Multidimensional discrete item response theory

mdirt fits a variety of item response models with discrete latent variables. These include, but are not limited to, latent class analysis, multidimensional latent class models, multidimensional discrete latent class models, DINA/DINO models, grade of measurement models, C-RUM, and so on. If response models are not defined explicitly then customized models can defined using the createItem function.

Usage

mdirt(
  data,
  model,
  customTheta = NULL,
  structure = NULL,
  item.Q = NULL,
  nruns = 1,
  method = "EM",
  covdata = NULL,
  formula = NULL,
  itemtype = "lca",
  optimizer = "nlminb",
  return_max = TRUE,
  group = NULL,
  GenRandomPars = FALSE,
  verbose = TRUE,
  pars = NULL,
  technical = list(),
  ...
)

Arguments

data: a matrix or data.frame that consists of numerically ordered data, organized in the form of integers, with missing data coded as NA
model: number of mutually exclusive classes to fit, or alternatively a more specific mirt.model definition (which reflects the so-called Q-matrix). Note that when using a mirt.model, the order with which the syntax factors/attributes are defined are associated with the columns in the customTheta input
customTheta: input passed to technical = list(customTheta = ...), but is included directly in this function for convenience. This input is most interesting for discrete latent models because it allows customized patterns of latent classes (i.e., defines the possible combinations of the latent attribute profile). The default builds the pattern customTheta = diag(model), which is the typical pattern for the traditional latent class analysis whereby class membership mutually distinct and exhaustive. See thetaComb for a quick method to generate a matrix with all possible combinations
structure: an R formula allowing the profile probability patterns (i.e., the structural component of the model) to be fitted according to a log-linear model. When NULL, all profile probabilities (except one) will be estimated. Use of this input requires that the customTheta input is supplied, and that the column names in this matrix match the names found within this formula
item.Q: a list of item-level Q-matrices indicating how the respective categories should be modeled by the underlying attributes. Each matrix must represent a $K_i \times A$ matrix, where $K_i$ represents the number of categories for the ith item, and $A$ is the number of attributes included in the Theta matrix; otherwise, a value ofNULL will default to a matrix consisting of 1's for each $K_i \times A$ element except for the first row, which contains only 0's for proper identification. Incidentally, the first row of each matrix must contain only 0's so that the first category represents the reference category for identification
nruns: a numeric value indicating how many times the model should be fit to the data when using random starting values. If greater than 1, GenRandomPars is set to true by default
method: estimation method. Can be 'EM' or 'BL' (see mirt for more details)
covdata: a data.frame of data used for latent regression models
formula: an R formula (or list of formulas) indicating how the latent traits can be regressed using external covariates in covdata. If a named list of formulas is supplied (where the names correspond to the latent trait/attribute names in model) then specific regression effects can be estimated for each factor. Supplying a single formula will estimate the regression parameters for all latent variables by default
itemtype: a vector indicating the itemtype associated with each item. For discrete models this is limited to only 'lca' or items defined using a createItem definition
optimizer: optimizer used for the M-step, set to 'nlminb' by default. See mirt for more details
return_max: logical; when nruns > 1, return the model that has the most optimal maximum likelihood criteria? If FALSE, returns a list of all the estimated objects
group: a factor variable indicating group membership used for multiple group analyses
GenRandomPars: logical; use random starting values
verbose: logical; turn on messages to the R console
pars: used for modifying starting values; see mirt for details
technical: list of lower-level inputs. See mirt for details
...: additional arguments to be passed to the estimation engine. See mirt for more details and examples

Details

Posterior classification accuracy for each response pattern may be obtained via the fscores function. The summary() function will display the category probability values given the class membership, which can also be displayed graphically with plot(), while coef() displays the raw coefficient values (and their standard errors, if estimated). Finally, anova() is used to compare nested models, while M2 and itemfit may be used for model fitting purposes.

'lca' model definition

The latent class IRT model with two latent classes has the form

$$P(x = k|\theta_1, \theta_2, a1, a2) = \frac{exp(a1 \theta_1 + a2 \theta_2)}{ \sum_j^K exp(a1 \theta_1 + a2 \theta_2)}$$

where the $\theta$ values generally take on discrete points (such as 0 or 1). For proper identification, the first category slope parameters ($a1$ and $a2$) are never freely estimated. Alternatively, supplying a different grid of $\theta$ values will allow the estimation of similar models (multidimensional discrete models, grade of membership, etc.). See the examples below.

When the item.Q for is utilized, the above equation can be understood as

$$P(x = k|\theta_1, \theta_2, a1, a2) = \frac{exp(a1 \theta_1 Q_{j1} + a2 \theta_2 Q_{j2})}{ \sum_j^K exp(a1 \theta_1 Q_{j1} + a2 \theta_2 Q_{j2})}$$

where by construction Q is a $K_i \times A$ matrix indicating whether the category should be modeled according to the latent class structure. For the standard latent class model, the Q-matrix has as many rows as categories, as many columns as the number of classes/attributes modeled, and consist of 0's in the first row and 1's elsewhere. This of course can be over-written by passing an alternative item.Q definition for each respective item.

References

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

Proctor, C. H. (1970). A probabilistic formulation and statistical analysis for Guttman scaling. Psychometrika, 35, 73-78. doi:10.18637/jss.v048.i06

Author

Phil Chalmers rphilip.chalmers@gmail.com

Examples


# LSAT6 dataset
dat <- expand.table(LSAT6)

# fit with 2-3 latent classes
(mod2 <- mdirt(dat, 2))
#> 
Iteration: 1, Log-Lik: -3209.317, Max-Change: 2.67790
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#> 
#> Call:
#> mdirt(data = dat, model = 2)
#> 
#> Latent class model with 2 classes and 2 profiles.
#> Converged within 1e-04 tolerance after 363 EM iterations.
#> mirt version: 1.42.1 
#> M-step optimizer: nlminb 
#> EM acceleration: Ramsay
#> Latent density type: discrete
#> 
#> Log-likelihood = -2467.408
#> Estimated parameters: 11 
#> AIC = 4956.816
#> BIC = 5010.802; SABIC = 4975.865
#> G2 (20) = 22.74, p = 0.3018, RMSEA = 0.012
if (FALSE) { # \dontrun{
(mod3 <- mdirt(dat, 3))
summary(mod2)
residuals(mod2)
residuals(mod2, type = 'exp')
anova(mod2, mod3)
M2(mod2)
itemfit(mod2)

# generate classification plots
plot(mod2)
plot(mod2, facet_items = FALSE)
plot(mod2, profile = TRUE)

# available for polytomous data
mod <- mdirt(Science, 2)
summary(mod)
plot(mod)
plot(mod, profile=TRUE)

# classification based on response patterns
fscores(mod2, full.scores = FALSE)

# classify individuals either with the largest posterior probability.....
fs <- fscores(mod2)
head(fs)
classes <- 1:2
class_max <- classes[apply(apply(fs, 1, max) == fs, 1, which)]
table(class_max)

# ... or by probability sampling (i.e., plausible value draws)
class_prob <- apply(fs, 1, function(x) sample(1:2, 1, prob=x))
table(class_prob)

# plausible value imputations for stochastic classification in both classes
pvs <- fscores(mod2, plausible.draws=10)
tabs <- lapply(pvs, function(x) apply(x, 2, table))
tabs[[1]]


# fit with random starting points (run in parallel to save time)
if(interactive()) mirtCluster()
mod <- mdirt(dat, 2, nruns=10)

#--------------------------
# Grade of measurement model

# define a custom Theta grid for including a 'fuzzy' class membership
(Theta <- matrix(c(1, 0, .5, .5, 0, 1), nrow=3 , ncol=2, byrow=TRUE))
(mod_gom <- mdirt(dat, 2, customTheta = Theta))
summary(mod_gom)

#-----------------
# Multidimensional discrete latent class model

dat <- key2binary(SAT12,
     key = c(1,4,5,2,3,1,2,1,3,1,2,4,2,1,5,3,4,4,1,4,3,3,4,1,3,5,1,3,1,5,4,5))

# define Theta grid for three latent classes
(Theta <- thetaComb(0:1, 3))
(mod_discrete <- mdirt(dat, 3, customTheta = Theta))
summary(mod_discrete)

# Located latent class model
model <- mirt.model('C1 = 1-32
                     C2 = 1-32
                     C3 = 1-32
                     CONSTRAIN = (1-32, a1), (1-32, a2), (1-32, a3)')
(mod_located <- mdirt(dat, model, customTheta = diag(3)))
summary(mod_located)

#-----------------
### DINA model example
# generate some suitable data for a two dimensional DINA application
#     (first columns are intercepts)
set.seed(1)
Theta <- expand.table(matrix(c(1,0,0,0,
                               1,1,0,0,
                               1,0,1,0,
                               1,1,1,1), 4, 4, byrow=TRUE),
                      freq = c(200,200,100,500))
a <- matrix(c(rnorm(15, -1.5, .5), rlnorm(5, .2, .3), numeric(15), rlnorm(5, .2, .3),
              numeric(15), rlnorm(5, .2, .3)), 15, 4)

guess <- plogis(a[11:15,1]) # population guess
slip <- 1 - plogis(rowSums(a[11:15,])) # population slip

dat <- simdata(a, Theta=Theta, itemtype = 'lca')

# first column is the intercept, 2nd and 3rd are attributes
theta <- cbind(1, thetaComb(0:1, 2))
theta <- cbind(theta, theta[,2] * theta[,3]) #DINA interaction of main attributes
model <- mirt.model('Intercept = 1-15
                     A1 = 1-5
                     A2 = 6-10
                     A1A2 = 11-15')

# last 5 items are DINA (first 10 are unidimensional C-RUMs)
DINA <- mdirt(dat, model, customTheta = theta)
coef(DINA, simplify=TRUE)
summary(DINA)
M2(DINA) # fits well (as it should)

cfs <- coef(DINA, simplify=TRUE)$items[11:15,]
cbind(guess, estguess = plogis(cfs[,1]))
cbind(slip, estslip = 1 - plogis(rowSums(cfs)))


### DINO model example
theta <- cbind(1, thetaComb(0:1, 2))
# define theta matrix with negative interaction term
(theta <- cbind(theta, -theta[,2] * theta[,3]))

model <- mirt.model('Intercept = 1-15
                     A1 = 1-5, 11-15
                     A2 = 6-15
                     Yoshi = 11-15
                     CONSTRAIN = (11,a2,a3,a4), (12,a2,a3,a4), (13,a2,a3,a4),
                                 (14,a2,a3,a4), (15,a2,a3,a4)')

# last five items are DINOs (first 10 are unidimensional C-RUMs)
DINO <- mdirt(dat, model, customTheta = theta)
coef(DINO, simplify=TRUE)
summary(DINO)
M2(DINO) #doesn't fit as well, because not the generating model

## C-RUM (analogous to MIRT model)
theta <- cbind(1, thetaComb(0:1, 2))
model <- mirt.model('Intercept = 1-15
                     A1 = 1-5, 11-15
                     A2 = 6-15')

CRUM <- mdirt(dat, model, customTheta = theta)
coef(CRUM, simplify=TRUE)
summary(CRUM)

# good fit, but over-saturated (main effects for items 11-15 can be set to 0)
M2(CRUM)

#------------------
# multidimensional latent class model

dat <- key2binary(SAT12,
     key = c(1,4,5,2,3,1,2,1,3,1,2,4,2,1,5,3,4,4,1,4,3,3,4,1,3,5,1,3,1,5,4,5))

# 5 latent classes within 2 different sets of items
model <- mirt.model('C1 = 1-16
                     C2 = 1-16
                     C3 = 1-16
                     C4 = 1-16
                     C5 = 1-16
                     C6 = 17-32
                     C7 = 17-32
                     C8 = 17-32
                     C9 = 17-32
                     C10 = 17-32
                     CONSTRAIN = (1-16, a1), (1-16, a2), (1-16, a3), (1-16, a4), (1-16, a5),
                       (17-32, a6), (17-32, a7), (17-32, a8), (17-32, a9), (17-32, a10)')

theta <- diag(10) # defined explicitly. Otherwise, this profile is assumed
mod <- mdirt(dat, model, customTheta = theta)
coef(mod, simplify=TRUE)
summary(mod)

#------------------
# multiple group with constrained group probabilities
 dat <- key2binary(SAT12,
   key = c(1,4,5,2,3,1,2,1,3,1,2,4,2,1,5,3,4,4,1,4,3,3,4,1,3,5,1,3,1,5,4,5))
group <- rep(c('G1', 'G2'), each = nrow(SAT12)/2)
Theta <- diag(2)

# the latent class parameters are technically located in the (nitems + 1) location
model <- mirt.model('A1 = 1-32
                     A2 = 1-32
                     CONSTRAINB = (33, c1)')
mod <- mdirt(dat, model, group = group, customTheta = Theta)
coef(mod, simplify=TRUE)
summary(mod)


#------------------
# Probabilistic Guttman Model (Proctor, 1970)

# example analysis can also be found in the sirt package (see ?prob.guttman)
data(data.read, package = 'sirt')
head(data.read)

Theta <- matrix(c(1,0,0,0,
                  1,1,0,0,
                  1,1,1,0,
                  1,1,1,1), 4, byrow=TRUE)

model <- mirt.model("INTERCEPT = 1-12
                     C1 = 1,7,9,11
                     C2 = 2,5,8,10,12
                     C3 = 3,4,6")

mod <- mdirt(data.read, model, customTheta=Theta)
summary(mod)

M2(mod)
itemfit(mod)


} # }