Full-Information Item Bi-factor and Two-Tier Analysis
bfactor fits a confirmatory maximum likelihood two-tier/bifactor/testlet 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.
bfactor(
data,
model,
model2 = paste0("G = 1-", ncol(data)),
group = NULL,
quadpts = NULL,
invariance = "",
...
)data |
a |
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 |
group |
a factor variable indicating group membership used for multiple group analyses |
quadpts |
number of quadrature nodes to use after accounting for the reduced number of dimensions.
Scheme is the same as the one used in |
invariance |
see |
... |
additional arguments to be passed to the estimation engine. See |
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 two-tier model has a specific block diagonal covariance structure between the primary and secondary latent traits. Namely, the secondary latent traits are assumed to be orthogonal to all traits and have a fixed variance of 1, while the primary traits can be organized to vary and covary with other primary traits in the model.
Σ_{two-tier} = ≤ft(\begin{array}{cc} G & 0 \\ 0 & diag(S) \end{array} \right)
The bifactor model is a special case of the two-tier model when G above is a 1x1 matrix, and therefore only 1 primary factor is being modeled. Evaluation of the numerical integrals for the two-tier model requires only ncol(G) + 1 dimensions for integration since the S second order (or 'specific') factors require only 1 integration grid due to the dimension reduction technique.
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.
function returns an object of class SingleGroupClass
(SingleGroupClass-class) or MultipleGroupClass(MultipleGroupClass-class).
Phil Chalmers rphilip.chalmers@gmail.com
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. doi: 10.18637/jss.v048.i06
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.
## Not run:
###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 declared 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('2PL', 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)
#########
# testlet response model
#simulate data
set.seed(1234)
a <- matrix(0, 12, 4)
a[,1] <- rlnorm(12, .2, .3)
ind <- 1
for(i in 1:3){
a[ind:(ind+3),i+1] <- a[ind:(ind+3),1]
ind <- ind+4
}
print(a)
d <- rnorm(12, 0, .5)
sigma <- diag(c(1, .5, 1, .5))
dataset <- simdata(a,d,2000,itemtype=rep('2PL', 12),sigma=sigma)
# estimate by applying constraints and freeing the latent variances
specific <- c(rep(1,4),rep(2,4), rep(3,4))
model <- "G = 1-12
CONSTRAIN = (1, a1, a2), (2, a1, a2), (3, a1, a2), (4, a1, a2),
(5, a1, a3), (6, a1, a3), (7, a1, a3), (8, a1, a3),
(9, a1, a4), (10, a1, a4), (11, a1, a4), (12, a1, a4)
COV = S1*S1, S2*S2, S3*S3"
simmod <- bfactor(dataset, specific, model)
coef(simmod, simplify=TRUE)
#########
# 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('2PL', 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 <- '
G1 = 1-8
G2 = 9-16
COV = G1*G2'
#quadpts dropped for faster estimation, but not as precise
simmod <- bfactor(dataset, specific, model, quadpts = 9, TOL = 1e-3)
coef(simmod, simplify=TRUE)
summary(simmod)
itemfit(simmod, QMC=TRUE)
M2(simmod, QMC=TRUE)
residuals(simmod, QMC=TRUE)
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.