Nothing
R/IRT_info.R
In TestAnaAPP: A 'shiny' App for Test Analysis and Visualization
Defines functions
generateDIFInfo
generateCRMInfo
generateMIRTInfo
generateIRTInfo
generateIRTInfo <- function(Model) {
info <- NULL
if(Model == "Rasch model (1PL)" | Model == "Two parameters logistic model (2PL)" |
Model == "Three parameters logistic model (3PL)" | Model == "Four parameters logistic model (4PL)"){
info <- withMathJax(
helpText("The formula of item response theory model for dichotomous response data can be expressed as,"),
helpText("$$P(X_{ij}=1 | \\theta_{j}) = c_i+(u_i-c_i)\\frac{1}{1+e^{(-\\alpha_i(\\theta_j-\\beta_i))}}\\tag{1}$$"),
helpText("where:"),
helpText("\\(X_{ij}\\): The response of \\(j\\) (where \\(j = 1,2,3,...,J\\)) to item \\(i\\) (where \\(i = 1,2,3,...,I\\));"),
helpText("\\(P(X_{ij}=1 | \\theta_{j})\\): The probability of a correct response;"),
helpText("\\(\\theta_{j}\\): The latent trait (ability) of respondent \\(j\\);"),
helpText("\\(\\alpha_i\\): The discrimination parameter of item \\(i\\);"),
helpText("\\(\\beta_i\\): The difficulty parameter of item \\(i\\);"),
helpText("\\(c_i\\): The guessing parameter of item \\(i\\);"),
helpText("\\(u_i\\): The upper asymptote of item \\(i\\), and the slip parameter can calculated by \\(1-u_i\\);"),
helpText("The formula (1) is a four-parameter logistic model (4PL)."),
helpText("The 4PL model reduces to a three-parameter logistic model (3PL) when \\(u_i = 1\\):"),
helpText("$$P(X_{ij}=1 | \\theta_{j}) = c_i+\\frac{1-c_i}{1+e^{-\\alpha_i(\\theta_j-\\beta_i)}}\\tag{2}$$"),
helpText("It further reduces to a two-parameter logistic model (2PL) when \\(u_i = 1\\) and \\(c_i = 0\\):"),
helpText("$$P(X_{ij}=1 | \\theta_{j}) = \\frac{1}{1+e^{-\\alpha_i(\\theta_j-\\beta_i)}}\\tag{3}$$"),
helpText("Finally, the model simplifies to the Rasch model (1PL) when \\(u_i = 1\\), \\(c_i = 0\\), and \\(\\alpha_i = 1\\):"),
helpText("$$P(X_{ij}=1 | \\theta_{j}) = \\frac{1}{1+e^{-(\\theta_j-\\beta_i)}}\\tag{4}$$")
)
}else if(Model == "Partial credit model (PCM)" ){
info <- withMathJax(
helpText("The partial credit model (PCM) is a variant of the Rasch model (Masters, 1982) specifically designed to analyze response data with more than two categories."),
helpText("The item response function (IRF) for the PCM is represented by the following equation:"),
helpText("$$P(X_{ikj}=k | \\theta_{j}) = \\frac{e^{\\sum_{h=0}^{k}(\\theta_j-\\beta_{ih})}}{\\sum_{c=0}^{m_i}e^{\\sum_{h=0}^{c}(\\theta_j-\\beta_{ih})}}\\tag{1}$$"),
helpText("where:"),
helpText("\\(X_{ikj}\\): The response of \\(j\\) to item \\(i\\) in category \\(k\\);"),
helpText("\\(P(X_{ikj}=k | \\theta_{j})\\): The probability of subject \\(j\\) responds to item \\(i\\) with category \\(k\\);"),
helpText("\\(\\theta_{j}\\): The latent trait (ability) of respondent \\(j\\);"),
helpText("\\(\\beta_{ih}\\): The threshold parameter (difficulty) for category \\(k\\) on item \\(j\\);"),
helpText("\\(m_i\\): The number of categories for item \\(i\\).")
)
}else if(Model == "Generalized partial credit model (GPCM)"){
info <- withMathJax(
helpText("The generalized partial credit model (GPCM) is an extension of the partial credit model (PCM) that allows for different discrimination parameters for each category of an item."),
helpText("The item response function (IRF) for the GPCM is represented by the following equation:"),
helpText("$$P(X_{ikj}=k | \\theta_{j}) = \\frac{e^{\\sum_{h=0}^{k}[\\alpha_i(\\theta_j-\\beta_{ih})]}}{\\sum_{c=0}^{m_i}e^{\\sum_{h=0}^{c}[\\alpha_i(\\theta_j-\\beta_{ih})]}}\\tag{1}$$"),
helpText("where:"),
helpText("\\(X_{ikj}\\): The response of \\(j\\) to item \\(i\\) in category \\(k\\);"),
helpText("\\(P(X_{ikj}=k | \\theta_{j})\\): The probability of subject \\(j\\) responds to item \\(i\\) with category \\(k\\);"),
helpText("\\(\\theta_{j}\\): The latent trait (ability) of respondent \\(j\\);"),
helpText("\\(\\alpha_{i}\\): The discrimination parameter of item \\(i\\);"),
helpText("\\(\\beta_{ih}\\): The threshold parameter (difficulty) for category \\(k\\) on item \\(j\\);"),
helpText("\\(m_i\\): The number of categories for item \\(i\\).")
)
}else if(Model == "Graded response model (GRM)"){
info <- withMathJax(
helpText("The graded response model (GRM) estimates the cumulative probability of response categories \\(k\\) or higher."),
helpText("The cumulative response probability function for the GRM model can be expressed as:"),
helpText("$$P(X_{ij}{\\geq} k | \\theta_{j}) = \\frac{e^{[\\alpha_i(\\theta_j-\\beta_{ik})]}}{1+e^{[\\alpha_i(\\theta_j-\\beta_{ik})]}}\\tag{1}$$"),
helpText("where:"),
helpText("\\(X_{ikj}\\): The response of \\(j\\) to item \\(i\\) in category \\(k\\);"),
helpText("\\(P(X_{ij}{\\geq} k | \\theta_{j})\\): The probability of respondent \\(j\\) responds to item \\(i\\) with category \\(k\\) or a higher category;"),
helpText("\\(\\theta_{j}\\): The latent trait (ability) of respondent \\(j\\);"),
helpText("\\(\\alpha_i\\): The discrimination parameter of item \\(i\\);"),
helpText("\\(\\beta_{ih}\\): The threshold parameter (difficulty) for category \\(k\\) on item \\(i\\);"),
helpText("\\(m_i\\): The number of categories for item \\(i\\)."),
helpText("To calculate the probability of a specific response category, it is necessary to subtract the adjoining cumulative probabilities:"),
helpText("$$P(X_{ij}= k | \\theta_{j})=P(X_{ij}{\\geq} k | \\theta_{j})-P(X_{ij}{\\geq} k+1 | \\theta_{j})\\tag{2}$$"),
helpText("It is important to note that the GRM model, in contrast to the GPCM and PCM models, assumes the inequality \\(\\beta_{ik}{\\geq}{\\beta_{i(k-1)}}\\). This inequality indicates that the difficulty of higher response categories is consistently greater than that of lower categories.")
)
}
return(info)
}
generateMIRTInfo <- function(Model){
info <- NULL
if(Model == "Rasch model (1PL)" | Model == "Two parameters logistic model (2PL)" |
Model == "Three parameters logistic model (3PL)" | Model == "Four parameters logistic model (4PL)"){
info <- withMathJax(
helpText("The formula of multidimensional item response theory (MIRT) model for dichotomous response data can be expressed as,"),
helpText("$$P(X_{ij}=1 | \\theta_{jq}) = c_i+(u_i-c_i)\\frac{1}{1+e^{(-(\\alpha_{iq}\\theta_{jq}+d_i))}}\\tag{1}$$"),
helpText("where:"),
helpText("\\(X_{ij}\\): The response of \\(j\\) (where \\(j = 1,2,3,...,J\\)) to item \\(i\\) (where \\(i = 1,2,3,...,I\\));"),
helpText("\\(q (q = c(1,2,3,...,Q))\\): The dimension of the latent trait (ability);"),
helpText("\\(P(X_{ij}=1 | \\theta_{jq})\\): The probability of a correct response;"),
helpText("\\(\\theta_{jq}\\): The latent trait (ability) of respondent \\(j\\) on dimension \\(q\\);"),
helpText("\\(\\alpha_{iq}\\): The discrimination parameter of item \\(i\\) on dimension \\(q\\);"),
helpText("\\(d_i\\): The threshold parameter of item \\(i\\), which is associated with the item difficult parameter \\(\\beta_i\\) and item discrimination \\(\\alpha_{iq}\\);"),
helpText("\\(c_i\\): The guessing parameter of item \\(i\\);"),
helpText("\\(u_i\\): The upper asymptote of item \\(i\\), and slip parameter denoted as \\(1-u_i\\)."),
helpText("The formula (1) is a multidimensional four-parameter logistic model (4PL)."),
helpText("The model reduces to a multidimensional three-parameter logistic model (3PL) when \\(u_i = 1\\):"),
helpText("$$P(X_{ij}=1 | \\theta_{jq}) = c_i+(1-c_i)\\frac{1}{1 + e^{(-(\\alpha_{iq}\\theta_{jq} + d_i))}}\\tag{2}$$"),
helpText("It further reduces to a multidimensional two-parameter logistic model (2PL) when \\(u_i = 1\\) and \\(c_i = 0\\):"),
helpText("$$P(X_{ij} = 1 | \\theta_{jq}) = \\frac{1}{1 + e^{(-(\\alpha_{iq}\\theta_{jq} + d_i))}}\\tag{3}$$"),
helpText("Finally, the model simplifies to the multidimensional Rasch model (1PL) when \\(u_i = 1\\), \\(c_i = 0\\), and \\(\\alpha_{iq} = 1\\):"),
helpText("$$P(X_{ij} = 1 | \\theta_{jq}) = \\frac{1}{1+e^{(-(\\theta_{jq} - \\beta_i))}}\\tag{4}$$"),
helpText("Where \\(\\beta_i=-d_i\\) is the item difficulty parameter."),
helpText(" "),
helpText("It is important to note that the parameter \\(d_i\\) cannot represent item difficulty, and it functions solely as a threshold parameter. In order to achieve a similar interpretation of item parameters as the unidimensional IRT model, researchers (Reckase, 2009; Zhang & Stone, 2008) have proposed the multidimensional discrimination index \\(MDISC_i\\) (Multidimensional Discrimination) to evaluate discrimination in MIRT. The item difficulty parameter \\(\\beta_i\\) can be calculated using \\(MDISC_i\\) and \\(d_i\\)."),
helpText("$$MDISC_i = \\sqrt{\\sum_{q=1}^Q(\\alpha_{iq})^2}\\tag{5}$$"),
helpText("$$\\beta_i = \\frac{-d_i}{MDISC_{i}}\\tag{6}$$"),
helpText("TestAnaAPP calculates the item difficulty parameter \\(\\beta_i\\) using the formula (6) and the discrimination index \\(MDISC_i\\).")
)
}else if(Model == "Partial credit model (PCM)"){
info <- withMathJax(
helpText("The item parameters of the multidimensional partial credit model (MPCM) consist solely of difficulty parameters, denoted as \\(\\beta_{ih}\\). Here, \\(\\beta_{ih}\\) represents the difficulty for category \\(k\\) on item \\(i\\)."),
helpText("The item response function (IRF) of the MPCM model is as follows:"),
helpText("$$P(X_{ikj}=k | \\theta_{jq}) = \\frac{e^{\\sum_{h=0}^{k}(\\theta_{jq}-\\beta_{ih})}}{\\sum_{c=0}^{m_i}e^{\\sum_{h=0}^{c}(\\theta_{jq}-\\beta_{ih})}}\\tag{1}$$"),
helpText("where:"),
helpText("\\(X_{ikj}\\): The response of \\(j\\) to item \\(i\\) in category \\(k\\);"),
helpText("\\(q (q = c(1,2,3,...,Q))\\): The dimension of the latent trait (ability);"),
helpText("\\(P(X_{ikj}=k | \\theta_{jq})\\): The probability of subject \\(j\\) responds to item \\(i\\) with category \\(k\\);"),
helpText("\\(\\theta_{jq}\\): The latent trait (ability) of respondent \\(j\\) on dimension \\(q\\);"),
helpText("\\(\\beta_{ih}\\): The threshold parameter (difficulty) for category \\(k\\) on item \\(j\\);"),
helpText("\\(m_i\\): The number of categories for item \\(i\\).")
)
}else if(Model == "Generalized partial credit model (GPCM)"){
info <- withMathJax(
helpText("In the case of the multidimensional generalized partial credit model (MGPCM), the item response function (IRF) can be described as follows:"),
helpText("$$P(X_{ikj}=k | \\theta_{jq}) = \\frac{e^{\\sum_{h=0}^{k}[(\\alpha_{iq}\\theta_{jq}+d_{ih})]}}{\\sum_{c=0}^{m_i}e^{\\sum_{h=0}^{c}[(\\alpha_{iq}\\theta_{jq}+d_{ih})]}}\\tag{1}$$"),
helpText("where:"),
helpText("\\(X_{ikj}\\): The response of \\(j\\) to item \\(i\\) in category \\(k\\);"),
helpText("\\(q (q = c(1,2,3,...,Q))\\): The dimension of the latent trait (ability);"),
helpText("\\(P(X_{ikj}=k | \\theta_{jq})\\): The probability of subject \\(j\\) responds to item \\(i\\) with category \\(k\\);"),
helpText("\\(\\theta_{jq}\\): The latent trait (ability) of respondent \\(j\\) on dimension \\(q\\);"),
helpText("\\(\\alpha_{iq}\\): The discrimination parameter for category \\(k\\) on item \\(i\\);"),
helpText("\\(d_{ih}\\): The threshold parameter for category \\(k\\) on item \\(i\\);"),
helpText("\\(m_i\\): The number of categories for item \\(i\\)."),
helpText(" "),
helpText("It is important to note that the parameter \\(d_i\\) cannot represent item difficulty, and it functions solely as a threshold parameter. In order to achieve a similar interpretation of item parameters as the unidimensional IRT model, researchers (Reckase, 2009; Zhang & Stone, 2008) have proposed the multidimensional discrimination index \\(MDISC_i\\) (Multidimensional Discrimination) to evaluate discrimination in MIRT. The item difficulty parameter \\(\\beta_i\\) can be calculated using \\(MDISC_i\\) and \\(d_i\\)."),
helpText("$$MDISC_i = \\sqrt{\\sum_{q=1}^Q(\\alpha_{iq})^2}\\tag{2}$$"),
helpText("$$\\beta_i = \\frac{-d_i}{MDISC_{i}}\\tag{3}$$"),
helpText("TestAnaAPP calculates the item difficulty parameter \\(\\beta_i\\) using the formula (3) and the discrimination index \\(MDISC_i\\).")
)
}else if(Model == "Graded response model (GRM)"){
info <- withMathJax(
helpText("The multidimensional graded response model (MGRM) employs the following function to represent the cumulative response probability:"),
helpText("$$P(X_{ij}{\\geq} k | \\theta_{jq}) = \\frac{e^{[(\\alpha_{iq}\\theta_{jq}+d_{ik})]}}{1+e^{[(\\alpha_{iq}\\theta_{jq}+d_{ik})]}}\\tag{1}$$"),
helpText("where:"),
helpText("\\(X_{ikj}\\): The response of \\(j\\) to item \\(i\\) in category \\(k\\);"),
helpText("\\(q (q = c(1,2,3,...,Q))\\): The dimension of the latent trait (ability);"),
helpText("\\(P(X_{ij}{\\geq} k | \\theta_{jq})\\): The probability of respondent \\(j\\) responds to item \\(i\\) with category \\(k\\) or a higher category;"),
helpText("\\(\\theta_{jq}\\): The latent trait (ability) of respondent \\(j\\) on dimension \\(q\\);"),
helpText("\\(\\alpha_{iq}\\): The discrimination parameter for category \\(k\\) on item \\(i\\);"),
helpText("\\(d_{ik}\\): The threshold parameter for category \\(k\\) on item \\(i\\);"),
helpText("\\(m_i\\): The number of categories for item \\(i\\)."),
helpText(" "),
helpText("To calculate the probability of a specific response category, it is necessary to subtract the adjoining cumulative probabilities:"),
helpText("$$P(X_{ij}= k | \\theta_{jq})=P(X_{ij}{\\geq} k | \\theta_{jq})-P(X_{ij}{\\geq} k+1 | \\theta_{jq})\\tag{2}$$"),
helpText("In contrast to the MGPCM and MPCM models, MGRM model assumes the inequality \\(d_{ik}{\\geq}{d_{i(k-1)}}\\). This inequality indicates that the difficulty of higher response categories is consistently greater than that of lower categories."),
helpText(" "),
helpText("It is important to note that the parameter \\(d_i\\) cannot represent item difficulty, and it functions solely as a threshold parameter. In order to achieve a similar interpretation of item parameters as the unidimensional IRT model, researchers (Reckase, 2009; Zhang & Stone, 2008) have proposed the multidimensional discrimination index \\(MDISC_i\\) (Multidimensional Discrimination) to evaluate discrimination in MIRT. The item difficulty parameter \\(\\beta_i\\) can be calculated using \\(MDISC_i\\) and \\(d_i\\)."),
helpText("$$MDISC_i = \\sqrt{\\sum_{q=1}^Q(\\alpha_{iq})^2}\\tag{3}$$"),
helpText("$$\\beta_i = \\frac{-d_i}{MDISC_{i}}\\tag{4}$$"),
helpText("TestAnaAPP calculates the item difficulty parameter \\(\\beta_i\\) using the formula (4) and the discrimination index \\(MDISC_i\\).")
)
}
return(info)
}
generateCRMInfo <- function(){
info <- NULL
info <- withMathJax(
helpText("As shown in Zopluoglu C (2022), the probability that the observed score \\(X_{ij}\\) is greater than or equal to some value \\(x\\) is given by:"),
helpText("$$P(X_{ij} \\geq x | \\theta_i, a_j, b_j, \\alpha_j) = \\frac{1}{\\sqrt{2\\pi}} \\int_{-\\infty}^{v} e^{-\\frac{t^2}{2}} dt$$"),
helpText("where:"),
helpText("$$v = a_j(\\theta_i - b_j - \\frac{1}{\\alpha_j} \\ln \\frac{x_{ij}}{k_j - x_{ij}})$$"),
helpText("Here, \\(a\\) is the discrimination parameter, \\(b\\) is the difficulty parameter, and \\(\\alpha\\) is a scaling parameter. The scaling parameter \\(\\alpha\\) defines the transformation from the observed score scale to the \\(\\theta\\) scale. \\(k_j\\) is the maximum possible score for the item \\(j\\)."),
helpText("Next, the observed scores \\(X\\) are transformed to a random variable \\(Z\\) using the following equation:"),
helpText("$$Z_{ij} = \\ln\\left(\\frac{X_{ij}}{k_j - X_{ij}}\\right)$$"),
helpText("The conditional probability density function of the random variable \\(Z\\) is then given by:"),
helpText("$$f(z_{ij}|\\theta_i, a_j, b_j, \\alpha_j) = \\frac{a_j}{\\sqrt{2\\pi \\alpha_j}} \\exp \\left(-\\frac{[a_j(\\theta_i - b_j - \\frac{z_{ij}}{\\alpha_j}]^2)}{2}\\right)$$"),
helpText("This is a normal density function with a mean of \\(\\alpha(\\theta - b)\\) and a variance of \\(\\alpha^2/a^2\\).")
)
return(info)
}
generateDIFInfo <- function(method){
info <- NULL
if(method == "Mantel Haenszel"){
info <- withMathJax(
helpText("The Mantel-Haenszel (MH) proposed by Holland and Thayer (1988) as a method for detecting differential item functioning (DIF) in dichotomous items."),
helpText("Based on the analysis of contingency tables, the MH procedure compares the item performance of two groups (generically known as reference and focal groups) who were previously matched on the ability scale. The detailed steps of the MH procedure can be found in Holland and Thayer (1988)."),
helpText("The following rules have been developed for use by ETS testing programs to identify items with DIF (Hidalgo & LOPez-Pina, 2004; Zwick & Ercikan, 1989):"),
helpText("Type A items (negligible DIF): items with \\(\\left|\\triangle_{\\alpha{MH}}\\right| < 1\\)."),
helpText("Type B items (moderate DIF): items with \\(1\\leq\\left|\\triangle_{\\alpha{MH}}\\right| \\leq 1.5\\), and MH test is statistically significant."),
helpText("Type C items (large DIF): items with \\(\\left|\\triangle_{\\alpha{MH}}\\right| > 1.5\\), and MH test is statistically significant."),
helpText(" "),
helpText(" "),
helpText("Reference:", style = "font-weight: bold;"),
helpText("Holland, P. W., & Thayer, D. T. (1988). Differential item performance and Mantel-Haenszel procedure. In H. Wainer & H. I. Braun (Eds.), Test validity (pp. 129-145). Hillsdale, NJ: Lawrence Erlbaum."),
helpText("Hidalgo, M. D., & LOPez-Pina, J. A. (2004). Differential Item Functioning Detection and Effect Size: A Comparison between Logistic Regression and Mantel-Haenszel Procedures. Educational and Psychological Measurement, 64(6), 903-915. https://doi.org/10.1177/0013164403261769"),
helpText("Zwick, R., & Ercikan, K. (1989). Analysis of Differential Item Functioning in the NAEP History Assessment. Journal of Educational Measurement, 26(1), 55-66. https://doi.org/10.1111/j.1745-3984.1989.tb00318.x")
)
}else if(method == "Logistic Regression"){
info <- withMathJax(
helpText("Swaminathan and Rogers (1990) proposed the use of logistic regression in DIF detection for dichotomous items. It has been extended for polytomous items by other researchers (see Choi et al., 2011)."),
helpText("The detailed steps of the logistic regression procedure can be found in Swaminathan & Rogers (1990) and Choi et al. (2011)."),
helpText("Zumbo (1999) suggested several pseudo \\(R^2\\) statistics as magnitude measures and guidelines for classifying DIF as negligible (< 0.13), moderate (between 0.13 and 0.26), and large (> 0.26)."),
helpText("Based on lordif package, three pseudo \\(R^2\\) measures - Cox & Snell (Cox and Snell 1989), Nagelkerke (Nagelkerke 1991), and McFadden (Menard 2000) - are provided."),
helpText("In the following table, the '12' in the variable name represents the comparison between Model 1 and Model 2 in
the logistic regression method, which means the test of uniform DIF; '23' represents the comparison between Model 2
and Model 3, which means the test of non-uniform DIF; '13' represents the comparison between Model 1 and Model 3, which
means an overall test of 'total DIF effect'."),
helpText("The chi-square statistic of the likelihood ratio test may be greatly affected by the sample size. It is recommended to combine the P value of the chi-square test and the pseudo \\(R^2\\) statistic to comprehensively judge whether DIF exists.",style = "font-weight: bold;"),
helpText(" "),
helpText(" "),
helpText("Reference:", style = "font-weight: bold;"),
helpText("Choi, S. W., Gibbons, L. E., & Crane, P. K. (2011). lordif: An R Package for Detecting Differential Item Functioning Using Iterative Hybrid Ordinal Logistic Regression/Item Response Theory and Monte Carlo Simulations. Journal of Statistical Software, 39(8). https://doi.org/10.18637/jss.v039.i08"),
helpText("Cox DR, Snell EJ (1989). The Analysis of Binary Data. 2nd edition. Chapman and Hall, London."),
helpText("Menard S (2000). 'Coefficients of Determination for Multiple Logistic Regression Analysis.' The American Statistician, 54, 17-24."),
helpText("Nagelkerke NJD (1991). 'A Note on a General Definition of the Coefficient of Determination.' Biometrika, 78, 691-692."),
helpText("Swaminathan, H., & Rogers, H. J. (1990). Detecting Differential Item Functioning Using Logistic Regression Procedures. Journal of Educational Measurement, 27(4), 361-370. https://doi.org/10.1111/j.1745-3984.1990.tb00754.x"),
helpText("Zumbo BD (1999). A Handbook on the Theory and Methods of Diffierential Item Functioning (DIF): Logistic Regression Modeling as a Unitary Framework for Binary and Likert-Type (Ordinal) Item Scores. Directorate of Human Resources Research and Evaluation, Department of National Defense, Ottawa, ON.")
)
}else if(method == "SIBTEST"){
info <- withMathJax(
helpText("The SIBTEST method was proposed by Shealy and Stout (1993) as a method for detecting uniform differential item functioning (DIF) in dichotomous items."),
helpText("The detailed steps of the SIBTEST procedure can be found in Shealy and Stout (1993). "),
helpText("The following rules have been developed by Roussos & Stout (1996) to identify items with DIF (see also Magis et al. (2010)):"),
helpText("Type A items (negligible DIF): items with \\(|{\\beta}| < 0.059\\)."),
helpText("Type B items (moderate DIF): items with \\(0.059 \\leq \\left|{\\beta}\\right| \\leq 0.088\\)."),
helpText("Type C items (large DIF): items with \\(|{\\beta}| > 0.088\\)."),
helpText("Since the significance test results are greatly affected by the sample size, please make a judgment based on the \\(\\beta\\) and significance test results.", style = "font-weight: bold;"),
helpText(" "),
helpText(" "),
helpText("Reference:", style = "font-weight: bold;"),
helpText("Magis, D., Beland, S., Tuerlinckx, F., & De Boeck, P. (2010). A general framework and an R package for the detection of dichotomous differential item functioning. Behavior Research Methods, 42(3), 847-862. https://doi.org/10.3758/BRM.42.3.847"),
helpText("Roussos, L. A., & Stout, W. F. (1996). Simulation Studies of the Effects of Small Sample Size and Studied Item Parameters on SIBTEST and Mantel-Haenszel Type I Error Performance. Journal of Educational Measurement, 33(2), 215-230. https://doi.org/10.1111/j.1745-3984.1996.tb00490.x"),
helpText("Shealy, R., & Stout, W. (1993). A model-based standardization approach that separates true bias/DIF from group ability differences and detects test bias/DTF as well as item bias/DIF. Psychometrika, 58(2), 159-194. https://doi.org/10.1007/BF02294572")
)
}
}
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Defines functions generateDIFInfo generateCRMInfo generateMIRTInfo generateIRTInfo
generateIRTInfo <- function(Model) {
info <- NULL
if(Model == "Rasch model (1PL)" | Model == "Two parameters logistic model (2PL)" |
Model == "Three parameters logistic model (3PL)" | Model == "Four parameters logistic model (4PL)"){
info <- withMathJax(
helpText("The formula of item response theory model for dichotomous response data can be expressed as,"),
helpText("$$P(X_{ij}=1 | \\theta_{j}) = c_i+(u_i-c_i)\\frac{1}{1+e^{(-\\alpha_i(\\theta_j-\\beta_i))}}\\tag{1}$$"),
helpText("where:"),
helpText("\\(X_{ij}\\): The response of \\(j\\) (where \\(j = 1,2,3,...,J\\)) to item \\(i\\) (where \\(i = 1,2,3,...,I\\));"),
helpText("\\(P(X_{ij}=1 | \\theta_{j})\\): The probability of a correct response;"),
helpText("\\(\\theta_{j}\\): The latent trait (ability) of respondent \\(j\\);"),
helpText("\\(\\alpha_i\\): The discrimination parameter of item \\(i\\);"),
helpText("\\(\\beta_i\\): The difficulty parameter of item \\(i\\);"),
helpText("\\(c_i\\): The guessing parameter of item \\(i\\);"),
helpText("\\(u_i\\): The upper asymptote of item \\(i\\), and the slip parameter can calculated by \\(1-u_i\\);"),
helpText("The formula (1) is a four-parameter logistic model (4PL)."),
helpText("The 4PL model reduces to a three-parameter logistic model (3PL) when \\(u_i = 1\\):"),
helpText("$$P(X_{ij}=1 | \\theta_{j}) = c_i+\\frac{1-c_i}{1+e^{-\\alpha_i(\\theta_j-\\beta_i)}}\\tag{2}$$"),
helpText("It further reduces to a two-parameter logistic model (2PL) when \\(u_i = 1\\) and \\(c_i = 0\\):"),
helpText("$$P(X_{ij}=1 | \\theta_{j}) = \\frac{1}{1+e^{-\\alpha_i(\\theta_j-\\beta_i)}}\\tag{3}$$"),
helpText("Finally, the model simplifies to the Rasch model (1PL) when \\(u_i = 1\\), \\(c_i = 0\\), and \\(\\alpha_i = 1\\):"),
helpText("$$P(X_{ij}=1 | \\theta_{j}) = \\frac{1}{1+e^{-(\\theta_j-\\beta_i)}}\\tag{4}$$")
)
}else if(Model == "Partial credit model (PCM)" ){
info <- withMathJax(
helpText("The partial credit model (PCM) is a variant of the Rasch model (Masters, 1982) specifically designed to analyze response data with more than two categories."),
helpText("The item response function (IRF) for the PCM is represented by the following equation:"),
helpText("$$P(X_{ikj}=k | \\theta_{j}) = \\frac{e^{\\sum_{h=0}^{k}(\\theta_j-\\beta_{ih})}}{\\sum_{c=0}^{m_i}e^{\\sum_{h=0}^{c}(\\theta_j-\\beta_{ih})}}\\tag{1}$$"),
helpText("where:"),
helpText("\\(X_{ikj}\\): The response of \\(j\\) to item \\(i\\) in category \\(k\\);"),
helpText("\\(P(X_{ikj}=k | \\theta_{j})\\): The probability of subject \\(j\\) responds to item \\(i\\) with category \\(k\\);"),
helpText("\\(\\theta_{j}\\): The latent trait (ability) of respondent \\(j\\);"),
helpText("\\(\\beta_{ih}\\): The threshold parameter (difficulty) for category \\(k\\) on item \\(j\\);"),
helpText("\\(m_i\\): The number of categories for item \\(i\\).")
)
}else if(Model == "Generalized partial credit model (GPCM)"){
info <- withMathJax(
helpText("The generalized partial credit model (GPCM) is an extension of the partial credit model (PCM) that allows for different discrimination parameters for each category of an item."),
helpText("The item response function (IRF) for the GPCM is represented by the following equation:"),
helpText("$$P(X_{ikj}=k | \\theta_{j}) = \\frac{e^{\\sum_{h=0}^{k}[\\alpha_i(\\theta_j-\\beta_{ih})]}}{\\sum_{c=0}^{m_i}e^{\\sum_{h=0}^{c}[\\alpha_i(\\theta_j-\\beta_{ih})]}}\\tag{1}$$"),
helpText("where:"),
helpText("\\(X_{ikj}\\): The response of \\(j\\) to item \\(i\\) in category \\(k\\);"),
helpText("\\(P(X_{ikj}=k | \\theta_{j})\\): The probability of subject \\(j\\) responds to item \\(i\\) with category \\(k\\);"),
helpText("\\(\\theta_{j}\\): The latent trait (ability) of respondent \\(j\\);"),
helpText("\\(\\alpha_{i}\\): The discrimination parameter of item \\(i\\);"),
helpText("\\(\\beta_{ih}\\): The threshold parameter (difficulty) for category \\(k\\) on item \\(j\\);"),
helpText("\\(m_i\\): The number of categories for item \\(i\\).")
)
}else if(Model == "Graded response model (GRM)"){
info <- withMathJax(
helpText("The graded response model (GRM) estimates the cumulative probability of response categories \\(k\\) or higher."),
helpText("The cumulative response probability function for the GRM model can be expressed as:"),
helpText("$$P(X_{ij}{\\geq} k | \\theta_{j}) = \\frac{e^{[\\alpha_i(\\theta_j-\\beta_{ik})]}}{1+e^{[\\alpha_i(\\theta_j-\\beta_{ik})]}}\\tag{1}$$"),
helpText("where:"),
helpText("\\(X_{ikj}\\): The response of \\(j\\) to item \\(i\\) in category \\(k\\);"),
helpText("\\(P(X_{ij}{\\geq} k | \\theta_{j})\\): The probability of respondent \\(j\\) responds to item \\(i\\) with category \\(k\\) or a higher category;"),
helpText("\\(\\theta_{j}\\): The latent trait (ability) of respondent \\(j\\);"),
helpText("\\(\\alpha_i\\): The discrimination parameter of item \\(i\\);"),
helpText("\\(\\beta_{ih}\\): The threshold parameter (difficulty) for category \\(k\\) on item \\(i\\);"),
helpText("\\(m_i\\): The number of categories for item \\(i\\)."),
helpText("To calculate the probability of a specific response category, it is necessary to subtract the adjoining cumulative probabilities:"),
helpText("$$P(X_{ij}= k | \\theta_{j})=P(X_{ij}{\\geq} k | \\theta_{j})-P(X_{ij}{\\geq} k+1 | \\theta_{j})\\tag{2}$$"),
helpText("It is important to note that the GRM model, in contrast to the GPCM and PCM models, assumes the inequality \\(\\beta_{ik}{\\geq}{\\beta_{i(k-1)}}\\). This inequality indicates that the difficulty of higher response categories is consistently greater than that of lower categories.")
)
}
return(info)
}
generateMIRTInfo <- function(Model){
info <- NULL
if(Model == "Rasch model (1PL)" | Model == "Two parameters logistic model (2PL)" |
Model == "Three parameters logistic model (3PL)" | Model == "Four parameters logistic model (4PL)"){
info <- withMathJax(
helpText("The formula of multidimensional item response theory (MIRT) model for dichotomous response data can be expressed as,"),
helpText("$$P(X_{ij}=1 | \\theta_{jq}) = c_i+(u_i-c_i)\\frac{1}{1+e^{(-(\\alpha_{iq}\\theta_{jq}+d_i))}}\\tag{1}$$"),
helpText("where:"),
helpText("\\(X_{ij}\\): The response of \\(j\\) (where \\(j = 1,2,3,...,J\\)) to item \\(i\\) (where \\(i = 1,2,3,...,I\\));"),
helpText("\\(q (q = c(1,2,3,...,Q))\\): The dimension of the latent trait (ability);"),
helpText("\\(P(X_{ij}=1 | \\theta_{jq})\\): The probability of a correct response;"),
helpText("\\(\\theta_{jq}\\): The latent trait (ability) of respondent \\(j\\) on dimension \\(q\\);"),
helpText("\\(\\alpha_{iq}\\): The discrimination parameter of item \\(i\\) on dimension \\(q\\);"),
helpText("\\(d_i\\): The threshold parameter of item \\(i\\), which is associated with the item difficult parameter \\(\\beta_i\\) and item discrimination \\(\\alpha_{iq}\\);"),
helpText("\\(c_i\\): The guessing parameter of item \\(i\\);"),
helpText("\\(u_i\\): The upper asymptote of item \\(i\\), and slip parameter denoted as \\(1-u_i\\)."),
helpText("The formula (1) is a multidimensional four-parameter logistic model (4PL)."),
helpText("The model reduces to a multidimensional three-parameter logistic model (3PL) when \\(u_i = 1\\):"),
helpText("$$P(X_{ij}=1 | \\theta_{jq}) = c_i+(1-c_i)\\frac{1}{1 + e^{(-(\\alpha_{iq}\\theta_{jq} + d_i))}}\\tag{2}$$"),
helpText("It further reduces to a multidimensional two-parameter logistic model (2PL) when \\(u_i = 1\\) and \\(c_i = 0\\):"),
helpText("$$P(X_{ij} = 1 | \\theta_{jq}) = \\frac{1}{1 + e^{(-(\\alpha_{iq}\\theta_{jq} + d_i))}}\\tag{3}$$"),
helpText("Finally, the model simplifies to the multidimensional Rasch model (1PL) when \\(u_i = 1\\), \\(c_i = 0\\), and \\(\\alpha_{iq} = 1\\):"),
helpText("$$P(X_{ij} = 1 | \\theta_{jq}) = \\frac{1}{1+e^{(-(\\theta_{jq} - \\beta_i))}}\\tag{4}$$"),
helpText("Where \\(\\beta_i=-d_i\\) is the item difficulty parameter."),
helpText(" "),
helpText("It is important to note that the parameter \\(d_i\\) cannot represent item difficulty, and it functions solely as a threshold parameter. In order to achieve a similar interpretation of item parameters as the unidimensional IRT model, researchers (Reckase, 2009; Zhang & Stone, 2008) have proposed the multidimensional discrimination index \\(MDISC_i\\) (Multidimensional Discrimination) to evaluate discrimination in MIRT. The item difficulty parameter \\(\\beta_i\\) can be calculated using \\(MDISC_i\\) and \\(d_i\\)."),
helpText("$$MDISC_i = \\sqrt{\\sum_{q=1}^Q(\\alpha_{iq})^2}\\tag{5}$$"),
helpText("$$\\beta_i = \\frac{-d_i}{MDISC_{i}}\\tag{6}$$"),
helpText("TestAnaAPP calculates the item difficulty parameter \\(\\beta_i\\) using the formula (6) and the discrimination index \\(MDISC_i\\).")
)
}else if(Model == "Partial credit model (PCM)"){
info <- withMathJax(
helpText("The item parameters of the multidimensional partial credit model (MPCM) consist solely of difficulty parameters, denoted as \\(\\beta_{ih}\\). Here, \\(\\beta_{ih}\\) represents the difficulty for category \\(k\\) on item \\(i\\)."),
helpText("The item response function (IRF) of the MPCM model is as follows:"),
helpText("$$P(X_{ikj}=k | \\theta_{jq}) = \\frac{e^{\\sum_{h=0}^{k}(\\theta_{jq}-\\beta_{ih})}}{\\sum_{c=0}^{m_i}e^{\\sum_{h=0}^{c}(\\theta_{jq}-\\beta_{ih})}}\\tag{1}$$"),
helpText("where:"),
helpText("\\(X_{ikj}\\): The response of \\(j\\) to item \\(i\\) in category \\(k\\);"),
helpText("\\(q (q = c(1,2,3,...,Q))\\): The dimension of the latent trait (ability);"),
helpText("\\(P(X_{ikj}=k | \\theta_{jq})\\): The probability of subject \\(j\\) responds to item \\(i\\) with category \\(k\\);"),
helpText("\\(\\theta_{jq}\\): The latent trait (ability) of respondent \\(j\\) on dimension \\(q\\);"),
helpText("\\(\\beta_{ih}\\): The threshold parameter (difficulty) for category \\(k\\) on item \\(j\\);"),
helpText("\\(m_i\\): The number of categories for item \\(i\\).")
)
}else if(Model == "Generalized partial credit model (GPCM)"){
info <- withMathJax(
helpText("In the case of the multidimensional generalized partial credit model (MGPCM), the item response function (IRF) can be described as follows:"),
helpText("$$P(X_{ikj}=k | \\theta_{jq}) = \\frac{e^{\\sum_{h=0}^{k}[(\\alpha_{iq}\\theta_{jq}+d_{ih})]}}{\\sum_{c=0}^{m_i}e^{\\sum_{h=0}^{c}[(\\alpha_{iq}\\theta_{jq}+d_{ih})]}}\\tag{1}$$"),
helpText("where:"),
helpText("\\(X_{ikj}\\): The response of \\(j\\) to item \\(i\\) in category \\(k\\);"),
helpText("\\(q (q = c(1,2,3,...,Q))\\): The dimension of the latent trait (ability);"),
helpText("\\(P(X_{ikj}=k | \\theta_{jq})\\): The probability of subject \\(j\\) responds to item \\(i\\) with category \\(k\\);"),
helpText("\\(\\theta_{jq}\\): The latent trait (ability) of respondent \\(j\\) on dimension \\(q\\);"),
helpText("\\(\\alpha_{iq}\\): The discrimination parameter for category \\(k\\) on item \\(i\\);"),
helpText("\\(d_{ih}\\): The threshold parameter for category \\(k\\) on item \\(i\\);"),
helpText("\\(m_i\\): The number of categories for item \\(i\\)."),
helpText(" "),
helpText("It is important to note that the parameter \\(d_i\\) cannot represent item difficulty, and it functions solely as a threshold parameter. In order to achieve a similar interpretation of item parameters as the unidimensional IRT model, researchers (Reckase, 2009; Zhang & Stone, 2008) have proposed the multidimensional discrimination index \\(MDISC_i\\) (Multidimensional Discrimination) to evaluate discrimination in MIRT. The item difficulty parameter \\(\\beta_i\\) can be calculated using \\(MDISC_i\\) and \\(d_i\\)."),
helpText("$$MDISC_i = \\sqrt{\\sum_{q=1}^Q(\\alpha_{iq})^2}\\tag{2}$$"),
helpText("$$\\beta_i = \\frac{-d_i}{MDISC_{i}}\\tag{3}$$"),
helpText("TestAnaAPP calculates the item difficulty parameter \\(\\beta_i\\) using the formula (3) and the discrimination index \\(MDISC_i\\).")
)
}else if(Model == "Graded response model (GRM)"){
info <- withMathJax(
helpText("The multidimensional graded response model (MGRM) employs the following function to represent the cumulative response probability:"),
helpText("$$P(X_{ij}{\\geq} k | \\theta_{jq}) = \\frac{e^{[(\\alpha_{iq}\\theta_{jq}+d_{ik})]}}{1+e^{[(\\alpha_{iq}\\theta_{jq}+d_{ik})]}}\\tag{1}$$"),
helpText("where:"),
helpText("\\(X_{ikj}\\): The response of \\(j\\) to item \\(i\\) in category \\(k\\);"),
helpText("\\(q (q = c(1,2,3,...,Q))\\): The dimension of the latent trait (ability);"),
helpText("\\(P(X_{ij}{\\geq} k | \\theta_{jq})\\): The probability of respondent \\(j\\) responds to item \\(i\\) with category \\(k\\) or a higher category;"),
helpText("\\(\\theta_{jq}\\): The latent trait (ability) of respondent \\(j\\) on dimension \\(q\\);"),
helpText("\\(\\alpha_{iq}\\): The discrimination parameter for category \\(k\\) on item \\(i\\);"),
helpText("\\(d_{ik}\\): The threshold parameter for category \\(k\\) on item \\(i\\);"),
helpText("\\(m_i\\): The number of categories for item \\(i\\)."),
helpText(" "),
helpText("To calculate the probability of a specific response category, it is necessary to subtract the adjoining cumulative probabilities:"),
helpText("$$P(X_{ij}= k | \\theta_{jq})=P(X_{ij}{\\geq} k | \\theta_{jq})-P(X_{ij}{\\geq} k+1 | \\theta_{jq})\\tag{2}$$"),
helpText("In contrast to the MGPCM and MPCM models, MGRM model assumes the inequality \\(d_{ik}{\\geq}{d_{i(k-1)}}\\). This inequality indicates that the difficulty of higher response categories is consistently greater than that of lower categories."),
helpText(" "),
helpText("It is important to note that the parameter \\(d_i\\) cannot represent item difficulty, and it functions solely as a threshold parameter. In order to achieve a similar interpretation of item parameters as the unidimensional IRT model, researchers (Reckase, 2009; Zhang & Stone, 2008) have proposed the multidimensional discrimination index \\(MDISC_i\\) (Multidimensional Discrimination) to evaluate discrimination in MIRT. The item difficulty parameter \\(\\beta_i\\) can be calculated using \\(MDISC_i\\) and \\(d_i\\)."),
helpText("$$MDISC_i = \\sqrt{\\sum_{q=1}^Q(\\alpha_{iq})^2}\\tag{3}$$"),
helpText("$$\\beta_i = \\frac{-d_i}{MDISC_{i}}\\tag{4}$$"),
helpText("TestAnaAPP calculates the item difficulty parameter \\(\\beta_i\\) using the formula (4) and the discrimination index \\(MDISC_i\\).")
)
}
return(info)
}
generateCRMInfo <- function(){
info <- NULL
info <- withMathJax(
helpText("As shown in Zopluoglu C (2022), the probability that the observed score \\(X_{ij}\\) is greater than or equal to some value \\(x\\) is given by:"),
helpText("$$P(X_{ij} \\geq x | \\theta_i, a_j, b_j, \\alpha_j) = \\frac{1}{\\sqrt{2\\pi}} \\int_{-\\infty}^{v} e^{-\\frac{t^2}{2}} dt$$"),
helpText("where:"),
helpText("$$v = a_j(\\theta_i - b_j - \\frac{1}{\\alpha_j} \\ln \\frac{x_{ij}}{k_j - x_{ij}})$$"),
helpText("Here, \\(a\\) is the discrimination parameter, \\(b\\) is the difficulty parameter, and \\(\\alpha\\) is a scaling parameter. The scaling parameter \\(\\alpha\\) defines the transformation from the observed score scale to the \\(\\theta\\) scale. \\(k_j\\) is the maximum possible score for the item \\(j\\)."),
helpText("Next, the observed scores \\(X\\) are transformed to a random variable \\(Z\\) using the following equation:"),
helpText("$$Z_{ij} = \\ln\\left(\\frac{X_{ij}}{k_j - X_{ij}}\\right)$$"),
helpText("The conditional probability density function of the random variable \\(Z\\) is then given by:"),
helpText("$$f(z_{ij}|\\theta_i, a_j, b_j, \\alpha_j) = \\frac{a_j}{\\sqrt{2\\pi \\alpha_j}} \\exp \\left(-\\frac{[a_j(\\theta_i - b_j - \\frac{z_{ij}}{\\alpha_j}]^2)}{2}\\right)$$"),
helpText("This is a normal density function with a mean of \\(\\alpha(\\theta - b)\\) and a variance of \\(\\alpha^2/a^2\\).")
)
return(info)
}
generateDIFInfo <- function(method){
info <- NULL
if(method == "Mantel Haenszel"){
info <- withMathJax(
helpText("The Mantel-Haenszel (MH) proposed by Holland and Thayer (1988) as a method for detecting differential item functioning (DIF) in dichotomous items."),
helpText("Based on the analysis of contingency tables, the MH procedure compares the item performance of two groups (generically known as reference and focal groups) who were previously matched on the ability scale. The detailed steps of the MH procedure can be found in Holland and Thayer (1988)."),
helpText("The following rules have been developed for use by ETS testing programs to identify items with DIF (Hidalgo & LOPez-Pina, 2004; Zwick & Ercikan, 1989):"),
helpText("Type A items (negligible DIF): items with \\(\\left|\\triangle_{\\alpha{MH}}\\right| < 1\\)."),
helpText("Type B items (moderate DIF): items with \\(1\\leq\\left|\\triangle_{\\alpha{MH}}\\right| \\leq 1.5\\), and MH test is statistically significant."),
helpText("Type C items (large DIF): items with \\(\\left|\\triangle_{\\alpha{MH}}\\right| > 1.5\\), and MH test is statistically significant."),
helpText(" "),
helpText(" "),
helpText("Reference:", style = "font-weight: bold;"),
helpText("Holland, P. W., & Thayer, D. T. (1988). Differential item performance and Mantel-Haenszel procedure. In H. Wainer & H. I. Braun (Eds.), Test validity (pp. 129-145). Hillsdale, NJ: Lawrence Erlbaum."),
helpText("Hidalgo, M. D., & LOPez-Pina, J. A. (2004). Differential Item Functioning Detection and Effect Size: A Comparison between Logistic Regression and Mantel-Haenszel Procedures. Educational and Psychological Measurement, 64(6), 903-915. https://doi.org/10.1177/0013164403261769"),
helpText("Zwick, R., & Ercikan, K. (1989). Analysis of Differential Item Functioning in the NAEP History Assessment. Journal of Educational Measurement, 26(1), 55-66. https://doi.org/10.1111/j.1745-3984.1989.tb00318.x")
)
}else if(method == "Logistic Regression"){
info <- withMathJax(
helpText("Swaminathan and Rogers (1990) proposed the use of logistic regression in DIF detection for dichotomous items. It has been extended for polytomous items by other researchers (see Choi et al., 2011)."),
helpText("The detailed steps of the logistic regression procedure can be found in Swaminathan & Rogers (1990) and Choi et al. (2011)."),
helpText("Zumbo (1999) suggested several pseudo \\(R^2\\) statistics as magnitude measures and guidelines for classifying DIF as negligible (< 0.13), moderate (between 0.13 and 0.26), and large (> 0.26)."),
helpText("Based on lordif package, three pseudo \\(R^2\\) measures - Cox & Snell (Cox and Snell 1989), Nagelkerke (Nagelkerke 1991), and McFadden (Menard 2000) - are provided."),
helpText("In the following table, the '12' in the variable name represents the comparison between Model 1 and Model 2 in
the logistic regression method, which means the test of uniform DIF; '23' represents the comparison between Model 2
and Model 3, which means the test of non-uniform DIF; '13' represents the comparison between Model 1 and Model 3, which
means an overall test of 'total DIF effect'."),
helpText("The chi-square statistic of the likelihood ratio test may be greatly affected by the sample size. It is recommended to combine the P value of the chi-square test and the pseudo \\(R^2\\) statistic to comprehensively judge whether DIF exists.",style = "font-weight: bold;"),
helpText(" "),
helpText(" "),
helpText("Reference:", style = "font-weight: bold;"),
helpText("Choi, S. W., Gibbons, L. E., & Crane, P. K. (2011). lordif: An R Package for Detecting Differential Item Functioning Using Iterative Hybrid Ordinal Logistic Regression/Item Response Theory and Monte Carlo Simulations. Journal of Statistical Software, 39(8). https://doi.org/10.18637/jss.v039.i08"),
helpText("Cox DR, Snell EJ (1989). The Analysis of Binary Data. 2nd edition. Chapman and Hall, London."),
helpText("Menard S (2000). 'Coefficients of Determination for Multiple Logistic Regression Analysis.' The American Statistician, 54, 17-24."),
helpText("Nagelkerke NJD (1991). 'A Note on a General Definition of the Coefficient of Determination.' Biometrika, 78, 691-692."),
helpText("Swaminathan, H., & Rogers, H. J. (1990). Detecting Differential Item Functioning Using Logistic Regression Procedures. Journal of Educational Measurement, 27(4), 361-370. https://doi.org/10.1111/j.1745-3984.1990.tb00754.x"),
helpText("Zumbo BD (1999). A Handbook on the Theory and Methods of Diffierential Item Functioning (DIF): Logistic Regression Modeling as a Unitary Framework for Binary and Likert-Type (Ordinal) Item Scores. Directorate of Human Resources Research and Evaluation, Department of National Defense, Ottawa, ON.")
)
}else if(method == "SIBTEST"){
info <- withMathJax(
helpText("The SIBTEST method was proposed by Shealy and Stout (1993) as a method for detecting uniform differential item functioning (DIF) in dichotomous items."),
helpText("The detailed steps of the SIBTEST procedure can be found in Shealy and Stout (1993). "),
helpText("The following rules have been developed by Roussos & Stout (1996) to identify items with DIF (see also Magis et al. (2010)):"),
helpText("Type A items (negligible DIF): items with \\(|{\\beta}| < 0.059\\)."),
helpText("Type B items (moderate DIF): items with \\(0.059 \\leq \\left|{\\beta}\\right| \\leq 0.088\\)."),
helpText("Type C items (large DIF): items with \\(|{\\beta}| > 0.088\\)."),
helpText("Since the significance test results are greatly affected by the sample size, please make a judgment based on the \\(\\beta\\) and significance test results.", style = "font-weight: bold;"),
helpText(" "),
helpText(" "),
helpText("Reference:", style = "font-weight: bold;"),
helpText("Magis, D., Beland, S., Tuerlinckx, F., & De Boeck, P. (2010). A general framework and an R package for the detection of dichotomous differential item functioning. Behavior Research Methods, 42(3), 847-862. https://doi.org/10.3758/BRM.42.3.847"),
helpText("Roussos, L. A., & Stout, W. F. (1996). Simulation Studies of the Effects of Small Sample Size and Studied Item Parameters on SIBTEST and Mantel-Haenszel Type I Error Performance. Journal of Educational Measurement, 33(2), 215-230. https://doi.org/10.1111/j.1745-3984.1996.tb00490.x"),
helpText("Shealy, R., & Stout, W. (1993). A model-based standardization approach that separates true bias/DIF from group ability differences and detects test bias/DTF as well as item bias/DIF. Psychometrika, 58(2), 159-194. https://doi.org/10.1007/BF02294572")
)
}
}
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