PerFit (version 1.3.1)

lzpoly: lzpoly person-fit statistic

Description

Compute the lzpoly (Drasgow, Levine, and Williams, 1985) person-fit statistic.

Usage

lzpoly(matrix, Ncat, 
       NA.method = "NPModel", 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

Simple imputation method to use in case missing values exist. 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 sl

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".

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 $I$xNcat 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).

Details

Statistic lzpoly is the natural extension of lz to polytomously scores items. In this case the user can choose one from three possible IRT models to fit the data: The partial credit model (IRT.PModel="PCM"), the generalized partial credit model (IRT.PModel="GPCM"), or the graded response model (IRT.PModel="GRM"). Ability parameters can be estimated by means of one of three methods: Empirical Bayes (Ability.PModel="EB"), expected a posteriori (Ability.PModel="EAP"), or multiple imputation (Ability.PModel="MI").

Both item and ability parameters may be provided as function parameters (IP and Ability, respectively). If IP is provided then Ability must also be provided. The reason is that the estimation of the ability parameters is done via the function factor.scores from the ltm package, and this procedure requires an object of a specific class (gpcm or grm) containing the estimated item parameters (i.e., providing a matrix of item parameters to IP is not sufficient).

Aberrant response behavior is (potentially) indicated by small values of lzpoly (i.e., in the left tail of the sampling distribution).

Missing values in matrix are imputed by one of three single imputation methods: 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).

References

Drasgow, F., Levine, M. V., and Williams, E. A. (1985) Appropriateness measurement with polychotomous item response models and standardized indices. British Journal of Mathematical and Statistical Psychology, 38(1), 67--86.

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

Magis, D., Raiche, G., and Beland, S. (2012) A didactic presentation of Snijders's l[sub]z[/sub] index of person fit with emphasis on response model selection and ability estimation. Journal of Educational and Behavioral Statistics, 37(1), 57--81.

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

Molenaar, I. W., and Hoijtink, H. (1990) The many null distributions of person fit indices. Psychometrika, 55(1), 75--106.

Snijders, T. B. (2001) Asymptotic null distribution of person fit statistics with estimated person parameter. Psychometrika, 66(3), 331--342.

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

lz,lzstar

Examples

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

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