# raschmix

##### Finite Mixtures of Rasch Models

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

- Keywords
- item response, Rasch model, mixture model

##### Usage

```
raschmix(formula, data, k, subset, weights, scores = c("saturated", "meanvar"),
restricted = FALSE, nrep = 3, cluster = NULL, control = NULL, verbose = TRUE,
drop = TRUE, unique = FALSE, which = NULL, reltol = 1e-10, deriv = "sum",
hessian = FALSE, restart = TRUE, model = NULL, gradtol = reltol, …)
```FLXMCrasch(formula = . ~ ., scores = "saturated", delta = NULL, nonExtremeProb = 1,
ref = 1, reltol = 1e-10, deriv = "sum", hessian = FALSE,
restart = 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 parameter.- restricted
Logical. Should the score distributions be restricted to being equal across components? See Frick et al. (2015) for details.

- nrep
Number of runs of the EM algorithm.

- 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.- 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.

- ref
Reference category for the saturated score model.

- reltol, gradtol, deriv, hessian
Control parameters passed to

`RaschModel.fit`

for the M-step. The`gradtol`

argument is deprecated and`reltol`

should be used instead.- restart
Logical. Should the estimation of the item parameters be restarted in each iteration? If FALSE, the estimates from the previous M-step are used as starting values.

- delta
Parameters of score model. If

`NULL`

, a score model is estimated.- model
An object inheriting from class

`"FLXM"`

for the`flexmix`

-driver, as typically produced by`FLXMCrasch`

. By default`FLXMCrasch`

is called automatically with the parameters computed from`raschmix`

.- …
Currently not used.

##### Details

`raschmix`

is intended as a convenience interface to the
`stepFlexmix`

function from the flexmix package
(Leisch 2004, Gr<U+2E821BA4> Leisch 2008). The formula argument of
raschmix is used to describe the model in terms of both items and
concomitant variables, if any. On the left-hand side of the formula
the item are specified, either as a matrix `y`

or as single items
`y1 + y2 + y3 + …`

. On the right-hand side, the concomitant
variables are specified. If no concomitant variables are to be
included in the model, the right-hand side of the is just written as
`~ 1`

. See Frick et al. (2012) for a detailed introduction.

`raschmix`

processes this model description and calls
`stepFlexmix`

with the suitable driver
`FLXMCrasch`

. Usually, the driver does not need to be called by
itself, but it is of course also possible to call
`stepFlexmix`

directly with this driver to fit
Rasch mixture models.

The Rasch mixture model with saturated score distribution as proposed by Rost (1990) is also known as “Mixed Rasch Model”. The mean-variance score distribution was suggested by Rost and von Davier (1995). A more recent extension is the restricted score specification by Frick et al. (2015) who also provide an extensive comparison using Monte Carlo studies.

##### 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).

##### References

Frick, H., Strobl, C., Leisch, F., and Zeileis, A. (2012).
Flexible Rasch Mixture Models with Package psychomix.
*Journal of Statistical Software*, **48**(7), 1--25.
http://www.jstatsoft.org/v48/i07/.

Frick, H., Strobl, C., and Zeileis, A. (2015).
Rasch Mixture Models for DIF Detection: A Comparison of Old and New
Score Specifications.
*Educational and Psychological Measurement*, **75**(2), 208--234.
doi:10.1177/0013164414536183.

Gr<U+2EB200AE>, 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

##### Examples

```
# NOT RUN {
##########
## Data ##
##########
## simulate response from Rost's scenario 2 (with 2 latent classes)
suppressWarnings(RNGversion("3.5.0"))
set.seed(1)
r2 <- simRaschmix(design = "rost2")
## plus informative and non-informative concomitants
d <- data.frame(
x1 = rbinom(nrow(r2), prob = c(0.4, 0.6)[attr(r2, "cluster")], size = 1),
x2 = rnorm(nrow(r2))
)
d$resp <- r2
## fit model with 2 latent classes (here the number is known a priori)
m <- raschmix(r2, k = 2, scores = "saturated")
summary(m)
## see below for examples which do not use this a priori information
## (these take a little longer to compute)
# }
# NOT RUN {
####################################################
## 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, "difficulty")
## graphical comparison
plot(m1b, pos = "top")
for(i in 1:2) lines(attr(r2, "difficulty")[,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, "cluster"))
# }
# NOT RUN {
# }
# NOT RUN {
################################################################
## Rasch mixture model with mean/variance score distribution ##
## (Rost & von Davier, 1995) ##
################################################################
## more parsimonious parameterization,
## fit multinomial logit model for score probabilities
## 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, "difficulty")[,i], lty = 2, type = "b")
## results from non-parametric and parametric specification
## essentially identical
max(abs(worth(m1b) - worth(m2b, component = 2:1)))
# }
# NOT RUN {
###########################
## Concomitant variables ##
###########################
## employ concomitant variables (x1 = informative, x2 = not)
# }
# NOT RUN {
## 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")
# }
# NOT RUN {
##########
## Misc ##
##########
## note: number of clusters can either be chosen directly
## or directly selected via AIC (or BIC, ICL)
# }
# NOT RUN {
raschmix(r2, k = 2)
raschmix(r2, k = 1:3, which = "AIC")
# }
```

*Documentation reproduced from package psychomix, version 1.1-8, License: GPL-2 | GPL-3*