bfactor: Full-Information Item Bi-factor and Two-Tier Analysis

Description

bfactor fits a confirmatory maximum likelihood two-tier/bifactor model to dichotomous and polytomous data under the item response theory paradigm. The IRT models are fit using a dimensional reduction EM algorithm so that regardless of the number of specific factors estimated the model only uses the number of factors in the second-tier structure plus 1. For the bifactor model the maximum number of dimensions is only 2 since the second-tier only consists of a ubiquitous unidimensional factor. See mirt for appropriate methods to be used on the objects returned from the estimation.

Usage

bfactor(data, model, model2 = mirt.model(paste0("G = 1-", ncol(data))),
  SE = FALSE, SE.type = "crossprod", group = NULL, verbose = TRUE, ...)

Arguments

data

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

model

a numeric vector specifying which factor loads on which item. For example, if for a 4 item test with two specific factors, the first specific factor loads on the first two items and the second specific factor on the last two, then the vector is

model2

a two-tier model specification object defined by mirt.model(). By default the model will fit a unidimensional model in the second-tier, and therefore be equivalent to the bifactor model

group

a factor variable indicating group membership used for multiple group analyses

logical; calculate information matrix and standard errors?

SE.type

type of standard errors to calculate. See mirt for details

verbose

logical; print observed log-likelihood value at each iteration?

...

additional arguments to be passed to the main estimation function. See mirt for more details

Details

bfactor follows the item factor analysis strategy explicated by Gibbons and Hedeker (1992), Gibbons et al. (2007), and Cai (2010). Nested models may be compared via an approximate chi-squared difference test or by a reduction in AIC or BIC (accessible via anova). See mirt for more details regarding the IRT estimation approach used in this package. The default is to use 21 quadrature for each dimensions, but this can be over-written by passing a

quadpts = #} argument.
Note: for multiple group two-tier analyses only the
second-tier means and variances should be freed since the
specific factors are not treated independently due to the
dimension reduction technique.

###load SAT12 and compute bifactor model with 3 specific factors 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)) specific <- c(2,3,2,3,3,2,1,2,1,1,1,3,1,3,1,2,1,1,3,3,1,1,3,1,3,3,1,3,2,3,1,2) mod1 <- bfactor(data, specific) summary(mod1) itemplot(mod1, 18, drop.zeros = TRUE) #drop the zero slopes to allow plotting

###Try with fixed guessing parameters added guess <- rep(.1,32) mod2 <- bfactor(data, specific, guess = guess) coef(mod2) anova(mod1, mod2)

## don't estimate specific factor for item 32 specific[32] <- NA mod3 <- bfactor(data, specific) anova(mod1, mod3)

# same, but decalred manually (not run) #sv <- mod2values(mod1) #sv$value[220] <- 0 #parameter 220 is the 32 items specific slope #sv$est[220] <- FALSE #mod3 <- bfactor(data, specific, pars = sv) #with excellent starting values

######### # mixed itemtype example

#simulate data a <- matrix(c( 1,0.5,NA, 1,0.5,NA, 1,0.5,NA, 1,0.5,NA, 1,0.5,NA, 1,0.5,NA, 1,0.5,NA, 1,NA,0.5, 1,NA,0.5, 1,NA,0.5, 1,NA,0.5, 1,NA,0.5, 1,NA,0.5, 1,NA,0.5),ncol=3,byrow=TRUE)

d <- matrix(c( -1.0,NA,NA, -1.5,NA,NA, 1.5,NA,NA, 0.0,NA,NA, 2.5,1.0,-1, 3.0,2.0,-0.5, 3.0,2.0,-0.5, 3.0,2.0,-0.5, 2.5,1.0,-1, 2.0,0.0,NA, -1.0,NA,NA, -1.5,NA,NA, 1.5,NA,NA, 0.0,NA,NA),ncol=3,byrow=TRUE) items <- rep('dich', 14) items[5:10] <- 'graded'

sigma <- diag(3) dataset <- simdata(a,d,2000,itemtype=items,sigma=sigma)

specific <- c(rep(1,7),rep(2,7)) simmod <- bfactor(dataset, specific) coef(simmod)

######### # Two-tier model

#simulate data set.seed(1234) a <- matrix(c( 0,1,0.5,NA,NA, 0,1,0.5,NA,NA, 0,1,0.5,NA,NA, 0,1,0.5,NA,NA, 0,1,0.5,NA,NA, 0,1,NA,0.5,NA, 0,1,NA,0.5,NA, 0,1,NA,0.5,NA, 1,0,NA,0.5,NA, 1,0,NA,0.5,NA, 1,0,NA,0.5,NA, 1,0,NA,NA,0.5, 1,0,NA,NA,0.5, 1,0,NA,NA,0.5, 1,0,NA,NA,0.5, 1,0,NA,NA,0.5),ncol=5,byrow=TRUE)

d <- matrix(rnorm(16)) items <- rep('dich', 16)

sigma <- diag(5) sigma[1,2] <- sigma[2,1] <- .4 dataset <- simdata(a,d,2000,itemtype=items,sigma=sigma)

specific <- c(rep(1,5),rep(2,6),rep(3,5)) model <- mirt.model(' G1 = 1-8 G2 = 9-16 COV = G1*G2')

simmod <- bfactor(dataset, specific, model, quadpts = 9, TOL = 1e-3) coef(simmod) summary(simmod)

[object Object] Cai, L. (2010). A two-tier full-information item factor analysis model with applications. Psychometrika, 75, 581-612.

Chalmers, R., P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48(6), 1-29.

Gibbons, R. D., & Hedeker, D. R. (1992). Full-information Item Bi-Factor Analysis. Psychometrika, 57, 423-436.

Gibbons, R. D., Darrell, R. B., Hedeker, D., Weiss, D. J., Segawa, E., Bhaumik, D. K., Kupfer, D. J., Frank, E., Grochocinski, V. J., & Stover, A. (2007). Full-Information item bifactor analysis of graded response data. Applied Psychological Measurement, 31, 4-19

anova-method, coef-method, summary-method, residuals-method, plot-method, expand.table, key2binary, mirt.model, mirt, bfactor, multipleGroup, mixedmirt, wald, itemplot, fscores, M2, extract.item, iteminfo, testinfo, probtrace, boot.mirt, imputeMissing, itemfit, mod2values, simdata, createItem models