# Conditional Maximum Likelihood Estimation for the Linear Logistic Partial Credit Model Conditional maximum likelihood estimation for the linear logistic partial credit model (Molenaar, 1995; Andersen, 1995; Fischer, 1995). The `immer_cml` function allows for known integer discrimination parameters like in the one-parameter logistic model (Verhelst & Glas, 1995). ```r immer_cml(dat, weights=NULL, W=NULL, b_const=NULL, par_init=NULL, a=NULL, irtmodel=NULL, normalization="first", nullcats="zeroprob", diff=FALSE, use_rcpp=FALSE, ...) ## S3 method for class 'immer_cml' summary(object, digits=3, file=NULL, ...) ## S3 method for class 'immer_cml' logLik(object,...) ## S3 method for class 'immer_cml' anova(object,...) ## S3 method for class 'immer_cml' coef(object,...) ## S3 method for class 'immer_cml' vcov(object,...) ``` ## Arguments - `dat`: Data frame with item responses - `weights`: Optional vector of sample weights - `W`: Design matrix $\bold{W}$ for linear logistic partial credit model. Every row corresponds to a parameter for item $i$ in category $h$ - `b_const`: Optional vector of parameter constants $b_{0ih}$ which can be used for parameter fixings. - `par_init`: Optional vector of initial parameter estimates - `a`: Optional vector of integer item discriminations - `irtmodel`: Type of item response model. `irtmodel="PCM"` and `irtmodel="PCM2"` follow the conventions of the `TAM` package. - `normalization`: The type of normalization in partial credit models. Can be `"first"` for the first item or `"sum"` for a sum constraint. - `nullcats`: A string indicating whether categories with zero frequencies should have a probability of zero (by fixing the constant parameter to a large value of `99`). - `diff`: Logical indicating whether the difference algorithm should be used. See `psychotools::elementary_symmetric_functions` for details. - `use_rcpp`: Logical indicating whether `Rcpp` package should be used for computation. - ``...``: Further arguments to be passed to `stats::optim`. - `object`: Object of class `immer_cml` - `digits`: Number of digits after decimal to be rounded. - `file`: Name of a file in which the output should be sunk ## Details The partial credit model can be written as $$ P(X_{pi}=h ) \propto \exp( a_i h \theta_p - b_{ih}) $$ where the item-category parameters $b_{ih}$ are linearly decomposed according to $$ b_{ih}=\sum_{v} w_{ihv} \beta_v + b_{0ih} $$ with unknown basis parameters $\beta_v$ and fixed values $w_{ihv}$ of the design matrix $\bold{W}$ (specified in `W`) and constants $b_{0ih}$ (specified in `b_const`). ## Returns List with following entries: - **item**: Data frame with item-category parameters - **b**: Item-category parameters $b_{ih}$ - **coefficients**: Estimated basis parameters $\beta_{v}$ - **vcov**: Covariance matrix of basis parameters $\beta_{v}$ - **par_summary**: Summary for basis parameters - **loglike**: Value of conditional log-likelihood - **deviance**: Deviance - **result_optim**: Result from optimization in `stats::optim` - **W**: Used design matrix $\bold{W}$ - **b_const**: Used constant vector $b_{0ih}$ - **par_init**: Used initial parameters - **suffstat**: Sufficient statistics - **score_freq**: Score frequencies - **dat**: Used dataset - **used_persons**: Used persons - **NP**: Number of missing data patterns - **N**: Number of persons - **I**: Number of items - **maxK**: Maximum number of categories per item - **K**: Maximum score of all items - **npars**: Number of estimated parameters - **pars_info**: Information of definition of item-category parameters $b_{ih}$ - **parm_index**: Parameter indices - **item_index**: Item indices - **score**: Raw score for each person ## References Andersen, E. B. (1995). Polytomous Rasch models and their estimation. In G. H. Fischer & I. W. Molenaar (Eds.). **Rasch Models** (pp. 39--52). New York: Springer. Fischer, G. H. (1995). The linear logistic test model. In G. H. Fischer & I. W. Molenaar (Eds.). **Rasch Models** (pp. 131--156). New York: Springer. Molenaar, I. W. (1995). Estimation of item parameters. In G. H. Fischer & I. W. Molenaar (Eds.). **Rasch Models** (pp. 39--52). New York: Springer. Verhelst, N. D. &, Glas, C. A. W. (1995). The one-parameter logistic model. In G. H. Fischer & I. W. Molenaar (Eds.). **Rasch Models** (pp. 215--238). New York: Springer. ## See Also For CML estimation see also the `eRm` and `psychotools` packages and the functions `eRm::RM` and `psychotools::raschmodel` for the Rasch model and `eRm::PCM` and `psychotools::pcmodel` for the partial credit model. See `eRm::LLTM` for the linear logistic test model and `eRm::LPCM` for the linear logistic partial credit model in the `eRm` package for CML implementations. The `immer_cml` function makes use of `psychotools::elementary_symmetric_functions`. For CML estimation with sample weights see also the `RM.weights` package. ## Examples ```r ############################################################################# # EXAMPLE 1: Dichotomous data data.read ############################################################################# library(sirt) library(psychotools) library(TAM) library(CDM) data(data.read, package="sirt") dat <- data.read I <- ncol(dat) #---------------------------------------------------------------- #--- Model 1: Rasch model, setting first item difficulty to zero mod1a <- immer::immer_cml( dat=dat) summary(mod1a) logLik(mod1a) # extract log likelihood coef(mod1a) # extract coefficients ## Not run: library(eRm) # estimate model in psychotools package mod1b <- psychotools::raschmodel(dat) summary(mod1b) logLik(mod1b) # estimate model in eRm package mod1c <- eRm::RM(dat, sum0=FALSE) summary(mod1c) mod1c$etapar # compare estimates of three packages cbind( coef(mod1a), coef(mod1b), mod1c$etapar ) #---------------------------------------------------------------- #-- Model 2: Rasch model sum normalization mod2a <- immer::immer_cml( dat=dat, normalization="sum") summary(mod2a) # compare estimation in TAM mod2b <- tam.mml( dat, constraint="items" ) summary(mod2b) mod2b$A[,2,] #---------------------------------------------------------------- #--- Model 3: some fixed item parameters # fix item difficulties of items 1,4,8 # define fixed parameters in constant parameter vector b_const <- rep(0,I) fix_items <- c(1,4,8) b_const[ fix_items ] <- c( -2.1, .195, -.95 ) # design matrix W <- matrix( 0, nrow=12, ncol=9) W[ cbind( setdiff( 1:12, fix_items ), 1:9 ) ] <- 1 colnames(W) <- colnames(dat)[ - fix_items ] # estimate model mod3 <- immer::immer_cml( dat=dat, W=W, b_const=b_const) summary(mod3) #---------------------------------------------------------------- #--- Model 4: One parameter logistic model # estimate non-integer item discriminations with 2PL model I <- ncol(dat) mod4a <- sirt::rasch.mml2( dat, est.a=1:I ) summary(mod4a) a <- mod4a$item$a # extract (non-integer) item discriminations # estimate integer item discriminations ranging from 1 to 3 a_integer <- immer::immer_opcat( a, hmean=2, min=1, max=3 ) # estimate one-parameter model with fixed integer item discriminations mod4 <- immer::immer_cml( dat=dat, a=a_integer ) summary(mod4) #---------------------------------------------------------------- #--- Model 5: Linear logistic test model # define design matrix W <- matrix( 0, nrow=12, ncol=5 ) colnames(W) <- c("B","C", paste0("Pos", 2:4)) rownames(W) <- colnames(dat) W[ 5:8, "B" ] <- 1 W[ 9:12, "C" ] <- 1 W[ c(2,6,10), "Pos2" ] <- 1 W[ c(3,7,11), "Pos3" ] <- 1 W[ c(4,8,12), "Pos4" ] <- 1 # estimation with immer_cml mod5a <- immer::immer_cml( dat, W=W ) summary(mod5a) # estimation in eRm package mod5b <- eRm::LLTM( dat, W=W ) summary(mod5b) # compare models 1 and 5 by a likelihood ratio test anova( mod1a, mod5a ) ############################################################################# # EXAMPLE 2: Polytomous data | data.Students ############################################################################# data(data.Students,package="CDM") dat <- data.Students dat <- dat[, grep("act", colnames(dat) ) ] dat <- dat[1:400,] # select a subdataset dat <- dat[ rowSums( 1 - is.na(dat) ) > 1, ] # remove persons with less than two valid responses #---------------------------------------------------------------- #--- Model 1: Partial credit model with constraint on first parameter mod1a <- immer::immer_cml( dat=dat ) summary(mod1a) # compare pcmodel function from psychotools package mod1b <- psychotools::pcmodel( dat ) summary(mod1b) # estimation in eRm package mod1c <- eRm::PCM( dat, sum0=FALSE ) # -> subjects with only one valid response must be removed summary(mod1c) #---------------------------------------------------------------- #-- Model 2: Partial credit model with sum constraint on item difficulties mod2a <- immer::immer_cml( dat=dat, irtmodel="PCM2", normalization="sum") summary(mod2a) # compare with estimation in TAM mod2b <- TAM::tam.mml( dat, irtmodel="PCM2", constraint="items") summary(mod2b) #---------------------------------------------------------------- #-- Model 3: Partial credit model with fixed integer item discriminations mod3 <- immer::immer_cml( dat=dat, normalization="first", a=c(2,2,1,3,1) ) summary(mod3) ############################################################################# # EXAMPLE 3: Polytomous data | Extracting the structure of W matrix ############################################################################# data(data.mixed1, package="sirt") dat <- data.mixed1 # use non-exported function "lpcm_data_prep" to extract the meaning # of the rows in W which are contained in value "pars_info" res <- immer:::lpcm_data_prep( dat, weights=NULL, a=NULL ) pi2 <- res$pars_info # create design matrix with some restrictions on item parameters W <- matrix( 0, nrow=nrow(pi2), ncol=2 ) colnames(W) <- c( "P2", "P3" ) rownames(W) <- res$parnames # joint item parameter for items I19 and I20 fixed at zero # item parameter items I21 and I22 W[ 3:10, 1 ] <- pi2$cat[ 3:10 ] # item parameters I23, I24 and I25 W[ 11:13, 2] <- 1 # estimate model with design matrix W mod <- immer::immer_cml( dat, W=W) summary(mod) ############################################################################# # EXAMPLE 4: Partial credit model with raters ############################################################################# data(data.immer07) dat <- data.immer07 #*** reshape dataset for one variable dfr1 <- immer::immer_reshape_wideformat( dat$I1, rater=dat$rater, pid=dat$pid ) #-- extract structure of design matrix res <- immer:::lpcm_data_prep( dat=dfr1[,-1], weights=NULL, a=NULL) pars_info <- res$pars_info # specify design matrix for partial credit model and main rater effects # -> set sum of all rater effects to zero W <- matrix( 0, nrow=nrow(pars_info), ncol=3+2 ) rownames(W) <- rownames(pars_info) colnames(W) <- c( "Cat1", "Cat2", "Cat3", "R1", "R2" ) # define item parameters W[ cbind( pars_info$index, pars_info$cat ) ] <- 1 # define rater parameters W[ paste(pars_info$item)=="R1", "R1" ] <- 1 W[ paste(pars_info$item)=="R2", "R2" ] <- 1 W[ paste(pars_info$item)=="R3", c("R1","R2") ] <- -1 # set parameter of first category to zero for identification constraints W <- W[,-1] # estimate model mod <- immer::immer_cml( dfr1[,-1], W=W) summary(mod) ############################################################################# # EXAMPLE 5: Multi-faceted Rasch model | Estimation with a design matrix ############################################################################# data(data.immer07) dat <- data.immer07 #*** reshape dataset dfr1 <- immer::immer_reshape_wideformat( dat[, paste0("I",1:4) ], rater=dat$rater, pid=dat$pid ) #-- structure of design matrix res <- immer:::lpcm_data_prep( dat=dfr1[,-1], weights=NULL, a=NULL) pars_info <- res$pars_info #--- define design matrix for multi-faceted Rasch model with only main effects W <- matrix( 0, nrow=nrow(pars_info), ncol=3+2+2 ) parnames <- rownames(W) <- rownames(pars_info) colnames(W) <- c( paste0("I",1:3), paste0("Cat",1:2), paste0("R",1:2) ) #+ define item effects for (ii in c("I1","I2","I3") ){ ind <- grep( ii, parnames ) W[ ind, ii ] <- pars_info$cat[ind ] } ind <- grep( "I4", parnames ) W[ ind, c("I1","I2","I3") ] <- -pars_info$cat[ind ] #+ define step parameters for (cc in 1:2 ){ ind <- which( pars_info$cat==cc ) W[ ind, paste0("Cat",1:cc) ] <- 1 } #+ define rater effects for (ii in c("R1","R2") ){ ind <- grep( ii, parnames ) W[ ind, ii ] <- pars_info$cat[ind ] } ind <- grep( "R3", parnames ) W[ ind, c("R1","R2") ] <- -pars_info$cat[ind ] #--- estimate model with immer_cml mod1 <- immer::immer_cml( dfr1[,-1], W=W, par_init=rep(0,ncol(W) ) ) summary(mod1) #--- comparison with estimation in TAM resp <- dfr1[,-1] mod2 <- TAM::tam.mml.mfr( resp=dat[,-c(1:2)], facets=dat[, "rater", drop=FALSE ], pid=dat$pid, formulaA=~ item + step + rater ) summary(mod2) ## End(Not run) ```

immer_cml function