A Monte Carlo simulation study of sample size requirements for the Graded Response Model

The graded response model (GRM) is commonly used in psychometrics to analyze ordinal response data. Despite its growing application in scale development and validation, sample size recommendations-such as those provided by the COSMIN guidelines (e.g., n ≥ 1000)-are often based on expert consensus rather than empirical validation. Furthermore, the extent to which the number of items (J) and the number of response categories (K) contribute to parameter estimation accuracy remains insufficiently explored.

We conducted a Monte Carlo simulation to examine how three design conditions-sample size (n = 500-1500), number of items (J = 5-50), and a number of response categories (K = 4-7)-influence the estimation accuracy of the latent trait parameter ([Formula: see text]) and the item discrimination parameter (a) under the GRM. For each condition, we generated a large population dataset based on predefined distributions for [Formula: see text], a, and b, and then randomly drew samples (n) for estimation. The GRM was fitted using the EM algorithm. Estimation accuracy was evaluated using root mean squared error (RMSE), FPC-corrected RMSE, and Pearson's correlation coefficient between true and estimated [Formula: see text] values.

The RMSE of the discrimination parameter a decreased with increasing sample size (n) and number of items (J), while the effect of K was negligible. In contrast, the RMSE of [Formula: see text] was primarily influenced by J, with only minor effects from n and K. Notably, the Pearson correlation between true and estimated [Formula: see text] values consistently exceeded r = .98 across all conditions, suggesting high ordinal fidelity even with small samples. Increasing J beyond approximately 30 yielded diminishing returns in RMSE reduction.

Our findings suggest that sample size recommendations for GRM should be flexibly tailored to the measurement goal. For accurate estimation of [Formula: see text], a sufficiently large number of items (e.g., J ≥ 30) can compensate for smaller sample sizes (n ≈ 500), whereas precise estimation of a requires larger samples (n ≥ 1000). The impact of increasing K was limited, indicating that additional response categories may not always enhance parameter recovery. These results provide empirically grounded guidance to support efficient and purpose-specific measurement designs in GRM applications.