Item focussed Trees for the Identification of Items in Differential Item Functioning

Description

A function to estimate item focussed trees for simultaneous selection of items and variables that induce DIF (Differential Item Functioning) in dichotomous or polytomous items. DIF detection can be based on the Rasch Model (dichotomous case), the Logistic Regression Approach (dichotomous case) or the Partial Credit Model (polytomous case). The basic method of item focussed recursive partitioning in Rasch Models is described in Tutz and Berger (2015).

Usage

DIFtree(Y, X, model = c("Rasch", "Logistic", "PCM"), type = c("udif",
  "dif", "nudif"), alpha = 0.05, nperm = 1000, trace = FALSE,
  penalize = FALSE, ...)

## S3 method for class 'DIFtree'
print(x, ...)

Arguments

Details

The methods require 0/1 coded answers on binary items ("Rasch" and "Logistic") or categorical answers on polytomous items ("PCM"). Items with DIF are gradually identified by recursive partitioning.

For "Rasch" one yields a model with linear predictors

eta_{pi}=theta_p-tr_i(x_p),

where theta_p correspond to the ability and x_p correspond to the covariate vector of person p.

For "Logistic" one yields a model with linear predictors

• Uniform DIF, type="udif"

  eta_{pi}=S_p beta_i+tr_i(x_p),

  where S_p corresponds to the test score and x_p corresponds to the covariate vector of person p.

• DIF and Non-Uniform DIF, type="dif", "nudif"

  eta_{pi}=tr_i(x_p)+tr_i(S_p,x_p),

  where S_p corresponds to the test score and x_p corresponds to the covariate vector of person p.

For "PCM" one yields a model with linear predictors

eta_{pir}=theta_p-tr_{ir}(x_p),

where theta_p correspond to the ability and x_p correspond to the covariate vector of person p.

Significance of each split is verified by permutation tests. The result of the permutation tests can strongly depend on the number of permutations nperm. In the case of pure terminal nodes estimates of the model do not exist. If penalize=TRUE a small ridge penalty is added during estimation to ensure existence of all parameters.

Value

Object of class "DIFtree". An object of class "DIFtree" is a list containing the following components:

Author(s)

Moritz Berger <moritz.berger@imbie.uni-bonn.de>
http://www.imbie.uni-bonn.de/personen/dr-moritz-berger/

References

Berger, Moritz and Tutz, Gerhard (2016): Detection of Uniform and Non-Uniform Differential Item Functioning by Item Focussed Trees, Journal of Educational and Behavioral Statistics 41(6), 559-592.

Bollmann, Stella, Berger, Moritz & Tutz, Gerhard (2018): Item-Focussed Trees for the Detection of Differential Item Functioning in Partial Credit Models, Educational and Psychological Measurement 78(5), 781-804.

Swaminathan, Hariharan and Rogers, H Jane (1990): Detecting differential item functioning using logistic regression procedures, Journal of Educational Measurements 27(4), 361-370.

Tutz, Gerhard and Berger, Moritz (2016): Item focussed Trees for the Identification of Items in Differential Item Functioning, Psychometrika 81(3), 727-750.

See Also

plot.DIFtree, predict.DIFtree, summary.DIFtree

Examples

data(data_sim_Rasch)
data(data_sim_PCM)
 
Y1 <- data_sim_Rasch[,1]
X1 <- data_sim_Rasch[,-1]

Y2 <- data_sim_PCM[,1]
X2 <- data_sim_PCM[,-1]
 
## Not run: 
 
mod1 <- DIFtree(Y=Y1,X=X1,model="Logistic",type="udif",alpha=0.05,nperm=1000,trace=TRUE)
print(mod1)

mod2 <- DIFtree(Y=Y2,X=X2,model="PCM",alpha=0.05,nperm=100,trace=TRUE)
print(mod2)

## End(Not run)

Simulated Data Set with Polytomous Items

Description

The data set is simulated from a Partial Credit Model where some items exhibit differential item functioning. Existing differences in item difficulties are simulated by step-functions. The true, simulated DIF structure is described in Bollmann et al. (2017), Section 4.3.

Usage

data(data_sim_PCM)

Format

A data frame containing 500 observations on 4 variables:

Y matrix with categorical responses (3-point scale)

x1 binary covariate

x2 ordinal covariate

x3 numeric covariate

References

Bollmann, Stella, Berger, Moritz & Tutz, Gerhard (2018): Item-Focussed Trees for the Detection of Differential Item Functioning in Partial Credit Models, Educational and Psychological Measurement 78(5), 781-804.

Examples

data(data_sim_PCM)
 
Y <- data_sim_PCM[,1]
X <- data_sim_PCM[,-1]

apply(Y,2,table)
summary(X)
  
 

Simulated Data Set with Dichotomous Items

Description

The data set is simulated from a Rasch model where some items exhibit differential item functioning. Existing differences in item difficulties are simulated by step-functions. The true, simulated DIF structure is described in Tutz and Berger (2015), Section 4.2.

Usage

data(data_sim_Rasch)

Format

A data frame containing 500 observations on 5 variables:

Y matrix with binary 0/1 response for 20 items

x1 binary covariate 1

x2 metric covariate 1

x3 binary covariate 2

x4 metric covariate 2

References

Berger, Moritz and Tutz, Gerhard (2016): Detection of Uniform and Non-Uniform Differential Item Functioning by Item Focussed TreesJournal of Educational and Behavioral Statistics 41(6), 559-592.

Tutz, Gerhard and Berger, Moritz (2016): Item focussed Trees for the Identification of Items in Differential Item Functioning, Psychometrika 81(3), 727-750.

Examples

data(data_sim_Rasch)
 
Y <- data_sim_Rasch[,1]
X <- data_sim_Rasch[,-1]

hist(rowSums(Y), breaks = 0:19 + 0.5)
summary(X)
  
 

Plotting of Item focussed Trees

Description

Visualization of trees for items with DIF identified by item focussed recursive partitioning in dichotomous or polytomous items.

Usage

## S3 method for class 'DIFtree'
plot(x, item, component = "intercept", cex.lines = 2,
  cex.branches = 1, cex.coefs = 1, cex.main = 1, title = NULL, ...)

Arguments

Author(s)

Moritz Berger <moritz.berger@imbie.uni-bonn.de>
http://www.imbie.uni-bonn.de/personen/dr-moritz-berger/

References

Berger, Moritz and Tutz, Gerhard (2016): Detection of Uniform and Non-Uniform Differential Item Functioning by Item Focussed Trees, Journal of Educational and Behavioral Statistics 41(6), 559-592.

Bollmann, Stella, Berger, Moritz & Tutz, Gerhard (2018): Item-Focussed Trees for the Detection of Differential Item Functioning in Partial Credit Models, Educational and Psychological Measurement 78(5), 781-804.

Tutz, Gerhard and Berger, Moritz (2016): Item focussed Trees for the Identification of Items in Differential Item Functioning, Psychometrika 81(3), 727-750.

See Also

DIFtree, predict.DIFtree, summary.DIFtree

Examples

data(data_sim_Rasch)
 
Y <- data_sim_Rasch[,1]
X <- data_sim_Rasch[,-1]
 
## Not run: 
 
mod <- DIFtree(Y=Y,X=X,model="Logistic",type="udif",alpha=0.05,nperm=1000,trace=TRUE)
 
plot(mod,item=1)

## End(Not run)

Prediction from fitted Item focussed Trees

Description

The function returns predictions of item parameters obtained by item focussed recursive partitioning in dichotomous or polytomous items.

Usage

## S3 method for class 'DIFtree'
predict(object, item, newdata, ...)

Arguments

Details

For "Rasch" model the function returns the predicted item difficulty. For "Logistic" models the function returns the predicted intercept and/or slope. For "PCM" the function returns the predicted threshold parameters.

Author(s)

Moritz Berger <moritz.berger@imbie.uni-bonn.de>
http://www.imbie.uni-bonn.de/personen/dr-moritz-berger/

References

Berger, Moritz and Tutz, Gerhard (2016): Detection of Uniform and Non-Uniform Differential Item Functioning by Item Focussed Trees, Journal of Educational and Behavioral Statistics 41(6), 559-592.

Bollmann, Stella, Berger, Moritz & Tutz, Gerhard (2018): Item-Focussed Trees for the Detection of Differential Item Functioning in Partial Credit Models, Educational and Psychological Measurement 78(5), 781-804.

Tutz, Gerhard and Berger, Moritz (2016): Item focussed Trees for the Identification of Items in Differential Item Functioning, Psychometrika 81(3), 727-750.

See Also

DIFtree, plot.DIFtree, summary.DIFtree

Examples

data(data_sim_Rasch)
 
Y <- data_sim_Rasch[,1]
X <- data_sim_Rasch[,-1]

Xnew <- data.frame("x1"=c(0,1),"x2"=c(-1.1,2.5),"x3"=c(1,0),"x4"=c(-0.2,0.7))
 
## Not run: 
 
mod <- DIFtree(Y=Y,X=X,model="Logistic",type="udif",alpha=0.05,nperm=1000,trace=TRUE)
 
predict(mod,item=1,Xnew)

## End(Not run)

Summary for fitted Item focussed Trees

Description

The function takes an object of class "DIFtree" and returns an useful summary with an overiew of all executed splits during the estimation procedure.

Usage

## S3 method for class 'DIFtree'
summary(object, ...)

## S3 method for class 'summary.DIFtree'
print(x, ...)

Arguments

Value

Object of class "summary.DIFtree". An object of class "summary.DIFtree" is a list containing the following components:

Author(s)

Moritz Berger <moritz.berger@imbie.uni-bonn.de>
http://www.imbie.uni-bonn.de/personen/dr-moritz-berger/

References

Berger, Moritz and Tutz, Gerhard (2016): Detection of Uniform and Non-Uniform Differential Item Functioning by Item Focussed Trees, Journal of Educational and Behavioral Statistics 41(6), 559-592.

Bollmann, Stella, Berger, Moritz & Tutz, Gerhard (2018): Item-Focussed Trees for the Detection of Differential Item Functioning in Partial Credit Models, Educational and Psychological Measurement 78(5), 781-804.

Tutz, Gerhard and Berger, Moritz (2016): Item focussed Trees for the Identification of Items in Differential Item Functioning, Psychometrika 81(3), 727-750.

See Also

DIFtree, plot.DIFtree, predict.DIFtree

Examples

data(data_sim_Rasch)
 
Y <- data_sim_Rasch[,1]
X <- data_sim_Rasch[,-1]
 
## Not run: 
 
mod <- DIFtree(Y=Y,X=X,model="Logistic",type="udif",alpha=0.05,nperm=1000,trace=TRUE)
 
summary(mod)

## End(Not run)