Statistical & Clinical Analysis
Rasch Model (1PL IRT) Formula: Correct-Response Probability & Fisher Item Information
Evaluate Computerized Adaptive Testing (CAT — NCLEX, GMAT, GRE) item response probability P = 1 / (1 + EXP(-(theta - b))) and Fisher Item Information I(theta) = P * (1 - P) using Georg Rasch’s 1960 1-Parameter Logistic model.
Specific Objectivity on a common logit ruler: why CAT engines select items where b = theta (I_max = 0.25)!
Danish mathematician Georg Rasch (1960, Probabilistic Models for Some Intelligence and Attainment Tests) placed examinee latent ability theta (in logits) and exam item difficulty b (in logits) on the EXACT same additive interval scale: log-odds LN(P / (1 - P)) = theta - b, or P(X = 1 | theta, b) = 1 / (1 + EXP(-(theta - b)))!
Furthermore, Fisher Item Information for a dichotomous Rasch item equals I(theta) = P * (1 - P), which reaches its global maximum of 0.50 * 0.50 = 0.2500 precisely when item difficulty matches examinee ability (b = theta -> P = 0.5000)—the mathematical principle driving every Computerized Adaptive Test (CAT) item selector! When ability theta = +1.25 logits faces an item of difficulty b = +0.45 logits (gap +0.80 logits), P = 0.6900 and Fisher Information I = 0.2139!
| Examinee Ability theta vs. Item Difficulty b (Logits) | Correct-Answer Probability P(theta) | Rasch Fisher Information P*(1 - P) |
|---|---|---|
| theta = +1.25, b = +0.45 (+0.80 logit advantage) | 0.6900 (69.00%) | 0.2139 |
| Optimal CAT Match: theta = +0.45, b = +0.45 (0 gap) | 0.5000 (50.00%) | 0.2500 (Maximum Information!) |
| Out-of-Scale Extreme |theta| > 10 logits | Numerical Guard | "Logit Parameter Outside [-10, +10] Scale" |
Rasch 1PL Correct-Response Probability P = 1 / (1 + EXP(-(theta - b)))
Computes the logistic probability of a correct response from person ability theta (B2) and item difficulty b (B3) on the [-10, +10] logit scale.
=IF(OR(ABS(B2) > 10, ABS(B3) > 10), "Logit Parameter Outside [-10, +10] Scale", ROUND(1 / (1 + EXP(-(B2 - B3))), 4))Rasch Dichotomous Fisher Item Information I(theta) = P * (1 - P)
Evaluates Fisher information contributed toward shrinking SE(theta) = 1 / SQRT(SUM(I_j)).
=IF(OR(ABS(B2) > 10, ABS(B3) > 10), "Logit Parameter Outside [-10, +10] Scale", ROUND((1 / (1 + EXP(-(B2 - B3)))) * (1 - 1 / (1 + EXP(-(B2 - B3)))), 4))Try the example
Use the original values or change them to test your understanding.