mirt (version 0.8.0)

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. This can also be understood as 'expalanatory IRT' if only fixed effects are modeled, or multilevel/mixed IRT if random and fixed effects are included. The method uses the MH-RM algorithm exclusively.

Usage

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

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 right sided R formula for specifying the fixed effect (aka 'explanatory') predictors from covdata and itemdesign. To estimate the intercepts for each item the keyword items is reserved and automatically

random

a right sided formula or list of formulas containing crossed random effects of the form v1 + ... v_n | G, where G is the grouping variable and v_n are random numeric predictors within each group. If no in

itemtype

same as itemtype in mirt, expect does not support the following item types: c('PC2PL', 'PC3PL', '2PLNRM', '3PLNRM', '3PLuNRM', '4PLNRM')

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). By default a

constrain

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

pars

used for parameter starting values. See mirt for more detail

return.design

logical; return the design matrices before they have (potentially) been reassigned?

...

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

Details

For dichotomous response models, mixedmirt follows the general form $$P(x = 1|\theta, \psi) = g + \frac{(u - g)}{1 + exp(-1 * [\mathbf{\theta a} + \mathbf{X \beta} + \mathbf{Z \gamma}])}$$ where X is a design matrix with associated $\beta$ fixed effect coefficients, and Z is a design matrix with associated $\gamma$ random effects. For simplicity and easier interpretation, the unique item intercept values typically found in $\mathbf{X \beta}$ are extracted and reassigned within mirt's 'intercept' parameters (e.g., 'd'). To observe how the design matrices are structured prior to reassignment and estimation pass the argument return.design = TRUE. Polytomous IRT models follow a similar format except the item intercepts are automatically estimated internally, rendering the items argument in the fixed formula redundent and therefore must be ommited from the specification. If there are a mixture of dichotomous and polytomous items the intercepts for the dichotomous models are also estimated for consistency. To simulate maximum a posteriori estimates for the random effects use the randef function.

See Also

randef, calcLogLik

Examples

#make some data
set.seed(1234)
N <- 750
a <- matrix(rlnorm(10,.3,1),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)
#use cl for parallel computing
library(parallel)
cl <- makeCluster(detectCores())

#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)
mod1 <- mixedmirt(data, covdata, model, fixed = ~ 0 + group + items, cl=cl)
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 = ~ 0 + group + items, itemtype = '2PL', cl=cl)
anova(mod1, mod1b) #much better with 2PL models using all criteria (as expected, given simdata pars)

#continuous predictor and interaction model with group
mod2 <- mixedmirt(data, covdata, model, fixed = ~ 0 + group + pseudoIQ, cl=cl)
mod3 <- mixedmirt(data, covdata, model, fixed = ~ 0 + group * pseudoIQ, cl=cl)
summary(mod2)
anova(mod1b, mod2)
anova(mod2, mod3)

#view fixed design matrix with and without unique item level intercepts
withint <- mixedmirt(data, covdata, model, fixed = ~ 0 + items + group, return.design = TRUE)
withoutint <- mixedmirt(data, covdata, model, fixed = ~ 0 + group, return.design = TRUE)

#notice that in result above, the intercepts 'items1 to items 10' were reassigned to 'd'
head(withint$X)
tail(withint$X)
head(withoutint$X) #no intercepts design here to be reassigned into item intercepts
tail(withoutint$X)

###################################################
##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))))

#notice that the 'fixed = ~ ... + items' argument is ommited
LLTM <- mixedmirt(data, model = model, fixed = ~ 0 + itemorder, itemdesign = itemdesign, cl=cl)
summary(LLTM)
coef(LLTM)
wald(LLTM)
L <- c(-1, 1, 0)
wald(LLTM, L) #first half different from second

#compare to items with estimated slopes (2PL)
twoPL <- mixedmirt(data, model = model, fixed = ~ 0 + itemorder, itemtype = '2PL',
                   itemdesign = itemdesign, cl=cl)
anova(twoPL, LLTM) #much better fit
summary(twoPL)
coef(twoPL)

wald(twoPL)
L <- matrix(0, 1, 34)
L[1, 1] <- 1
L[1, 2] <- -1
wald(twoPL, L) #n.s.
#twoPL model better than LLTM, and don't draw the incorrect conclusion that the first
#    half of the test is any easier/harder than the last

##LLTM with item error term
LLTMwithError <- mixedmirt(data, model = model, fixed = ~ 0 + itemorder, random = ~ 1|items,
    itemdesign = itemdesign, cl=cl)
summary(LLTMwithError)
#large item level variance after itemorder is regressed; not a great predictor of item difficulty
coef(LLTMwithError)

###################################################
### random effects
#make the number of groups much larger
covdata$group <- factor(rep(paste0('G',1:50), each = N/50))

#random groups
rmod1 <- mixedmirt(data, covdata, 1, fixed = ~ 0 + items, random = ~ 1|group, cl=cl)
summary(rmod1)
coef(rmod1)

#random groups and random items
rmod2 <- mixedmirt(data, covdata, 1, random = list(~ 1|group, ~ 1|items), cl=cl)
summary(rmod2)
eff <- randef(rmod2) #estimate random effects

#random slopes with fixed intercepts (suppressed correlation)
rmod3 <- mixedmirt(data, covdata, 1, fixed = ~ 0 + items, random = ~ -1 + pseudoIQ|group, cl=cl)
summary(rmod3)
(eff <- randef(rmod3))

###################################################
### Polytomous example

#make an arbitrary group difference
covdat <- data.frame(group = rep(c('m', 'f'), nrow(Science)/2))

#partial credit model
mod <- mixedmirt(Science, covdat, model=1, fixed = ~ 0 + group)
coef(mod)

#gpcm to estimate slopes
mod2 <- mixedmirt(Science, covdat, model=1, fixed = ~ 0 + group,
                 itemtype = 'gpcm')
summary(mod2)
anova(mod, mod2)

#graded model
mod3 <- mixedmirt(Science, covdat, model=1, fixed = ~ 0 + group,
                 itemtype = 'graded')
coef(mod3)