| Specification | Parameters of interest | Rasch | 2PL |
|---|---|---|---|
| Fixed effects — the \(\theta_p\) are \(P\) unknown parameters | \((\boldsymbol\theta, \boldsymbol\beta)\), and \(\boldsymbol\lambda\) under the 2PL | \(\beta_1 = 0\) | \(\beta_1 = 0\) and \(\lambda_1 = 1\) |
| Random effects — \(\theta_p \sim G\), \(G\) known up to scale \(\sigma\) | \((\boldsymbol\beta, \sigma)\), and \(\boldsymbol\lambda\) under the 2PL | none within this known-shape family — \(\sigma\) is identified by a continuous strictly increasing bivariate-probability map under the stated regularity conditions | \(\lambda_1 = 1\) |
| Random effects — \(\theta_p \sim G\), \(G\) with free location \(\mu\) and scale \(\sigma\) | \((\boldsymbol\beta, \mu, \sigma)\), and \(\boldsymbol\lambda\) under the 2PL | \(\beta_1 = 0\) | \(\lambda_1 = 1\) and \(\beta_1 = 0\); requires \(I \ge 3\) |
| Semiparametric — \(G\) itself an unknown parameter | \((\boldsymbol\beta, G)\) | \(\beta_1 = 0\) identifies \(\boldsymbol\beta\) and \(I+1\) response-score functionals; full \(G\) and its variance are not generally identified (Chapter 16) | open problem |
4 Identification
Reviewer 2 wrote that the Rasch model is not identifiable without constraints, that the usual \(\theta_p \sim N(0,1)\) fixes location and scale, and that a model estimating the mean and variance of the latent distribution — or replacing it with a Dirichlet process mixture — needs additional constraints that the manuscript never states. The objection is correct as an objection. It is also, as this chapter shows, based on a picture of the problem that the identification literature does not support in one important parametric case: when the shape of the random-effects distribution is known and only its scale is unknown, the Rasch scale is identified from the data, not fixed by convention. This does not extend to an unknown semiparametric \(G\), whose finite-item identification failure is more specific than “needs more constraints.”
Getting this right matters beyond answering a review. A model that is not identified does not estimate what a paper says it estimates, and a recovered distribution \(G\) that is only partially identified supports only some of the statements one might want to make about it. Chapter 16 returns to the semiparametric case with the Dirichlet process in hand; this chapter establishes what identification is and settles the parametric models.
4.1 What identification is
A statistical model attaches to each parameter value \(\boldsymbol\alpha \in A\) a sampling distribution \(P\{\cdot\,;\boldsymbol\alpha\}\) over the observable data.
Definition 4.1 Identification. Restated from San Martín (2016, § 8.2.2)
The parameter \(\boldsymbol\alpha\) is identified by the observations if the map \(\boldsymbol\alpha \mapsto P\{\cdot\,;\boldsymbol\alpha\}\) is injective: for \(\boldsymbol\alpha, \boldsymbol\alpha' \in A\),
\[ P\{E;\boldsymbol\alpha\} = P\{E;\boldsymbol\alpha'\} \ \text{ for every event } E \quad\Longrightarrow\quad \boldsymbol\alpha = \boldsymbol\alpha' . \tag{4.1}\]
Identification is a property of the model rather than of a realized sample. Equation 4.1 quantifies over sampling distributions, not over data sets. San Martín, following Koopmans, puts the point sharply: exact knowledge of the family of sampling distributions cannot be derived from any finite number of observations, and is “the limit approachable but not attainable by extended observation.” Hypothesising that knowledge anyway is what separates the problems of statistical inference, which arise from the variability of finite samples, from the problems of identification, which are the limits inference cannot pass even with infinite data.
There is always some function of the parameters that indexes the sampling distributions one-to-one; the identification question is whether the parameters one wants are a bijective function of it. This is not a technicality but the method of proof: San Martín’s results are obtained by constructing the identified parametrization first and then asking what restrictions make the parameters of interest recoverable from it. The consequence is stated in his discussion and is worth carrying: the restrictions so obtained are necessary as well as sufficient, which is more than a convention-based argument can deliver.
4.2 Location
Take the Rasch model Equation 3.2, and add a constant to every person and every item.
Proposition 4.1 Location indeterminacy. Derived here; no originality claim
For any \(c \in \mathbb{R}\), the parameter values \((\theta_p + c, \beta_i + c)\) and \((\theta_p, \beta_i)\) induce the same sampling distribution.
Proof. \((\theta_p + c) - (\beta_i + c) = \theta_p - \beta_i\), so every \(\pi_{pi}\) is unchanged; by local independence the distribution of \(\mathbf{U}\) is a function of the \(\pi_{pi}\) alone. ∎
The proof is one line and the content is not: the data cannot distinguish “everyone is one logit more able and every item one logit harder” from the original, because the model only ever speaks about differences. Fixing the origin is a choice of unit, like choosing where zero sits on a temperature scale.
4.3 Scale
The Rasch and 2PL models differ in their scale orbit, which determines the restrictions needed in the rest of Part II.
Proposition 4.2 Scale indeterminacy in the 2PL, and its absence in the Rasch model. Derived here; no originality claim
Under the 2PL, for any \(c > 0\) the values \((c\theta_p,\, \lambda_i/c,\, c\beta_i)\) induce the same sampling distribution as \((\theta_p, \lambda_i, \beta_i)\). Under the Rasch model, where \(\lambda_i \equiv 1\) is fixed, no such transformation exists.
Proof. \((\lambda_i/c)(c\theta_p - c\beta_i) = \lambda_i(\theta_p - \beta_i)\). Under the Rasch model the same rescaling would require \(\lambda_i \mapsto 1/c\), which leaves the model unless \(c = 1\). ∎
So the 2PL has a two-dimensional indeterminacy and the Rasch model a one-dimensional one. The 2PL needs two restrictions where the Rasch model needs one, and — this is the part that matters later — the second restriction is what pins the scale on which \(G\) lives. Under the Rasch model the unit is fixed by the model itself, which is why the variance of \(G\) can be a free parameter with a meaning; under the 2PL a convention has to supply it. Chapter 16 shows what that costs when \(G\) is nonparametric.
4.4 What actually needs restricting, and when
The usual account — “we set \(\theta_p \sim N(0,1)\) to fix location and scale” — treats the two indeterminacies as a single problem to be solved by fiat. The identification results do not support that reading. What must be restricted depends on how the person parameters are specified, and in one case of practical importance the answer is nothing.
Read the rows in order.
Fixed effects. The \(\theta_p\) are \(P\) unknown parameters. The identified parametrization is \(\{F(\theta_p - \beta_i)\}\), identified because each \(U_{pi}\) is Bernoulli and the \(U_{pi}\) are mutually independent; the parameters of interest are recovered from it once one item is used as the origin. The Rasch model needs \(\beta_1 = 0\); the 2PL needs \(\beta_1 = 0\) and \(\lambda_1 = 1\).
Random effects, a known shape \(G\) up to scale. Here \(\theta_p=\sigma X_p\), where the \(X_p\) are i.i.d. from a fixed, known distribution \(G\) with its origin and unit fixed, and the parameters of interest are \((\boldsymbol\beta,\sigma)\). Under the regularity conditions in San Martín (2016, § 8.5.1), the Rasch model needs no additional restriction. For each item, the marginal probability \(\delta_i=\Pr\{U_{pi}=1\}\) is identified, and
\[ p(\sigma,\beta)=\int F(\sigma x-\beta)\,G(dx) \]
is continuous and strictly decreasing in \(\beta\). Thus \(\beta_i\) is determined by \((\sigma,\delta_i)\). Substitution into the bivariate probability \(\delta_{12}=\Pr\{U_{p1}=1,U_{p2}=1\}\) produces a map of \(\sigma\) that San Martín proves is continuous and strictly increasing when \(I\ge2\). Its inverse identifies \(\sigma\), after which the item difficulties are identified.
This is a theorem about a fixed known shape with one unknown scale parameter. It does not identify the variance of an arbitrary unknown \(G\), and it does not justify treating a Dirichlet-process-mixture variance as data-identified.
Random effects, \(G\) with free location and scale. Now \(\mu\) is free as well, and one restriction is needed to place the origin: \(\beta_1 = 0\) for the Rasch model, \(\beta_1 = 0\) and \(\lambda_1 = 1\) for the 2PL, which additionally requires \(I \ge 3\).
What the submitted manuscript says. The manuscript does not discuss identification, and Reviewer 2 is right that it should. But the reviewer’s own framing — that \(\theta_p \sim N(0,1)\) is “typically used to fix the location and scale of the latent trait”, so that estimating \(\mu_\theta\) and \(\sigma^2_\theta\) leaves the model unidentified — is also not what the results say.
Setting \(\sigma^2_\theta=1\) in the known-shape random-effects Rasch family does not remove an indeterminacy; in that family \(\sigma\) is identified by the strictly monotone bivariate-probability map. It instead restricts an identified parameter and can misspecify the family. For an unknown semiparametric \(G\), however, finite-item data do not identify the full distribution or its variance. The revision must state which of these two models it means before making a scale claim.
4.5 Location constraints act on different models
Several location conventions are used, but they do not all constrain the same statistical object. The fixed-effects likelihood contains realized person parameters; the marginal random-effects likelihood contains their distribution \(G\) instead.
Theorem 4.1 Fixed-effects location orbit. Derived here; no originality claim
In the fixed-effects Rasch likelihood, the transformations \((\boldsymbol\theta,\boldsymbol\beta)\mapsto (\boldsymbol\theta+c\mathbf1_P,\boldsymbol\beta+c\mathbf1_I)\) form a location orbit. Each of \(\beta_1=0\), \(\sum_i\beta_i=0\), and \(\sum_p\theta_p=0\) selects exactly one member of that orbit. The selected likelihood, fitted probabilities, and within-person or within-item differences are invariant across the three representatives.
Proof. Substituting the orbit into each constraint gives one linear equation in \(c\) with a unique solution. Proposition 4.1 makes the likelihood constant on the orbit, and all within-type differences cancel \(c\). ∎
Theorem 4.2 Random-effects location orbit. Derived here; no originality claim
Let \(G_c\) be the law of \(\Theta+c\) when \(\Theta\sim G\). In the marginal Rasch model, \((\boldsymbol\beta,G)\) and \((\boldsymbol\beta+c\mathbf1_I,G_c)\) induce the same response distribution. An item anchor, item centering, or a finite shift-equivariant location constraint on \(G\) selects one representative. The realized constraint \(\sum_p\theta_p=0\) is not a constraint on this marginal parameter orbit.
Proof. After the change of variable \(t=\theta+c\), every term depends on \(t-(\beta_i+c)=\theta-\beta_i\), so the marginal response probabilities are unchanged. The first three constraints again determine one \(c\) when their location functionals exist. The realized sum is a function of latent draws, not of \((\boldsymbol\beta,G)\), and therefore does not select a marginal-model parameter. ∎
These likelihood equivalences do not make prior or posterior models interchangeable. A prior must either be transformed with the orbit or deliberately select a representative; holding a different prior fixed can change posterior geometry and summaries even when the likelihood quotient is the same.
Interpretation of the recovered \(G\). Fixing \(\mu_\theta = 0\) makes the person distribution centred by construction, so its estimated mean carries no information. Fixing \(\sum_i \beta_i = 0\) leaves \(\mu_\theta\) free and estimable, and it is then a statement about where this population sits relative to this item set. If the recovered distribution is the object of interest — and in this book it usually is — the second is the informative choice.
Posterior geometry. A sum-to-zero constraint on items is compatible with a free population location. Imposing a realized person-centering constraint inside a model that also assigns a population distribution changes the latent-draw model and can create a redundancy with its location hyperparameter.
A constraint has to be representable in the model as specified. A Dirichlet process prior on \(G\) cannot be made to satisfy \(\sum_p \theta_p = 0\), because the \(\theta_p\) are exchangeable draws and the constraint is a property of the realized set. Chapter 16 makes this precise; it is the reason the companion simulation constrains items rather than abilities.
4.6 A proper posterior is not identification
If the joint prior is proper and the likelihood is integrable under it, the posterior is proper even when the likelihood has an unidentified direction. It may have a mean, a variance, and a credible interval, and none of that establishes identification. Conditional on an identified likelihood orbit, the posterior distribution along the orbit is proportional to the corresponding conditional prior because the likelihood is constant there. The data may still inform identified combinations, so the marginal posterior of an unidentified coordinate need not literally equal its marginal prior when the prior couples the two directions.
Diagnostics do not catch this problem reliably: a model with a proper prior on an unidentified direction can converge cleanly. Moreover, the conditional credible interval along an unidentified orbit is prior-driven; it should not be presented as information supplied by the observations.
The defensible options are to impose a constraint that selects an identified representative, or to sample a deliberately expanded model and report only invariant, identified functionals after rescaling. Both require the transformation, prior, and reported estimand to be explicit. A proper prior can regularize or select a convention; it cannot by itself prove that the likelihood identifies the selected coordinate.
4.7 Two results that should be better known
San Martín’s discussion closes with two remarks that bear directly on how this literature is usually written, and both are stated here because the manuscript, like most papers, implicitly assumes the opposite of the first.
Fixed-effects identification does not imply random-effects identification. There is a heuristic in the psychometric literature that it does — San Martín attributes the rule to Adams et al. (1997) and Adams and Wu (2007) — and the results summarized in Table 4.1 refute it: the two specifications require different restrictions, and neither set follows from the other. By extension the heuristic fails for semiparametric versions too. Since almost every applied paper reasons about identification informally, and informally usually means by this heuristic, the point deserves more circulation than it has.
Identification analysis presumes the model is true, which makes it a design requirement. An identification restriction is not only a statement about parameters; it is sometimes a statement about what the test must contain. San Martín’s example is the 1PL-G model, whose identification requires a guessing parameter fixed at zero — which means the instrument must include an item nobody can get right by guessing. A model should be applied to data that, by design, satisfies its identification restrictions, and that is a fact about test construction rather than about estimation.
4.8 What is coming, and what is open
The semiparametric case — \(G\) itself an unknown parameter — is the one this book’s subject requires, and it is deferred to Chapter 16 because it needs the Dirichlet process first. Under the anchored Rasch model, the item parameters and \(I+1\) response-score functionals of \(G\) are identified from the response-pattern distribution, but the full \(G\) is not. Its variance is not generally among the identified finite-item functionals, so later reliability claims cannot silently substitute a recovered DPM variance for an identified population quantity.
One caveat belongs here rather than there, because it qualifies this book’s whole scope. Under TD-3 this book carries the 2PL, and the companion simulation fits Dirichlet process mixtures under both models. The identification of the semiparametric 2PL is an open problem; San Martín’s Table 8.1 marks it with question marks and his discussion says so directly. Accordingly, every DPM×2PL analysis in this project must pre-specify its location and scale constraints, label distributional recovery as convention-dependent and exploratory, and report sensitivity to alternative valid rescalings. Reliability must be defined as an invariant response-distribution functional or explicitly labelled as a convention-dependent parameter summary. No 2PL DPM result is called identified unless a separate theorem establishes it.
Chapter 26 returns to this boundary after the Rasch semiparametric result is in hand. It states the 2PL location-scale orbit, the software’s constraint pair, and exactly why the finite-item Rasch theorem is not extended by analogy.
4.9 Sources and provenance
This chapter is built on San Martín’s (2016) chapter in the Handbook of Item Response Theory, which is the reference treatment: Definition 4.1 is his § 8.2.2, the fixed-effects results are § 8.4, the random-effects results including the \(\sigma\)-from-bivariate-probabilities argument are § 8.5.1, Table 4.1 condenses his Table 8.1, and the two remarks of Section 4.7 are his § 8.7. The epistemological framing of identification as a limit not attainable by extended observation is Koopmans’s, quoted there.
Proposition 4.1, Proposition 4.2, Theorem 4.1, and Theorem 4.2 are elementary orbit arguments derived here without originality claims. San Martín establishes identification under specific fixed- and random-effects restrictions; he does not state the former four-way theorem that an earlier version of this chapter attributed to him. The split above keeps conditional fixed-effects likelihood, marginal random-effects likelihood, and prior/posterior models distinct.
The semiparametric results previewed in Section 4.8 are San Martín et al. (2011), reported in San Martín (2016, § 8.6) as his Theorems 8.4 and 8.5; Chapter 16 takes them from the primary source.
Section 4.6 is standard and is stated without novelty claim. Its claims are conditional on a proper joint prior and a finite prior predictive normalizing constant; the orbit statement is about the conditional prior along a flat likelihood direction, not an unconditional identity between marginal prior and posterior distributions.
San Martín, Ernesto. 2016. “Identification of Item Response Theory Models.” Chap. 8 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/b19166-8.
San Martín, Ernesto, Alejandro Jara, Jean-Marie Rolin, and Michel Mouchart. 2011. “On the Bayesian Nonparametric Generalization of IRT-Type Models.” Psychometrika 76 (3): 385–409. https://doi.org/10.1007/s11336-011-9213-9.