| Estimator | Weight f in eq. 6.2 | Bias with items known | Finite at a perfect or zero score | Information used |
|---|---|---|---|---|
| ML | a positive constant | \(-J/(2\mathcal{J}^2)\), order \(I^{-1}\) | no | within-person only |
| WLE | w with d log w / d theta = J / (2 J-info) | \(o(I^{-1})\) under Warm’s conditions; same first-order asymptotic variance as ML | yes | within-person only |
| MAP (Bayesian modal) | a prior density for theta | ML bias \(+\,(\partial\ln f/\partial\theta)/\mathcal{J}\) | yes | within-person and population |
| EAP (posterior mean) | not a member: the posterior mean, not a mode | shrunk toward \(\mu_\theta\); minimizes squared-error loss | yes | within-person and population |
6 Estimating Person Parameters
Reviewer 2 asked for the mathematical expression of \(\hat\theta_p\) and pointed out that the manuscript’s claim — that ability can be estimated by maximum likelihood — holds only when the item parameters are known, whereas practice substitutes estimates. Both points are correct, and this chapter answers them.
It also does something the usual treatment does not. Maximum likelihood, Bayesian modal estimation, and Warm’s weighted likelihood estimator are normally presented as three competing methods. They are one estimating equation with three choices of a single function, and seeing that makes the differences between them exact rather than impressionistic. The unification is Warm’s own, in the paper that introduced the third.
6.1 One equation, three estimators
Fix the item parameters for the moment. Write \(P_i(\theta)\) for the probability of a correct response to item \(i\), \(Q_i = 1 - P_i\), and \(P_i' = \partial P_i/\partial\theta\). The person’s likelihood is \(L(\mathbf{u}_p \mid \theta) = \prod_i P_i^{u_{pi}}Q_i^{1-u_{pi}}\), and maximizing it means solving
\[ \frac{\partial \ln L}{\partial\theta} = \sum_{i=1}^{I}\frac{(u_{pi} - P_i)P_i'}{P_i Q_i} = 0 . \tag{6.1}\]
Under the Rasch model \(P_i' = P_iQ_i\), so Equation 6.1 collapses to \(r_p = \sum_i P_i(\hat\theta_p)\): choose the ability at which the expected number correct equals the observed number correct. That is the explicit expression Reviewer 2 asked for, and it is worth stating in that form because it makes the estimator’s meaning plain.
Now consider maximizing not the likelihood but \(f(\theta)L(\mathbf{u}_p \mid \theta)\) for some positive function \(f\). The estimating equation becomes
\[ \sum_{i=1}^{I}\frac{(u_{pi} - P_i)P_i'}{P_i Q_i} \;+\; \frac{\partial \ln f(\theta)}{\partial\theta} = 0 . \tag{6.2}\]
Equation 6.2 is the whole family, and three choices of \(f\) name three estimators.
| \(f(\theta)\) | Estimator | What it is |
|---|---|---|
| a positive constant | ML | Equation 6.2 reduces to Equation 6.1 |
| a prior density for \(\theta\) | MAP (Bayesian modal, BME) | \(f \cdot L\) is proportional to the posterior |
| \(w(\theta)\) with \(\partial\ln w/\partial\theta = J/(2\mathcal{J})\) | WLE | removes the leading bias term |
So the switch from likelihood to posterior that Reviewer 1 found abrupt is not a switch of optimization template. It is a choice of \(f\) in one equation. Whether \(f\) represents a population prior or a test-specific bias-reducing weight is an inferential distinction even when the resulting objective has the same algebraic form.
6.2 Where the bias comes from, and how WLE removes it
Lord (1983) gives the asymptotic bias of the ML ability estimate, of order \(I^{-1}\):
\[ \operatorname{BIAS}(\hat\theta^{\mathrm{ML}}) = \frac{-J}{2\mathcal{J}^2}, \qquad \mathcal{J} = \sum_i \frac{P_i'^2}{P_iQ_i}, \qquad J = \sum_i \frac{P_i'P_i''}{P_iQ_i}. \tag{6.3}\]
For a prior \(f\), the bias of the corresponding estimator picks up one extra term,
\[ \operatorname{BIAS}(\theta^*) = \operatorname{BIAS}(\hat\theta^{\mathrm{ML}}) + \frac{\partial \ln f(\theta)/\partial\theta}{\mathcal{J}}, \tag{6.4}\]
which for a standard normal prior is \(-\theta/\mathcal{J}\) — shrinkage toward zero, visible as bias. Warm’s step is to read Equation 6.4 as an equation to be solved rather than a quantity to be reported: set the right-hand side to zero and solve for \(f\). Substituting Equation 6.3 gives
\[ \frac{\partial \ln w(\theta)}{\partial\theta} = \frac{J}{2\mathcal{J}}, \tag{6.5}\]
and the resulting estimating equation is
\[ \sum_{i=1}^{I}\frac{(u_{pi} - P_i)P_i'}{P_i Q_i} + \frac{J}{2\mathcal{J}} = 0 . \tag{6.6}\]
Theorem 6.1 Warm’s bias correction. Restated from Warm (1989, eqs. 6–10, Theorem, and Appendix)
Assume the item parameters are known; \(\theta\) is restricted to a bounded interval; the item response probabilities are continuous and bounded away from zero and one; five derivatives exist and are bounded; and test length grows by replication in Warm’s stated asymptotic scheme. Then the estimator solving Equation 6.6 has \(\operatorname{BIAS}(\hat\theta^{\mathrm{WLE}})=o(I^{-1})\), is asymptotically normal, and has the same first-order asymptotic variance \(\mathcal{J}^{-1}+o(I^{-1})\) as ML.
Warm proves the theorem in his appendix under these restrictive conditions and reports Hinkley’s suggestion that the method can extend to consistent estimating equations with a computable bias. That suggestion is not a replacement for model-specific regularity checks. The theorem concerns interior replicated-item asymptotics, not finite tests at their all-correct or all-incorrect boundaries.
For the 1PL and 2PL, WLE is algebraically Jeffreys-MAP although Warm’s construction and interpretation are not Bayesian. For these models Warm shows \(w(\theta)=\mathcal{J}(\theta)^{1/2}\), the Jeffreys weight. Maximizing \(w(\theta)L(\theta)\) therefore has exactly the same mode equation as using that test-specific Jeffreys weight as a prior. The algebra cannot support a claim that WLE is “not MAP.” Warm’s point is inferential: no population distribution for \(\theta\) is assumed, and \(w\) is a function of the administered items. He gives the direct comparison: two populations taking one test use the same WLE weight, while one population taking two tests uses different weights—the reverse of how a population prior behaves.
The variance equality is first-order asymptotic rather than a finite-sample identity. Under Theorem 6.1’s conditions WLE and ML have the same leading variance term. Their finite-test variances can differ, especially near the scale boundary. WLE deliberately uses only the within-person likelihood and a test-derived adjustment, not population information about the person. That makes it the natural contrast to EAP; it does not make every finite-sample risk component equal.
6.3 The 2PL specialization
For the 2PL,
\[ P_i(\theta)=\operatorname{logit}^{-1}\{\lambda_i(\theta-\beta_i)\}, \qquad P_i'(\theta)=\lambda_iP_iQ_i, \]
so the ML score and information are
\[ S_{2\mathrm{PL}}(\theta)=\sum_{i=1}^I\lambda_i(u_{pi}-P_i), \qquad \mathcal{J}_{2\mathrm{PL}}(\theta)=\sum_{i=1}^I\lambda_i^2P_iQ_i. \tag{6.7}\]
The Warm numerator specializes to
\[ J_{2\mathrm{PL}}(\theta) =\sum_{i=1}^I\lambda_i^3P_iQ_i(1-2P_i), \]
and the 2PL WLE solves
\[ \sum_{i=1}^I\lambda_i(u_{pi}-P_i) +\frac{J_{2\mathrm{PL}}(\theta)}{2\mathcal{J}_{2\mathrm{PL}}(\theta)}=0. \tag{6.8}\]
Thus discrimination weights the score, enters information quadratically, and enters the Warm bias numerator cubically. When all \(\lambda_i=1\), these expressions reduce to the Rasch equations above. Warm’s known-item and asymptotic conditions apply to both.
6.4 Where there is no estimate at all
Under the Rasch model, Equation 6.1 has no finite solution when \(r_p=0\) or \(r_p=I\). The same boundary result holds for the 2PL when every \(\lambda_i>0\): an all-correct likelihood increases with \(\theta\), and an all-incorrect likelihood decreases. The ML estimate is \(+\infty\) or \(-\infty\).
On a twenty-item screening test in a class of sixty this is not an edge case. It is the students the screening is for.
The estimators respond differently, and the difference is diagnostic of what each is doing. MAP is finite at the boundary because the prior supplies curvature the likelihood lacks. For the ordinary logistic Rasch and 2PL models, the \(\mathcal{J}(\theta)^{1/2}\) weight yields finite WLE roots for extreme response patterns. ML is not finite, and practices such as bounding the estimate, dropping extreme scores, or assigning a conventional value are analytic decisions that should be reported. No software-default claim is needed for the mathematical comparison.
6.5 Plug-in scoring and calibration uncertainty
Everything so far fixed the item parameters. A common two-stage workflow calibrates items and then scores persons:
\[ \hat\theta_p = \hat\theta_p(\hat{\boldsymbol\psi}), \]
substituting the calibration estimates into Equation 6.1 or Equation 6.6 as though they were the truth. This is the plug-in estimator. The inverse root information \(\mathcal{J}(\hat\theta_p)^{-1/2}\) is a local/asymptotic standard error conditional on the plugged-in item values. Whether it understates an unconditional standard error, and by how much, depends on how calibration and scoring samples are related.
Proposition 6.1 Independent-calibration delta method. Derived here; no originality claim
Let \(\boldsymbol\psi\) collect all calibrated item parameters, let \(S_p(\theta,\boldsymbol\psi)=0\) define a regular ability estimator, and suppose \(\hat{\boldsymbol\psi}\) comes from an independent calibration sample with covariance \(V_\psi\). Write
\[ \mathbf{g}_\psi =-\left(\frac{\partial S_p}{\partial\theta}\right)^{-1} \frac{\partial S_p}{\partial\boldsymbol\psi}. \]
Then the first-order variance is
\[ \operatorname{Var}\{\hat\theta_p(\hat{\boldsymbol\psi})\} \approx \mathcal{J}(\theta_p)^{-1} +\mathbf{g}_\psi^\top V_\psi\mathbf{g}_\psi. \tag{6.9}\]
Proof. The implicit-function theorem gives the displayed sensitivity \(\mathbf{g}_\psi\). A first-order Taylor expansion around the true item parameters, followed by independence of the calibration estimate and scoring response, adds the quadratic delta-method term to the known-item conditional variance. ∎
For the Rasch ML score, the sensitivity to difficulty \(\beta_j\) is \(g_{\beta_j}=P_jQ_j/\mathcal{J}\). The 2PL has sensitivities to both difficulties and discriminations, so \(\boldsymbol\psi=(\boldsymbol\beta,\boldsymbol\lambda)\). If calibration and scoring samples overlap, the independence step fails and cross-covariance terms can change the result. A stacked estimating-equation sandwich, a design-respecting bootstrap, or a joint posterior is then required.
The size of the calibration term cannot be inferred from the number of persons being scored. It depends on the calibration sample size and design, item information, calibration/scoring overlap, constraints, and the target estimand. A scoring condition with \(P=20\) signals a possible risk; it does not determine the magnitude or direction of the error by itself.
Warm’s theorem does not cover this. Theorem 6.1 is proved, in Warm’s words, “assuming the item parameters are known constants.” The \(O(I^{-1})\) bias that WLE removes is the bias of the ideal estimator. Whatever bias the plug-in step introduces is a separate matter that WLE does not address, and treating a WLE as unbiased in an operational setting conflates the two.
This is the honest form of Reviewer 2’s third bullet. The claim is not that plug-in estimation is wrong. The claim is that its uncertainty target must be named. Chapters 7 and 8 call inverse-information MSEM and reliability conditional on the calibrated item values. A calibration-aware unconditional reliability replaces that error component with one that includes Equation 6.9 or its overlap-aware analogue; the two coefficients answer different design questions.
6.6 The four estimators
The last column is the one that matters for Part VI. ML and WLE use only the responses of the person being scored; MAP and EAP additionally use the population. That is not a defect of either pair — it is the difference between asking “what does this person’s test say about them” and “what is my best guess about this person.” Chapter 17 shows that the two questions have different answers because they are different questions, and Part VI builds estimators for a third and fourth.
6.7 Sources and provenance
The unification of Section 6.1 is Warm’s (1989), and this chapter follows his development closely because it is clearer than the usual three-methods presentation: his equations (3) through (5) give the family, (6) is Lord’s (1983) bias expression, (7) is Lord’s (1984) result for the Bayesian modal estimator under a standard normal prior, (8) is the conjecture that the bias of the general member picks up one extra term, and (9) and (10) solve for the weight and state the estimating equation. Theorem 6.1 is his Theorem, proved in his appendix; the remark on generality is Hinkley’s, reported there.
One step of that chain is worth spelling out, because a reader who opens Lord will not find Equation 6.3 on the page. Lord derives the bias in the three-parameter parameterization, as his equations (27) to (29) on p. 237: \(B_1(\hat\theta) = I^{-2}\sum_i A_i I_i(\phi_i - \tfrac12)\) with \(\phi_i = (P_i - c_i)/(1 - c_i)\) and \(I_i = P_i'^2/(P_iQ_i)\), and he notes there that \(B_1\) is of order \(n^{-1}\) because \(I\) is of order \(n\) — which is Equation 6.3’s order claim in his notation. Setting \(c_i = 0\) and using \(P_i' = A_iP_iQ_i\) and \(P_i'' = A_i^2P_iQ_i(Q_i - P_i)\) turns his sum into \(-J/2\), so the guessing-free form this chapter displays is Warm’s rendering of Lord’s result rather than a formula Lord himself writes. The equivalence was checked by hand against pp. 236–237, where Lord’s (20) also supplies the \(\mathcal{J}\) used here.
The observation that \(w(\theta)\) coincides with the Jeffreys prior is Warm’s, citing Jeffreys (1961) and Akaike (1978); so is the disclaimer that WLE is nonetheless not Bayesian in derivation or population interpretation. Algebraically, maximizing the weighted likelihood is the Jeffreys-weight MAP objective for the 1PL and 2PL. Warm makes the test dependence explicit through his two-populations/one-test and one-population/two-tests comparison; the administered-test argument is his.
The score equation and its Rasch simplification follow Baker and Kim (2004, ch. 3). Proposition 6.1 is a first-order implicit-function and delta-method derivation given here without an originality claim. It assumes independent calibration and scoring; the overlap alternatives are stated because a marginal law-of-total-variance identity alone does not supply the required covariance structure.
Baker, Frank B., and Seock-Ho Kim. 2004. Item Response Theory: Parameter Estimation Techniques. 2nd ed. Marcel Dekker.
Lord, Frederic M. 1983. “Unbiased Estimators of Ability Parameters, of Their Variance, and of Their Parallel-Forms Reliability.” Psychometrika 48 (2): 233–45. https://doi.org/10.1007/BF02294018.
Warm, Thomas A. 1989. “Weighted Likelihood Estimation of Ability in Item Response Theory.” Psychometrika 54 (3): 427–50. https://doi.org/10.1007/BF02294627.