# bfactor: Full-Information Item Bi-factor and Two-Tier Analysis In mirt: Multidimensional Item Response Theory

## Description

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.

## Usage

 1 2 3 4 5 6 7 8 9 bfactor( data, model, model2 = paste0("G = 1-", ncol(data)), group = NULL, quadpts = NULL, invariance = "", ... ) 

## 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 c(1,1,2,2). For items that should only load on the second-tier factors (have no specific component) NA values may be used as place-holders. These numbers will be translated into a format suitable for mirt.model(), combined with the definition in model2, with the letter 'S' added to the respective factor number model2 a two-tier model specification object defined by mirt.model() or a string to be passed to 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 quadpts number of quadrature nodes to use after accounting for the reduced number of dimensions. Scheme is the same as the one used in mirt, however it is in regards to the reduced dimensions (e.g., a bifactor model has 2 dimensions to be integrated) invariance see multipleGroup for details, however, the specific factor variances and means will be constrained according to the dimensional reduction algorithm ... additional arguments to be passed to the estimation engine. See mirt for more details and examples

## 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 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.

## Value

function returns an object of class SingleGroupClass (SingleGroupClass-class) or MultipleGroupClass(MultipleGroupClass-class).

## Author(s)

Phil Chalmers rphilip.chalmers@gmail.com

## References

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

Bradlow, E.T., Wainer, H., & Wang, X. (1999). A Bayesian random effects model for testlets. Psychometrika, 64, 153-168.

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.

Wainer, H., Bradlow, E.T., & Wang, X. (2007). Testlet response theory and its applications. New York, NY: Cambridge University Press.

mirt
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 ## 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) ######### # General testlet response model (Wainer, 2007) #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) # Constrained testlet model (Bradlow, 1999) model2 <- "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), (GROUP, COV_22, COV_33, COV_44) COV = S1*S1, S2*S2, S3*S3" simmod2 <- bfactor(dataset, specific, model2) coef(simmod2, simplify=TRUE) anova(simmod, simmod2) ######### # 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)