Generating an artificial item response dataset

Description

This function generates an artificial item response dataset allowing various options.

Usage

DataGeneration(
  seed = 1,
  N = 2000,
  nitem_D = 0,
  nitem_P = 0,
  nitem_C = 0,
  model_D = "2PL",
  model_P = "GPCM",
  latent_dist = "Normal",
  item_D = NULL,
  item_P = NULL,
  item_C = NULL,
  theta = NULL,
  prob = 0.5,
  d = 1.7,
  sd_ratio = 1,
  m = 0,
  s = 1,
  a_l = 0.8,
  a_u = 2.5,
  b_m = NULL,
  b_sd = NULL,
  c_l = 0,
  c_u = 0.2,
  categ = 5,
  possible_ans = c(0.1, 0.3, 0.5, 0.7, 0.9)
)

Arguments

Value

This function returns a list of several objects:

Author(s)

Seewoo Li cu@yonsei.ac.kr

References

Li, S. (2021). Using a two-component normal mixture distribution as a latent distribution in estimating parameters of item response models. Journal of Educational Evaluation, 34(4), 759-789.

Examples

# Dichotomous item responses

Alldata <- DataGeneration(N = 500,
                          nitem_D = 10)


# Polytomous item responses

Alldata <- DataGeneration(N = 1000,
                          nitem_P = 10)


# Mixed-format items

Alldata <- DataGeneration(N = 1000,
                          nitem_D = 20,
                          nitem_P = 10)

# Continuous items

AllData <- DataGeneration(N = 1000,
                          nitem_C = 10)

# Dataset from non-normal latent density using two-component Gaussian mixture distribution

Alldata <- DataGeneration(N=1000,
                          nitem_P = 10,
                          latent_dist = "2NM",
                          d = 1.664,
                          sd_ratio = 2,
                          prob = 0.3)

Item and ability parameters estimation for continuous response items

Description

This function estimates IRT item and ability parameters when all items are scored continuously. Based on Bock & Aitkin's (1981) marginal maximum likelihood and EM algorithm (EM-MML), this function provides several latent distribution estimation algorithms which could free the normality assumption on the latent variable. If the normality assumption is violated, application of these latent distribution estimation methods could reflect non-normal characteristics of the unknown true latent distribution, thereby providing more accurate parameter estimates (Li, 2021; Woods & Lin, 2009; Woods & Thissen, 2006).

Usage

IRTest_Cont(
  data,
  range = c(-6, 6),
  q = 121,
  initialitem = NULL,
  ability_method = "EAP",
  latent_dist = "Normal",
  max_iter = 200,
  threshold = 1e-04,
  bandwidth = "SJ-ste",
  h = NULL
)

Arguments

Details

The probability of a response u=x, where 0<u<1 (see Martinez, 2023)

P(u=x | a, b, \nu) = \frac{1}{B(\mu\nu, \,\nu(1-\mu))} u^{\mu\nu-1} (1-u)^{\nu(1-\mu)-1}

where \mu = \frac{e^{a(\theta -b)}}{1+e^{a(\theta -b)}}.

Latent distribution estimation methods

1) Empirical histogram method

P(\theta=X_k)=A(X_k)

where k=1, 2, ..., q, X_k is the location of the kth quadrature point, and A(X_k) is a value of probability mass function evaluated at X_k. Empirical histogram method thus has q-1 parameters.

2) Two-component Gaussian mixture distribution

P(\theta=X)=\pi \phi(X; \mu_1, \sigma_1)+(1-\pi) \phi(X; \mu_2, \sigma_2)

where \phi(X; \mu, \sigma) is the value of a Gaussian component with mean \mu and standard deviation \sigma evaluated at X.

3) Davidian curve method

P(\theta=X)=\left\{\sum_{\lambda=0}^{h}{{m}_{\lambda}{X}^{\lambda}}\right\}^{2}\phi(X; 0, 1)

where h corresponds to the argument h and determines the degree of the polynomial.

4) Kernel density estimation method

P(\theta=X)=\frac{1}{Nh}\sum_{j=1}^{N}{K\left(\frac{X-\theta_j}{h}\right)}

where N is the number of examinees, \theta_j is jth examinee's ability parameter, h is the bandwidth which corresponds to the argument bandwidth, and K( \cdot ) is a kernel function. The Gaussian kernel is used in this function.

5) Log-linear smoothing method

P(\theta=X_{q})=\exp{\left(\beta_{0}+\sum_{m=1}^{h}{\beta_{m}X_{q}^{m}}\right)}

where h is the hyper parameter which determines the smoothness of the density, and \theta can take total Q finite values (X_1, \dots ,X_q, \dots, X_Q).

Value

This function returns a list of several objects:

Author(s)

Seewoo Li cu@yonsei.ac.kr

References

Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46(4), 443-459.

Casabianca, J. M., & Lewis, C. (2015). IRT item parameter recovery with marginal maximum likelihood estimation using loglinear smoothing models. Journal of Educational and Behavioral Statistics, 40(6), 547-578.

Li, S. (2021). Using a two-component normal mixture distribution as a latent distribution in estimating parameters of item response models. Journal of Educational Evaluation, 34(4), 759-789.

Li, S. (2022). The effect of estimating latent distribution using kernel density estimation method on the accuracy and efficiency of parameter estimation of item response models [Master's thesis, Yonsei University, Seoul]. Yonsei University Library.

Martinez, A. J. (2023). Beta item factor analysis for asymmetric, bounded, and continuous item response data. OSF. DOI:10.31234/osf.io/tp8sx.

Mislevy, R. J. (1984). Estimating latent distributions. Psychometrika, 49(3), 359-381.

Mislevy, R. J., & Bock, R. D. (1985). Implementation of the EM algorithm in the estimation of item parameters: The BILOG computer program. In D. J. Weiss (Ed.). Proceedings of the 1982 item response theory and computerized adaptive testing conference (pp. 189-202). University of Minnesota, Department of Psychology, Computerized Adaptive Testing Conference.

Woods, C. M., & Lin, N. (2009). Item response theory with estimation of the latent density using Davidian curves. Applied Psychological Measurement, 33(2), 102-117.

Woods, C. M., & Thissen, D. (2006). Item response theory with estimation of the latent population distribution using spline-based densities. Psychometrika, 71(2), 281-301.

Examples

# Generating a continuous item response data
data <- DataGeneration(N = 1000, nitem_C = 10)$data_C

# Analysis
M1 <- IRTest_Cont(data, max_iter = 3) # increase `max_iter` in real analyses.

Item and ability parameters estimation for dichotomous items

Description

This function estimates IRT item and ability parameters when all items are scored dichotomously. Based on Bock & Aitkin's (1981) marginal maximum likelihood and EM algorithm (EM-MML), this function provides several latent distribution estimation algorithms which could free the normality assumption on the latent variable. If the normality assumption is violated, application of these latent distribution estimation methods could reflect non-normal characteristics of the unknown true latent distribution, and, thus, could provide more accurate parameter estimates (Li, 2021; Woods & Lin, 2009; Woods & Thissen, 2006).

Usage

IRTest_Dich(
  data,
  model = "2PL",
  range = c(-6, 6),
  q = 121,
  initialitem = NULL,
  ability_method = "EAP",
  latent_dist = "Normal",
  max_iter = 200,
  threshold = 1e-04,
  bandwidth = "SJ-ste",
  h = NULL
)

Arguments

Details

The probabilities for a correct response (u=1)

1) One-parameter logistic (1PL) model

P(u=1|\theta, b)=\frac{\exp{(\theta-b)}}{1+\exp{(\theta-b)}}

2) Two-parameter logistic (2PL) model

P(u=1|\theta, a, b)=\frac{\exp{(a(\theta-b))}}{1+\exp{(a(\theta-b))}}

3) Three-parameter logistic (3PL) model

P(u=1|\theta, a, b, c)=c + (1-c)\frac{\exp{(a(\theta-b))}}{1+\exp{(a(\theta-b))}}

Latent distribution estimation methods

1) Empirical histogram method

P(\theta=X_k)=A(X_k)

where k=1, 2, ..., q, X_k is the location of the kth quadrature point, and A(X_k) is a value of probability mass function evaluated at X_k. Empirical histogram method thus has q-1 parameters.

2) Two-component Gaussian mixture distribution

P(\theta=X)=\pi \phi(X; \mu_1, \sigma_1)+(1-\pi) \phi(X; \mu_2, \sigma_2)

where \phi(X; \mu, \sigma) is the value of a Gaussian component with mean \mu and standard deviation \sigma evaluated at X.

3) Davidian curve method

P(\theta=X)=\left\{\sum_{\lambda=0}^{h}{{m}_{\lambda}{X}^{\lambda}}\right\}^{2}\phi(X; 0, 1)

where h corresponds to the argument h and determines the degree of the polynomial.

4) Kernel density estimation method

P(\theta=X)=\frac{1}{Nh}\sum_{j=1}^{N}{K\left(\frac{X-\theta_j}{h}\right)}

where N is the number of examinees, \theta_j is jth examinee's ability parameter, h is the bandwidth which corresponds to the argument bandwidth, and K( \cdot ) is a kernel function. The Gaussian kernel is used in this function.

5) Log-linear smoothing method

P(\theta=X_{q})=\exp{\left(\beta_{0}+\sum_{m=1}^{h}{\beta_{m}X_{q}^{m}}\right)}

where h is the hyper parameter which determines the smoothness of the density, and \theta can take total Q finite values (X_1, \dots ,X_q, \dots, X_Q).

Value

This function returns a list of several objects:

Author(s)

Seewoo Li cu@yonsei.ac.kr

References

Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46(4), 443-459.

Casabianca, J. M., & Lewis, C. (2015). IRT item parameter recovery with marginal maximum likelihood estimation using loglinear smoothing models. Journal of Educational and Behavioral Statistics, 40(6), 547-578.

Li, S. (2021). Using a two-component normal mixture distribution as a latent distribution in estimating parameters of item response models. Journal of Educational Evaluation, 34(4), 759-789.

Li, S. (2022). The effect of estimating latent distribution using kernel density estimation method on the accuracy and efficiency of parameter estimation of item response models [Master's thesis, Yonsei University, Seoul]. Yonsei University Library.

Mislevy, R. J. (1984). Estimating latent distributions. Psychometrika, 49(3), 359-381.

Mislevy, R. J., & Bock, R. D. (1985). Implementation of the EM algorithm in the estimation of item parameters: The BILOG computer program. In D. J. Weiss (Ed.). Proceedings of the 1982 item response theory and computerized adaptive testing conference (pp. 189-202). University of Minnesota, Department of Psychology, Computerized Adaptive Testing Conference.

Woods, C. M., & Lin, N. (2009). Item response theory with estimation of the latent density using Davidian curves. Applied Psychological Measurement, 33(2), 102-117.

Woods, C. M., & Thissen, D. (2006). Item response theory with estimation of the latent population distribution using spline-based densities. Psychometrika, 71(2), 281-301.

Examples

# A preparation of dichotomous item response data

data <- DataGeneration(N=500,
                       nitem_D = 10)$data_D

# Analysis

M1 <- IRTest_Dich(data)

Item and ability parameters estimation for a mixed-format item response data

Description

This function estimates IRT item and ability parameters when a test consists of mixed-format items (i.e., a combination of dichotomous and polytomous items). In educational context, the combination of these two item formats takes an advantage; Dichotomous item format expedites scoring and is conducive to cover broad domain, while Polytomous item format (e.g., free response item) encourages students to exert complex cognitive skills (Lee et al., 2020). Based on Bock & Aitkin's (1981) marginal maximum likelihood and EM algorithm (EM-MML), this function incorporates several latent distribution estimation algorithms which could free the normality assumption on the latent variable. If the normality assumption is violated, application of these latent distribution estimation methods could reflect some features of the unknown true latent distribution, and, thus, could provide more accurate parameter estimates (Li, 2021; Woods & Lin, 2009; Woods & Thissen, 2006).

Usage

IRTest_Mix(
  data_D,
  data_P,
  model_D = "2PL",
  model_P = "GPCM",
  range = c(-6, 6),
  q = 121,
  initialitem_D = NULL,
  initialitem_P = NULL,
  ability_method = "EAP",
  latent_dist = "Normal",
  max_iter = 200,
  threshold = 1e-04,
  bandwidth = "SJ-ste",
  h = NULL
)

Arguments

Details

Dichotomous: the probabilities for a correct response (u=1)

1) One-parameter logistic (1PL) model

P(u=1|\theta, b)=\frac{\exp{(\theta-b)}}{1+\exp{(\theta-b)}}

2) Two-parameter logistic (2PL) model

P(u=1|\theta, a, b)=\frac{\exp{(a(\theta-b))}}{1+\exp{(a(\theta-b))}}

3) Three-parameter logistic (3PL) model

P(u=1|\theta, a, b, c)=c + (1-c)\frac{\exp{(a(\theta-b))}}{1+\exp{(a(\theta-b))}}

Polytomous: the probability for scoring u=k (i.e., k=0, 1, ..., m; m \ge 2)

1) Partial credit model (PCM)

P(u=0|\theta, b_1, ..., b_{m})=\frac{1}{1+\sum_{c=1}^{m}{\exp{\left[\sum_{v=1}^{c}{a(\theta-b_v)}\right]}}}

P(u=1|\theta, b_1, ..., b_{m})=\frac{\exp{(\theta-b_1)}}{1+\sum_{c=1}^{m}{\exp{\left[\sum_{v=1}^{c}{\theta-b_v}\right]}}}

\vdots

P(u=m|\theta, b_1, ..., b_{m})=\frac{\exp{\left[\sum_{v=1}^{m}{\theta-b_v}\right]}}{1+\sum_{c=1}^{m}{\exp{\left[\sum_{v=1}^{c}{\theta-b_v}\right]}}}

2) Generalized partial credit model (GPCM)

P(u=0|\theta, a, b_1, ..., b_{m})=\frac{1}{1+\sum_{c=1}^{m}{\exp{\left[\sum_{v=1}^{c}{a(\theta-b_v)}\right]}}}

P(u=1|\theta, a, b_1, ..., b_{m})=\frac{\exp{(a(\theta-b_1))}}{1+\sum_{c=1}^{m}{\exp{\left[\sum_{v=1}^{c}{a(\theta-b_v)}\right]}}}

\vdots

P(u=m|\theta, a, b_1, ..., b_{m})=\frac{\exp{\left[\sum_{v=1}^{m}{a(\theta-b_v)}\right]}}{1+\sum_{c=1}^{m}{\exp{\left[\sum_{v=1}^{c}{a(\theta-b_v)}\right]}}}

3) Graded response model (GRM)

P(u=0|\theta, a, b_1, ..., b_{m})=1-\frac{1}{1+\exp{\left[-a(\theta-b_1)\right]}}

P(u=1|\theta, a, b_1, ..., b_{m})=\frac{1}{1+\exp{\left[-a(\theta-b_1)\right]}}-\frac{1}{1+\exp{\left[-a(\theta-b_2)\right]}}

\vdots

P(u=m|\theta, a, b_1, ..., b_{m})=\frac{1}{1+\exp{\left[-a(\theta-b_m)\right]}}-0

Latent distribution estimation methods

1) Empirical histogram method

P(\theta=X_k)=A(X_k)

where k=1, 2, ..., q, X_k is the location of the kth quadrature point, and A(X_k) is a value of probability mass function evaluated at X_k. Empirical histogram method thus has q-1 parameters.

2) Two-component Gaussian mixture distribution

P(\theta=X)=\pi \phi(X; \mu_1, \sigma_1)+(1-\pi) \phi(X; \mu_2, \sigma_2)

where \phi(X; \mu, \sigma) is the value of a Gaussian component with mean \mu and standard deviation \sigma evaluated at X.

3) Davidian curve method

P(\theta=X)=\left\{\sum_{\lambda=0}^{h}{{m}_{\lambda}{X}^{\lambda}}\right\}^{2}\phi(X; 0, 1)

where h corresponds to the argument h and determines the degree of the polynomial.

4) Kernel density estimation method

P(\theta=X)=\frac{1}{Nh}\sum_{j=1}^{N}{K\left(\frac{X-\theta_j}{h}\right)}

where N is the number of examinees, \theta_j is jth examinee's ability parameter, h is the bandwidth which corresponds to the argument bw, and K( \bullet ) is a kernel function. The Gaussian kernel is used in this function.

5) Log-linear smoothing method

P(\theta=X_{q})=\exp{\left(\beta_{0}+\sum_{m=1}^{h}{\beta_{m}X_{q}^{m}}\right)}

where h is the hyper parameter which determines the smoothness of the density, and \theta can take total Q finite values (X_1, \dots ,X_q, \dots, X_Q).

Value

This function returns a list of several objects:

Author(s)

Seewoo Li cu@yonsei.ac.kr

References

Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46(4), 443-459.

Casabianca, J. M., & Lewis, C. (2015). IRT item parameter recovery with marginal maximum likelihood estimation using loglinear smoothing models. Journal of Educational and Behavioral Statistics, 40(6), 547-578.

Lee, W. C., Kim, S. Y., Choi, J., & Kang, Y. (2020). IRT Approaches to Modeling Scores on Mixed-Format Tests. Journal of Educational Measurement, 57(2), 230-254.

Li, S. (2021). Using a two-component normal mixture distribution as a latent distribution in estimating parameters of item response models. Journal of Educational Evaluation, 34(4), 759-789.

Li, S. (2022). The effect of estimating latent distribution using kernel density estimation method on the accuracy and efficiency of parameter estimation of item response models [Master's thesis, Yonsei University, Seoul]. Yonsei University Library.

Mislevy, R. J. (1984). Estimating latent distributions. Psychometrika, 49(3), 359-381.

Mislevy, R. J., & Bock, R. D. (1985). Implementation of the EM algorithm in the estimation of item parameters: The BILOG computer program. In D. J. Weiss (Ed.). Proceedings of the 1982 item response theory and computerized adaptive testing conference (pp. 189-202). University of Minnesota, Department of Psychology, Computerized Adaptive Testing Conference.

Woods, C. M., & Lin, N. (2009). Item response theory with estimation of the latent density using Davidian curves. Applied Psychological Measurement, 33(2), 102-117.

Woods, C. M., & Thissen, D. (2006). Item response theory with estimation of the latent population distribution using spline-based densities. Psychometrika, 71(2), 281-301.

Examples

# A preparation of mixed-format item response data

Alldata <- DataGeneration(N=1000,
                          nitem_D = 5,
                          nitem_P = 3)

DataD <- Alldata$data_D   # item response data for the dichotomous items
DataP <- Alldata$data_P   # item response data for the polytomous items

# Analysis

M1 <- IRTest_Mix(DataD, DataP)

Item and ability parameters estimation for polytomous items

Description

This function estimates IRT item and ability parameters when all items are scored polytomously. Based on Bock & Aitkin's (1981) marginal maximum likelihood and EM algorithm (EM-MML), this function provides several latent distribution estimation algorithms which could free the normality assumption on the latent variable. If the normality assumption is violated, application of these latent distribution estimation methods could reflect non-normal characteristics of the unknown true latent distribution, and, thus, could provide more accurate parameter estimates (Li, 2021; Woods & Lin, 2009; Woods & Thissen, 2006).

Usage

IRTest_Poly(
  data,
  model = "GPCM",
  range = c(-6, 6),
  q = 121,
  initialitem = NULL,
  ability_method = "EAP",
  latent_dist = "Normal",
  max_iter = 200,
  threshold = 1e-04,
  bandwidth = "SJ-ste",
  h = NULL
)

Arguments

Details

The probability for scoring u=k (i.e., k=0, 1, ..., m; m \ge 2)

1) Partial credit model (PCM)

P(u=0|\theta, b_1, ..., b_{m})=\frac{1}{1+\sum_{c=1}^{m}{\exp{\left[\sum_{v=1}^{c}{a(\theta-b_v)}\right]}}}

P(u=1|\theta, b_1, ..., b_{m})=\frac{\exp{(\theta-b_1)}}{1+\sum_{c=1}^{m}{\exp{\left[\sum_{v=1}^{c}{\theta-b_v}\right]}}}

\vdots

P(u=m|\theta, b_1, ..., b_{m})=\frac{\exp{\left[\sum_{v=1}^{m}{\theta-b_v}\right]}}{1+\sum_{c=1}^{m}{\exp{\left[\sum_{v=1}^{c}{\theta-b_v}\right]}}}

2) Generalized partial credit model (GPCM)

P(u=0|\theta, a, b_1, ..., b_{m})=\frac{1}{1+\sum_{c=1}^{m}{\exp{\left[\sum_{v=1}^{c}{a(\theta-b_v)}\right]}}}

P(u=1|\theta, a, b_1, ..., b_{m})=\frac{\exp{(a(\theta-b_1))}}{1+\sum_{c=1}^{m}{\exp{\left[\sum_{v=1}^{c}{a(\theta-b_v)}\right]}}}

\vdots

P(u=m|\theta, a, b_1, ..., b_{m})=\frac{\exp{\left[\sum_{v=1}^{m}{a(\theta-b_v)}\right]}}{1+\sum_{c=1}^{m}{\exp{\left[\sum_{v=1}^{c}{a(\theta-b_v)}\right]}}}

3) Graded response model (GRM)

P(u=0|\theta, a, b_1, ..., b_{m})=1-\frac{1}{1+\exp{\left[-a(\theta-b_1)\right]}}

P(u=1|\theta, a, b_1, ..., b_{m})=\frac{1}{1+\exp{\left[-a(\theta-b_1)\right]}}-\frac{1}{1+\exp{\left[-a(\theta-b_2)\right]}}

\vdots

P(u=m|\theta, a, b_1, ..., b_{m})=\frac{1}{1+\exp{\left[-a(\theta-b_m)\right]}}-0

Latent distribution estimation methods

1) Empirical histogram method

P(\theta=X_k)=A(X_k)

where k=1, 2, ..., q, X_k is the location of the kth quadrature point, and A(X_k) is a value of probability mass function evaluated at X_k. Empirical histogram method thus has q-1 parameters.

2) Two-component Gaussian mixture distribution

P(\theta=X)=\pi \phi(X; \mu_1, \sigma_1)+(1-\pi) \phi(X; \mu_2, \sigma_2)

where \phi(X; \mu, \sigma) is the value of a Gaussian component with mean \mu and standard deviation \sigma evaluated at X.

3) Davidian curve method

P(\theta=X)=\left\{\sum_{\lambda=0}^{h}{{m}_{\lambda}{X}^{\lambda}}\right\}^{2}\phi(X; 0, 1)

where h corresponds to the argument h and determines the degree of the polynomial.

4) Kernel density estimation method

P(\theta=X)=\frac{1}{Nh}\sum_{j=1}^{N}{K\left(\frac{X-\theta_j}{h}\right)}

where N is the number of examinees, \theta_j is jth examinee's ability parameter, h is the bandwidth which corresponds to the argument bw, and K( \bullet ) is a kernel function. The Gaussian kernel is used in this function.

5) Log-linear smoothing method

P(\theta=X_{q})=\exp{\left(\beta_{0}+\sum_{m=1}^{h}{\beta_{m}X_{q}^{m}}\right)}

where h is the hyper parameter which determines the smoothness of the density, and \theta can take total Q finite values (X_1, \dots ,X_q, \dots, X_Q).

Value

This function returns a list of several objects:

Author(s)

Seewoo Li cu@yonsei.ac.kr

References

Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46(4), 443-459.

Casabianca, J. M., & Lewis, C. (2015). IRT item parameter recovery with marginal maximum likelihood estimation using loglinear smoothing models. Journal of Educational and Behavioral Statistics, 40(6), 547-578.

Li, S. (2021). Using a two-component normal mixture distribution as a latent distribution in estimating parameters of item response models. Journal of Educational Evaluation, 34(4), 759-789.

Li, S. (2022). The effect of estimating latent distribution using kernel density estimation method on the accuracy and efficiency of parameter estimation of item response models [Master's thesis, Yonsei University, Seoul]. Yonsei University Library.

Mislevy, R. J. (1984). Estimating latent distributions. Psychometrika, 49(3), 359-381.

Mislevy, R. J., & Bock, R. D. (1985). Implementation of the EM algorithm in the estimation of item parameters: The BILOG computer program. In D. J. Weiss (Ed.). Proceedings of the 1982 item response theory and computerized adaptive testing conference (pp. 189-202). University of Minnesota, Department of Psychology, Computerized Adaptive Testing Conference.

Woods, C. M., & Lin, N. (2009). Item response theory with estimation of the latent density using Davidian curves. Applied Psychological Measurement, 33(2), 102-117.

Woods, C. M., & Thissen, D. (2006). Item response theory with estimation of the latent population distribution using spline-based densities. Psychometrika, 71(2), 281-301.

Examples

# Preparation of dichotomous item response data

data <- DataGeneration(N=1000,
                       nitem_P = 8)$data_P

# Analysis

M1 <- IRTest_Poly(data)

Ability parameter estimation with fixed item parameters

Description

Ability parameter estimation when item responses and item parameters are given. This function can be useful in ability parameter estimation is adaptive testing.

Usage

adaptive_test(
  response,
  item,
  model = "dich",
  ability_method = "EAP",
  quad = NULL,
  prior = NULL
)

Arguments

Value

Author(s)

Seewoo Li cu@yonsei.ac.kr

Examples

# dichotomous

response <- c(1,1,0)
item <- matrix(
  c(
      1, -0.5,   0,
    1.5,   -1,   0,
    1.2,    0, 0.2
  ), nrow = 3, byrow = TRUE
)
adaptive_test(response, item, model = "dich", ability_method = "WLE")


# polytomous

response <- c(1,2,0)
item <- matrix(
    c(
      1, -0.5, 0.5,
    1.5,   -1,   0,
    1.2,    0, 0.4
    ), nrow = 3, byrow = TRUE
  )
adaptive_test(response, item, model="GPCM", ability_method = "WLE")


# mixed-format test

response <- list(c(0,0,0),c(2,2,1))
item <- list(
  matrix(
    c(
        1, -0.5, 0,
      1.5,   -1, 0,
      1.2,    0, 0
    ), nrow = 3, byrow = TRUE
  ),
  matrix(
    c(
        1, -0.5, 0.5,
      1.5,   -1,   0,
      1.2,    0, 0.4
    ), nrow = 3, byrow = TRUE
  )
)
adaptive_test(response, item, model = "GPCM", ability_method = "WLE")


# continuous response

response <- c(0.88, 0.68, 0.21)
item <- matrix(
  c(
    1, -0.5, 10,
    1.5,   -1,  8,
    1.2,    0, 11
  ), nrow = 3, byrow = TRUE
)
adaptive_test(response, item, model = "cont", ability_method = "WLE")

Model comparison

Description

Model comparison

Usage

## S3 method for class 'IRTest'
anova(...)

Arguments

Value

Model-fit indices and results of likelihood ratio test (LRT).

Author(s)

Seewoo Li cu@yonsei.ac.kr

Selecting the best model

Description

Selecting the best model

Usage

best_model(..., criterion = "HQ")

Arguments

Value

The best model and model-fit indices.

Author(s)

Seewoo Li cu@yonsei.ac.kr

A recommendation for category collapsing of items based on item parameters

Description

In a polytomous item, one or more score categories may not have the highest probability among the categories in an acceptable \theta range. In this case, the category may possibly be regarded as redundant in a psychometric point of view and can be collapsed into another score category. This function returns a recommendation for a recategorization scheme based on item parameters.

Usage

cat_clps(item.matrix, range = c(-4, 4), increment = 0.005)

Arguments

Value

A list of recommended recategorization for each item.

Author(s)

Seewoo Li cu@yonsei.ac.kr

Extract Model Coefficients

Description

A generic function which extracts model coefficients from objects returned by modeling functions.

Usage

## S3 method for class 'IRTest'
coef(object, complete = TRUE, ...)

Arguments

Value

Coefficients extracted from the model (object).

Extract Standard Errors of Model Coefficients

Description

Standard errors of model coefficients calculated by using Fisher information functions.

Usage

coef_se(object, complete = TRUE)

Arguments

Value

Standard errors extracted from the model (object).

Re-parameterized two-component normal mixture distribution

Description

Probability density for the re-parameterized two-component normal mixture distribution.

Usage

dist2(x, prob = 0.5, d = 0, sd_ratio = 1, overallmean = 0, overallsd = 1)

Arguments

Details

The overall mean and overall standard deviation obtained from original parameters;

1) Overall mean (\bar{\mu})

\bar{\mu}=\pi\mu_1 + (1-\pi)\mu_2

2) Overall standard deviation (\bar{\sigma})

\bar{\sigma}=\sqrt{\pi\sigma_{1}^{2}+(1-\pi)\sigma_{2}^{2}+\pi(1-\pi)(\mu_2-\mu_1)^2}

Value

The evaluated probability density value(s).

Author(s)

Seewoo Li cu@yonsei.ac.kr

References

Li, S. (2021). Using a two-component normal mixture distribution as a latent distribution in estimating parameters of item response models. Journal of Educational Evaluation, 34(4), 759-789.

Examples

# Evaluated density
dnst <- dist2(seq(-6,6,.1), prob = 0.3, d = 1, sd_ratio=0.5)

# Plot of the density
plot(seq(-6,6,.1), dnst)

Estimated factor scores

Description

Factor scores of examinees.

Usage

factor_score(x, ability_method = "EAP", quad = NULL, prior = NULL)

Arguments

Value

Author(s)

Seewoo Li cu@yonsei.ac.kr

Examples

# A preparation of dichotomous item response data

data <- DataGeneration(N=500, nitem_D = 10)$data_D

# Analysis

M1 <- IRTest_Dich(data)

# Item fit statistics

factor_score(M1, ability_method = "MLE")

Item information function

Description

Item information function

Usage

inform_f_item(x, test, item = 1, type = "d")

Arguments

Value

A vector of the evaluated item information values.

Author(s)

Seewoo Li cu@yonsei.ac.kr

Test information function

Description

Test information function

Usage

inform_f_test(x, test)

Arguments

Value

A vector of test information values of the same length as x.

Author(s)

Seewoo Li cu@yonsei.ac.kr

Item fit diagnostics

Description

This function analyzes and reports item-fit test results.

Usage

item_fit(x, bins = 10, bin.center = "mean")

Arguments

Details

Bock's \chi^{2} (1960) or Yen's Q_{1} (1981) is currently available.

Value

This function returns a matrix of item-fit test results.

Author(s)

Seewoo Li cu@yonsei.ac.kr

References

Bock, R.D. (1960), Methods and applications of optimal scaling. Chapel Hill, NC: L.L. Thurstone Psychometric Laboratory.

Yen, W. M. (1981). Using simulation results to choose a latent trait model. Applied Psychological Measurement, 5(2), 245–262.

Examples

# A preparation of dichotomous item response data

data <- DataGeneration(N=500,
                       nitem_D = 10)$data_D

# Analysis

M1 <- IRTest_Dich(data)

# Item fit statistics

item_fit(M1)

Latent density function

Description

Density function of the estimated latent distribution with mean and standard deviation equal to 0 and 1, respectively.

Usage

latent_distribution(x, model.fit)

Arguments

Value

The evaluated values of the PDF, a length of which equals to that of x.

Examples

# Data generation and model fitting
data <- DataGeneration(N=1000,
                       nitem_D = 15,
                       latent_dist = "2NM",
                       d = 1.664,
                       sd_ratio = 2,
                       prob = 0.3)$data_D

M1 <- IRTest_Dich(data = data, latent_dist = "KDE")

# Plotting the latent distribution
ggplot2::ggplot()+
  ggplot2::stat_function(fun=latent_distribution, args=list(M1))+
  ggplot2::lims(x=c(-6,6), y=c(0,0.5))

Extract Log-Likelihood

Description

Extract Log-Likelihood

Usage

## S3 method for class 'IRTest'
logLik(object, ...)

Arguments

Value

Extracted log-likelihood.

Recovering original parameters of two-component Gaussian mixture distribution from re-parameterized values

Description

Recovering original parameters of two-component Gaussian mixture distribution from re-parameterized values

Usage

original_par_2GM(
  prob = 0.5,
  d = 0,
  sd_ratio = 1,
  overallmean = 0,
  overallsd = 1
)

Arguments

Details

Original two-component Gaussian mixture distribution

f(x)=\pi\times \phi(x | \mu_1, \sigma_1)+(1-\pi)\times \phi(x | \mu_2, \sigma_2)

, where \phi is a Gaussian component.

Re-parameterized two-component Gaussian mixture distribution

f(x)=2GM(x|\pi, \delta, \zeta, \bar{\mu}, \bar{\sigma})

, where \bar{\mu} is overall mean and \bar{\sigma} is overall standard deviation of the distribution.

The original parameters retrieved from re-parameterized values

1) Mean of the first Gaussian component (m1).

\mu_1=-(1-\pi)\delta\bar{\sigma}+\bar{\mu}

2) Mean of the second Gaussian component (m2).

\mu_2=\pi\delta\bar{\sigma}+\bar{\mu}

3) Standard deviation of the first Gaussian component (s1).

\sigma_1^2=\bar{\sigma}^2\left(\frac{1-\pi(1-\pi)\delta^2}{\pi+(1-\pi)\zeta^2}\right)

4) Standard deviation of the second Gaussian component (s2).

\sigma_2^2=\bar{\sigma}^2\left(\frac{1-\pi(1-\pi)\delta^2}{\frac{1}{\zeta^2}\pi+(1-\pi)}\right)=\zeta^2\sigma_1^2

Value

This function returns a vector of length 4: c(m1,m2,s1,s2).

Author(s)

Seewoo Li cu@yonsei.ac.kr

References

Li, S. (2021). Using a two-component normal mixture distribution as a latent distribution in estimating parameters of item response models. Journal of Educational Evaluation, 34(4), 759-789.

Plot of the estimated latent distribution

Description

This function draws a plot of the estimated latent distribution (the population distribution of the latent variable).

Usage

## S3 method for class 'IRTest'
plot(x, ...)

Arguments

Value

A plot of estimated latent distribution.

Author(s)

Seewoo Li cu@yonsei.ac.kr

Examples

# Data generation and model fitting

data <- DataGeneration(N=1000,
                       nitem_D = 15,
                       latent_dist = "2NM",
                       d = 1.664,
                       sd_ratio = 2,
                       prob = 0.3)$data_D

M1 <- IRTest_Dich(data = data, latent_dist = "KDE")

# Plotting the latent distribution

plot(x = M1, linewidth = 1, color = 'red') +
  ggplot2::lims(x = c(-6, 6), y = c(0, .5))

Plot of item response functions

Description

This function draws item response functions of an item of the fitted model.

Usage

plot_item(x, item.number = 1, type = NULL)

Arguments

Value

This function returns a plot of item response functions.

Author(s)

Seewoo Li cu@yonsei.ac.kr

Examples

# A preparation of dichotomous item response data

data <- DataGeneration(N=500, nitem_D = 10)$data_D

# Analysis

M1 <- IRTest_Dich(data)

# Plotting item response function

plot_item(M1, item.number = 1)

Printing the result

Description

This function prints the summarized information.

Usage

## S3 method for class 'IRTest'
print(x, ...)

Arguments

Value

Printed texts on the console recommending the usage of summary function and the direct access to the details using "$" sign.

Author(s)

Seewoo Li cu@yonsei.ac.kr

Examples

data <- DataGeneration(N=1000, nitem_P = 8)$data_P

M1 <- IRTest_Poly(data = data, latent_dist = "KDE")

M1

Printing the summary

Description

This function prints the summarized information.

Usage

## S3 method for class 'IRTest_summary'
print(x, ...)

Arguments

Value

Summarized texts on the console.

Author(s)

Seewoo Li cu@yonsei.ac.kr

Examples

data <- DataGeneration(N=1000, nitem_P = 8)$data_P

M1 <- IRTest_Poly(data = data,
                  latent_dist = "2NM")

summary(M1)

Recategorization of data using a new categorization scheme

Description

With a recategorization scheme as an input, this function implements recategorization for the input data.

Usage

recategorize(data, new_cat)

Arguments

Value

Recategorized data

Author(s)

Seewoo Li cu@yonsei.ac.kr

Examples

# Preparation of dichotomous item response data

data <- DataGeneration(N=1000,
                       nitem_P = 8)$data_P

# Analysis

M1 <- IRTest_Poly(data)

# Recommendation of category collapsing

new_cat <- cat_clps(M1$par_est)

# Recategorization of data

recategorize(data, new_cat)

Marginal reliability coefficient of IRT

Description

Marginal reliability coefficient of IRT

Usage

reliability(x)

Arguments

Details

Reliability coefficient on summed-score scale

In accordance with the concept of reliability in classical test theory (CTT), this function calculates the IRT reliability coefficients.

The basic concept and formula of the reliability coefficient can be expressed as follows (Kim & Feldt, 2010):

An observed score of Item i, X_i, is decomposed as the sum of a true score T_i and an error e_i. Then, with the assumption of \sigma_{T_{i}e_{j}}=\sigma_{e_{i}e_{j}}=0, the reliability coefficient of a test is defined as;

\rho_{TX}=\rho_{XX^{'}}=\frac{\sigma_{T}^{2}}{\sigma_{X}^{2}}=\frac{\sigma_{T}^{2}}{\sigma_{T}^{2}+\sigma_{e}^{2}}=1-\frac{\sigma_{e}^{2}}{\sigma_{X}^{2}}

See May and Nicewander (1994) for the specific formula used in this function.

Reliability coefficient on \theta scale

For the coefficient on the \theta scale, this function calculates the parallel-forms reliability (Green et al., 1984; Kim, 2012):

\rho_{\hat{\theta} \hat{\theta}^{'}} =\frac{\sigma_{E\left(\hat{\theta}\mid \theta \right )}^{2}}{\sigma_{E\left(\hat{\theta}\mid \theta \right )}^{2}+E\left( \sigma_{\hat{\theta}|\theta}^{2} \right)} =\frac{1}{1+E\left(I\left(\hat{\theta}\right)^{-1}\right)}

This assumes that \sigma_{E\left(\hat{\theta}\mid \theta \right )}^{2}=\sigma_{\theta}^{2}=1. Although the formula is often employed in several IRT studies and applications, the underlying assumption may not be true.

Value

Estimated marginal reliability coefficients.

Author(s)

Seewoo Li cu@yonsei.ac.kr

References

Green, B.F., Bock, R.D., Humphreys, L.G., Linn, R.L., & Reckase, M.D. (1984). Technical guidelines for assessing computerized adaptive tests. Journal of Educational Measurement, 21(4), 347–360.

Kim, S. (2012). A note on the reliability coefficients for item response model-based ability estimates. Psychometrika, 77(1), 153-162.

Kim, S., Feldt, L.S. (2010). The estimation of the IRT reliability coefficient and its lower and upper bounds, with comparisons to CTT reliability statistics. Asia Pacific Education Review, 11, 179–188.

May, K., Nicewander, A.W. (1994). Reliability and information functions for percentile ranks. Journal of Educational Measurement, 31(4), 313-325.

Examples

data <- DataGeneration(N=500, nitem_D = 10)$data_D

# Analysis

M1 <- IRTest_Dich(data)


# Reliability coefficients
reliability(M1)

Summary of the results

Description

This function summarizes the output (e.g., convergence of the estimation algorithm, number of parameters, model-fit, and estimated latent distribution).

Usage

## S3 method for class 'IRTest'
summary(object, ...)

Arguments

Value

Summarized information.

Examples

data <- DataGeneration(N=1000, nitem_P = 8)$data_P

M1 <- IRTest_Poly(data = data, latent_dist = "KDE")

summary(M1)