psychomix (version 0.1-1)

raschmix: Finite Mixtures of Rasch Models

Description

Fit finte mixtures of Rasch models for item response data via conditional maximum likelihood with the EM algorithm.

Usage

raschmix(formula, data, k, subset, weights, scores = c("saturated", "meanvar"),
         nrep = 3, cluster = NULL, control = list(minprior = 0),
         verbose = TRUE, drop = TRUE, unique = FALSE, which = NULL,
	 gradtol = 1e-6, deriv = "sum", hessian = FALSE, ...)
FLXMCrasch(formula = . ~ ., scores = c("saturated", "meanvar"),
  nonExtremeProb = 1, gradtol = 1e-6, deriv = "sum", hessian = FALSE, ...)
simRaschmix(nobs = 1800, itemp = NULL, mean = NULL, sd = NULL,
            design = c("rost1", "rost2", "rost3", "cont1",
                       "cont1-2", "cont2", "cont2-2"),
            extremes = FALSE, attributes = TRUE)

Arguments

formula

Symbolic description of the model (of type y ~ 1 or y ~ x).

data, subset

Arguments controlling formula processing.

k

A vector of integers indicating the number of components of the finite mixture; passed in turn to the k argument of stepFlexmix.

weights

An optional vector of weights to be used in the fitting process; passed in turn to the weights argument of flexmix.

scores

Indicates which model should be fitted for the score probabilities: either a saturated model with a separate parameter for each score probability, or, for meanvar, a multinomial logit model with a location and a scale

nrep

Number of runs for the starting values for the EM algorithm (if cluster = "mrm") or number of runs of the EM algorithm itself.

cluster

Either a matrix with k columns of initial cluster membership probabilities for each observation; or a factor or integer vector with the initial cluster assignments of observations at the start of the EM algorithm. If clu

control

An object of class "FLXcontrol" or a named list; controls the EM algorithm and passed in turn to the control argument of flexmix.

verbose

A logical; if TRUE progress information is shown for different starts of the EM algorithm.

drop

A logical; if TRUE and k is of length 1, then a single raschmix object is returned instead of a stepRaschmix object.

unique

A logical; if TRUE, then unique() is called on the result; for details see stepFlexmix.

which

number of model to get if k is a vector of integers longer than one. If character, interpreted as number of components or name of an information criterion.

nonExtremeProb

A numeric giving the probability of scoring either none or all items.

gradtol, deriv, hessian

Control parameters passed to RaschModel.fit for the M-step.

nobs

Number of observations.

itemp

Optional item parameters for designs "cont1", "cont1-2", "cont2", and "cont2-2". Given as a matrix with each column representing one latent class.

mean, sd

Parameters of the normal distribution used to generate person parameters.

design

Type of data generating process.

extremes

Logical. Should observations with none or all items solved be included in the data?

attributes

Logical. Should the true group membership as well as true item and person parameters be attached to the data as attributes "group", "item", and "person"?

...

Currently not used.

Value

• Either an object of class "raschmix" containing the best model with respect to the log-likelihood (if k is a scalar) or the one selected according to which (if specified and k is a vector of integers longer than 1) or an object of class "stepRaschmix" (if which is not specified and k is a vector of integers longer than 1).

encoding

latin1

Details

Internally stepFlexmix is called with suitable arguments to fit the finite mixture model with the EM algorithm.

FLXMCrasch is the flexmix-driver for the Rasch mixture models with saturated score distribution as proposed by Rost (1990), also known as Mixed Rasch Model.

For the design argument of simRaschmix, "rost1", "rost2", and "rost3" refer to the 3 data generating processes (dgps) introduced in Rost (1990). The other arguments refer to similar dgps with the same sets of item parameters but the person parameters drawn from normal distributions. "cont1" is the counterpart to "rost1" with the person parameters drawn from a standard normal distribution. "cont1-2" is similar, but the person parameters stem from two different normal distributions with means 2 and -2. "cont2" and "cont2-2" are the counterparts to "rost2". For "cont2" for all observations with the same item parameters, the same normal distribution is used to generate person parameters. For "cont2-2", in each group regarding the item parameters extra heterogenity is introduced via sampling the person parameters from two different normal distributions (with means 2 and -2), similar to dgp "cont1-2".

References

Gr�n, B., and Leisch, F. (2008). FlexMix Version 2: Finite Mixtures with Concomitant Variables and Varying and Constant Parameters. Journal of Statistical Software, 28(4), 1--35. http://www.jstatsoft.org/v28/i04/.

Leisch, F. (2004). FlexMix: A General Framework for Finite Mixture Models and Latent Class Regression in R. Journal of Statistical Software, 11(8), 1--18. http://www.jstatsoft.org/v11/i08/.

Rost, J. (1990). Rasch Models in Latent Classes: An Integration of Two Approaches to Item Analysis. Applied Psychological Measurement, 14(3), 271--282.

Rost, J., and von Davier, M. (1995). Mixture Distribution Rasch Models. In Fischer, G.H., and Molenaar, I.W. (eds.), Rasch Models: Foundations, Recent Developments, and Applications, chapter 14, pp. 257--268. Springer-Verlag, New York.

See Also

flexmix, stepFlexmix

Examples

##########
## Data ##
##########

## simulate response from Rost's scenario 2
set.seed(1)
r2 <- simRaschmix(design = "rost2")

## plus informative and noninformative concomitants
d <- data.frame(
  x1 = rbinom(nrow(r2), prob = c(0.4, 0.6)[attr(r2, "group")], size = 1),
  x2 = rnorm(nrow(r2))
)
d$resp <- r2


####################################################
## Rasch mixture model with saturated score model ##
## (Rost, 1990)                                   ##
####################################################

## fit models for k = 1, 2, 3
m1 <- raschmix(r2, k = 1:3, score = "saturated")
## equivalently: m1 <- raschmix(resp ~ 1, data = d, k = 1:3, score = "saturated")

## inspect results
m1
plot(m1)

## select best BIC model
BIC(m1)
m1b <- getModel(m1, which = "BIC")
summary(m1b)

## compare estimated with true item parameters
parameters(m1b, "item") ##  9 items, item_1 = 0
worth(m1b)              ## 10 items, sum = 0
attr(r2, "item")

## graphical comparison
plot(m1b, pos = "top")
for(i in 1:2) lines(attr(r2, "item")[,i], lty = 2, type = "b")

## extract estimated raw score probabilities
## (approximately equal across components and roughly uniform)
scoreProbs(m1b)

## note: parameters() and worth() take "component" argument
parameters(m1b, "item",  component = 2)
parameters(m1b, "score", component = 1)
worth(m1b, component = 2:1)

## inspect posterior probabilities
histogram(m1b)
head(posterior(m1b)) ## for first observations only

## compare resulting clusters with true groups
table(model = clusters(m1b), true = attr(r2, "group"))

## optionally: leverage mRm package for faster computation of
## starting values
library("mRm")
## fit 2-component model
m1b_mrm <- raschmix(r2, k = 2, score = "saturated", cluster = "mrm")
## essentially identical to previous solution
table(clusters(m1b), clusters(m1b_mrm))   
worth(m1b) - worth(m1b_mrm)

################################################################
##  Rasch mixture model with mean/variance score distribution ##
## (Rost & von Davier, 1995)                                  ##
################################################################

## more parsimonious parametrization,
## fit multinomial logit model for score probabilites

## fit models and select best BIC
m2 <- raschmix(r2, k = 1:3, score = "meanvar")
plot(m2)
m2b <- getModel(m2, which = "BIC")

## compare number of estimated parameters
dim(parameters(m2b)) 
dim(parameters(m1b)) 

## graphical comparison with true parameters
plot(m2b, pos = "top")
for(i in 1:2) lines(attr(r2, "item")[,i], lty = 2, type = "b")

## results from non-parametric and parametric specification
## essentially identical
max(abs(worth(m1b) - worth(m2b, component = 2:1)))


###########################
## Concomitant variables ##
###########################

## employ concomitant variables (x1 = informative, x2 = not)
## fit model
cm2 <- raschmix(resp ~ x1 + x2, data = d, k = 2:3, score = "meanvar")

## BIC selection
rbind(m2 = BIC(m2), cm2 = c(NA, BIC(cm2)))
cm2b <- getModel(cm2, which = "BIC")

## concomitant coefficients
parameters(cm2b, which = "concomitant")


##########
## Misc ##
##########

## note: number of clusters can either be chosen directly
## or directly selected via AIC (or BIC, ICL)
raschmix(r2, k = 2)
raschmix(r2, k = 1:3, which = "AIC")