Graded Response Model (GRM)

The 2PL model is designed for binary (right/wrong or yes/no) items and has one threshold per item. The GRM extends this to items with m ordered categories by assigning m−1 thresholds — one for each boundary between adjacent categories — while retaining a single discrimination parameter per item. Each boundary has its own logistic curve; the 2PL is the special case where m = 2.

Both handle ordered polytomous items, but they parameterize the response process differently. The GRM models cumulative probabilities — the chance of responding at or above each category — using a two-parameter logistic form with a common discrimination per item. The PCM (and its generalization, the GPCM) models step probabilities — the chance of choosing category k over k−1 — and is more natural for sequentially scored tasks such as essay rubrics or performance assessments.

Out-of-order thresholds, sometimes called threshold reversals or disordering, mean that two adjacent categories are not being reliably distinguished by respondents. The standard remedy is to collapse the disordered categories into a single category and re-run the calibration. Persistent disordering may indicate that the response format has too many categories for the construct being measured.

A minimum of around 200 respondents is commonly cited as a practical floor, but the required sample grows with the number of items, the number of response categories, and low item discriminations. Items with five or more categories and moderate discrimination may need 300 to 500 respondents for stable parameter recovery. Simulation studies or pilot calibrations can clarify requirements for a specific instrument.