R2noharm: Estimation of a NOHARM Analysis from within R

In sirt: Supplementary Item Response Theory Models

Estimation of a NOHARM Analysis from within R

Description

This function enables the estimation of a NOHARM analysis (Fraser & McDonald, 1988; McDonald, 1982a, 1982b, 1997) from within R. NOHARM estimates a compensatory multidimensional factor analysis for dichotomous response data. Arguments of this function strictly follow the rules of the NOHARM manual (see Fraser & McDonald, 2012; Lee & Lee, 2016).

Usage

R2noharm(dat=NULL,pm=NULL, n=NULL, model.type, weights=NULL, dimensions=NULL,
      guesses=NULL, noharm.path, F.pattern=NULL, F.init=NULL,
      P.pattern=NULL, P.init=NULL, digits.pm=4, writename=NULL,
      display.fit=5,  dec=".", display=TRUE)

## S3 method for class 'R2noharm'
summary(object, logfile=NULL, ...)

Arguments

dat	An N \times I data frame of item responses for N subjects and I items
pm	A matrix or a vector containing product-moment correlations
n	Sample size. This value must only be included if pm is provided.
model.type	Can be "EFA" (exploratory factor analysis) or "CFA" (confirmatory factor analysis).
weights	Optional vector of student weights
dimensions	Number of dimensions in exploratory factor analysis
guesses	An optional vector of fixed guessing parameters of length I. In case of the default NULL, all guessing parameters are set to zero.
noharm.path	Local path where the NOHARM 4 command line 64-bit version is located.
F.pattern	Pattern matrix for F (I \times D)
F.init	Initial matrix for F (I \times D)
P.pattern	Pattern matrix for P (D \times D)
P.init	Initial matrix for P (D \times D)
digits.pm	Number of digits after decimal separator which are used for estimation
writename	Name for NOHARM input and output files
display.fit	How many digits (after decimal separator) should be used for printing results on the R console?
dec	Decimal separator ("." or ",")
display	Display output?
object	Object of class R2noharm
logfile	File name if the summary should be sunk into a file
...	Further arguments to be passed

Details

NOHARM estimates a multidimensional compensatory item response model with the probit link function \Phi. For item responses X_{pi} of person p on item i the model equation is defined as

P( X_{pi}=1 | \bold{\theta}_p )=c_i + ( 1 - c_i ) \Phi( f_{i0} + f_{i1} \theta_{p1} + ... + f_{iD} \theta_{pD} )

where F=(f_{id}) is a loading matrix and P the covariance matrix of \bold{\theta}_p. The guessing parameters c_i must be provided as fixed values.

For the definition of F and P matrices, please consult the NOHARM manual.

This function needs the 64-bit command line version which can be downloaded from (some links may be broken in the meantime)

http://noharm.niagararesearch.ca/nh4cldl.html
https://noharm.software.informer.com/4.0/
https://cehs.unl.edu/edpsych/software-urls-and-other-interesting-sites/

Value

A list with following entries

tanaka	Tanaka index
rmsr	RMSR statistic
N.itempair	Sample sizes of pairwise item observations
pm	Product moment matrix
weights	Used student weights
guesses	Fixed guessing parameters
residuals	Residual covariance matrix
final.constants	Vector of final constants
thresholds	Threshold parameters
uniquenesses	Item uniquenesses
loadings.theta	Matrix of loadings in theta parametrization (common factor parametrization)
factor.cor	Covariance matrix of factors
difficulties	Item difficulties (for unidimensional models)
discriminations	Item discriminations (for unidimensional models)
loadings	Loading matrix (latent trait parametrization)
model.type	Used model type
Nobs	Number of observations
Nitems	Number of items
modtype	Model type according to the NOHARM specification (see NOHARM manual)
F.init	Initial loading matrix for F
F.pattern	Pattern loading matrix for F
P.init	Initial covariance matrix for P
P.pattern	Pattern covariance matrix for P
dat	Original data frame
systime	System time
noharm.path	Used NOHARM directory
digits.pm	Number of digits in product moment matrix
dec	Used decimal symbol
display.fit	Number of digits for fit display
dimensions	Number of dimensions
chisquare	Statistic \chi^2
Nestpars	Number of estimated parameters
df	Degrees of freedom
chisquare_df	Ratio \chi^2 / df
rmsea	RMSEA statistic
p.chisquare	Significance for \chi^2 statistic

Note

Possible errors often occur due to wrong dec specification.

References

Fraser, C., & McDonald, R. P. (1988). NOHARM: Least squares item factor analysis. Multivariate Behavioral Research, 23, 267-269. https://doi.org/10.1207/s15327906mbr2302_9

Fraser, C., & McDonald, R. P. (2012). NOHARM 4 Manual.
http://noharm.niagararesearch.ca/nh4man/nhman.html.

Lee, J. J., & Lee, M. K. (2016). An overview of the normal ogive harmonic analysis robust method (NOHARM) approach to item response theory. Tutorials in Quantitative Methods for Psychology, 12(1), 1-8. https://doi.org/10.20982/tqmp.12.1.p001

McDonald, R. P. (1982a). Linear versus nonlinear models in item response theory. Applied Psychological Measurement, 6(4), 379-396. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1177/014662168200600402")}

McDonald, R. P. (1982b). Unidimensional and multidimensional models for item response theory. I.R.T., C.A.T. conference, Minneapolis, 1982, Proceedings.

McDonald, R. P. (1997). Normal-ogive multidimensional model. In W. van der Linden & R. K. Hambleton (1997): Handbook of modern item response theory (pp. 257-269). New York: Springer. http://dx.doi.org/10.1007/978-1-4757-2691-6

See Also

For estimating standard errors see R2noharm.jackknife.

For EAP person parameter estimates see R2noharm.EAP.

For an R implementation of the NOHARM model see noharm.sirt.

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Data data.noharm18 with 18 items
#############################################################################

# load data
data(data.noharm18)
dat <- data.noharm18
I <- ncol(dat) # number of items

# locate noharm.path
noharm.path <- "c:/NOHARM"

#****************************************
# Model 1: 1-dimensional Rasch model (1-PL model)

# estimate one factor variance
P.pattern <- matrix( 1, ncol=1, nrow=1 )
P.init <- P.pattern
# fix all entries in the loading matrix to 1
F.pattern <- matrix( 0, nrow=I, ncol=1 )
F.init <- 1 + 0*F.pattern       #
# estimate model
mod <- sirt::R2noharm( dat=dat, model.type="CFA",
           F.pattern=F.pattern, F.init=F.init, P.pattern=P.pattern,
           P.init=P.init, writename="ex1__1dim_1pl",
       noharm.path=noharm.path, dec="," )
# summary
summary(mod, logfile="ex1__1dim_1pl__SUMMARY")
# jackknife
jmod <- sirt::R2noharm.jackknife( mod, jackunits=20 )
summary(jmod, logfile="ex1__1dim_1pl__JACKKNIFE")
# compute factor scores (EAPs)
emod <- sirt::R2noharm.EAP(mod)

#*****-----
# Model 1b: Include student weights in estimation
N <- nrow(dat)
weights <- stats::runif( N, 1, 5 )
mod1b <- sirt::R2noharm( dat=dat, model.type="CFA",  weights=weights,
            F.pattern=F.pattern, F.init=F.init, P.pattern=P.pattern,
            P.init=P.init, writename="ex1__1dim_1pl_w",
            noharm.path=noharm.path, dec="," )
summary(mod1b)

#****************************************
# Model 2: 1-dimensional 2PL Model

# set trait variance equal to 1
P.pattern <- matrix( 0, ncol=1, nrow=1 )
P.init <- 1+0*P.pattern
# loading matrix
F.pattern <- matrix( 1, nrow=I, ncol=1 )
F.init <- 1 + 0*F.pattern

mod <- sirt::R2noharm( dat=dat, model.type="CFA",
            F.pattern=F.pattern, F.init=F.init, P.pattern=P.pattern,
            P.init=P.init, writename="ex2__1dim_2pl",
            noharm.path=noharm.path, dec="," )

summary(mod)
jmod <- sirt::R2noharm.jackknife( mod, jackunits=20 )
summary(jmod)

#****************************************
# Model 3: 1-dimensional 3PL Model with fixed guessing parameters

# set trait variance equal to 1
P.pattern <- matrix( 0, ncol=1, nrow=1 )
P.init <- 1+0*P.pattern
# loading matrix
F.pattern <- matrix( 1, nrow=I, ncol=1 )
F.init <- 1 + 0*F.pattern       #
# fix guessing parameters equal to .2 (for all items)
guesses <- rep( .1, I )

mod <- sirt::R2noharm( dat=dat, model.type="CFA",
          F.pattern=F.pattern, F.init=F.init, P.pattern=P.pattern,
          P.init=P.init, guesses=guesses,
          writename="ex3__1dim_3pl", noharm.path=noharm.path, dec=","  )
summary(mod)
jmod <- sirt::R2noharm.jackknife( mod, jackunits=20 )
summary(jmod)

#****************************************
# Model 4: 3-dimensional Rasch model

# estimate one factor variance
P.pattern <- matrix( 1, ncol=3, nrow=3 )
P.init <- .8*P.pattern
diag(P.init) <- 1
# fix all entries in the loading matrix to 1
F.init <- F.pattern <- matrix( 0, nrow=I, ncol=3 )
F.init[1:6,1] <- 1
F.init[7:12,2] <- 1
F.init[13:18,3] <- 1

mod <- sirt::R2noharm( dat=dat, model.type="CFA",
          F.pattern=F.pattern, F.init=F.init, P.pattern=P.pattern,
          P.init=P.init, writename="ex4__3dim_1pl",
          noharm.path=noharm.path, dec="," )
# write output from R console in a file
summary(mod, logfile="ex4__3dim_1pl__SUMMARY.Rout")

jmod <- sirt::R2noharm.jackknife( mod, jackunits=20 )
summary(jmod)

# extract factor scores
emod <- sirt::R2noharm.EAP(mod)

#****************************************
# Model 5: 3-dimensional 2PL model

# estimate one factor variance
P.pattern <- matrix( 1, ncol=3, nrow=3 )
P.init <- .8*P.pattern
diag(P.init) <- 0
# fix all entries in the loading matrix to 1
F.pattern <- matrix( 0, nrow=I, ncol=3 )
F.pattern[1:6,1] <- 1
F.pattern[7:12,2] <- 1
F.pattern[13:18,3] <- 1
F.init <- F.pattern

mod <- sirt::R2noharm( dat=dat, model.type="CFA",
          F.pattern=F.pattern, F.init=F.init, P.pattern=P.pattern,
          P.init=P.init, writename="ex5__3dim_2pl",
          noharm.path=noharm.path, dec="," )
summary(mod)
# use 50 jackknife units with 4 persons within a unit
jmod <- sirt::R2noharm.jackknife( mod, jackunits=seq( 1:50, each=4 ) )
summary(jmod)

#****************************************
# Model 6: Exploratory Factor Analysis with 3 factors

mod <- sirt::R2noharm( dat=dat, model.type="EFA",  dimensions=3,
           writename="ex6__3dim_efa", noharm.path=noharm.path,dec=",")
summary(mod)

jmod <- sirt::R2noharm.jackknife( mod, jackunits=20 )

#############################################################################
# EXAMPLE 2: NOHARM manual Example A
#############################################################################

# See NOHARM manual: http://noharm.niagararesearch.ca/nh4man/nhman.html
# The following text and data is copied from this manual.
#
# In the first example, we demonstrate how to prepare the input for a 2-dimensional
# model using exploratory analysis. Data from a 9 item test were collected from
# 200 students and the 9x9 product-moment matrix of the responses was computed.
#
# Our hypothesis is for a 2-dimensional model with no guessing,
# i.e., all guesses are equal to zero. However, because we are unsure of any
# particular pattern for matrix F, we wish to prescribe an exploratory analysis, i.e.,
# set EX=1. Also, we will content ourselves with letting the program supply all
# initial values.
#
# We would like both the sample product-moment matrix and the residual matrix to
# be included in the output.

# scan product-moment matrix copied from the NOHARM manual
pm <- scan()
     0.8967
     0.2278 0.2356
     0.6857 0.2061 0.7459
     0.8146 0.2310 0.6873 0.8905
     0.4505 0.1147 0.3729 0.4443 0.5000
     0.7860 0.2080 0.6542 0.7791 0.4624 0.8723
     0.2614 0.0612 0.2140 0.2554 0.1914 0.2800 0.2907
     0.7549 0.1878 0.6236 0.7465 0.4505 0.7590 0.2756 0.8442
     0.6191 0.1588 0.5131 0.6116 0.3845 0.6302 0.2454 0.6129 0.6879

ex2 <- sirt::R2noharm( pm=pm, n=200, model.type="EFA", dimensions=2,
         noharm.path=noharm.path, writename="ex2_noharmExA", dec=",")
summary(ex2)

#############################################################################
# EXAMPLE 3: NOHARM manual Example B
#############################################################################

# See NOHARM manual: http://noharm.niagararesearch.ca/nh4man/nhman.html
# The following text and data is copied from this manual.

# Suppose we have the product-moment matrix of data from 125 students on 9 items.
# Our hypothesis is for 2 dimensions with simple structure. In this case,
# items 1 to 5 have coefficients of theta which are to be estimated for one
# latent trait but are to be fixed at zero for the other one.
# For the latent trait for which items 1 to 5 have zero coefficients,
# items 6 to 9 have coefficients which are to be estimated. For the other
# latent trait, items 6 to 9 will have zero coefficients.
# We also wish to estimate the correlation between the latent traits,
# so we prescribe P as a 2x2 correlation matrix.
#
# Our hypothesis prescribes that there was no guessing involved, i.e.,
# all guesses are equal to zero. For demonstration purposes,
# let us not have the program print out the sample product-moment matrix.
# Also let us not supply any starting values but, rather, use the defaults
# supplied by the program.

pm <- scan()
    0.930
    0.762 0.797
    0.541 0.496 0.560
    0.352 0.321 0.261 0.366
    0.205 0.181 0.149 0.110 0.214
    0.858 0.747 0.521 0.336 0.203 0.918
    0.773 0.667 0.465 0.308 0.184 0.775 0.820
    0.547 0.474 0.347 0.233 0.132 0.563 0.524 0.579
    0.329 0.290 0.190 0.140 0.087 0.333 0.308 0.252 0.348

I <- 9    # number of items
# define loading matrix
F.pattern <- matrix(0,I,2)
F.pattern[1:5,1] <- 1
F.pattern[6:9,2] <- 1
F.init <- F.pattern
# define covariance matrix
P.pattern <- matrix(1,2,2)
diag(P.pattern) <- 0
P.init <- 1+P.pattern

ex3 <- sirt::R2noharm( pm=pm, n=125,, model.type="CFA",
           F.pattern=F.pattern, F.init=F.init, P.pattern=P.pattern,
           P.init=P.init, writename="ex3_noharmExB",
           noharm.path=noharm.path, dec="," )
summary(ex3)

#############################################################################
# EXAMPLE 4: NOHARM manual Example C
#############################################################################

data(data.noharmExC)
# See NOHARM manual: http://noharm.niagararesearch.ca/nh4man/nhman.html
# The following text and data is copied from this manual.

# In this example, suppose that from 300 respondents we have item
# responses scored dichotomously, 1 or 0, for 8 items.
#
# Our hypothesis is for a unidimensional model where all eight items
# have coefficients of theta which are to be estimated.
# Suppose that since the items were multiple choice with 5 options each,
# we set the fixed guesses all to 0.2 (not necessarily good reasoning!)
#
# Let's supply initial values for the coefficients of theta (F matrix)
# as .75 for items 1 to 4 and .6 for items 5 to 8.

I <- 8
guesses <- rep(.2,I)
F.pattern <- matrix(1,I,1)
F.init <- F.pattern
F.init[1:4,1] <- .75
F.init[5:8,1] <- .6
P.pattern <- matrix(0,1,1)
P.init <- 1 + 0 * P.pattern

ex4 <- sirt::R2noharm( dat=data.noharmExC,, model.type="CFA",
           guesses=guesses, F.pattern=F.pattern, F.init=F.init,
           P.pattern=P.pattern, P.init=P.init, writename="ex3_noharmExC",
           noharm.path=noharm.path, dec="," )
summary(ex4)

# modify F pattern matrix
# f11=f51 (since both have equal pattern values of 2),
# f21=f61 (since both have equal pattern values of 3),
# f31=f71 (since both have equal pattern values of 4),
# f41=f81 (since both have equal pattern values of 5).
F.pattern[ c(1,5) ] <- 2
F.pattern[ c(2,6) ] <- 3
F.pattern[ c(3,7) ] <- 4
F.pattern[ c(4,8) ] <- 5
F.init <- .5+0*F.init

ex4a <- sirt::R2noharm( dat=data.noharmExC,, model.type="CFA",
           guesses=guesses, F.pattern=F.pattern, F.init=F.init,
           P.pattern=P.pattern, P.init=P.init, writename="ex3_noharmExC1",
           noharm.path=noharm.path, dec="," )
summary(ex4a)

## End(Not run)

sirt documentation built on May 29, 2024, 8:43 a.m.