createItem {mirt}

Create a user defined item with correct generic functions

Description

Initializes the proper S4 class and methods necessary for mirt functions to use in estimation. To use the defined objects pass to the mirt(..., customItems = list()) command, and ensure that the classes are properly labelled and unique in the list. Additionally, the input mirt(..., customItemsData = list()) can also be included to specify additional item-level information to better recycle custom-item definitions (e.g., for supplying varying Q-matrices), where the list input must have the same length as the number of items. For further examples regarding how this function can be used for fitting unfolding-type models see Liu and Chalmers (2018).

Usage

createItem(
  name,
  par,
  est,
  P,
  gr = NULL,
  hss = NULL,
  gen = NULL,
  lbound = NULL,
  ubound = NULL,
  derivType = "Richardson",
  derivType.hss = "Richardson",
  bytecompile = TRUE
)

Arguments

name

a character indicating the item class name to be defined

par

a named vector of the starting values for the parameters

est

a logical vector indicating which parameters should be freely estimated by default

P

the probability trace function for all categories (first column is category 1, second category two, etc). First input contains a vector of all the item parameters, the second input must be a matrix called Theta, the third input must be the number of categories called ncat, and (optionally) a fourth argument termed itemdata may be included containing further users specification information. The last optional input is to be utilized within the estimation functions such as mirt via the list input customItemsData to more naturally recycle custom-item definitions. Therefore, these inputs must be of the form

function(par, Theta, ncat){...}

or

function(par, Theta, ncat, itemdata){...}

to be valid; however, the names of the arguements is not relavent.

Finally, this function must return a matrix object of category probabilities, where the columns represent each respective category

gr

gradient function (vector of first derivatives) of the log-likelihood used in estimation. The function must be of the form gr(x, Theta), where x is the object defined by createItem() and Theta is a matrix of latent trait parameters. Tabulated (EM) or raw (MHRM) data are located in the x@dat slot, and are used to form the complete data log-likelihood. If not specified a numeric approximation will be used

hss

Hessian function (matrix of second derivatives) of the log-likelihood used in estimation. If not specified a numeric approximation will be used (required for the MH-RM algorithm only). The input is identical to the gr argument

gen

a function used when GenRandomPars = TRUE is passed to the estimation function to generate random starting values. Function must be of the form function(object) ... and must return a vector with properties equivalent to the par object. If NULL, parameters will remain at the defined starting values by default

lbound

optional vector indicating the lower bounds of the parameters. If not specified then the bounds will be set to -Inf

ubound

optional vector indicating the lower bounds of the parameters. If not specified then the bounds will be set to Inf

derivType

if the gr term is not specified this type will be used to obtain the gradient numerically or symbolically. Default is the 'Richardson' extrapolation method; see numerical_deriv for details and other options. If 'symbolic' is supplied then the gradient will be computed using a symbolical approach (potentially the most accurate method, though may fail depending on how the P function was defined)

derivType.hss

if the hss term is not specified this type will be used to obtain the Hessian numerically. Default is the 'Richardson' extrapolation method; see numerical_deriv for details and other options. If 'symbolic' is supplied then the Hessian will be computed using a symbolical approach (potentially the most accurate method, though may fail depending on how the P function was defined)

bytecompile

logical; where applicable, byte compile the functions provided? Default is TRUE to provide

Details

The summary() function will not return proper standardized loadings since the function is not sure how to handle them (no slopes could be defined at all!). Instead loadings of .001 are filled in as place-holders.

Author(s)

Phil Chalmers [e-mail]

References

Chalmers, R., P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48(6), 1-29. doi:10.18637/jss.v048.i06

Liu, C.-W. and Chalmers, R. P. (2018). Fitting item response unfolding models to Likert-scale data using mirt in R. PLoS ONE, 13, 5. doi:10.1371/journal.pone.0196292

Examples

## Not run: 

name <- 'old2PL'
par <- c(a = .5, b = -2)
est <- c(TRUE, TRUE)
P.old2PL <- function(par,Theta, ncat){
     a <- par[1]
     b <- par[2]
     P1 <- 1 / (1 + exp(-1*a*(Theta - b)))
     cbind(1-P1, P1)
}

x <- createItem(name, par=par, est=est, P=P.old2PL)

# So, let's estimate it!
dat <- expand.table(LSAT7)
sv <- mirt(dat, 1, c(rep('2PL',4), 'old2PL'), customItems=list(old2PL=x), pars = 'values')
tail(sv) #looks good
mod <- mirt(dat, 1, c(rep('2PL',4), 'old2PL'), customItems=list(old2PL=x))
coef(mod)
mod2 <- mirt(dat, 1, c(rep('2PL',4), 'old2PL'), customItems=list(old2PL=x), method = 'MHRM')
coef(mod2)

# same definition as above, but using symbolic derivative computations
# (can be more accurate/stable)
xs <- createItem(name, par=par, est=est, P=P.old2PL, derivType = 'symbolic')
mod <- mirt(dat, 1, c(rep('2PL',4), 'old2PL'), customItems=list(old2PL=xs))
coef(mod, simplify=TRUE)

# several secondary functions supported
M2(mod, calcNull=FALSE)
itemfit(mod)
fscores(mod, full.scores=FALSE)
plot(mod)

# fit the same model, but specify gradient function explicitly (use of a browser() may be helpful)
gr <- function(x, Theta){
     # browser()
     a <- x@par[1]
     b <- x@par[2]
     P <- probtrace(x, Theta)
     PQ <- apply(P, 1, prod)
     r_P <- x@dat / P
     grad <- numeric(2)
     grad[2] <- sum(-a * PQ * (r_P[,2] - r_P[,1]))
     grad[1] <- sum((Theta - b) * PQ * (r_P[,2] - r_P[,1]))

     ## check with internal numerical form to be safe
     # numerical_deriv(x@par[x@est], mirt:::EML, obj=x, Theta=Theta)
     grad
}

x <- createItem(name, par=par, est=est, P=P.old2PL, gr=gr)
mod <- mirt(dat, 1, c(rep('2PL',4), 'old2PL'), customItems=list(old2PL=x))
coef(mod, simplify=TRUE)

### non-linear
name <- 'nonlin'
par <- c(a1 = .5, a2 = .1, d = 0)
est <- c(TRUE, TRUE, TRUE)
P.nonlin <- function(par,Theta, ncat=2){
     a1 <- par[1]
     a2 <- par[2]
     d <- par[3]
     P1 <- 1 / (1 + exp(-1*(a1*Theta + a2*Theta^2 + d)))
     cbind(1-P1, P1)
}

x2 <- createItem(name, par=par, est=est, P=P.nonlin)

mod <- mirt(dat, 1, c(rep('2PL',4), 'nonlin'), customItems=list(nonlin=x2))
coef(mod)

### nominal response model (Bock 1972 version)
Tnom.dev <- function(ncat) {
   T <- matrix(1/ncat, ncat, ncat - 1)
   diag(T[-1, ]) <-  diag(T[-1, ]) - 1
   return(T)
}

name <- 'nom'
par <- c(alp=c(3,0,-3),gam=rep(.4,3))
est <- rep(TRUE, length(par))
P.nom <- function(par, Theta, ncat){
   alp <- par[1:(ncat-1)]
   gam <- par[ncat:length(par)]
   a <- Tnom.dev(ncat) %*% alp
   c <- Tnom.dev(ncat) %*% gam
   z <- matrix(0, nrow(Theta), ncat)
   for(i in 1:ncat)
       z[,i] <- a[i] * Theta + c[i]
   P <- exp(z) / rowSums(exp(z))
   P
}

nom1 <- createItem(name, par=par, est=est, P=P.nom)
nommod <- mirt(Science, 1, 'nom1', customItems=list(nom1=nom1))
coef(nommod)
Tnom.dev(4) %*% coef(nommod)[[1]][1:3] #a
Tnom.dev(4) %*% coef(nommod)[[1]][4:6] #d


## End(Not run)