# Finite Mixtures of Rasch Models Fit finite mixtures of Rasch models for item response data via conditional maximum likelihood with the EM algorithm. latin1 ```r 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, Grün and 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. ## Returns 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. tools:::Rd_expr_doi("10.18637/jss.v048.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. Grün, 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. tools:::Rd_expr_doi("10.18637/jss.v028.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. tools:::Rd_expr_doi("10.18637/jss.v011.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`, `simRaschmix` ## Examples ```r ########## ## 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) #################################################### ## 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")) ################################################################ ## 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))) ########################### ## 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") ## End(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") ## End(Not run) ```

raschmix function