R: Simulate response patterns

simdata {mirt}

R Documentation

Simulate response patterns

Description

Simulates response patterns for compensatory and noncompensatory MIRT models from multivariate normally distributed factor (\theta) scores, or from a user input matrix of \theta's.

Usage

simdata(
  a,
  d,
  N,
  itemtype,
  sigma = NULL,
  mu = NULL,
  guess = 0,
  upper = 1,
  nominal = NULL,
  t = NULL,
  Theta = NULL,
  gpcm_mats = list(),
  returnList = FALSE,
  model = NULL,
  equal.K = TRUE,
  which.items = NULL,
  mins = 0,
  lca_cats = NULL,
  prob.list = NULL
)

Arguments

`a`	a matrix/vector of slope parameters. If slopes are to be constrained to zero then use `NA` or simply set them equal to 0
`d`	a matrix/vector of intercepts. The matrix should have as many columns as the item with the largest number of categories, and filled empty locations with `NA`. When a vector is used the test is assumed to consist only of dichotomous items (because only one intercept per item is provided). When `itemtype = 'lca'` intercepts will not be used
`N`	sample size
`itemtype`	a character vector of length `nrow(a)` (or 1, if all the item types are the same) specifying the type of items to simulate. Inputs can either be the same as the inputs found in the `itemtype` argument in `mirt` or the internal classes defined by the package. Typical `itemtype` inputs that are passed to `mirt` are used then these will be converted into the respective internal classes automatically. If the internal class of the object is specified instead, the inputs can be `'dich', 'graded', 'gpcm', 'sequential', 'nominal', 'nestlogit', 'partcomp', 'gumm'`, or `'lca'`, for dichotomous, graded, generalized partial credit, sequential, nominal, nested logit, partially compensatory, generalized graded unfolding model, and latent class analysis model. Note that for the gpcm, nominal, and nested logit models there should be as many parameters as desired categories, however to parametrized them for meaningful interpretation the first category intercept should equal 0 for these models (second column for `'nestlogit'`, since first column is for the correct item traceline). For nested logit models the 'correct' category is always the lowest category (i.e., == 1). It may be helpful to use `mod2values` on data-sets that have already been estimated to understand the itemtypes more intimately
`sigma`	a covariance matrix of the underlying distribution. Default is the identity matrix. Used when `Theta` is not supplied
`mu`	a mean vector of the underlying distribution. Default is a vector of zeros. Used when `Theta` is not supplied
`guess`	a vector of guessing parameters for each item; only applicable for dichotomous items. Must be either a scalar value that will affect all of the dichotomous items, or a vector with as many values as to be simulated items
`upper`	same as `guess`, but for upper bound parameters
`nominal`	a matrix of specific item category slopes for nominal models. Should be the dimensions as the intercept specification with one less column, with `NA` in locations where not applicable. Note that during estimation the first slope will be constrained to 0 and the last will be constrained to the number of categories minus 1, so it is best to set these as the values for the first and last categories as well
`t`	matrix of t-values for the 'ggum' itemtype, where each row corresponds to a given item. Also determines the number of categories, where `NA` can be used for non-applicable categories
`Theta`	a user specified matrix of the underlying ability parameters, where `nrow(Theta) == N` and `ncol(Theta) == ncol(a)`. When this is supplied the `N` input is not required
`gpcm_mats`	a list of matrices specifying the scoring scheme for generalized partial credit models (see `mirt` for details)
`returnList`	logical; return a list containing the data, item objects defined by `mirt` containing the population parameters and item structure, and the latent trait matrix `Theta`? Default is FALSE
`model`	a single group object, typically returned by functions such as `mirt` or `bfactor`. Supplying this will render all other parameter elements (excluding the `Theta`, `N`, `mu`, and `sigma` inputs) redundant (unless explicitly provided). This input can therefore be used to create parametric bootstrap data whereby plausible data implied by the estimated model can be generated and evaluated
`equal.K`	logical; when a `model` input is supplied, should the generated data contain the same number of categories as the original data indicated by `extract.mirt(model, 'K')`? Default is TRUE, which will redrawn data until this condition is satisfied
`which.items`	an integer vector used to indicate which items to simulate when a `model` input is included. Default simulates all items
`mins`	an integer vector (or single value to be used for each item) indicating what the lowest category should be. If `model` is supplied then this will be extracted from `slot(mod, 'Data')$mins`, otherwise the default is 0
`lca_cats`	a vector indicating how many categories each lca item should have. If not supplied then it is assumed that 2 categories should be generated for each item
`prob.list`	an optional list containing matrix/data.frames of probabilities values for each category to be simulated. This is useful when creating customized probability functions to be sampled from

Details

Returns a data matrix simulated from the parameters, or a list containing the data, item objects, and Theta matrix.

Author(s)

Phil Chalmers rphilip.chalmers@gmail.com

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

Reckase, M. D. (2009). Multidimensional Item Response Theory. New York: Springer.

Examples

### Parameters from Reckase (2009), p. 153

set.seed(1234)

a <- matrix(c(
 .7471, .0250, .1428,
 .4595, .0097, .0692,
 .8613, .0067, .4040,
1.0141, .0080, .0470,
 .5521, .0204, .1482,
1.3547, .0064, .5362,
1.3761, .0861, .4676,
 .8525, .0383, .2574,
1.0113, .0055, .2024,
 .9212, .0119, .3044,
 .0026, .0119, .8036,
 .0008, .1905,1.1945,
 .0575, .0853, .7077,
 .0182, .3307,2.1414,
 .0256, .0478, .8551,
 .0246, .1496, .9348,
 .0262, .2872,1.3561,
 .0038, .2229, .8993,
 .0039, .4720, .7318,
 .0068, .0949, .6416,
 .3073, .9704, .0031,
 .1819, .4980, .0020,
 .4115,1.1136, .2008,
 .1536,1.7251, .0345,
 .1530, .6688, .0020,
 .2890,1.2419, .0220,
 .1341,1.4882, .0050,
 .0524, .4754, .0012,
 .2139, .4612, .0063,
 .1761,1.1200, .0870),30,3,byrow=TRUE)*1.702

d <- matrix(c(.1826,-.1924,-.4656,-.4336,-.4428,-.5845,-1.0403,
  .6431,.0122,.0912,.8082,-.1867,.4533,-1.8398,.4139,
  -.3004,-.1824,.5125,1.1342,.0230,.6172,-.1955,-.3668,
  -1.7590,-.2434,.4925,-.3410,.2896,.006,.0329),ncol=1)*1.702

mu <- c(-.4, -.7, .1)
sigma <- matrix(c(1.21,.297,1.232,.297,.81,.252,1.232,.252,1.96),3,3)

dataset1 <- simdata(a, d, 2000, itemtype = '2PL')
dataset2 <- simdata(a, d, 2000, itemtype = '2PL', mu = mu, sigma = sigma)

#mod <- mirt(dataset1, 3, method = 'MHRM')
#coef(mod)

## No test: 

### Unidimensional graded response model with 5 categories each

a <- matrix(rlnorm(20,.2,.3))

# for the graded model, ensure that there is enough space between the intercepts,
# otherwise closer categories will not be selected often (minimum distance of 0.3 here)
diffs <- t(apply(matrix(runif(20*4, .3, 1), 20), 1, cumsum))
diffs <- -(diffs - rowMeans(diffs))
d <- diffs + rnorm(20)

dat <- simdata(a, d, 500, itemtype = 'graded')
# mod <- mirt(dat, 1)

### An example of a mixed item, bifactor loadings pattern with correlated specific factors

a <- matrix(c(
.8,.4,NA,
.4,.4,NA,
.7,.4,NA,
.8,NA,.4,
.4,NA,.4,
.7,NA,.4),ncol=3,byrow=TRUE)

d <- matrix(c(
-1.0,NA,NA,
 1.5,NA,NA,
 0.0,NA,NA,
0.0,-1.0,1.5,  #the first 0 here is the recommended constraint for nominal
0.0,1.0,-1, #the first 0 here is the recommended constraint for gpcm
2.0,0.0,NA),ncol=3,byrow=TRUE)

nominal <- matrix(NA, nrow(d), ncol(d))
# the first 0 and last (ncat - 1) = 2 values are the recommended constraints
nominal[4, ] <- c(0,1.2,2)

sigma <- diag(3)
sigma[2,3] <- sigma[3,2] <- .25
items <- c('2PL','2PL','2PL','nominal','gpcm','graded')

dataset <- simdata(a,d,2000,items,sigma=sigma,nominal=nominal)

#mod <- bfactor(dataset, c(1,1,1,2,2,2), itemtype=c(rep('2PL', 3), 'nominal', 'gpcm','graded'))
#coef(mod)

#### Convert standardized factor loadings to slopes

F2a <- function(F, D=1.702){
    h2 <- rowSums(F^2)
    a <- (F / sqrt(1 - h2)) * D
    a
}

(F <- matrix(c(rep(.7, 5), rep(.5,5))))

##       [,1]
##  [1,]  0.7
##  [2,]  0.7
##  [3,]  0.7
##  [4,]  0.7
##  [5,]  0.7
##  [6,]  0.5
##  [7,]  0.5
##  [8,]  0.5
##  [9,]  0.5
## [10,]  0.5

(a <- F2a(F))

##            [,1]
##  [1,] 1.6682937
##  [2,] 1.6682937
##  [3,] 1.6682937
##  [4,] 1.6682937
##  [5,] 1.6682937
##  [6,] 0.9826502
##  [7,] 0.9826502
##  [8,] 0.9826502
##  [9,] 0.9826502
## [10,] 0.9826502

d <- rnorm(10)

dat <- simdata(a, d, 5000, itemtype = '2PL')
mod <- mirt(dat, 1)
coef(mod, simplify=TRUE)$items

##                a1           d g u
## Item_1  1.5912124  0.09329250 0 1
## Item_2  1.6611479  0.77704003 0 1
## Item_3  1.6100289  0.04348744 0 1
## Item_4  1.5251147 -0.39866322 0 1
## Item_5  1.7126206 -0.69888431 0 1
## Item_6  0.9352647 -1.71863199 0 1
## Item_7  0.9211469  0.73522267 0 1
## Item_8  0.9893392 -1.25501247 0 1
## Item_9  0.9999014  0.41687594 0 1
## Item_10 0.9416579 -1.00423353 0 1

summary(mod)

##            F1    h2
## Item_1  0.683 0.466
## Item_2  0.698 0.488
## Item_3  0.687 0.472
## Item_4  0.667 0.445
## Item_5  0.709 0.503
## Item_6  0.482 0.232
## Item_7  0.476 0.227
## Item_8  0.503 0.253
## Item_9  0.507 0.257
## Item_10 0.484 0.234
## 
## SS loadings:  3.577 
## Proportion Var:  0.358 
## 
## Factor correlations: 
## 
##    F1
## F1  1

mod2 <- mirt(dat, 'F1 = 1-10
                   CONSTRAIN = (1-5, a1), (6-10, a1)')
summary(mod2)

##            F1    h2
## Item_1  0.689 0.475
## Item_2  0.689 0.475
## Item_3  0.689 0.475
## Item_4  0.689 0.475
## Item_5  0.689 0.475
## Item_6  0.491 0.241
## Item_7  0.491 0.241
## Item_8  0.491 0.241
## Item_9  0.491 0.241
## Item_10 0.491 0.241
## 
## SS loadings:  3.576 
## Proportion Var:  0.358 
## 
## Factor correlations: 
## 
##    F1
## F1  1

anova(mod2, mod)

##           AIC    SABIC       HQ      BIC    logLik    X2 df   p
## mod2 58058.43 58098.51 58085.85 58136.64 -29017.22             
## mod  58068.01 58134.80 58113.69 58198.35 -29014.01 6.425  8 0.6

#### Convert classical 3PL paramerization into slope-intercept form
nitems <- 50
as <- rlnorm(nitems, .2, .2)
bs <- rnorm(nitems, 0, 1)
gs <- rbeta(nitems, 5, 17)

# convert first item (only intercepts differ in resulting transformation)
traditional2mirt(c('a'=as[1], 'b'=bs[1], 'g'=gs[1], 'u'=1), cls='3PL')

##         a1          d          g          u 
##  1.2795115 -0.1107008  0.2525144  1.0000000

# convert all difficulties to intercepts
ds <- numeric(nitems)
for(i in 1:nitems)
   ds[i] <- traditional2mirt(c('a'=as[i], 'b'=bs[i], 'g'=gs[i], 'u'=1),
                             cls='3PL')[2]

dat <- simdata(as, ds, N=5000, guess=gs, itemtype = '3PL')

# estimate with beta prior for guessing parameters
# mod <- mirt(dat, model="Theta = 1-50
#                         PRIOR = (1-50, g, expbeta, 5, 17)", itemtype = '3PL')
# coef(mod, simplify=TRUE, IRTpars=TRUE)$items
# data.frame(as, bs, gs, us=1)


#### Unidimensional nonlinear factor pattern

theta <- rnorm(2000)
Theta <- cbind(theta,theta^2)

a <- matrix(c(
.8,.4,
.4,.4,
.7,.4,
.8,NA,
.4,NA,
.7,NA),ncol=2,byrow=TRUE)
d <- matrix(rnorm(6))
itemtype <- rep('2PL',6)

nonlindata <- simdata(a=a, d=d, itemtype=itemtype, Theta=Theta)

#model <- '
#F1 = 1-6
#(F1 * F1) = 1-3'
#mod <- mirt(nonlindata, model)
#coef(mod)

#### 2PLNRM model for item 4 (with 4 categories), 2PL otherwise

a <- matrix(rlnorm(4,0,.2))

# first column of item 4 is the intercept for the correct category of 2PL model,
#    otherwise nominal model configuration
d <- matrix(c(
-1.0,NA,NA,NA,
 1.5,NA,NA,NA,
 0.0,NA,NA,NA,
 1, 0.0,-0.5,0.5),ncol=4,byrow=TRUE)

nominal <- matrix(NA, nrow(d), ncol(d))
nominal[4, ] <- c(NA,0,.5,.6)

items <- c(rep('2PL',3),'nestlogit')

dataset <- simdata(a,d,2000,items,nominal=nominal)

#mod <- mirt(dataset, 1, itemtype = c('2PL', '2PL', '2PL', '2PLNRM'), key=c(NA,NA,NA,0))
#coef(mod)
#itemplot(mod,4)

# return list of simulation parameters
listobj <- simdata(a,d,2000,items,nominal=nominal, returnList=TRUE)
str(listobj)

## List of 3
##  $ itemobjects:List of 4
##   ..$ :Formal class 'dich' [package "mirt"] with 23 slots
##   .. .. ..@ par          : num [1:4] 0.863 -1 -999 999
##   .. .. .. ..- attr(*, "na.action")= 'omit' int [1:3] 3 4 5
##   .. .. ..@ SEpar        : num(0) 
##   .. .. ..@ parnames     : chr(0) 
##   .. .. ..@ est          : logi(0) 
##   .. .. ..@ dps          :function ()  
##   .. .. ..@ dps2         :function ()  
##   .. .. ..@ constr       : logi(0) 
##   .. .. ..@ itemclass    : int(0) 
##   .. .. ..@ parnum       : num(0) 
##   .. .. ..@ nfact        : int 1
##   .. .. ..@ nfixedeffects: num(0) 
##   .. .. ..@ fixed.design : num[0 , 0 ] 
##   .. .. ..@ dat          : num[0 , 0 ] 
##   .. .. ..@ ncat         : int 2
##   .. .. ..@ gradient     : num(0) 
##   .. .. ..@ hessian      : num[0 , 0 ] 
##   .. .. ..@ itemtrace    : num[0 , 0 ] 
##   .. .. ..@ lbound       : num(0) 
##   .. .. ..@ ubound       : num(0) 
##   .. .. ..@ any.prior    : logi(0) 
##   .. .. ..@ prior.type   : int(0) 
##   .. .. ..@ prior_1      : num(0) 
##   .. .. ..@ prior_2      : num(0) 
##   ..$ :Formal class 'dich' [package "mirt"] with 23 slots
##   .. .. ..@ par          : num [1:4] 1.25 1.5 -999 999
##   .. .. .. ..- attr(*, "na.action")= 'omit' int [1:3] 3 4 5
##   .. .. ..@ SEpar        : num(0) 
##   .. .. ..@ parnames     : chr(0) 
##   .. .. ..@ est          : logi(0) 
##   .. .. ..@ dps          :function ()  
##   .. .. ..@ dps2         :function ()  
##   .. .. ..@ constr       : logi(0) 
##   .. .. ..@ itemclass    : int(0) 
##   .. .. ..@ parnum       : num(0) 
##   .. .. ..@ nfact        : int 1
##   .. .. ..@ nfixedeffects: num(0) 
##   .. .. ..@ fixed.design : num[0 , 0 ] 
##   .. .. ..@ dat          : num[0 , 0 ] 
##   .. .. ..@ ncat         : int 2
##   .. .. ..@ gradient     : num(0) 
##   .. .. ..@ hessian      : num[0 , 0 ] 
##   .. .. ..@ itemtrace    : num[0 , 0 ] 
##   .. .. ..@ lbound       : num(0) 
##   .. .. ..@ ubound       : num(0) 
##   .. .. ..@ any.prior    : logi(0) 
##   .. .. ..@ prior.type   : int(0) 
##   .. .. ..@ prior_1      : num(0) 
##   .. .. ..@ prior_2      : num(0) 
##   ..$ :Formal class 'dich' [package "mirt"] with 23 slots
##   .. .. ..@ par          : num [1:4] 1.02 0 -999 999
##   .. .. .. ..- attr(*, "na.action")= 'omit' int [1:3] 3 4 5
##   .. .. ..@ SEpar        : num(0) 
##   .. .. ..@ parnames     : chr(0) 
##   .. .. ..@ est          : logi(0) 
##   .. .. ..@ dps          :function ()  
##   .. .. ..@ dps2         :function ()  
##   .. .. ..@ constr       : logi(0) 
##   .. .. ..@ itemclass    : int(0) 
##   .. .. ..@ parnum       : num(0) 
##   .. .. ..@ nfact        : int 1
##   .. .. ..@ nfixedeffects: num(0) 
##   .. .. ..@ fixed.design : num[0 , 0 ] 
##   .. .. ..@ dat          : num[0 , 0 ] 
##   .. .. ..@ ncat         : int 2
##   .. .. ..@ gradient     : num(0) 
##   .. .. ..@ hessian      : num[0 , 0 ] 
##   .. .. ..@ itemtrace    : num[0 , 0 ] 
##   .. .. ..@ lbound       : num(0) 
##   .. .. ..@ ubound       : num(0) 
##   .. .. ..@ any.prior    : logi(0) 
##   .. .. ..@ prior.type   : int(0) 
##   .. .. ..@ prior_1      : num(0) 
##   .. .. ..@ prior_2      : num(0) 
##   ..$ :Formal class 'nestlogit' [package "mirt"] with 24 slots
##   .. .. ..@ correctcat   : int 1
##   .. .. ..@ par          : num [1:10] 0.771 1 -999 999 0 ...
##   .. .. ..@ SEpar        : num(0) 
##   .. .. ..@ parnames     : chr(0) 
##   .. .. ..@ est          : logi(0) 
##   .. .. ..@ dps          :function ()  
##   .. .. ..@ dps2         :function ()  
##   .. .. ..@ constr       : logi(0) 
##   .. .. ..@ itemclass    : int(0) 
##   .. .. ..@ parnum       : num(0) 
##   .. .. ..@ nfact        : int 1
##   .. .. ..@ nfixedeffects: num(0) 
##   .. .. ..@ fixed.design : num[0 , 0 ] 
##   .. .. ..@ dat          : num[0 , 0 ] 
##   .. .. ..@ ncat         : int 4
##   .. .. ..@ gradient     : num(0) 
##   .. .. ..@ hessian      : num[0 , 0 ] 
##   .. .. ..@ itemtrace    : num[0 , 0 ] 
##   .. .. ..@ lbound       : num(0) 
##   .. .. ..@ ubound       : num(0) 
##   .. .. ..@ any.prior    : logi(0) 
##   .. .. ..@ prior.type   : int(0) 
##   .. .. ..@ prior_1      : num(0) 
##   .. .. ..@ prior_2      : num(0) 
##  $ data       : num [1:2000, 1:4] 1 1 1 0 0 1 0 1 0 1 ...
##   ..- attr(*, "dimnames")=List of 2
##   .. ..$ : NULL
##   .. ..$ : chr [1:4] "Item_1" "Item_2" "Item_3" "Item_4"
##  $ Theta      : num [1:2000, 1] -0.399 1.931 1.621 -0.424 1.343 ...

# generate dataset from converged model
mod <- mirt(Science, 1, itemtype = c(rep('gpcm', 3), 'nominal'))
sim <- simdata(model=mod, N=1000)
head(sim)

##      Comfort Work Future Benefit
## [1,]       4    3      3       4
## [2,]       3    2      3       3
## [3,]       3    2      4       2
## [4,]       4    3      4       3
## [5,]       3    3      2       3
## [6,]       2    3      3       2

Theta <- matrix(rnorm(100))
sim <- simdata(model=mod, Theta=Theta)
head(sim)

##      Comfort Work Future Benefit
## [1,]       3    3      4       4
## [2,]       4    2      3       2
## [3,]       3    2      3       2
## [4,]       3    3      3       3
## [5,]       3    3      3       3
## [6,]       3    3      4       3

# alternatively, define a suitable object with functions from the mirtCAT package
# help(generate.mirt_object)
library(mirtCAT)

## Loading required package: shiny

nitems <- 50
a1 <- rlnorm(nitems, .2,.2)
d <- rnorm(nitems)
g <- rbeta(nitems, 20, 80)
pars <- data.frame(a1=a1, d=d, g=g)
head(pars)

##          a1          d         g
## 1 1.1271598 -0.6479570 0.2637693
## 2 1.3539630 -1.4832170 0.1903528
## 3 1.1640507  0.3390045 0.1782070
## 4 0.9568311 -1.3578148 0.1389409
## 5 1.0394247  0.3835373 0.2361323
## 6 1.1479152  1.3522255 0.1990177

obj <- generate.mirt_object(pars, '3PL')
dat <- simdata(N=200, model=obj)

#### 10 item GGUMs test with 4 categories each
a <- rlnorm(10, .2, .2)
b <- rnorm(10) #passed to d= input, but used as the b parameters
diffs <- t(apply(matrix(runif(10*3, .3, 1), 10), 1, cumsum))
t <- -(diffs - rowMeans(diffs))

dat <- simdata(a, b, 1000, 'ggum', t=t)
apply(dat, 2, table)

##   Item_1 Item_2 Item_3 Item_4 Item_5 Item_6 Item_7 Item_8 Item_9 Item_10
## 0    477    496    674    332    438    417    558    458    475     513
## 1    299    309    218    388    364    365    288    320    314     305
## 2    153    155     79    232    158    175    109    167    162     138
## 3     71     40     29     48     40     43     45     55     49      44

# mod <- mirt(dat, 1, 'ggum')
# coef(mod)

######
# prob.list example

# custom probability function that returns a matrix
fun <- function(a, b, theta){
    P <- 1 / (1 + exp(-a * (theta-b)))
    cbind(1-P, P)
}

set.seed(1)
theta <- matrix(rnorm(100))
prob.list <- list()
nitems <- 5
a <- rlnorm(nitems, .2, .2); b <- rnorm(nitems, 0, 1/2)
for(i in 1:nitems) prob.list[[i]] <- fun(a[i], b[i], theta)
str(prob.list)

## List of 5
##  $ : num [1:100, 1:2] 0.836 0.68 0.865 0.317 0.645 ...
##  $ : num [1:100, 1:2] 0.771 0.554 0.813 0.179 0.509 ...
##  $ : num [1:100, 1:2] 0.75 0.569 0.788 0.239 0.532 ...
##  $ : num [1:100, 1:2] 0.737 0.503 0.785 0.146 0.457 ...
##  $ : num [1:100, 1:2] 0.828 0.669 0.858 0.308 0.634 ...

dat <- simdata(prob.list=prob.list)
head(dat)

##      Item_1 Item_2 Item_3 Item_4 Item_5
## [1,]      0      0      0      1      1
## [2,]      0      0      0      1      0
## [3,]      0      0      0      0      0
## [4,]      1      1      0      0      0
## [5,]      1      1      0      0      0
## [6,]      0      1      0      1      0

# prob.list input is useful when defining custom items as well
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)

prob.list[[1]] <- x@P(x@par, theta)


## End(No test)

[Package mirt version 1.43 Index]