C.Sato, Cstar person-fit statistics

Description

Computes the caution statistic C.Sato and the modified caution statistic Cstar.

Usage

C.Sato(matrix,
       NA.method = "Pairwise", Save.MatImp = FALSE, 
       IP = NULL, IRT.PModel = "2PL", Ability = NULL, Ability.PModel = "ML",
       mu = 0, sigma = 1)

Cstar(matrix,
     NA.method = "Pairwise", Save.MatImp = FALSE, 
     IP = NULL, IRT.PModel = "2PL", Ability = NULL, Ability.PModel = "ML",
     mu = 0, sigma = 1)

Arguments

Details

The C.Sato statistic (also refered to as C in the literature) was proposed by Sato (1975):

C.Sato = 1-cov(Xn,p)/cov(Xn*,p),

where Xn is the 0-1 response vector of respondent n, p is the vector of item proportions-correct, and Xn* is the so-called Guttman vector containing correct answers for the easiest items (i.e., with the largest proportion-correct values) only. C.Sato is zero for Guttman vectors and its value tends to increase for response vectors that depart from the group's answering pattern, hence warning the researcher to be cautious about interpreting such item scores. Therefore, (potentially) aberrant response behavior is indicated by large values of C.Sato (i.e., in the right tail of the sampling distribution).

Harnisch and Linn (1981) proposed a modified version of the caution statistic which bounds the caution statistic between 0 and 1 (also referred to as C* or MCI in the literature):

Cstar = [cov(Xn*,p)-cov(Xn,p)] / [cov(Xn*,p)-cov(Xn',p)],

where Xn' is the reversed Guttman vector containing correct answers for the hardest items (i.e., with the smallest proportion-correct values) only. Cstar is sensitive to the so-called Guttman errors. A Guttman error is a pair of scores (0,1), where the 0-score pertains to the easiest item and the 1-score pertains to the hardest item. Cstar ranges between 0 (perfect Guttman vector) and 1 (reversed Guttman error), thus larger values indicate potential aberrant response behavior.

These statistics are not computed for rows of matrix that consist of only 0s or only 1s (NA values are returned instead).

Missing values in matrix are dealt with by means of pairwise elimination by default. Alternatively, single imputation is also available. Three single imputation methods exist: Hotdeck imputation (NA.method = "Hotdeck"), nonparametric model imputation (NA.method = "NPModel"), and parametric model imputation (NA.method = "PModel"); see Zhang and Walker (2008).

• Hotdeck imputation replaces missing responses of an examinee ('recipient') by item scores from the examinee which is closest to the recipient ('donor'), based on the recipient's nonmissing item scores. The similarity between nonmissing item scores of recipients and donors is based on the sum of absolute differences between the corresponding item scores. The donor's response pattern is deemed to be the most similar to the recipient's response pattern in the group, so item scores of the former are used to replace the corresponding missing values of the latter. When multiple donors are equidistant to a recipient, one donor is randomly drawn from the set of all donors.

• The nonparametric model imputation method is similar to the hotdeck imputation, but item scores are generated from Bernoulli distributions with probabilities defined by donors with similar total score than the recipient (based on all items except the NAs).

• The parametric model imputation method is similar to the hotdeck imputation, but item scores are generated from Bernoulli distributions with probabilities estimated by means of parametric IRT models (IRT.PModel = "1PL", "2PL", or "3PL"). Item parameters (IP) and ability parameters (Ability) may be provided for this purpose (otherwise the algorithm finds estimates for these parameters).

Value

An object of class "PerFit", which is a list with 12 elements:

Author(s)

Jorge N. Tendeiro j.n.tendeiro@rug.nl

References

Harnisch, D. L., and Linn, R. L. (1981) Analysis of item response patterns: Questionable test data and dissimilar curriculum practices. Journal of Educational Measurement, 18(3), 133–146.

Karabatsos, G. (2003) Comparing the Aberrant Response Detection Performance of Thirty-Six Person-Fit Statistics. Applied Measurement In Education, 16(4), 277–298.

Meijer, R. R., and Sijtsma, K. (2001) Methodology review: Evaluating person fit. Applied Psychological Measurement, 25(2), 107–135.

Sato, T. (1975) The construction and interpretation of S-P tables. Tokyo: Meiji Tosho.

Zhang, B., and Walker, C. M. (2008) Impact of missing data on person-model fit and person trait estimation. Applied Psychological Measurement, 32(6), 466–479.

See Also

Ht

Examples

# Load the inadequacy scale data (dichotomous item scores):
data(InadequacyData)

# Compute the C.Sato scores:
C.out <- C.Sato(InadequacyData)

# Compute the Cstar scores:
Cstar.out <- Cstar(InadequacyData)

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(PerFit)
Loading required package: ltm
Loading required package: MASS
Loading required package: msm
Loading required package: polycor
Loading required package: mvtnorm
Loading required package: sfsmisc
Loading required package: mirt
Loading required package: stats4
Loading required package: lattice

Attaching package: 'mirt'

The following object is masked from 'package:ltm':

    Science

> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/PerFit/C.index.Rd_%03d_medium.png", width=480, height=480)
> ### Name: The caution statistic
> ### Title: C.Sato, Cstar person-fit statistics
> ### Aliases: Cstar C.Sato
> ### Keywords: univar
> 
> ### ** Examples
> 
> # Load the inadequacy scale data (dichotomous item scores):
> data(InadequacyData)
> 
> # Compute the C.Sato scores:
> C.out <- C.Sato(InadequacyData)
> 
> # Compute the Cstar scores:
> Cstar.out <- Cstar(InadequacyData)
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>