equating coefficient A. - **B**: equating coefficient B. - **varAB**: covariance matrix of the equating coefficients. - **commonitem**: list of length 2 containing the names of common item parameters and the names of the common items selected for equating. - **suffixes**: suffixes used to identify the forms. - **ni**: number of common items. - **nis**: number of common items selected for equating. - **forms**: names of equated forms. - **method**: the equating method used. - **itmp**: number of item parameters of the IRT model. ## References Battauz, M. (2015). equateIRT: An R Package for IRT Test Equating. **Journal of Statistical Software**, 68 , 1--22. Kolen, M.J. and Brennan, R.L. (2014). **Test equating, scaling, and linking: methods and practices**, 3nd ed., New York: Springer Ogasawara, H. (2000). Asymptotic standard errors of IRT equating coefficients using moments. **Economic Review (Otaru University of Commerce)**, 51 , 1--23. Ogasawara, H. (2001). Standard Errors of Item Response Theory Equating/Linking by Response Function Methods. **Applied Psychological Measurement**, 25 , 53--67. ## Author(s) Michela Battauz ## See Also `eqc`, `itm`, `modIRT`, `summary.eqc` ## Examples ```r # the following code can be used to start from item response data ## 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) # two-parameter logistic model # direct equating coefficients between forms 2 and 3 using the Haebara method l23 <- direc(mods = mod2pl, which = c(2,3), method = "Haebara") summary(l23) ## End(Not run) # =========================================================================== # the following code uses item parameter estimates previously obtained # three-parameter logistic model # direct equating coefficients between forms 1 and 2 using the Stocking-Lord method data(est3pl) test <- paste("test", 1:5, sep = "") mod3pl <- modIRT(coef = est3pl$coef, var = est3pl$var, names = test, display = FALSE) l12 <- direc(mods = mod3pl, which = c("test1", "test2"), method = "Stocking-Lord") summary(l12) # two-parameter logistic model # direct equating coefficients between forms 1 and 5 using the Haebara method data(est2pl) test <- paste("test", 1:5, sep = "") mod2pl <- modIRT(coef = est2pl$coef, var = est2pl$var, names = test, display = FALSE) l15 <- direc(mods = mod2pl, which = c(1,5), method = "Haebara") summary(l15) # Rasch model # direct equating coefficients between forms 5 and 4 using the mean-mean method data(estrasch) test <- paste("test", 1:5, sep = "") modrasch <- modIRT(coef = estrasch$coef, var = estrasch$var, names = test, display = FALSE) l54 <- direc(mods = modrasch, which = c(5,4), method = "mean-mean") summary(l54) ```

direc function