Ht person-fit statistic

Description

Sijtsma's Ht person-fit statistic.

Usage

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

Arguments

Details

Sijtsma (1986) adapted a statistic introduced by Mokken (1971) that originally allowed assessing the scalability of an item to the Guttman (1944, 1950) model. The same statistic was applied by Sijtsma to the transposed data in order to detect respondents that would not comply with the Guttman model. Assume, without loss of generality, that the rows of the data matrix are ordered by increasing order of total score Sn (n = 1,..., N). The statistic formula is given by the ratio

Ht=Cov(Xn,R(n)) / Cov.max(Xn,R(n)),

where R(n) is the vector of total item scores computed excluding respondent n and the denominator is the maximum covariance given the marginal. Hence, Ht is actually similar to Sato's C.Sato.

Ht is maximum 1 for respondent n when no respondent with a total score smaller/larger than Sn can answer an item correctly/incorrectly that respondent n has answered incorrectly/correctly, respectively. Ht equals zero when the average covariance of the response pattern of respondent n with the other response patterns equals zero. Hence, (potentially) aberrant response behavior is indicated by small values of Ht (i.e., in the left tail of the sampling distribution). The Ht statistic was shown to perform relatively well in several simulation studies (Karabatsos, 2003; Sijtsma, 1986; Sijtsma and Meijer, 1992, Tendeiro and Meijer, 2014).

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

Guttman, L. (1944) A basis for scaling qualitative data. American Sociological Review, 9, 139-150.

Guttman, L. (1950) The basis for scalogram analysis. In S. A. Stouffer, L. Guttman, E. A. Suchman, P. F. Lazarsfeld, S. A. Star & J. A. Claussen (Eds.), Measurement and precision (pp. 60-90). Princeton NJ: Princeton University Press.

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

Mokken, R. J. (1971) A theory and procedure of scale analysis. Berlin, Germany: De Gruyter.

Sijtsma, K. (1986) A coefficient of deviance of response patterns. Kwantitatieve Methoden, 7, 131–145.

Sijtsma, K., and Meijer, R. R. (1992) A method for investigating the intersection of item response functions in Mokken's nonparametric IRT model. Applied Psychological Measurement, 16(2), 149-157.

Tendeiro, J. N., and Meijer, R. R. (2014) Detection of Invalid Test Scores: The Usefulness of Simple Nonparametric Statistics. Journal of Educational Measurement, 51(3), 239-259.

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

C.Sato

Examples

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

# Compute the Ht scores:
Ht.out <- Ht(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.
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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/Ht.Rd_%03d_medium.png", width=480, height=480)
> ### Name: Ht statistic
> ### Title: Ht person-fit statistic
> ### Aliases: Ht
> ### Keywords: univar
> 
> ### ** Examples
> 
> # Load the inadequacy scale data (dichotomous item scores):
> data(InadequacyData)
> 
> # Compute the Ht scores:
> Ht.out <- Ht(InadequacyData)
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>