R: Item and ability parameters estimation for polytomous items

IRTest_Poly {IRTest}

R Documentation

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

`data`	A matrix or data frame of item responses coded as `0, 1, ..., m` for the `m+1` category item. Rows and columns indicate examinees and items, respectively.
`model`	A character value for an IRT model to be applied. Currently, `PCM`, `GPCM`, and `GRM` are available. The default is `"GPCM"`.
`range`	Range of the latent variable to be considered in the quadrature scheme. The default is from `-6` to `6`: `c(-6, 6)`.
`q`	A numeric value that represents the number of quadrature points. The default value is 121.
`initialitem`	A matrix of initial item parameter values for starting the estimation algorithm. The default value is `NULL`.
`ability_method`	The ability parameter estimation method. The available options are Expected a posteriori (`EAP`) and Maximum Likelihood Estimates (`MLE`). The default is `EAP`.
`latent_dist`	A character string that determines latent distribution estimation method. Insert `"Normal"`, `"normal"`, or `"N"` for the normality assumption on the latent distribution, `"EHM"` for empirical histogram method (Mislevy, 1984; Mislevy & Bock, 1985), `"2NM"` or `"Mixture"` for using two-component Gaussian mixture distribution (Li, 2021; Mislevy, 1984), `"DC"` or `"Davidian"` for Davidian-curve method (Woods & Lin, 2009), `"KDE"` for kernel density estimation method (Li, 2022), and `"LLS"` for log-linear smoothing method (Casabianca & Lewis, 2015). The default value is set to `"Normal"` to follow the convention.
`max_iter`	A numeric value that determines the maximum number of iterations in the EM-MML. The default value is 200.
`threshold`	A numeric value that determines the threshold of EM-MML convergence. A maximum item parameter change is monitored and compared with the threshold. The default value is 0.0001.
`bandwidth`	A character value that can be used if `latent_dist = "KDE"`. This argument determines the bandwidth estimation method for `"KDE"`. The default value is `"SJ-ste"`. See `density` for available options.
`h`	A natural number less than or equal to 10 if `latent_dist = "DC" or "LLS"`. This argument determines the complexity of the distribution.

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:

`par_est`	The item parameter estimates.
`se`	The asymptotic standard errors for item parameter estimates.
`fk`	The estimated frequencies of examinees at quadrature points.
`iter`	The number of EM-MML iterations elapsed for the convergence.
`quad`	The location of quadrature points.
`diff`	The final value of the monitored maximum item parameter change.
`Ak`	The estimated discrete latent distribution. It is discrete (i.e., probability mass function) by the quadrature scheme.
`Pk`	The posterior probabilities of examinees at quadrature points.
`theta`	The estimated ability parameter values. If `ability_method = "MLE"`. If an examinee receives a maximum or minimum score for all items, the function returns `\pmInf`.
`theta_se`	Standard error of ability estimates. The asymptotic standard errors for `ability_method = "MLE"` (the function returns `NA` for all or none correct answers). The standard deviations of the posterior distributions for `ability_method = "MLE"`.
`logL`	The deviance (i.e., -2logL).
`density_par`	The estimated density parameters.
`Options`	A replication of input arguments and other information.

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)

[Package IRTest version 2.0.0 Index]