| tmt_rm {tmt} | R Documentation |
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",
...
)
Arguments
dat |
a matrix of dichotomous (0/1) data or a list of the function |
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. |
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.
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 |
Author(s)
Jan Steinfeld
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. (accepted). Conditional maximum likelihood estimation in probability-based multistage designs. Behaviormetrika, xx(x), xxx-xxx.
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.
See Also
Examples
# 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)
## Not run:
############################################################################
# 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)
## End(Not run)