tmt_rm: Estimation (CML) ot the Rasch model with or without multistage designs.

Description

The tmt_rm function estimates the Rasch model. If the data are collected based on a multistage design (see Zwitser and Maris, 2015) the specific multistage design mstdesign has to be submitted.

Usage

tmt_rm(
  dat,
  mstdesign = NULL,
  weights = NULL,
  start = NULL,
  sum0 = TRUE,
  se = TRUE,
  optimization = "nlminb",
  ...
)

Value

List with following entries

betapar: Estimated item difficulty parameters (if sum0=FALSE, than the first item is set to 0)
se.beta: Standard errors of the estimated item parameters
loglik: Conditional log-likelihood of the model
df: Number of estimated parameters
N: Number of Persons
I: Number of items
data_orig: Submitted data frame with item responses
data: Used data frame with item responses
desmat: Design matrix
convergence: Convergence criterion
iterations: Number of iterations
hessian: Hessian-Matrix
model: Used model ((mst) for Rasch model with multistage design)
call: Submitted arguments for the function (matched call)
designelements: If the multistage version is requested, the preprocessed design is returned, otherwise NULL
mstdesign: If the multistage version is requested, the submitted design is returned, otherwise NULL

Arguments

dat: a matrix of dichotomous (0/1) data or a list of the function tmt_designsim
mstdesign: Model for the multistage design, if CML estimation without multistage designs is required, than leave the default value
weights: is optional for the weights of cases
start: Vector of start values. If no vector is provided, the start values will be automatic generated
sum0: logical: If the item parameters should be normed to 'sum = 0' as recommended by Glas (2016, p. 208). Otherwise sum0=FALSE
se: logical: should the standard error should be estimated?
optimization: character: Per default 'nlminb' is used but 'optim' is also supported.
...: optional further arguments for optim and nlminb use control = list() with arguments.

Author

Jan Steinfeld

Details

According to Glas (1988) <doi:10.3102/10769986013001045> CML estimation of item parameters is biased if the data is collected in multistage designs and this design is not considered. Zwitser and Maris (2015) <doi:10.1007/s11336-013-9369-6> propose to use an additional design matrix to fragment the elementary symmetric function. Their approach is implemented in this package. MST designs with a probabilistic instead of a deterministic routing rule (see, e.g. Chen, Yamamoto, & von Davier, 2014 <doi:10.1201/b16858>) are not estimated with this method, therefore the proposed solouting is again modified by Steinfeld and Robitzsch (2021) <doi:10.31234/osf.io/ew27f> which is also integrated into this package.

References

Baker, F. B., & Harwell, M. R. (1996). Computing elementary symmetric functions and their derivatives: A didactic. Applied Psychological Measurement, 20(2), 169-192.
Baker, F. B., & Kim, S. H. (2004). Item response theory: Parameter estimation techniques. CRC Press.
Chen, H., Yamamoto, K., & von Davier, M. (2014). Controlling multistage testing exposure rates in international large-scale assessments. In A. Yan, A. A. von Davier, & C. Lewis (Eds.), Computerized Multistage Testing: Theory and Applications (pp. 391-409). New York: CRC Press. https://doi.org/10.1201/b16858
Fischer, G. H., & Molenaar, I. W. (Eds.). (2012). Rasch models: Foundations, recent developments, and applications. Springer Science & Business Media.
Formann, A. K. (1986). A note on the computation of the second-order derivatives of the elementary symmetric functions in the Rasch model. Psychometrika, 51(2), 335-339.
Glas, C.A.W. (1988). The Rasch model and multistage testing. Journal of Educational Statistics, 13(1), 45-52.
Glas, C.A.W. (2016). Maximum-Likelihood Estimation. In van der Linden, W.J. (Ed.), Handbook of Item Response Theory: Volume two: Statistical tools. (pp. 197 - 236). New York: CRC Press.
Rasch, G. (1960). Probabalistic models for some intelligence and attainment tests. Danmarks paedagogiske institut.
Steinfeld, J., & Robitzsch, A. (2024). Conditional maximum likelihood estimation in probability-based multistage designs. Behaviormetrika, 51(2), 617-634.
Steinfeld, J., Robitzsch, A. (2023). Estimating item parameters in multistage designs with the tmt package in R. Quantitative and Computational Methods in Behavioral Science, 3, e10087. https://doi.org/10.5964/qcmb.10087
Steinfeld, J., & Robitzsch, A. (2021). Item parameter estimation in multistage designs: A comparison of different estimation approaches for the Rasch model. Psych, 3(3), 279-307. https://doi.org/10.3390/psych3030022
Verhelst, N.D., Glas, C.A.W., & van der Sluis, A. (1984). Estimation Problems in the Rasch-Model: The Basic Symmetric Functions. Computational Statistics Quarterly, 1(3), 245-262.
Zwitser, R. J., & Maris, G. (2015). Conditional statistical inference with multistage testing designs. Psychometrika, 80(1), 65-84.

Examples

Run this code

# example for tmt_rm
#############################################################################
# Example-1 simple Rasch model 
#############################################################################
dat <- tmt:::sim.rm(theta = 100, b = 10, seed = 1111)
dat.rm <- tmt_rm(dat = dat)
summary(dat.rm)

#############################################################################
# Example-1 for multistage-design
#############################################################################
mstdesign <- "
  M1 =~ c(i1, i2, i3, i4, i5)
  M2 =~ c(i6, i7, i8, i9, i10)
  M3 =~ c(i11, i12, i13, i14, i15)

  # define path
  p1 := M2(0,2) + M1
  p2 := M2(3,5) + M3
"

items <- seq(-1,1,length.out = 15)
names(items) <- paste0("i",1:15)
persons = 1000

dat <- tmt_sim(mstdesign = mstdesign, 
  items = items, persons = persons)
dat.rm <- tmt_rm(dat = dat, mstdesign = mstdesign)
summary(dat.rm)

if (FALSE) {
  ############################################################################
  # Example-2 simple Rasch model 
  ############################################################################
  dat <- tmt:::sim.rm(theta = 100, b = 10, seed = 1111)
  dat.rm <- tmt_rm(dat = dat)
  summary(dat.rm)

  ############################################################################
  # Example-2 for multistage-design
  ############################################################################
  # also using 'paste' is possible
  mstdesign <- "
    M1 =~ paste0('i',1:5)
    M2 =~ paste0('i',6:10)
    M3 =~ paste0('i',11:15)
    M4 =~ paste0('i',16:20)
    M5 =~ paste0('i',21:25)
    M6 =~ paste0('i',26:30)

    # define path
    p1 := M4(0,2) + M2(0,2) + M1
    p2 := M4(0,2) + M2(3,5) + M3
    p3 := M4(3,5) + M5(0,2) + M3
    p4 := M4(3,5) + M5(3,5) + M6
  "
  items <- seq(-1,1,length.out = 30)
  names(items) <- paste0("i",1:30)
  persons = 1000
  dat <- tmt_sim(mstdesign = mstdesign, 
    items = items, persons = persons)
  dat.rm <- tmt_rm(dat = dat, mstdesign = mstdesign)
  summary(dat.rm)

    ############################################################################
  # Example-3 for cumulative multistage-design
  ############################################################################
  # also using 'paste' is possible
  mstdesign <- "
    M1  =~ paste0('i',21:30)
    M2  =~ paste0('i',11:20)
    M3  =~ paste0('i', 1:10)
    M4  =~ paste0('i',31:40)
    M5  =~ paste0('i',41:50)
    M6  =~ paste0('i',51:60)
    
    # define path
    p1 := M1(0, 5) += M2( 0,10) += M3
    p2 := M1(0, 5) += M2(11,15) += M4
    p3 := M1(6,10) += M5( 6,15) += M4
    p4 := M1(6,10) += M5(16,20) += M6
    "
  items <- seq(-1,1,length.out = 60)
  names(items) <- paste0("i",1:60)
  persons = 1000
  dat <- tmt_sim(mstdesign = mstdesign, 
    items = items, persons = persons)
  dat.rm <- tmt_rm(dat = dat, mstdesign = mstdesign)
  summary(dat.rm)

}

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