# Joint Maximum Likelihood Estimation for the Partial Credit Model with a Design Matrix for Item Parameters and $\varepsilon$-Adjustment Bias Correction Estimates the partial credit model with a design matrix for item parameters with joint maximum likelihood (JML). The $\varepsilon$-adjustment bias correction is implemented with reduces bias of the JML estimation method (Bertoli-Barsotti, Lando & Punzo, 2014). ```r immer_jml(dat, A=NULL, maxK=NULL, center_theta=TRUE, b_fixed=NULL, weights=NULL, irtmodel="PCM", pid=NULL, rater=NULL, eps=0.3, est_method="eps_adj", maxiter=1000, conv=1e-05, max_incr=3, incr_fac=1.1, maxiter_update=10, maxiter_line_search=6, conv_update=1e-05, verbose=TRUE, use_Rcpp=TRUE, shortcut=TRUE) ## S3 method for class 'immer_jml' summary(object, digits=3, file=NULL, ...) ## S3 method for class 'immer_jml' logLik(object, ...) ## S3 method for class 'immer_jml' IRT.likelihood(object, theta=seq(-9,9,len=41), ...) ``` ## Arguments - `dat`: Data frame with polytomous item responses $0,1,\ldots, K$ - `A`: Design matrix (items $\times$ categories $\times$ basis parameters). Entries for categories are for $1,\ldots,K$ - `maxK`: Optional vector with maximum category per item - `center_theta`: Logical indicating whether the trait estimates should be centered - `b_fixed`: Matrix with fixed $b$ parameters - `irtmodel`: Specified item response model. Can be one of the two partial credit model parametrizations `PCM` and `PCM2`. - `weights`: Optional vector of sampling weights - `pid`: Person identifier - `rater`: Optional rater identifier - `eps`: Adjustment parameter $\varepsilon$ - `est_method`: Estimation method. Can be `'eps_adj'` for the $\varepsilon$-adjustment, `'jml'` for the JML without bias correction and `'jml_bc'` for JML with bias correction. - `maxiter`: Maximum number of iterations - `conv`: Convergence criterion - `max_incr`: Maximum increment - `incr_fac`: Factor for shrinking increments from `max_incr` in every iteration - `maxiter_update`: Maximum number of iterations for parameter updates - `maxiter_line_search`: Maximum number of iterations within line search - `conv_update`: Convergence criterion for updates - `verbose`: Logical indicating whether convergence progress should be displayed - `use_Rcpp`: Logical indicating whether `Rcpp` package should be used for computation. - `shortcut`: Logical indicating whether a computational shortcut should be used for efficiency reasons - `object`: Object of class `immer_jml` - `digits`: Number of digits after decimal to print - `file`: Name of a file in which the output should be sunk - `theta`: Grid of $\theta$ values - ``...``: Further arguments to be passed. ## Details The function uses the partial credit model as $P(X_i=h | \theta ) \propto \exp( h \theta - b_{ih} )$ with $b_{ih}=\sum_l a_{ihl} \xi_l$ where the values $a_{ihl}$ are included in the design matrix `A` and $\xi_l$ denotes basis item parameters. The adjustment parameter $\varepsilon$ is applied to the sum score as the sufficient statistic for the person parameter. In more detail, extreme scores $S_p=0$ (minimum score) or $S_p=M_p$ (maximum score) are adjusted to $S_p^\ast=\varepsilon$ or $S_p^\ast=M_p - \varepsilon$, respectively. Therefore, the adjustment possesses more influence on parameter estimation for datasets with a small number of items. ## Returns List with following entries - **b**: Item parameters $b_{ih}$ - **theta**: Person parameters - **theta_se**: Standard errors for person parameters - **xsi**: Basis parameters - **xsi_se**: Standard errors for bias parameters - **probs**: Predicted item response probabilities - **person**: Data frame with person scores - **dat_score**: Scoring matrix - **score_pers**: Sufficient statistics for persons - **score_items**: Sufficient statistics for items - **loglike**: Log-likelihood value ## References Bertoli-Barsotti, L., Lando, T., & Punzo, A. (2014). Estimating a Rasch Model via fuzzy empirical probability functions. In D. Vicari, A. Okada, G. Ragozini & C. Weihs (Eds.). **Analysis and Modeling of Complex Data in Behavioral and Social Sciences**, Springer. ## See Also See `TAM::tam.jml` for joint maximum likelihood estimation. The $varepsilon$-adjustment is also implemented in `sirt::mle.pcm.group`. ## Examples ```r ############################################################################# # EXAMPLE 1: Rasch model ############################################################################# data(data.read, package="sirt") dat <- data.read #--- Model 1: Rasch model with JML and epsilon-adjustment mod1a <- immer::immer_jml(dat) summary(mod1a) ## Not run: #- JML estimation, only handling extreme scores mod1b <- immer::immer_jml( dat, est_method="jml") summary(mod1b) #- JML estimation with (I-1)/I bias correction mod1c <- immer::immer_jml( dat, est_method="jml_bc" ) summary(mod1c) # compare different estimators round( cbind( eps=mod1a$xsi, JML=mod1b$xsi, BC=mod1c$xsi ), 2 ) #--- Model 2: LLTM by defining a design matrix for item difficulties A <- array(0, dim=c(12,1,3) ) A[1:4,1,1] <- 1 A[5:8,1,2] <- 1 A[9:12,1,3] <- 1 mod2 <- immer::immer_jml(dat, A=A) summary(mod2) ############################################################################# # EXAMPLE 2: Partial credit model ############################################################################# library(TAM) data(data.gpcm, package="TAM") dat <- data.gpcm #-- JML estimation in TAM mod0 <- TAM::tam.jml(resp=dat, bias=FALSE) summary(mod0) # extract design matrix A <- mod0$A A <- A[,-1,] #-- JML estimation mod1 <- immer::immer_jml(dat, A=A, est_method="jml") summary(mod1) #-- JML estimation with epsilon-adjusted bias correction mod2 <- immer::immer_jml(dat, A=A, est_method="eps_adj") summary(mod2) ############################################################################# # EXAMPLE 3: Rating scale model with raters | Use design matrix from TAM ############################################################################# data(data.ratings1, package="sirt") dat <- data.ratings1 facets <- dat[,"rater", drop=FALSE] resp <- dat[,paste0("k",1:5)] #* Model 1: Rating scale model in TAM formulaA <- ~ item + rater + step mod1 <- TAM::tam.mml.mfr(resp=resp, facets=facets, formulaA=formulaA, pid=dat$idstud) summary(mod1) #* Model 2: Same model estimated with JML resp0 <- mod1$resp A0 <- mod1$A[,-1,] mod2 <- immer::immer_jml(dat=resp0, A=A0, est_method="eps_adj") summary(mod2) ## End(Not run) ```

immer_jml function