U3poly person-fit statistic

Description

Generalization of van der Flier's U3 person-fit statistic to polytomously scored items.

Usage

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U3poly(matrix, Ncat, 
       NA.method = "Pairwise", Save.MatImp = FALSE, 
       IP = NULL, IRT.PModel = "GRM", Ability = NULL, Ability.PModel = "EAP")

Arguments

matrix

A data matrix of polytomous item scores: Persons as rows, items as columns, item scores are integers between 0 and (Ncat-1), missing values allowed.

Ncat

Number of answer options for each item.

NA.method

Method to deal with missing values. The default is pairwise elimination ("Pairwise"). Alternatively, simple imputation methods are also available. The options available are "Hotdeck", "NPModel" (default), and "PModel".

Save.MatImp

Logical. Save (imputted) data matrix to file? Default is FALSE.

IP

Matrix with previously estimated item parameters: One row per item. The first (Ncat-1) columns contain the between-categories threshold parameters (for the GRM) or the item step difficulties (for the PCM and the GPCM). The last, Ncat-th, column has the slopes.

In case no item parameters are available then IP=NULL.

IRT.PModel

Specify the IRT model to use in order to estimate the item parameters (only if IP=NULL). The options available are "PCM", "GPCM", and "GRM" (default).

Ability

Vector with previoulsy estimated latent ability parameters, one per respondent, following the order of the row index of matrix.

In case no ability parameters are available then Ability=NULL.

Ability.PModel

Specify the method to use in order to estimate the latent ability parameters (only if Ability=NULL). The options available are "EB", "EAP" (default), and "MI".

Details

Emons (2008) generalized the U3 statistic (van der Flier, 1980, 1982) to polytomous items. The idea is based on the so-called item-step difficulty, which is the probability of moving from answer category (c) to answer category (c+1) (c=0,...,Ncat-2).

U3poly varies from 0 (no misfit) through 1 (extreme misfit). Hence, increasingly large U3poly values provide stronger indications of answering misfit.

The number of answer options, Ncat, is the same for all items.

U3poly reduces to U3 when Ncat=2.

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 multinomial 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 multinomial distributions with probabilities estimated by means of parametric IRT models (IRT.PModel = "PCM", "GPCM", or "GRM"). 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:

PFscores

A list of length N (number of respondents) with the values of the person-fit statistic.

PFstatistic

The person-fit statistic used.

PerfVects

Not applicable.

ID.all0s

Not applicable.

ID.all1s

Not applicable.

matrix

The data matrix after imputation of missing values was performed (if applicable).

Ncat

The number of response categories.

IRT.PModel

The parametric IRT model used.

IP

The IxNcat matrix of estimated item parameters.

Ability.PModel

The method used to estimate abilities used.

Ability

The vector of N estimated ability parameters.

NAs.method

The imputation method used (if applicable).

Author(s)

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

References

Emons, W. M. (2008) Nonparametric person-fit analysis of polytomous item scores. Applied Psychological Measurement, 32(3), 224–247.

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.

van der Flier, H. (1980) Vergelijkbaarheid van individuele testprestaties [Comparability of individual test performance]. Lisse: The Netherlands.

van der Flier, H. (1982) Deviant response patterns and comparability of test scores. Journal of Cross-Cultural Psychology, 13(3), 267–298.

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

U3, ZU3

Examples

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# Load the physical functioning data (polytomous item scores):
data(PhysFuncData)

# Compute the U3poly scores:
U3poly.out <- U3poly(PhysFuncData,Ncat=3)