| Method | Incidental parameters | Assumption about \(G\) | Model and identification scope | Asymptotic qualification | 2PL status |
|---|---|---|---|---|---|
| Joint ML (JML) | maximized over, alongside the items | none | Rasch or identified 2PL; anchors still required | inconsistent for fixed \(I\); consistent under Haberman’s joint-growth conditions | available with location/scale constraints |
| Rasch conditional ML (CML) | conditioned away exactly, using \(r_p\) | none | Rasch only; exact person-total conditioning | item-consistent as \(P \to \infty\) with fixed \(I\), under regularity | no ordinary Rasch analogue (Proposition 3.1) |
| Restricted marginal ML | integrated against a specified parametric \(G\) | a specified family, usually Gaussian | Rasch or identified parametric 2PL | consistency requires correct \(G\) family, identification, and regularity | available for an identified parametric 2PL |
| Unrestricted marginal ML | integrated against a freely estimated mixing distribution | no parametric shape; a finite-support maximizer | finite-item Rasch identifies anchored items and finitely many \(G\) functionals, not full \(G\) | no categorical full-\(G\) consistency claim; unrestricted estimation can be unsatisfactory | semiparametric identification open |
| Joint Bayesian marginal model | integrated jointly under priors on items and \(G\) | a prior on a parametric or semiparametric \(G\) | a proper posterior does not establish likelihood identification | concentration claims require a separately identified model and prior-support conditions | DPM×2PL recovery is exploratory and convention-dependent |
5 Estimating Item Parameters
Reviewer 1 objected that the manuscript switches from maximum likelihood to Bayesian estimation without saying why, and particularly that it gives the item difficulties a prior without explanation. The objection is fair, and the answer is not a preference. It is that the person parameters are a nuisance whose number grows with the sample. They can be maximized over, conditioned away, or integrated out; splitting integration by how \(G\) is specified and whether priors are assigned yields the five approaches compared here.
This chapter sets out the four, and it does so with an eye on a question the next chapters need answered: what does each method assume about \(G\)? That is the axis along which they differ most, and it is the axis this book cares about.
5.1 The incidental-parameters problem
Write the joint likelihood over all persons and items,
\[ L(\boldsymbol\theta, \boldsymbol\beta \mid \mathbf{u}) = \prod_{p=1}^{P}\prod_{i=1}^{I} \pi_{pi}^{u_{pi}}(1-\pi_{pi})^{1-u_{pi}}, \tag{5.1}\]
and maximize it over everything at once. This is joint maximum likelihood (JML), and it is the obvious thing to do. It does not work, and the reason is structural rather than computational.
The item parameters are structural: there are \(I\) of them and \(I\) stays fixed as more people take the test. The person parameters are incidental: there is one per person, so their number grows with the sample. Ordinary maximum-likelihood asymptotics assume the parameter count is fixed while information accumulates, and here it is not — every new person brings \(I\) new observations and one new parameter, so the information per parameter never grows.
Theorem 5.1 Inconsistency of joint maximum likelihood. Restated from Haberman (2016, § 9.2.1), reporting Andersen (1973, pp. 66–69); the general phenomenon is Neyman and Scott’s (1948)
Under the Rasch model with \(I\) fixed and \(P \to \infty\), the JML estimator of the item parameters does not converge to the truth. In the classical two-item case, the estimated difficulty difference converges almost surely to twice its true value.
The two-item case is worth carrying because it makes the failure concrete: the estimator is not noisy, it is systematically wrong by a factor, and no amount of additional data fixes it. The familiar \((I-1)/I\) correction applied in practice is an attempt to undo this factor and is a patch on a structural problem rather than a solution to it.
Haberman adds a qualification that the usual telling omits, and it matters for how severely one should judge JML. If \(I\) grows as well as \(P\) — specifically if \(\log(I)/P \to 0\) with the parameters uniformly bounded — then JML is consistent for both the item and the person parameters. The inconsistency is a fixed-\(I\) phenomenon. This theorem does not supply a finite-sample threshold at which the bias becomes small; claims about a particular 10- or 60-item design require separate simulation or empirical evidence.
5.2 Conditioning it away
Theorem 3.1 said that under the Rasch model \(r_p\) is sufficient for \(\theta_p\). Sufficiency means the conditional distribution of \(\mathbf{U}_p\) given \(r_p\) does not depend on \(\theta_p\) at all, so conditioning on the total score removes the incidental parameters exactly rather than approximately:
\[ L_C(\boldsymbol\beta \mid \mathbf{u}, \mathbf{r}) = \prod_{p=1}^{P} \Pr\{\mathbf{U}_p = \mathbf{u}_p \mid r_p; \boldsymbol\beta\}. \tag{5.2}\]
Maximizing Equation 5.2 is conditional maximum likelihood (CML), and the resulting estimator of \(\boldsymbol\beta\) is consistent as \(P \to \infty\) with \(I\) fixed.
The property that matters for this book is not consistency but what CML does not require: Equation 5.2 contains no distribution for \(\theta\). The person parameters have not been integrated out against an assumed \(G\); they have been conditioned away. JML also specifies no \(G\), but it is inconsistent in the fixed-\(I\) regime. Unrestricted marginal methods avoid a parametric shape for \(G\) but inherit identification qualifications. CML’s distinctive conjunction is no model for \(G\), exact elimination of person parameters, and fixed-\(I\) consistency for Rasch item parameters. That makes it the clean conditional reference for measuring the cost of a distributional assumption.
The price is computational and, more importantly, structural. The conditional probabilities involve elementary symmetric functions of the item parameters, whose evaluation is delicate for long tests. And Proposition 3.1 says the whole construction is unavailable under the unrestricted 2PL in its ordinary Rasch form: the parameter-indexed weighted score cannot serve as a parameter-free person total. A 2PL analysis can still be used as a generalization or sensitivity analysis, but it lacks this exact Rasch conditional anchor.
5.3 Integrating it away
The third option treats the \(\theta_p\) as draws from a distribution and integrates:
\[ L_M(\boldsymbol\beta, G \mid \mathbf{u}) = \prod_{p=1}^{P} \int L(\theta, \boldsymbol\beta \mid \mathbf{u}_p)\, dG(\theta). \tag{5.3}\]
This is marginal maximum likelihood (MML). The incidental parameters are gone, the method accommodates fixed \(I\), and restricted MML extends to an identified parametric 2PL once location and scale constraints are imposed. Consistency additionally requires correct specification of the chosen family for \(G\), identification, and the usual regularity conditions; it is not a property of the integral alone.
What it costs is stated in Equation 5.3 itself: \(G\) appears, and something must be said about it. Two things can be said, and the distinction is Haberman’s.
Restricted marginal estimation fixes a parametric family — almost always \(G = N(\mu_\theta, \sigma^2_\theta)\) — and estimates its parameters alongside \(\boldsymbol\beta\). This is what “MML” means in ordinary usage and what the manuscript does. The integral is then computed by Gauss–Hermite quadrature and the whole maximized by EM: treating the \(\theta_p\) as missing data, the E step forms the posterior of each person’s trait under current parameters and the M step maximizes the expected complete log-likelihood, which separates by item. That algorithm is Bock and Aitkin’s (1981), and it is what made MML practical.
Unrestricted marginal estimation leaves \(G\) completely free and maximizes Equation 5.3 over \(\boldsymbol\beta\) and over the space of distribution functions jointly. The maximizer is a discrete distribution on finitely many support points. This is nonparametric MML, and it is the frequentist counterpart of what Part V does with a prior over distributions — Chapter 14 traces it and Chapter 15 develops the Bayesian route. It is worth naming here because the two are usually presented as unrelated traditions when they are two answers to one question. It is not a license for categorical consistency claims: Haberman warns that unrestricted estimation can be unsatisfactory, and finite-item Rasch data identify only anchored item parameters and finitely many functionals of \(G\), not the full mixing distribution. For a semiparametric 2PL, identification remains open.
5.4 Giving the items a prior
The fifth approach is the manuscript’s, and it is a Bayesian version of marginalization. Under this book’s preferred sum-to-zero item constraint, independent priors are placed on unconstrained auxiliaries and then centered:
\[ \widetilde\beta_i \overset{\text{iid}}{\sim} N(0,\sigma^2_\beta), \qquad \beta_i=\widetilde\beta_i-\frac1I\sum_{j=1}^I\widetilde\beta_j, \qquad \sum_i\beta_i=0. \tag{5.4}\]
The induced prior on \(\boldsymbol\beta\) is exchangeable on the constrained subspace but is not iid: centering makes the difficulties dependent. Hyperpriors are placed on \(\sigma^2_\beta\) and on \(G\)’s parameters, and the joint posterior is reported. An anchor-based parameterization could instead put priors on the free item coordinates, but the two prior constructions should not be conflated.
Chapter 2 established that this is not where the distributional assumption on \(\theta\) enters — restricted MML makes that assumption too. Equation 5.4 is the genuinely additional regularization model, and Reviewer 1 was right to ask what it buys.
Partial pooling across items. An item answered correctly by everyone has no finite unregularized difficulty estimate; under a proper version of Equation 5.4 it has a finite posterior, shrunk toward the constrained item center. The size of that effect is a prior-sensitivity question.
Coherent propagation of uncertainty. In a two-stage procedure the item estimates are treated as known when persons are scored, which understates the uncertainty in \(\hat\theta_p\) — Chapter 6 makes this precise and it is the substance of Reviewer 2’s third bullet. Fitting everything jointly carries the item uncertainty into the person posterior automatically.
A route to a flexible \(G\). Once \(G\) is a parameter with a prior, replacing that prior with one over distributions is a change of prior rather than a change of method. That is what makes the semiparametric extension of Part V natural in this framework and awkward in the others.
The price is a second modelling assumption. Equation 5.4 is exchangeable over item labels before centering and regularizes the constrained vector toward a common center. For a test assembled from a blueprint with deliberate difficulty targets, that is best treated as a regularization device whose influence is checked, not automatically as a literal model of item writing.
5.5 Five approaches, side by side
The \(G\) column is only one axis. The model/scope and identification columns prevent a shape-free mixing distribution from being read as an identified full \(G\), and they keep the open semiparametric-2PL problem visible. Part V weakens parametric shape assumptions; it does not thereby solve identification.
5.6 Sources and provenance
The incidental-parameters framing and Theorem 5.1 follow Haberman’s (2016) chapter, which is the reference treatment: the JML inconsistency and the two-item example are his § 9.2.1, reporting Andersen (1973, pp. 66–69), and the consistency result under growing \(I\) is his citation of his own earlier work. The distinction between restricted and unrestricted marginal estimation is his §§ 9.4–9.5, and the terminology is worth adopting because the usual vocabulary has no name for the difference. Neyman and Scott (1948) named the general phenomenon and are cited through Haberman under rule R3b; the primary is not held.
The EM algorithm for restricted marginal estimation is Bock and Aitkin’s (1981). CML’s development follows Debelak et al. (2022, sec. 3.2), and the sufficiency it rests on is Theorem 3.1.
The argument of Section 5.4 is assembled here rather than taken from a source. Its three components — partial pooling, uncertainty propagation, and the route to a flexible \(G\) — are individually standard; the claim that the third is what makes the Bayesian framework the natural home for a semiparametric \(G\) is this book’s framing, and it is argued rather than asserted in Chapter 15 and Chapter 16.
Bock, R. Darrell, and Murray Aitkin. 1981. “Marginal Maximum Likelihood Estimation of Item Parameters: Application of an EM Algorithm.” Psychometrika 46 (4): 443–59. https://doi.org/10.1007/BF02293801.
Debelak, Rudolf, Carolin Strobl, and Matthew D. Zeigenfuse. 2022. An Introduction to the Rasch Model with Examples in r. Chapman; Hall/CRC.
Haberman, Shelby J. 2016. “Models with Nuisance and Incidental Parameters.” Chap. 9 in Handbook of Item Response Theory, Volume Two: Statistical Tools, edited by Wim J. van der Linden. Chapman; Hall/CRC. https://doi.org/10.1201/b19166-9.
Neyman, Jerzy, and Elizabeth L. Scott. 1948. “Consistent Estimates Based on Partially Consistent Observations.” Econometrica 16 (1): 1–32. https://doi.org/10.2307/1914288.