mirt (version 0.5.0)

mixedmirt: Mixed effects modeling for MIRT models

Description

mixedmirt fits MIRT models using FIML estimation to dichotomous and polytomous IRT models conditional on fixed and random effect of person and item level covariates. The method uses the MH-RM algorithm exclusively. The D scaling parameter is automatically fixed to 1 so that all coefficients can be interpreted on the exponential metric.

Usage

mixedmirt(data, covdata = NULL, model, fixed = ~1,
    random = NULL, itemtype = NULL, itemdesign = NULL,
    fixed.constrain = FALSE, constrain = NULL, pars = NULL,
    ...)

Arguments

data

a matrix or data.frame that consists of numerically ordered data, with missing data coded as NA

covdata

a data.frame that consists of the nrow(data) by K 'person level' fixed and random predictors

model

an object returned from confmirt.model() declaring how the factor model is to be estimated. See confmirt.model for more details

fixed

a standard R formula for specifying the fixed effect predictors from covdata and itemdesign. By default constraints are not imposed, so the fixed person effects are not equal accross items, but this can be enabled using

random

a formula similar to the nlme random variable sepcifications for declaring the random slope and intercept predictors. Not currently available, but will be available some time in the future

itemtype

same as itemtype in mirt

itemdesign

a data.frame object used to create a design matrix for the items, where each nrow(itemdesign) == nitems and the number of columns is equal to the number of fixed effect predictors (i.e., item intercepts). If the intput consists of v

fixed.constrain

logical; constrain the fixed person effects to be equal accross items? Disable this when modelling item level covariates and apply the constraints manually

constrain

a list indicating parameter equality constrains. See mirt for more detail

pars

used for parameter starting values. See mirt for more detail

...

additinonal arguments to be passed to the MH-RM estimation engine. See confmirt for more detail

Examples

#make some data
set.seed(1234)
N <- 750
a <- matrix(rlnorm(10,.2,.5),10,1)
d <- matrix(rnorm(10), 10)
Theta <- matrix(sort(rnorm(N)))
pseudoIQ <- Theta * 5 + 100  + rnorm(N, 0 , 5)
group <- factor(rep(c('G1','G2','G3'), each = N/3))
data <- simdata(a,d,N, itemtype = rep('dich',10), Theta=Theta)
covdata <- data.frame(group, pseudoIQ)

#specify IRT model
model <- confmirt.model()
     Theta = 1-10


#model with no person predictors
mod0 <- mirt(data, model, itemtype = 'Rasch')
#group as a fixed effect predictor (aka, uniform dif) and equal effect for all items with
#   fixed.constrain = TRUE
mod1 <- mixedmirt(data, covdata, model, fixed = ~ group, itemtype = 'Rasch', fixed.constrain = TRUE)
anova(mod0, mod1)
summary(mod1)
coef(mod1)

#same model as above in lme4
wide <- data.frame(id=1:nrow(data),data,covdata)
long <- reshape2::melt(wide, id.vars = c('id', 'group', 'pseudoIQ'))
library(lme4)
lmod0 <- lmer(value ~ 0 + variable + (1|id), long, family = binomial)
lmod1 <- lmer(value ~ 0 + group + variable + (1|id), long, family = binomial)
anova(lmod0, lmod1)

#model using 2PL items instead of Rasch
mod1b <- mixedmirt(data, covdata, model, fixed = ~ group, itemtype = '2PL', fixed.constrain = TRUE)
anova(mod1, mod1b) #much better with 2PL models using all criteria (as expected, given simdata pars)

#global nonuniform dif (group interaction with latent variable)
(dif <- mixedmirt(data, covdata, model, fixed = ~ group * Theta, fixed.constrain = TRUE))
anova(mod1b, dif)
#free the interaction terms for each item to detect dif (less power for each item though)
sv <- mixedmirt(data, covdata, model, fixed = ~ group * Theta, pars = 'values')
constrain <- list(sv$parnum[sv$name == 'groupG2'], sv$parnum[sv$name == 'groupG3']) # main effects
itemdif <- mixedmirt(data, covdata, model, fixed = ~ group * Theta, fixed.constrain = FALSE,
     constrain=constrain)
anova(dif, itemdif)

#continuous predictor and interaction model with group
mod2 <- mixedmirt(data, covdata, model, fixed = ~ group + pseudoIQ, fixed.constrain = TRUE)
mod3 <- mixedmirt(data, covdata, model, fixed = ~ group * pseudoIQ, fixed.constrain = TRUE)
summary(mod2)
anova(mod1b, mod2)
anova(mod2, mod3)

###########
##LLTM, and 2PL version of LLTM
data(SAT12)
data <- 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))
model <- confmirt.model()
Theta = 1-32


itemdesign <- data.frame(itemorder = factor(c(rep('easier', 16), rep('harder', 16))))

LLTM <- mixedmirt(data, model = model, fixed = ~ itemorder, itemtype = 'Rasch', itemdesign = itemdesign)
coef(LLTM)
wald(LLTM)
L <- matrix(c(-1, 1), 1)
wald(LLTM, L) #first half different from second

#compare to standard items with estimated slopes (2PL)?
twoPL <- mixedmirt(data, model = model, fixed = ~ itemorder, itemtype = '2PL', itemdesign = itemdesign)
coef(twoPL)
wald(twoPL)
L <- matrix(0, 1, 34)
L[1, 1] <- 1
L[1, 18] <- -1
wald(twoPL, L) #n.s.
anova(twoPL, LLTM)
#twoPL model better than LLTM, and don't draw the (spurious?) conclusion that the first
#    half of the test is any easier/harder than the last

### Similar example, but with simulated data instead and using numeric item desing matrix

set.seed(1234)
N <- 750
a <- matrix(rep(1,10))
d <- matrix(c(rep(-1,5), rep(1,5)))
Theta <- matrix(rnorm(N))
data <- simdata(a, d, N, itemtype = rep('dich',10), Theta=Theta, D=1)
itemdesign <- data.frame(itempred=rep(1, ncol(data)))
model <- confmirt.model()
   Theta = 1-10


sv <- mixedmirt(data, model = model, fixed = ~ itempred, pars = 'values',
                 itemtype = 'Rasch', itemdesign = itemdesign)
sv$value[sv$name == 'd'] <- 0
sv$est[sv$name == 'd'] <- FALSE

#make design such that the first 5 items are systematically more difficult than the last 5
constrain <- list()
constrain[[1]] <- sv$parnum[sv$name == 'itempred'][1:5]
constrain[[2]] <- sv$parnum[sv$name == 'itempred'][-c(1:5)]
mod <- mixedmirt(data, model = model, fixed = ~ itempred, pars = sv,
                 itemtype = 'Rasch', constrain = constrain, itemdesign = itemdesign)
coef(mod)
rasch <- mirt(data, 1, itemtype = 'Rasch', D=1)
anova(mod, rasch) #n.s., LLTM model a much better choice compared to Rasch