PerFit (version 1.3.1)

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

Arguments

matrix

Data matrix of dichotomous item scores: Persons as rows, items as columns, item scores are either 0 or 1, missing values allowed.

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, and three columns ([,1] item discrimination; [,2] item difficulty; [,3] lower-asymptote, also referred to as pseudo-guessing parameter). In case no item parameters are available then

IRT.PModel

Specify the IRT model to use in order to estimate the item parameters (only if IP=NULL). The options available are "1PL", "2PL" (default), and "3PL".

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 "ML" (default), "BM", and "WL".

mu

Mean of the apriori distribution. Only used when method="BM". Default is 0.

sigma

Standard deviation of the apriori distribution. Only used when method="BM". Default is 1.

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
  A message indicating whether perfect response vectors (all-0s or all-1s) were removed from the analysis.
• ID.all0s
  Row indices of all-0s response vectors removed from the analysis (if applicable).
• ID.all1s
  Row indices of all-1s response vectors removed from the analysis (if applicable).
• matrix
  The data matrix after imputation of missing values was performed (if applicable).
• Ncat
  The number of response categories (2 in this case).
• IRT.PModel
  The parametric IRT model used in case NA.method="PModel", otherwise NULL.
• IP
  The $I$x3 matrix of estimated item parameters in case NA.method="PModel", otherwise NULL.
• Ability.PModel
  The method used to estimate abilities in case NA.method="PModel", otherwise NULL.
• Ability
  The vector of $N$ estimated ability parameters in case NA.method="PModel", otherwise NULL.
• NAs.method
  The imputation method used (if applicable).

Details

The C.Sato statistic (also refered to as C in the literature) was proposed by Sato (1975): $$C.Sato = 1-\frac{cov(x_n,p)}{cov(x_n^*,p)},$$ where $x_n$ is the 0-1 response vector of respondent $n$, $p$ is the vector of item proportions-correct, and $x_n^*$ 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): $$Cstar = \frac{cov(x_n^*,p)-cov(x_n,p)}{cov(x_n^*,p)-cov(x_n',p)},$$ where $x_n'$ 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 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 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).

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)