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 = "",
  ...
)

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

Value

function returns an object of class SingleGroupClass

(SingleGroupClass-class) or MultipleGroupClass(MultipleGroupClass-class).

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.

$$\Sigma_{two-tier} = \left(\begin{array}{cc} G & 0 \\ 0 & diag(S) \end{array} \right)$$

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.

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.

Author

Phil Chalmers rphilip.chalmers@gmail.com

Examples


# \donttest{

### 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)
#>              G      S1      S2      S3      h2
#> Item.1  0.4078  0.0000  0.2273  0.0000 0.21799
#> Item.2  0.6195  0.0000  0.0000  0.3391 0.49881
#> Item.3  0.5573  0.0000 -0.0744  0.0000 0.31612
#> Item.4  0.2808  0.0000  0.0000  0.3101 0.17503
#> Item.5  0.4779  0.0000  0.0000  0.2544 0.29311
#> Item.6  0.5341  0.0000  0.2724  0.0000 0.35948
#> Item.7  0.4741  0.4210  0.0000  0.0000 0.40199
#> Item.8  0.3537  0.0000  0.2732  0.0000 0.19971
#> Item.9  0.2181  0.5321  0.0000  0.0000 0.33068
#> Item.10 0.4849  0.3792  0.0000  0.0000 0.37895
#> Item.11 0.6442  0.3321  0.0000  0.0000 0.52536
#> Item.12 0.0699  0.0000  0.0000  0.1592 0.03023
#> Item.13 0.5219  0.2743  0.0000  0.0000 0.34764
#> Item.14 0.4791  0.0000  0.0000  0.4563 0.43776
#> Item.15 0.5985  0.2402  0.0000  0.0000 0.41590
#> Item.16 0.3885  0.0000  0.2049  0.0000 0.19292
#> Item.17 0.6636  0.1177  0.0000  0.0000 0.45423
#> Item.18 0.7163  0.0775  0.0000  0.0000 0.51903
#> Item.19 0.4520  0.0000  0.0000  0.0198 0.20466
#> Item.20 0.6578  0.0000  0.0000  0.1792 0.46478
#> Item.21 0.2806  0.3451  0.0000  0.0000 0.19787
#> Item.22 0.7025 -0.0281  0.0000  0.0000 0.49425
#> Item.23 0.3236  0.0000  0.0000  0.2657 0.17536
#> Item.24 0.5848  0.1030  0.0000  0.0000 0.35266
#> Item.25 0.3732  0.0000  0.0000  0.3297 0.24799
#> Item.26 0.6430  0.0000  0.0000  0.2124 0.45854
#> Item.27 0.7374  0.1554  0.0000  0.0000 0.56792
#> Item.28 0.5256  0.0000  0.0000  0.0758 0.28199
#> Item.29 0.4185  0.0000  0.7071  0.0000 0.67516
#> Item.30 0.2455  0.0000  0.0000 -0.0959 0.06946
#> Item.31 0.8333 -0.0872  0.0000  0.0000 0.70202
#> Item.32 0.0780  0.0000  0.0165  0.0000 0.00635
#> 
#> SS loadings:  8.435 1.03 0.748 0.781 
#> Proportion Var:  0.264 0.032 0.023 0.024 
#> 
#> Factor correlations: 
#> 
#>    G S1 S2 S3
#> G  1         
#> S1 0  1      
#> S2 0  0  1   
#> S3 0  0  0  1
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)
#> $Item.1
#>        a1 a2    a3 a4      d   g u
#> par 1.225  0 0.624  0 -1.822 0.1 1
#> 
#> $Item.2
#>        a1 a2 a3    a4     d   g u
#> par 1.721  0  0 0.954 0.171 0.1 1
#> 
#> $Item.3
#>        a1 a2     a3 a4      d   g u
#> par 2.415  0 -0.459  0 -2.602 0.1 1
#> 
#> $Item.4
#>        a1 a2 a3    a4      d   g u
#> par 0.745  0  0 0.695 -0.989 0.1 1
#> 
#> $Item.5
#>        a1 a2 a3    a4     d   g u
#> par 1.048  0  0 0.603 0.419 0.1 1
#> 
#> $Item.6
#>       a1 a2    a3 a4      d   g u
#> par 3.06  0 0.501  0 -5.002 0.1 1
#> 
#> $Item.7
#>        a1    a2 a3 a4     d   g u
#> par 1.121 0.839  0  0 1.373 0.1 1
#> 
#> $Item.8
#>        a1 a2    a3 a4      d   g u
#> par 1.956  0 1.443  0 -3.772 0.1 1
#> 
#> $Item.9
#>        a1    a2 a3 a4     d   g u
#> par 0.512 1.236  0  0 2.484 0.1 1
#> 
#> $Item.10
#>       a1    a2 a3 a4      d   g u
#> par 1.68 1.506  0  0 -1.031 0.1 1
#> 
#> $Item.11
#>        a1    a2 a3 a4     d   g u
#> par 1.655 0.842  0  0 5.441 0.1 1
#> 
#> $Item.12
#>        a1 a2 a3    a4      d   g u
#> par 0.129  0  0 0.364 -0.641 0.1 1
#> 
#> $Item.13
#>        a1    a2 a3 a4     d   g u
#> par 1.183 0.477  0  0 0.679 0.1 1
#> 
#> $Item.14
#>        a1 a2 a3    a4     d   g u
#> par 1.125  0  0 1.058 1.164 0.1 1
#> 
#> $Item.15
#>        a1    a2 a3 a4     d   g u
#> par 1.435 0.317  0  0 1.863 0.1 1
#> 
#> $Item.16
#>       a1 a2    a3 a4      d   g u
#> par 0.95  0 0.573  0 -0.783 0.1 1
#> 
#> $Item.17
#>        a1    a2 a3 a4     d   g u
#> par 1.547 0.059  0  0 4.112 0.1 1
#> 
#> $Item.18
#>        a1    a2 a3 a4      d   g u
#> par 2.731 0.094  0  0 -1.808 0.1 1
#> 
#> $Item.19
#>        a1 a2 a3    a4      d   g u
#> par 0.918  0  0 0.101 -0.001 0.1 1
#> 
#> $Item.20
#>        a1 a2 a3    a4     d   g u
#> par 1.456  0  0 0.593 2.501 0.1 1
#> 
#> $Item.21
#>        a1    a2 a3 a4    d   g u
#> par 0.596 0.493  0  0 2.49 0.1 1
#> 
#> $Item.22
#>        a1     a2 a3 a4     d   g u
#> par 1.554 -0.242  0  0 3.428 0.1 1
#> 
#> $Item.23
#>        a1 a2 a3    a4      d   g u
#> par 0.908  0  0 0.766 -1.488 0.1 1
#> 
#> $Item.24
#>        a1    a2 a3 a4     d   g u
#> par 1.379 0.001  0  0 1.132 0.1 1
#> 
#> $Item.25
#>       a1 a2 a3    a4      d   g u
#> par 1.03  0  0 1.094 -1.164 0.1 1
#> 
#> $Item.26
#>        a1 a2 a3    a4      d   g u
#> par 1.985  0  0 0.747 -0.663 0.1 1
#> 
#> $Item.27
#>        a1    a2 a3 a4     d   g u
#> par 1.909 0.348  0  0 2.642 0.1 1
#> 
#> $Item.28
#>        a1 a2 a3    a4      d   g u
#> par 1.213  0  0 0.142 -0.097 0.1 1
#> 
#> $Item.29
#>        a1 a2    a3 a4      d   g u
#> par 1.938  0 2.339  0 -2.209 0.1 1
#> 
#> $Item.30
#>        a1 a2 a3     a4      d   g u
#> par 0.479  0  0 -0.128 -0.527 0.1 1
#> 
#> $Item.31
#>        a1    a2 a3 a4     d   g u
#> par 3.173 -0.82  0  0 3.316 0.1 1
#> 
#> $Item.32
#>        a1 a2     a3 a4      d   g u
#> par 0.534  0 -0.053  0 -2.786 0.1 1
#> 
#> $GroupPars
#>     MEAN_1 MEAN_2 MEAN_3 MEAN_4 COV_11 COV_21 COV_31 COV_41 COV_22 COV_32
#> par      0      0      0      0      1      0      0      0      1      0
#>     COV_42 COV_33 COV_43 COV_44
#> par      0      1      0      1
#> 
anova(mod1, mod2)
#>           AIC    SABIC       HQ      BIC    logLik     X2 df   p
#> mod1 19062.10 19179.44 19226.42 19484.21 -9435.052              
#> mod2 19009.55 19126.88 19173.87 19431.65 -9408.775 52.553  0 NaN

## don't estimate specific factor for item 32
specific[32] <- NA
mod3 <- bfactor(data, specific)
anova(mod3, mod1)
#>           AIC    SABIC       HQ      BIC    logLik   X2 df     p
#> mod3 19060.12 19176.23 19222.73 19477.83 -9435.062              
#> mod1 19062.10 19179.44 19226.42 19484.21 -9435.052 0.02  1 0.886

# 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)
itemstats(dataset)
#> $overall
#>     N mean_total.score sd_total.score ave.r  sd.r alpha
#>  2000           15.043          4.481 0.176 0.038 0.731
#> 
#> $itemstats
#>            N  mean    sd total.r total.r_if_rm alpha_if_rm
#> Item_1  2000 0.300 0.458   0.414         0.324       0.719
#> Item_2  2000 0.221 0.415   0.432         0.353       0.718
#> Item_3  2000 0.771 0.420   0.429         0.347       0.719
#> Item_4  2000 0.494 0.500   0.471         0.377       0.714
#> Item_5  2000 1.832 0.967   0.557         0.380       0.714
#> Item_6  2000 2.141 0.900   0.558         0.395       0.710
#> Item_7  2000 2.148 0.895   0.553         0.391       0.711
#> Item_8  2000 2.157 0.881   0.509         0.342       0.718
#> Item_9  2000 1.846 0.975   0.577         0.402       0.711
#> Item_10 2000 1.319 0.749   0.542         0.408       0.708
#> Item_11 2000 0.312 0.464   0.434         0.345       0.718
#> Item_12 2000 0.226 0.418   0.395         0.313       0.721
#> Item_13 2000 0.778 0.416   0.373         0.289       0.723
#> Item_14 2000 0.498 0.500   0.445         0.349       0.717
#> 
#> $proportions
#>             0     1     2     3
#> Item_1  0.700 0.300    NA    NA
#> Item_2  0.779 0.221    NA    NA
#> Item_3  0.229 0.771    NA    NA
#> Item_4  0.506 0.494    NA    NA
#> Item_5  0.116 0.218 0.384 0.281
#> Item_6  0.086 0.085 0.430 0.399
#> Item_7  0.082 0.091 0.424 0.402
#> Item_8  0.080 0.082 0.439 0.399
#> Item_9  0.119 0.206 0.384 0.291
#> Item_10 0.172 0.338 0.490    NA
#> Item_11 0.688 0.312    NA    NA
#> Item_12 0.774 0.226    NA    NA
#> Item_13 0.222 0.778    NA    NA
#> Item_14 0.502 0.498    NA    NA
#> 

specific <- c(rep(1,7),rep(2,7))
simmod <- bfactor(dataset, specific)
coef(simmod)
#> $Item_1
#>        a1    a2 a3      d g u
#> par 0.964 0.397  0 -1.031 0 1
#> 
#> $Item_2
#>        a1    a2 a3      d g u
#> par 1.223 0.662  0 -1.688 0 1
#> 
#> $Item_3
#>       a1   a2 a3     d g u
#> par 1.14 0.58  0 1.574 0 1
#> 
#> $Item_4
#>        a1    a2 a3      d g u
#> par 1.058 0.508  0 -0.031 0 1
#> 
#> $Item_5
#>        a1    a2 a3    d1   d2     d3
#> par 0.898 0.612  0 2.439 0.86 -1.144
#> 
#> $Item_6
#>        a1    a2 a3    d1    d2     d3
#> par 1.012 0.532  0 2.847 1.935 -0.524
#> 
#> $Item_7
#>        a1    a2 a3    d1    d2     d3
#> par 1.007 0.679  0 2.971 1.978 -0.506
#> 
#> $Item_8
#>        a1 a2    a3    d1    d2     d3
#> par 0.855  0 0.642 2.879 1.984 -0.502
#> 
#> $Item_9
#>        a1 a2    a3    d1    d2    d3
#> par 0.999  0 0.492 2.421 0.904 -1.11
#> 
#> $Item_10
#>        a1 a2    a3    d1     d2
#> par 1.081  0 0.661 2.023 -0.042
#> 
#> $Item_11
#>        a1 a2    a3      d g u
#> par 1.057  0 0.499 -0.997 0 1
#> 
#> $Item_12
#>        a1 a2    a3     d g u
#> par 0.991  0 0.706 -1.57 0 1
#> 
#> $Item_13
#>        a1 a2    a3     d g u
#> par 0.886  0 0.465 1.497 0 1
#> 
#> $Item_14
#>        a1 a2    a3      d g u
#> par 0.958  0 0.401 -0.012 0 1
#> 
#> $GroupPars
#>     MEAN_1 MEAN_2 MEAN_3 COV_11 COV_21 COV_31 COV_22 COV_32 COV_33
#> par      0      0      0      1      0      0      1      0      1
#> 

#########
# 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)
#>            [,1]      [,2]     [,3]      [,4]
#>  [1,] 0.8503394 0.8503394 0.000000 0.0000000
#>  [2,] 1.3274088 1.3274088 0.000000 0.0000000
#>  [3,] 1.6910208 1.6910208 0.000000 0.0000000
#>  [4,] 0.6042850 0.6042850 0.000000 0.0000000
#>  [5,] 1.3892130 0.0000000 1.389213 0.0000000
#>  [6,] 1.4216480 0.0000000 1.421648 0.0000000
#>  [7,] 1.0279618 0.0000000 1.027962 0.0000000
#>  [8,] 1.0366667 0.0000000 1.036667 0.0000000
#>  [9,] 1.0311394 0.0000000 0.000000 1.0311394
#> [10,] 0.9351846 0.0000000 0.000000 0.9351846
#> [11,] 1.0584888 0.0000000 0.000000 1.0584888
#> [12,] 0.9052755 0.0000000 0.000000 0.9052755
d <- rnorm(12, 0, .5)
sigma <- diag(c(1, .5, 1, .5))
dataset <- simdata(a,d,2000,itemtype=rep('2PL', 12),sigma=sigma)
itemstats(dataset)
#> $overall
#>     N mean_total.score sd_total.score ave.r  sd.r alpha
#>  2000                6          2.929 0.175 0.068 0.717
#> 
#> $itemstats
#>            N  mean    sd total.r total.r_if_rm alpha_if_rm
#> Item_1  2000 0.426 0.495   0.438         0.287       0.708
#> Item_2  2000 0.502 0.500   0.560         0.425       0.689
#> Item_3  2000 0.571 0.495   0.575         0.445       0.686
#> Item_4  2000 0.502 0.500   0.383         0.224       0.716
#> Item_5  2000 0.464 0.499   0.549         0.413       0.690
#> Item_6  2000 0.436 0.496   0.561         0.428       0.688
#> Item_7  2000 0.440 0.497   0.500         0.356       0.698
#> Item_8  2000 0.693 0.462   0.474         0.339       0.701
#> Item_9  2000 0.511 0.500   0.481         0.334       0.701
#> Item_10 2000 0.456 0.498   0.465         0.316       0.704
#> Item_11 2000 0.458 0.498   0.459         0.309       0.705
#> Item_12 2000 0.540 0.498   0.475         0.327       0.702
#> 
#> $proportions
#>             0     1
#> Item_1  0.575 0.426
#> Item_2  0.498 0.502
#> Item_3  0.430 0.571
#> Item_4  0.498 0.502
#> Item_5  0.536 0.464
#> Item_6  0.564 0.436
#> Item_7  0.559 0.440
#> Item_8  0.308 0.693
#> Item_9  0.488 0.511
#> Item_10 0.543 0.456
#> Item_11 0.541 0.458
#> Item_12 0.460 0.540
#> 

# 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)
#> $items
#>            a1    a2    a3    a4      d g u
#> Item_1  0.794 0.794 0.000 0.000 -0.359 0 1
#> Item_2  1.544 1.544 0.000 0.000  0.011 0 1
#> Item_3  1.762 1.762 0.000 0.000  0.479 0 1
#> Item_4  0.544 0.544 0.000 0.000  0.011 0 1
#> Item_5  1.386 0.000 1.386 0.000 -0.244 0 1
#> Item_6  1.497 0.000 1.497 0.000 -0.449 0 1
#> Item_7  0.853 0.000 0.853 0.000 -0.312 0 1
#> Item_8  0.953 0.000 0.953 0.000  1.101 0 1
#> Item_9  0.981 0.000 0.000 0.981  0.058 0 1
#> Item_10 0.913 0.000 0.000 0.913 -0.217 0 1
#> Item_11 0.868 0.000 0.000 0.868 -0.204 0 1
#> Item_12 0.966 0.000 0.000 0.966  0.206 0 1
#> 
#> $means
#>  G S1 S2 S3 
#>  0  0  0  0 
#> 
#> $cov
#>    G    S1    S2    S3
#> G  1 0.000 0.000 0.000
#> S1 0 0.452 0.000 0.000
#> S2 0 0.000 1.135 0.000
#> S3 0 0.000 0.000 0.432
#> 

# 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)
#> $items
#>            a1    a2    a3    a4      d g u
#> Item_1  0.744 0.744 0.000 0.000 -0.360 0 1
#> Item_2  1.453 1.453 0.000 0.000  0.010 0 1
#> Item_3  1.664 1.664 0.000 0.000  0.482 0 1
#> Item_4  0.509 0.509 0.000 0.000  0.011 0 1
#> Item_5  1.541 0.000 1.541 0.000 -0.241 0 1
#> Item_6  1.670 0.000 1.670 0.000 -0.445 0 1
#> Item_7  0.968 0.000 0.968 0.000 -0.313 0 1
#> Item_8  1.075 0.000 1.075 0.000  1.098 0 1
#> Item_9  0.927 0.000 0.000 0.927  0.059 0 1
#> Item_10 0.854 0.000 0.000 0.854 -0.218 0 1
#> Item_11 0.813 0.000 0.000 0.813 -0.205 0 1
#> Item_12 0.908 0.000 0.000 0.908  0.207 0 1
#> 
#> $means
#>  G S1 S2 S3 
#>  0  0  0  0 
#> 
#> $cov
#>    G    S1    S2    S3
#> G  1 0.000 0.000 0.000
#> S1 0 0.667 0.000 0.000
#> S2 0 0.000 0.667 0.000
#> S3 0 0.000 0.000 0.667
#> 
anova(simmod2, simmod)
#>              AIC    SABIC       HQ      BIC   logLik     X2 df     p
#> simmod2 30256.59 30317.19 30308.00 30396.61 -15103.3                
#> simmod  30248.79 30314.24 30304.32 30400.02 -15097.4 11.795  2 0.003


#########
# 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)
itemstats(dataset)
#> $overall
#>     N mean_total.score sd_total.score ave.r  sd.r alpha
#>  2000            7.086          3.077 0.108 0.058 0.662
#> 
#> $itemstats
#>            N  mean    sd total.r total.r_if_rm alpha_if_rm
#> Item_1  2000 0.288 0.453   0.378         0.241       0.650
#> Item_2  2000 0.571 0.495   0.422         0.276       0.646
#> Item_3  2000 0.705 0.456   0.381         0.245       0.650
#> Item_4  2000 0.133 0.340   0.289         0.183       0.656
#> Item_5  2000 0.601 0.490   0.393         0.246       0.650
#> Item_6  2000 0.587 0.492   0.419         0.274       0.646
#> Item_7  2000 0.379 0.485   0.444         0.304       0.642
#> Item_8  2000 0.378 0.485   0.400         0.256       0.649
#> Item_9  2000 0.386 0.487   0.392         0.246       0.650
#> Item_10 2000 0.322 0.467   0.400         0.261       0.648
#> Item_11 2000 0.402 0.490   0.455         0.315       0.640
#> Item_12 2000 0.318 0.466   0.414         0.278       0.646
#> Item_13 2000 0.368 0.482   0.423         0.281       0.645
#> Item_14 2000 0.498 0.500   0.424         0.277       0.646
#> Item_15 2000 0.669 0.471   0.394         0.254       0.649
#> Item_16 2000 0.482 0.500   0.444         0.300       0.642
#> 
#> $proportions
#>             0     1
#> Item_1  0.713 0.288
#> Item_2  0.430 0.571
#> Item_3  0.295 0.705
#> Item_4  0.867 0.133
#> Item_5  0.400 0.601
#> Item_6  0.413 0.587
#> Item_7  0.621 0.379
#> Item_8  0.622 0.378
#> Item_9  0.614 0.386
#> Item_10 0.678 0.322
#> Item_11 0.598 0.402
#> Item_12 0.681 0.318
#> Item_13 0.632 0.368
#> Item_14 0.502 0.498
#> Item_15 0.330 0.669
#> Item_16 0.518 0.482
#> 

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)
#> $items
#>            a1    a2    a3    a4    a5      d g u
#> Item_1  0.965 0.000 0.385 0.000 0.000 -1.100 0 1
#> Item_2  1.076 0.000 0.550 0.000 0.000  0.363 0 1
#> Item_3  0.898 0.000 0.592 0.000 0.000  1.068 0 1
#> Item_4  0.896 0.000 0.710 0.000 0.000 -2.293 0 1
#> Item_5  0.892 0.000 0.848 0.000 0.000  0.526 0 1
#> Item_6  1.013 0.000 0.000 0.413 0.000  0.435 0 1
#> Item_7  1.162 0.000 0.000 0.451 0.000 -0.639 0 1
#> Item_8  0.945 0.000 0.000 0.609 0.000 -0.623 0 1
#> Item_9  0.000 0.831 0.000 0.371 0.000 -0.544 0 1
#> Item_10 0.000 0.925 0.000 0.610 0.000 -0.926 0 1
#> Item_11 0.000 1.142 0.000 0.495 0.000 -0.517 0 1
#> Item_12 0.000 0.978 0.000 0.000 0.634 -0.964 0 1
#> Item_13 0.000 1.108 0.000 0.000 0.437 -0.694 0 1
#> Item_14 0.000 1.004 0.000 0.000 0.321 -0.012 0 1
#> Item_15 0.000 0.916 0.000 0.000 0.758  0.897 0 1
#> Item_16 0.000 1.020 0.000 0.000 0.650 -0.096 0 1
#> 
#> $means
#> G1 G2 S1 S2 S3 
#>  0  0  0  0  0 
#> 
#> $cov
#>       G1    G2 S1 S2 S3
#> G1 1.000 0.412  0  0  0
#> G2 0.412 1.000  0  0  0
#> S1 0.000 0.000  1  0  0
#> S2 0.000 0.000  0  1  0
#> S3 0.000 0.000  0  0  1
#> 
summary(simmod)
#>            G1    G2    S1    S2    S3    h2
#> Item_1  0.484 0.000 0.193 0.000 0.000 0.271
#> Item_2  0.516 0.000 0.263 0.000 0.000 0.335
#> Item_3  0.446 0.000 0.294 0.000 0.000 0.285
#> Item_4  0.437 0.000 0.346 0.000 0.000 0.311
#> Item_5  0.425 0.000 0.404 0.000 0.000 0.343
#> Item_6  0.501 0.000 0.000 0.204 0.000 0.293
#> Item_7  0.551 0.000 0.000 0.214 0.000 0.349
#> Item_8  0.463 0.000 0.000 0.299 0.000 0.304
#> Item_9  0.000 0.431 0.000 0.192 0.000 0.222
#> Item_10 0.000 0.456 0.000 0.300 0.000 0.298
#> Item_11 0.000 0.541 0.000 0.235 0.000 0.348
#> Item_12 0.000 0.474 0.000 0.000 0.307 0.319
#> Item_13 0.000 0.533 0.000 0.000 0.210 0.329
#> Item_14 0.000 0.501 0.000 0.000 0.160 0.277
#> Item_15 0.000 0.441 0.000 0.000 0.365 0.328
#> Item_16 0.000 0.488 0.000 0.000 0.311 0.336
#> 
#> SS loadings:  1.839 1.88 0.476 0.359 0.395 
#> Proportion Var:  0.115 0.118 0.03 0.022 0.025 
#> 
#> Factor correlations: 
#> 
#>       G1 G2 S1 S2 S3
#> G1 1.000            
#> G2 0.412  1         
#> S1 0.000  0  1      
#> S2 0.000  0  0  1   
#> S3 0.000  0  0  0  1
itemfit(simmod, QMC=TRUE)
#>       item   S_X2 df.S_X2 RMSEA.S_X2 p.S_X2
#> 1   Item_1  7.103       9      0.000  0.626
#> 2   Item_2 13.326      10      0.013  0.206
#> 3   Item_3  8.332       9      0.000  0.501
#> 4   Item_4  8.531      10      0.000  0.577
#> 5   Item_5  7.170      10      0.000  0.709
#> 6   Item_6  3.967      10      0.000  0.949
#> 7   Item_7  8.350      10      0.000  0.595
#> 8   Item_8 16.010      10      0.017  0.099
#> 9   Item_9 17.529      10      0.019  0.063
#> 10 Item_10 12.058      10      0.010  0.281
#> 11 Item_11 13.567      10      0.013  0.194
#> 12 Item_12 13.907       9      0.017  0.126
#> 13 Item_13 11.144      10      0.008  0.346
#> 14 Item_14  7.852      10      0.000  0.643
#> 15 Item_15 14.142       9      0.017  0.117
#> 16 Item_16  5.926      10      0.000  0.821
M2(simmod, QMC=TRUE)
#>             M2 df         p RMSEA RMSEA_5   RMSEA_95      SRMSR      TLI CFI
#> stats 86.28163 87 0.5015988     0       0 0.01201603 0.01662365 1.000285   1
residuals(simmod, QMC=TRUE)
#> LD matrix (lower triangle) and standardized values.
#> 
#> Upper triangle summary:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>  -0.046  -0.011  -0.002  -0.001   0.012   0.041 
#> 
#>         Item_1 Item_2 Item_3 Item_4 Item_5 Item_6 Item_7 Item_8 Item_9 Item_10
#> Item_1      NA -0.011 -0.015 -0.007  0.016  0.003  0.006 -0.002  0.004  -0.009
#> Item_2   0.263     NA -0.002  0.005 -0.001  0.016 -0.022  0.021 -0.003  -0.006
#> Item_3   0.441  0.008     NA  0.014 -0.001  0.029 -0.017 -0.011 -0.021  -0.028
#> Item_4   0.086  0.054  0.376     NA -0.014 -0.023  0.021 -0.004  0.020  -0.040
#> Item_5   0.514  0.004  0.003  0.386     NA -0.028  0.011  0.014 -0.021   0.013
#> Item_6   0.015  0.483  1.630  1.038  1.588     NA -0.022 -0.009  0.031  -0.004
#> Item_7   0.077  0.996  0.590  0.852  0.258  0.992     NA -0.007 -0.004   0.013
#> Item_8   0.012  0.858  0.264  0.039  0.377  0.154  0.094     NA -0.020   0.008
#> Item_9   0.033  0.017  0.863  0.803  0.890  1.974  0.037  0.808     NA   0.001
#> Item_10  0.157  0.084  1.528  3.158  0.360  0.038  0.330  0.128  0.001      NA
#> Item_11  0.510  0.125  2.195  0.231  0.004  0.215  1.049  0.004  0.031   0.754
#> Item_12  2.017  0.253  1.865  0.388  0.005  0.417  0.074  0.090  0.443   0.042
#> Item_13  0.470  2.122  0.125  0.271  0.881  0.264  0.310  4.304  0.009   0.059
#> Item_14  0.101  1.546  0.165  0.006  0.296  0.004  1.672  0.765  3.341   0.066
#> Item_15  0.822  0.257  0.011  0.442  0.443  0.113  0.526  0.297  2.306   0.044
#> Item_16  0.097  0.627  1.486  0.127  0.445  0.011  0.732  0.061  0.007   0.944
#>         Item_11 Item_12 Item_13 Item_14 Item_15 Item_16
#> Item_1   -0.016  -0.032   0.015  -0.007   0.020  -0.007
#> Item_2    0.008   0.011  -0.033  -0.028  -0.011   0.018
#> Item_3    0.033   0.031   0.008  -0.009  -0.002   0.027
#> Item_4   -0.011   0.014  -0.012  -0.002  -0.015  -0.008
#> Item_5   -0.001   0.002  -0.021  -0.012   0.015   0.015
#> Item_6    0.010   0.014  -0.011   0.001  -0.008   0.002
#> Item_7    0.023   0.006   0.012   0.029  -0.016   0.019
#> Item_8   -0.001   0.007  -0.046  -0.020   0.012   0.006
#> Item_9    0.004  -0.015  -0.002   0.041  -0.034  -0.002
#> Item_10  -0.019   0.005   0.005   0.006  -0.005   0.022
#> Item_11      NA   0.009   0.000  -0.025   0.039  -0.011
#> Item_12   0.149      NA  -0.008   0.013  -0.002  -0.009
#> Item_13   0.000   0.140      NA   0.013  -0.012   0.003
#> Item_14   1.269   0.338   0.336      NA  -0.007  -0.023
#> Item_15   3.061   0.010   0.284   0.086      NA   0.010
#> Item_16   0.250   0.153   0.015   1.026   0.190      NA

# }