Analytical First-Order Bias Correction for Joint Maximum Likelihood Estimators in the Rasch Model

This function performs an analytical bias correction for the Joint Maximum Likelihood (JML) estimators of person (theta) and item (beta) parameters in the Rasch model.

Usage

biasCorrection(jmle_obj = NULL, theta, beta, X, I, ...)

Arguments

jmle_obj: optional An object of class "jmleIRT" as returned by
theta: Numeric vector of estimated person parameters $\hat{\theta}$.
beta: Numeric vector of estimated item parameters $\hat{\beta}$.
X: Numeric matrix of responses (persons by items), coded as 0/1, with NA allowed for missing responses.
I: Integer scalar representing the number of items (used for bias scaling).
...: further arguments, but currently unused.

Value

A named list containing bias-corrected parameter vectors:

theta: Numeric vector of bias-corrected person parameters.
beta: Numeric vector of bias-corrected item parameters.

Details

The JML estimators are well-known to suffer from incidental parameter bias when the number of items ($I$) is finite, resulting in biased parameter estimates. This bias is generally of order $1/I$.

The bias correction implemented here follows the first-order analytic correction approach, which approximates the bias term $B$ for each parameter as: $$ \hat{\theta}^{corr} = \hat{\theta} - \frac{1}{I} \hat{B}_{\theta}, \quad \hat{\beta}^{corr} = \hat{\beta} - \frac{1}{I} \hat{B}_{\beta} $$ where $\hat{B}$ is estimated based on the observed scores $u_i$ and the expected Fisher information $v_i$, calculated as $$ \hat{B}_i \approx \frac{E[v_i u_i]}{E[v_i^2]}. $$ In this implementation, scores and information are calculated based on the logistic form of the Rasch model, where $p_{pi} = \frac{e^{\theta_p - \beta_i}}{1 + e^{\theta_p - \beta_i}}$ is the probability that person $p$ correctly answers item $i$.

Missing responses (NA) in the data matrix $X$ are handled by omitting those entries.

Let $P$ denote the number of persons and $I$ the number of items. Let $X$ be a $P \times I$ matrix with entries $X_{pi} \in \{0,1\}$ (with NA allowed for missing responses).

The bias correction is inspired by the incidental parameters literature (Neyman & Scott, 1948; Lancaster, 2000; Arellano & Hahn, 2006), which shows that Maximum Likelihood estimators in models with many nuisance parameters (like person parameters in Rasch) are biased when the number of observations per parameter is limited.

This function applies a computationally efficient closed-form bias correction using the first and second derivatives of the Rasch model log-likelihood function, evaluated at the JML estimates.

The estimator reduces bias by estimating expected score and information terms: $$ u_{pi} = X_{pi} - p_{pi}, \quad v_{pi} = p_{pi} (1 - p_{pi}) $$ for person $p$ and item $i$, and then aggregating these across items or persons.

The bias terms for person $p$ and item $i$ are estimated as: $$ \hat{B}_{\theta,p} = \frac{\sum_i v_{pi} u_{pi}}{\sum_i v_{pi}^2}, \quad \hat{B}_{\beta,i} = \frac{\sum_p v_{pi} (p_{pi} - X_{pi})}{\sum_p v_{pi}^2} $$

This method is applicable when the number of items $I$ is fixed and moderate, and the number of persons $P$ is large.

References

- Neyman, J., & Scott, E. L. (1948). Consistent Estimates Based on Partially Consistent Observations. Econometrica, 16(1), 1-32. - Lancaster, T. (2000). The incidental parameter problem since 1948. Journal of Econometrics, 95(2), 391-413. - Arellano, M., & Hahn, J. (2006). Understanding Bias in Nonlinear Panel Models. Review of Economic Studies, 73(2), 557-578.

Examples

if (FALSE) { # \dontrun{
# Example data matrix with 2 persons and 4 items:
X <- matrix(c(1, 0, 1, NA, 1, 0, 1, 1), nrow = 2, byrow = TRUE)
theta <- c(0.5, -0.5)
beta <- c(-0.2, 0.1, 0.3, -0.1)
I <- ncol(X)
corrected <- biasCorrection(theta = theta, beta = beta, X = X, I = I)
print(corrected$theta)
print(corrected$beta)
} # }