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).
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
)This function returns a list of several objects:
The item parameter estimates.
The asymptotic standard errors for item parameter estimates.
The estimated frequencies of examinees at quadrature points.
The number of EM-MML iterations elapsed for the convergence.
The location of quadrature points.
The final value of the monitored maximum item parameter change.
The estimated discrete latent distribution. It is discrete (i.e., probability mass function) by the quadrature scheme.
The posterior probabilities of examinees at quadrature points.
The estimated ability parameter values. If ability_method = "MLE", the function returns \(\pm\)Inf for all or none correct answers.
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".
The deviance (i.e., -2logL).
The estimated density parameters.
A replication of input arguments and other information.
A matrix or data frame of item responses where responses are coded as 0 or 1. Rows and columns indicate examinees and items, respectively.
Range of the latent variable to be considered in the quadrature scheme.
The default is from -6 to 6: c(-6, 6).
A numeric value that represents the number of quadrature points. The default value is 121.
A matrix of initial item parameter values for starting the estimation algorithm. The default value is NULL.
The ability parameter estimation method.
The available options are Expected a posteriori (EAP), Maximum Likelihood Estimates (MLE), and weighted likelihood estimates (WLE).
The default is EAP.
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.
A numeric value that determines the maximum number of iterations in the EM-MML. The default value is 200.
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.
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.
A natural number less than or equal to 10 if latent_dist = "DC" or "LLS".
This argument determines the complexity of the distribution.
Seewoo Li cu@yonsei.ac.kr
$$P(u=x | a, b, \nu) = \frac{1}{B(\mu\nu, \,\nu(1-\mu))} u^{\mu\nu-1} (1-u)^{\nu(1-\mu)-1}$$
where \(\mu = \frac{e^{a(\theta -b)}}{1+e^{a(\theta -b)}}\).
1) Empirical histogram method $$P(\theta=X_k)=A(X_k)$$ where \(k=1, 2, ..., q\), \(X_k\) is the location of the \(k\)th 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 \(j\)th 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 (\(X_1, \dots ,X_q, \dots, X_Q\)).
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.
# \donttest{
# 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.
# }
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