Plot the nonparametric person response function with variability bands.
PRFplot(matrix, respID, h=.09, N.FPts=101,
VarBands=FALSE, VarBands.area=FALSE, alpha=.05,
Xlabel=NA, Xcex=1.5, Ylabel=NA, Ycex=1.5, title=NA, Tcex=1.5,
NA.method="Pairwise", Save.MatImp=FALSE,
IP = NULL, IRT.PModel = "2PL", Ability = NULL, Ability.PModel = "ML",
mu = 0, sigma = 1,
message = TRUE)Data matrix of dichotomous item scores: Persons as rows, items as columns, item scores are either 0 or 1, missing values allowed.
Vector specifying the respondents for whom PRFs are to be computed.
Bandwidth value. Default is 0.09.
Number of (equidistant) focal points in the [0,1] interval. Default is 101.
Logical: Draw the (1-alpha) variability bands? Default is FALSE.
Logical: Draw the area between the (1-alpha) variability bands? Default is FALSE.
Significance level for the variability bands. Default is 0.05.
Define label of x-axis, otherwise a default label is shown.
Font size of the label of x-axis.
Define label of y-axis, otherwise a default label is shown.
Font size of the label of y-axis.
Define the title of the plot, otherwise a default title is shown.
Font size of the title of the plot.
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".
Logical. Save (imputted) data matrix to file? Default is FALSE.
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 IP=NULL.
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".
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.
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".
Mean of the apriori distribution. Only used when method="BM". Default is 0.
Standard deviation of the apriori distribution. Only used when method="BM". Default is 1.
Display prompt message (one per person)? Default is TRUE.
The output is a list with three functional data objects of class fd (see package fda):
PRF.FDOFunctional data object of the PRFs for the entire sample.
VarBandsLow.FDOFunctional data object of the lower-bound of the variability bands for the entire sample.
VarBandsHigh.FDOFunctional data object of the upper-bound of the variability bands for the entire sample.
Function PRFplot displays the so-called nonparametric person response functions (PRFs; Emons, Sijtsma, and Meijer, 2004; Sijtsma and Meijer, 2001), for the respondents identified in respID. The PRF relates item difficulty (0-1 range on the x-axis) with the associated probability of correct response (on the y-axis). The PRF is typically nonincreasing, implying that the probability of answering increasingly difficult items should (at least) not increase. The code is based on nonparametric kernel smoothing (Emons et al., 2004). The value of the PRF at each focal point (representing a difficulty parameter between 0 and 1) is estimated as a weighted sum score, where scores pertaining to items with difficulty close to the focal point are given the largest weights. The weights are functions of the Gaussian kernel function. It is necessary to specify a bandwidth value (h) in order to compute the weights. The h value controls the trade-off between bias and sampling variation (Emons et al., 2004). Small h values reduce bias but increase variance, leading to PRFs that capture too much measurement error. Large h values, on the other hand, increase bias which renders PRFs that are often too flat, thus missing potentially relevant misfitting response behavior. Therefore, it is important to carefuly specify the value h. Emons et al. (2004, pp. 10-13), after a simulation study, advised that "h values between 0.07 and 0.11 are reasonable choices".
Moreover, variability bands of level alpha (0.05 by default) can also be added to the plot. These bands are computed following the jackknife procedure explained in Emons et al. (2004).
The PRFs and variability bands for each respondent are approximated by means of functional data objects (e.g., Ramsay, Hooker, and Graves, 2009), with the help of the fda package. This procedure follows two steps:
Compute a B-splines basis system. This basis consists of a set of (thirteen) piecewise polinomials, all of degree three/order four (i.e., cubic polinomial segments), with one knot per break point. This allows any two consecutive splines, sp1 and sp2, with common break point BP, verifying sp1(BP) = sp2(BP), sp1'(BP) = sp2'(BP), and sp1''(BP) = sp2''(BP). At 0 and 1 (extremes of the x-range), four (= order) knots are used.
Specify coefficients c for the B-splines basis system computed above and then create functional data objects. Based on smoothing using regression analysis (Ramsay et al., 2009, section 4.3).
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).
PRFplot returns three functional data objects (for the PRFs, lower-bound of the variability bands, and upper-band of the variability bands) for all respondents in the sample.
Emons, W. M., Sijtsma, K., and Meijer, R. R. (2004) Testing hypotheses about the person-response function in person-fit analysis. Multivariate Behavioral Research, 39(1), 1--35.
Ramsay, J. O., Hooker, G., and Graves, S. (2009) Functional data analysis with R and MATLAB. New York: US.
Sijtsma, K., and Meijer, R. R. (2001) The person response function as a tool in person-fit research. Psychometrika, 66(2), 191--207.
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.
# NOT RUN {
# Load the inadequacy scale data (dichotomous item scores):
data(InadequacyData)
# As an example, compute the Ht person-fit scores:
Ht.out <- Ht(InadequacyData)
Ht.out$PFscores
# Determine which respondents were flagged by Ht at 1% level:
set.seed(124) # To fix the random seed generator.
Ht.flagged <- flagged.resp(Ht.out, Blvl=.01, scores=FALSE)
Ht.flagged <- Ht.flagged$PFSscores[,1]
# Flagged respondents: 30 37 46 49 137 216 531.
# Plot the PRFs of the first three flagged respondents:
Flagged <- PRFplot(InadequacyData, respID=Ht.flagged[1:3])
# Plot the person response function of respondent 35 (not flagged as aberrant):
PRFplot(InadequacyData, respID=35)
# Plot the PRFs of all respondents:
plot(Flagged$PRF.FDO)
# }
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