Simulate response patterns

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.

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.

Author

Phil Chalmers rphilip.chalmers@gmail.com

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)

if (FALSE) { # \dontrun{

### 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))))
(a <- F2a(F))
d <- rnorm(10)

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

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

#### 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')

# 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)

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

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

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

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)

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)
# 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)

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

# 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)


} # }