Logistic Ordinal Regression Differential Item Functioning using IRT

Description

Analysis of Differential Item Functioning (DIF) for dichotomous and polytomous items, using an iterative hybrid of (ordinal) logistic regression and item response theory (IRT).

Details

Ordinal logistic regression (OLR) provides a flexible framework for detecting various types of DIF. Previous efforts extended the framework by substituting the matching variable based on sum scores with IRT based trait scores and by employing an iterative process of purifying the matching variable with the use of group-specific item parameters (Crane et. al., 2006). This package represents an effort to integrate both statistical and IRT procedures into a single program. A Monte Carlo simulation approach was incorporated to derive empirical threshold values for various DIF statistics and effect size measures.

Author(s)

Seung W. Choi, with contributions from Laura E. Gibbons and Paul K. Crane

Maintainer: Seung W. Choi <choi.phd@gmail.com>

References

Choi, S. W., Gibbons, L. E., Crane, P. K. (2011). lordif: An R Package for Detecting Differential Item Functioning Using Iterative Hybrid Ordinal Logistic Regression/Item Response Theory and Monte Carlo Simulations. Journal of Statistical Software, 39(8), 1-30. URL http://www.jstatsoft.org/v39/i08/.

Crane, P. K., Gibbons, L. E., Jolley, L., & van Belle, G. (2006). Differential item functioning analysis with ordinal logistic regression techniques: DIF detect and difwithpar. Medical Care, 44(11 Suppl 3), S115-S123.

See Also

mirt::mirt(), rms::lrm()

Examples

##load PROMIS Anxiety sample data (n=766)
## Not run: data(Anxiety)
##age : 0=younger than 65 or 1=65 or older
##gender: 0=Male or 1=Female
##education: 0=some college or higher 1=high school or lower
##run age-related DIF on all 29 items (takes about a minute)
## Not run: age.dif <- lordif(Anxiety[paste("R",1:29,sep="")], Anxiety$age) 
##with sample weights - produces the same results as above, i.e., with equal weights
## Not run: age.dif <- lordif(Anxiety[paste("R",1:29,sep="")], Anxiety$age, weights=rep(1/766,766)) 
##print output
## Not run: print(age.dif)
##print extended output
## Not run: summary(age.dif)
##generate plots for DIF items (reference group: <65)
## Not run: plot(age.dif,labels=c("Younger","Older"))
##run Monte Carlo simulations for threshold values
##this may take several minutes
## Not run: age.dif.MC<-montecarlo(age.dif,alpha=0.05,nr=100)
##print output
## Not run: print(age.dif.MC)
##print extended output
## Not run: summary(age.dif.MC)
##generate plots for Monte Carlo threshold values
## Not run: plot(age.dif.MC)
##run DFIT analysis
## Not run: age.dif.DFIT<-DFIT(age.dif)

A Measure of Anxiety

Description

The data contains responses from 766 people sampled from a general population to the PROMIS Anxiety scale (https://www.healthmeasures.net/explore-measurement-systems/promis/) composed of 29 Likert-type questions with a common rating scale (1=Never, 2=Rarely, 3=Sometimes, 4=Often, and 5=Always).

Usage

data(Anxiety)

Format

A data frame with 766 observations on the following 32 variables.

age

0=younger than 65 and 1=65 and older

gender

0=Male and 1=Female

education

0=some college or higher and 1=high school or lower

R1

I felt fearful

R2

I felt frightened

R3

It scared me when I felt nervous

R4

I felt anxious

R5

I felt like I needed help for my anxiety

R6

I was concerned about my mental health

R7

I felt upset

R8

I had a racing or pounding heart

R9

I was anxious if my normal routine was disturbed

R10

I had sudden feelings of panic

R11

I was easily startled

R12

I had trouble paying attention

R13

I avoided public places or activities

R14

I felt fidgety

R15

I felt something awful would happen

R16

I felt worried

R17

I felt terrified

R18

I worried about other people's reactions to me

R19

I found it hard to focus on anything other than my anxiety

R20

My worries overwhelmed me

R21

I had twitching or trembling muscles

R22

I felt nervous

R23

I felt indecisive

R24

Many situations made me worry

R25

I had difficulty sleeping

R26

I had trouble relaxing

R27

I felt uneasy

R28

I felt tense

R29

I had difficulty calming down

Source

https://www.healthmeasures.net/explore-measurement-systems/promis/

References

PROMIS Cooperative Group. Unpublished Manual for the Patient-Reported Outcomes Measurement Information System (PROMIS) Version 1.1. October, 2008: https://www.healthmeasures.net/explore-measurement-systems/promis/

Examples

  ## Not run: data(Anxiety)

calculates DFIT statistics

Description

Calculates DFIT statistics using an object of class "lordif"

Usage

  DFIT(obj)

Arguments

Details

Calculates DFIT statistics, including the compensatory differential item functioning (CDIF), the non-compensatory differential item functioning (NCDIF), and the differential test functioning (DTF), based on an object returned from lordif.

Value

Author(s)

Seung W. Choi <choi.phd@gmail.com>

References

Oshima, T., & Morris, S. (2008). Raju's differential functioning of items and tests (DFIT). Educational Measurement: Issues and Practice, 27, 43-50.

Raju, N. S., van der Linden, W. J., & Fleer, P. F., (1995). An IRT-based internal measure of test bias with application of differential item functioning. Applied Psychological Measurement, 19, 353-368.

See Also

lordif

Examples

##load PROMIS Anxiety sample data (n=766)
## Not run: data(Anxiety)
##age : 0=younger than 65 or 1=65 or older
##run age-related DIF on all 29 items (takes about a minute)
## Not run: age.DIF <- lordif(Anxiety[paste("R",1:29,sep="")],Anxiety$age)
##run DFIT
## Not run: age.DIF.DFIT <- DFIT(age.DIF)

calculates item response probabilities

Description

Calculates item response probabilities over a theta grid according to either the GRM or the GPCM.

Usage

  calcprob(ipar, theta, model = "GRM")

Arguments

Details

Calculates an array of item response probabilities according to either the Graded Response Model (GRM: Samejima, 1969) or the Generalized Partial Credit Model (GPCM: Muraki, 1992) over a grid of theta values. The two required input objects are ipar and theta. ipar is a data frame containing item parameters in the following order: a, cb1, cb2,..., cb(maxCat-1). Items may have different numbers of categories. The variable maxCAT is the maximum number of response categories across all items. theta is a vector containing a grid of theta values. The IRT model can be either "GRM" or "GPCM".

Value

Returns an array of item response probabilities of dimension, c(nq, ni, maxCAT-1), where nq is the length of the theta grid, ni is the number of items in ipar, i.e., nrow(ipar), and maxCAT is the maximum number of response categories across all items.

Author(s)

Seung W. Choi <choi.phd@gmail.com>

References

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16, 159-176.

See Also

probgrm, probgpcm

Examples

##item.par<-read.csv(fn,head=F,col.names=c("a","cb1","cb2","cb3","cb4"))
##theta <- seq(-4,4,.1)
## Not run: calcprob(item.par,theta,model="GPCM")

calculates EAP theta estimates and associated standard errors

Description

Calculates Expected A Posteriori (EAP) theta estimates and associated standard error estimates (posterior standard deviations).

Usage

  calctheta(ipar, resp.data, theta, prior.mean = 0, prior.sd = 1, model = "GRM")

Arguments

Details

Calculates EAP theta estimates and standard error estimates based on the input item parameters (ipar), the item response data (resp.data), and the IRT model specified ("GRM" or "GPCM").

Value

A list object with the following components

Note

Some missing item responses (NA) are allowed.

Author(s)

Seung W. Choi <choi.phd@gmail.com>

References

Bock, R. D. & Mislevy, R. J. (1982). Adaptive EAP Estimation of Ability in a Microcomputer Environment. Applied Psychological Measurement, 6(4), 431-444.

See Also

calcprob, probgrm, probgpcm

Examples

  ## Not run: calctheta(ipar,resp.data,model="GPCM")

collapses response categories

Description

Collapses response categories if cell frequencies by group are below a minimum threshold specified by minCell and returns a vector of recoded responses.

Usage

  collapse(resp, group, minCell)

Arguments

Details

Collapses item response categories in resp if the two-way contingency table (resp x group) has cell frequencies less than the minimum cell count threshold specified by minCell.

Value

a numeric vector of the same length as resp with collapsed/recoded values.

Note

The lowest item response category/score should be coded as 1 not 0 (e.g., 1, 2, 3, 4, 5; and not 0, 1, 2, 3, 4). There must be at least two unique categories after collapsing/recoding.

Author(s)

Seung W. Choi <choi.phd@gmail.com>

See Also

recode

Examples

  r1 <- c(1,1,2,1,1,2,2,1,2,2,1,2,2,1,1,2,1,2,2,3,3,1,2,3)
  gr <- c(0,0,0,1,1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1) 
  collapse(r1,gr,2) #minCell=2
  ## returns  c(1,1,2,1,1,2,2,1,2,2,1,2,2,1,1,2,1,2,2,2,2,1,2,2)
  ## response categories 2 and 3 are collapsed

performs Stocking-Lord Equating

Description

Computes linear transformation constants to equate a set of GRM/GPCM item parameters to a target scale using a test characteristic curve equating procedure (Stocking & Lord, 1983)

Usage

  equate(ipar.to, ipar.from, theta, model = "GRM", start.AK = c(1, 0), 
  lower.AK = c(0.5, -2), upper.AK = c(2, 2))

Arguments

Details

Computes linear transformation constants (A and K) that equate a set of item parameters (ipar.from) to the scale defined by a target item parameters (ipar.to) by minimizing the squared difference between the test characteristic curves (Stocking & Lord, 1983). The minimization is performed by the nlminb function (in stats).

Value

returns a vector of two elements, c(A, K) where A is a multiplicative constant and K is an additive constant

Note

The item parameters are assumed to be on the theta metric (0,1). The number of category threshold parameters may differ across items but not greater than (maxCat-1).

Author(s)

Seung W. Choi <choi.phd@gmail.com>

References

Stocking, M. L. & Lord, F. M. (1983). Developing a Common Metric in Item Response Theory. Applied Psychological Measurement, 7(2), 201-210.

See Also

tcc

Examples

##ipar.to is a data frame containing "target" item parameters
##ipar.from is a data frame containing "to-be-equated" item parameters
## Not run: AK <- equate(ipar.to,ipar.from)
#AK[1] contains the multiplicative constant
#AK[2] contains the additive constant

extracts IRT item parameters

Description

Extracts IRT item parameter estimates from an output returned from mirt::mirt()

Usage

  extract(ipar)

Arguments

Details

similar to the coef function in the mirt package

Value

a data frame containing item parameter estimates in the order of a, cb1, cb2,..., cb(maxCat-1).

Author(s)

Seung W. Choi <choi.phd@gmail.com>

See Also

lordif, mirt::mirt()

Examples

##calib.sparse <- mirt(sparse.matrix,1,itemtype="gpcm") 
## Not run: ipar.sparse <- extract(calib.sparse)

determines a cutoff threshold

Description

Determines an empirical cutoff value for statistics generated from a Monte Carlo simulation.

Usage

  getcutoff(stat, alpha, reverse)

Arguments

Details

Calculates the quantile value of statistics sampled from a Monte Carlo simulation. For example, when alpha = 0.1 the function determines the 1st percentile of the statistics. Conversely, if reverse is True, the function determines the 99th percentile, i.e., (1-alpha) x 100.

Value

returns a scalar corresponding to the quantile of the statistics determined by alpha or (1-alpha)

Author(s)

Seung W. Choi <choi.phd@gmail.com>

See Also

montecarlo, permute

Examples

#top 1 percent
getcutoff(runif(1000),0.01,TRUE)
#bottom 1 percent
getcutoff(runif(1000),0.01,FALSE)

performs Logistic Ordinal Regression Differential Item Functioning using IRT

Description

performs iterative hybrid ordinal logistic regression/IRT DIF

Usage

lordif(resp.data, group, selection = NULL, criterion = c("Chisqr", "R2", "Beta"), 
pseudo.R2 = c("McFadden", "Nagelkerke", "CoxSnell"), alpha = 0.01, beta.change = 0.1, 
R2.change = 0.02, maxIter = 10, minCell = 5, minTheta = -4, maxTheta = 4, inc = 0.1, 
control = list(), model = "GRM", anchor = NULL, MonteCarlo = FALSE, nr = 100, 
weights = NULL, normwt = TRUE)

Arguments

Details

Performs an ordinal (common odds-ratio) logistic regression differential item functioning (DIF) analysis using IRT theta estimates as the conditioning variable. The graded response model (GRM) or the generalized partial credit model (GPCM) is used for IRT trait estimation. Items flagged for DIF are treated as unique items and group-specific item parameters are obtained. Non-DIF items serve as anchor items to the initial single-group calibration. The procedure runs iteratively until the same set of items is flagged over two consecutive iterations, unless anchor items are specified.

Value

Returns an object (list) of class "lordif" with the following components:

Note

requires the mirt and rms packages

Author(s)

Seung W. Choi <choi.phd@gmail.com>

References

Choi, S. W., Gibbons, L. E., Crane, P. K. (2011). lordif: An R Package for Detecting Differential Item Functioning Using Iterative Hybrid Ordinal Logistic Regression/Item Response Theory and Monte Carlo Simulations. Journal of Statistical Software, 39(8), 1-30. URL http://www.jstatsoft.org/v39/i08/.

Crane, P. K., Gibbons, L. E., Jolley, L., and van Belle, G. (2006). Differential item functioning analysis with ordinal logistic regression techniques: DIF detect and difwithpar. Medical Care, 44(11 Suppl 3), S115-S123.

See Also

rundif

Examples

  ## Not run: data(Anxiety)
  ## Not run: resp.data <- Anxiety[paste("R",1:29,sep="")]
  ## Not run: age <- Anxiety$age
  ## Not run: age.DIF <- lordif(resp.data,age,model="GPCM",anchor=c(1:5,7,8,10,12:17,19:23,25:29))
  ## Not run: print(age.DIF)

performs Monte Carlo simulations to generate empirical distributions

Description

performs Monte Carlo simulations under no-DIF conditions to generate empirical distributions of statistics

Usage

montecarlo(obj, alpha = 0.01, nr = 100)

Arguments

Details

Simulated datasets are generated under no-DIF conditions and have the same dimensions as the empirical dataset. Group n-sizes and differences (impact) in theta estimates are preserved in the simulated datasets. Returns empirical distributions and thresholds for various statistics and effect size measures.

Value

Returns an object (list) of class "lordif.MC" with the following components:

Note

nr must be a large integer (e.g., 500) to generate smooth distributions.

Author(s)

Seung W. Choi <choi.phd@gmail.com>

References

Choi, S. W., Gibbons, L. E., Crane, P. K. (2011). lordif: An R Package for Detecting Differential Item Functioning Using Iterative Hybrid Ordinal Logistic Regression/Item Response Theory and Monte Carlo Simulations. Journal of Statistical Software, 39(8), 1-30. URL http://www.jstatsoft.org/v39/i08/.

See Also

permute, lordif

Examples

##load PROMIS Anxiety sample data (n=766)
## Not run: data(Anxiety)
##age : 0=younger than 65 or 1=65 or older
##run age-related DIF on all 29 items (takes about a minute)
## Not run: age.DIF <- lordif(Anxiety[paste("R",1:29,sep="")],Anxiety$age)
##the following takes several minutes
## Not run: age.DIF.MC <- montecarlo(age.DIF,alpha=0.01,nr=100)

performs permutation test for empirical cutoff thresholds

Description

performs permutation tests under no-DIF conditions to generate empirical distributions of DIF statistics

Usage

permute(obj, alpha = 0.01, nr = 100)

Arguments

Details

The vector of group designations is randomly shuffled nr times to estimate the sampling distribution of the statistics when the null hypothesis is true. Returns empirical distributions and thresholds for various statistics and effect size measures.

Value

Returns an object (list) of class "lordif.MC" with the following components:

Note

nr must be a large integer (e.g., 500) for smooth distributions.

Author(s)

Seung W. Choi <choi.phd@gmail.com>

References

Choi, S. W., Gibbons, L. E., Crane, P. K. (2011). lordif: An R Package for Detecting Differential Item Functioning Using Iterative Hybrid Ordinal Logistic Regression/Item Response Theory and Monte Carlo Simulations. Journal of Statistical Software, 39(8), 1-30. URL http://www.jstatsoft.org/v39/i08/.

See Also

montecarlo, lordif

Examples

##load PROMIS Anxiety sample data (n=766)
## Not run: data(Anxiety)
##age : 0=younger than 65 or 1=65 or older
##run age-related DIF on all 29 items (takes about a minute)
## Not run: age.DIF <- lordif(Anxiety[paste("R",1:29,sep="")],Anxiety$age)
##the following takes several minutes
## Not run: age.DIF.MC <- permute(age.DIF,alpha=0.01,nr=100)

Plot method for lordif class

Description

plots diagnostic graphs for items flagged for DIF

Usage

  ## S3 method for class 'lordif'
plot(x, labels = c("Reference", "Focal"), width = 7, height = 7, BW = FALSE, ...)

Arguments

Details

Generates the following graphs IF there is one or more DIF items: 1. Trait Distributions - density graphs for groups 2. Item True Score Functions - true score functions by theta for groups 3. Differences in Item True Score Functions - unsigned differences 4. Item Response Functions - item response function for groups 5. Impact (Weighted by Density) - unsigned differences weighted by theta distributions for the focal group 6. Comparison of TCCs by group based on all items and DIF items 7. Impact on Theta Estimates - theta estimates by group before and after accounting for DIF

Graphs 2-5 are generated for each DIF item Graphs 1, 6 and 7 are generated for the whole analysis

Value

Returns no object.

Note

Produces graphs only if x contains DIF items. No graphs if no DIF items are present.

Author(s)

Seung W. Choi <choi.phd@gmail.com>

References

Choi, S. W., Gibbons, L. E., Crane, P. K. (2011). lordif: An R Package for Detecting Differential Item Functioning Using Iterative Hybrid Ordinal Logistic Regression/Item Response Theory and Monte Carlo Simulations. Journal of Statistical Software, 39(8), 1-30. URL http://www.jstatsoft.org/v39/i08/.

See Also

lordif

Examples

##run lordif first
## Not run: age.dif <- lordif(Anxiety[paste("R",1:29,sep="")],Anxiety$age)
## Not run: plot(age.dif,labels=c("Younger","Older"),width=8,height=7,cex=0.8,lwd=1)

Plot method for Monte Carlo simulation output

Description

Produces plots for Monte Carlo output

Usage

  ## S3 method for class 'lordif.MC'
plot(x, mfrow = c(3, 1), width = 7, height = 7, BW = FALSE, ...)

Arguments

Details

Generates the following graphs: 1. thresholds for Chi-square probability for Model 1 vs. 2 2. thresholds for Chi-square probability for Model 1 vs. 3 3. thresholds for Chi-square probability for Model 2 vs. 3 4. pseudo R-square change from Model 1 to 2 5. pseudo R-square change from Model 1 to 3 6. pseudo R-square change from Model 2 to 3 7. proportional beta change from Model 1 to 2

Value

Returns no object.

Note

x is an object of class lordif.MC returned from montecarlo

Author(s)

Seung W. Choi <choi.phd@gmail.com>

References

Choi, S. W., Gibbons, L. E., Crane, P. K. (2011). lordif: An R Package for Detecting Differential Item Functioning Using Iterative Hybrid Ordinal Logistic Regression/Item Response Theory and Monte Carlo Simulations. Journal of Statistical Software, 39(8), 1-30. URL http://www.jstatsoft.org/v39/i08/.

See Also

lordif, montecarlo, permute

Examples

## s3 plot method for class 'lordif.MC':
## Not run: age.dif <- lordif(Anxiety[paste("R",1:29,sep="")],Anxiety$age)
## Not run: age.dif.MC <- montecarlo(age.dif,alpha=.05,nr=500)
## Not run: plot(age.dif.MC,mfrow=c(1,1),width=8,height=7)

calculates item response probabilities according to GPCM

Description

Calculates a matrix of item response probabilities over a grid of theta values for an item

Usage

  probgpcm(theta, a, cb)

Arguments

Details

The Generalized Partial Credit Model (Muraki, 1992) is assumed.

Value

Returns a matrix of item response probabilities. The first dimension corresponds to the length of theta.

Author(s)

Seung W. Choi <choi.phd@gmail.com>

References

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16, 159-176.

See Also

calcprob, probgrm

Examples

probgrm(seq(-4,4,.1), 1.5, c(-1.2,0.5,1.5))

calculates item response probabilities according to GRM

Description

Calculates a matrix of item response probabilities over a grid of theta values for an item

Usage

probgrm(theta, a, cb)

Arguments

Details

The Graded Response Model (Samejima, 1969) is assumed.

Value

Returns a matrix of item response probabilities. The first dimension corresponds to the length of theta.

Author(s)

Seung W. Choi <choi.phd@gmail.com>

References

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

See Also

calcprob, probgpcm

Examples

  probgrm(seq(-4,4,.1), 1.5, c(-1.2,0.5,1.5))

recodes item responses

Description

Recodes item responses as specified by original and modified

Usage

  recode(vec, original, modified)

Arguments

Details

vec, original, and modified must be of the same mode. original and modified must be of the same length.

Value

Returns a vector of the same length and mode as vec with recoded values.

Author(s)

Seung W. Choi <choi.phd@gmail.com>

Examples

x <- c(0,1,2,3,4,0,1,2,3,4)
y <- c(0,1,2,3,4)
z <- c(1,2,3,4,5)
recode(x,y,z)
##returns c(1,2,3,4,5,1,2,3,4,5)

runs ordinal logistic regression DIF

Description

Runs ordinal logistic regression DIF

Usage

rundif(item, resp, theta, gr, criterion, alpha, beta.change, pseudo.R2, R2.change, wt)

Arguments

Details

The argument item lists the column numbers of the data frame resp to be included in the analysis.

Value

Returns a list of the following components:

Author(s)

Seung W. Choi <choi.phd@gmail.com>

References

Choi, S. W., Gibbons, L. E., Crane, P. K. (2011). lordif: An R Package for Detecting Differential Item Functioning Using Iterative Hybrid Ordinal Logistic Regression/Item Response Theory and Monte Carlo Simulations. Journal of Statistical Software, 39(8), 1-30. URL http://www.jstatsoft.org/v39/i08/.

Crane, P. K., Gibbons, L. E., Jolley, L., and van Belle, G. (2006). Differential item functioning analysis with ordinal logistic regression techniques: DIF detect and difwithpar. Medical Care, 44(11 Suppl 3), S115-S123.

See Also

runolr, lordif

Examples

## Not run: rundif(item,resp,theta,gr)

runs ordinal logistic regression models

Description

Runs ordinal logistic regression models and produces DIF statistics and effect size measures

Usage

runolr(rv, ev, gr, wt)

Arguments

Details

Model 1: ev

Model 2: ev + gr

Model 3: ev*gr or equivalently ev + gr + ev*gr

Value

Returns a list of the following components:

Author(s)

Seung W. Choi <choi.phd@gmail.com>

References

Choi, S. W., Gibbons, L. E., Crane, P. K. (2011). lordif: An R Package for Detecting Differential Item Functioning Using Iterative Hybrid Ordinal Logistic Regression/Item Response Theory and Monte Carlo Simulations. Journal of Statistical Software, 39(8), 1-30. URL http://www.jstatsoft.org/v39/i08/.

Crane, P. K., Gibbons, L. E., Jolley, L., & van Belle, G. (2006). Differential item functioning analysis with ordinal logistic regression techniques: DIF detect and difwithpar. Medical Care, 44(11 Suppl 3), S115-S123.

See Also

rundif, lordif, rms::lrm()

Examples

## Not run: runolr(rv, ev, gr)

splits item response vectors of DIF items by group

Description

Splits item response vectors of DIF items into separate group-specific vectors.

Usage

  separate(resp, flag, gr)

Arguments

Details

To obtain group specific item calibrations, response vectors of DIF items are split into multiple vectors by group and treated as separate items.

Value

Returns a data frame with item response vectors for non-DIF items followed by separated item response vectors for DIF items

Author(s)

Seung W. Choi <choi.phd@gmail.com>

Examples

## Not run: separate(resp,flag,gr)

computes a test characteristic curve (tcc)

Description

computes a test characteristic curve (tcc) from input item parameters

Usage

  tcc(a, cb, theta, model = "GRM")

Arguments

Details

Either the graded response model (GRM) or the generalized partial credit model (GPCM) can be specified. The lowest response/score for each item is coded as 0 not 1.

Value

Retruns a vector of tcc values over a theta grid

Author(s)

Seung W. Choi <choi.phd@gmail.com>

See Also

probgrm, probgpcm

Examples

## Not run: tcc(a,cb,theta,model="GRM")
## Not run: tcc(a,cb,theta,model="GPCM")