2.1 Conditional precision as a test-theoretic estimand

In the operator-theoretic formulation, CSEM is a conditional norm of the projection residual. Let $T={P}_{\mathcal{G}}X=\mathbb{E}\left(X\mid \mathcal{G}\right)$ , and let $\varepsilon =X-T$ . The unconditional SEM is the global residual norm,

$$\begin{align*}\mathrm{SEM}(X)={\left\{\mathbb{E}\left({\varepsilon}^2\right)\right\}}^{1/2}.\end{align*}$$

The observed-score CSEM localizes this residual norm by conditioning on the observed score:

$$\begin{align*}{\mathrm{CSEM}}_X(x)={\left\{\mathbb{E}\left({\varepsilon}^2\mid X=x\right)\right\}}^{1/2}.\end{align*}$$

If observed-score intervals ${B}_1,\dots, {B}_K$ group respondents, the score-band CSEM is

$$\begin{align*}{\mathrm{CSEM}}_{B_k}(X)={\left\{\mathbb{E}\left({\varepsilon}^2\mid X\in {B}_k\right)\right\}}^{1/2}.\end{align*}$$

Equivalently,

$$\begin{align*}{\mathrm{CSEM}}_{B_k}^2(X)=\frac{\int_{B_k}{\mathrm{CSEM}}_X^2(x)\;d{P}_X(x)}{P_X\left({B}_k\right)}.\end{align*}$$

The unconditional SEM is recovered as the weighted average of the squared score-band CSEMs:

$$\begin{align*}{\mathrm{SEM}}^2(X)=\sum \limits_{k=1}^KP\left(X\in {B}_k\right){\mathrm{CSEM}}_{B_k}^2(X).\end{align*}$$

This result clarifies the test-theoretic status of conditional precision. As Zumbo (Reference Zumbo2026c) observed, score- and score-band-based CSEMs do not introduce new kinds of measurement error. They describe how the same projection residual varies when localized to observed-score values or observed-score intervals. The central question shifts from what would happen under indefinitely repeated identical administrations to which $\sigma$ -algebra defines systematic variation. Different conditioning structures encode different theoretical commitments about persons, contexts, occasions, constructs, and sources of error.

More generally, a conditional SEM replaces the unconditional second moment with a conditional one:

$$\begin{align*}{\mathrm{CSEM}}_{\mathcal{H}}(X)={\left\{\mathbb{E}\left({\varepsilon}^2\mid \mathcal{H}\right)\right\}}^{1/2},\end{align*}$$

where $\mathcal{H}$ specifies the information relative to which measurement precision is described. Score-conditional, trait-conditional, person-conditional, and multilevel standard errors need not represent different kinds of error. They can be understood as different conditionings of the same projection residual. From this perspective, the central psychometric question is not merely which SEM has been reported, but with respect to which information structure residual variation has been localized.

2.2 Observed-score conditional precision for individual score interpretation

The observed-score conditional SEM corresponds to the norm of the projection residual conditioned on an examinee’s observed score. In operator-theoretic terms,

$$\begin{align*}\mathrm{SEM}(x)=\sqrt{\mathbb{E}\left(\left.{\varepsilon}^2\right|\;X=x\right)},\end{align*}$$

where $\varepsilon =X-\mathbb{E}\left(X\mid \mathcal{G}\right)$ . This expression is identical to the observed-score conditional standard error of measurement, ${\mathrm{CSEM}}_X(x)$ , defined in the preceding section. In short, because the conditioning event is $X=x$ , this quantity is score-conditional rather than person-conditional; examinees with the same observed score therefore share the same value.

The purpose of the present subsection is therefore interpretive rather than definitional: it explains how this conditional precision measure is understood for an individual examinee within the operator-theoretic framework. In essence, $\mathrm{SEM}(x)$ is a conditional standard deviation that need not be constant across individuals. It depends on the examinee’s observed score rather than the examinee as an individual. Consequently, all examinees with the same observed score share the same observed-score conditional SEM. This formulation does not introduce a new notion of individual conditional precision. Rather, it provides a unified operator-theoretic framework in which individual-, score-, trait-, and context-specific precision are all expressed as conditional manifestations of the same projection residual. The contribution is therefore not a new concept of conditional precision but a common mathematical framework that unifies these quantities within a single test-theoretic formulation. Different conditioning structures yield distinct conditional manifestations of the same projection residual and, consequently, distinct conditional precision estimands.

In summary, the individual SEM, $\mathrm{SEM}(x)$ , is the length of the error component of the observed score vector, conditional on the examinee’s observed score. Consequently, examinees with different observed scores may have different CSEMs, whereas examinees with the same observed score receive the same observed-score conditional SEM. Like conditional precision measures in item response theory, this formulation permits measurement precision to vary across the score continuum. Its derivation, however, rests on the more general framework of conditional expectation and orthogonal projection rather than on a particular latent-variable model. More broadly, the operator-theoretic framework shows that individual-, score-, trait-, and context-specific precision all arise by conditioning the same projection residual on different information structures. Conditional expectation and orthogonal projection, therefore, provide a unified mathematical framework for defining and interpreting conditional precision estimands across test-theoretic formulations.

The observed-score conditional SEM considered here is one member of a broader class of conditional precision estimands. Other conditioning structures—for example, person-, trait-, or context-specific information—would define different conditional precision estimands within the same operator-theoretic framework. It is, therefore, notable that the operator-theoretic framework described herein is more general than this subsection currently suggests. The observed-score CSEM corresponds to conditioning on $\sigma (X).$ A genuinely person-specific precision measure would instead condition on an information structure containing person-level information, for example, using the notation from the preceding section, ${\mathcal{H}}_{\mathrm{person}}$ , which highlights the main theme that changing the conditioning structure changes the estimand.

2.3 Pooling adjacent low-frequency score groups

Pfadt et al. (Reference Pfadt, Molenaar, Hurks and Sijtsma2026) recommend pooling adjacent low-frequency score groups to stabilize estimation in sparse regions of the score distribution. As Zumbo (Reference Zumbo2026c) describes, this recommendation is statistically sensible, but it has a test-theoretic consequence: pooling changes the estimand. Once adjacent score groups are merged, the target is no longer pointwise score-conditional precision at (X = x), but precision conditional on membership in a score band. Thus, pooling does not merely improve estimation of the same quantity; it changes the quantity being estimated.

4.1 Practical implications for responsible score interpretation

When interpreting individual test scores, first determine the precision question you want to answer. Do you need precision for a specific observed score, a score range, a person, or a particular measurement context? This choice determines the conditional precision estimand. Next, select an estimation procedure that targets that estimand, and verify that its assumptions match your testing situation. Finally, interpret the reported SEM or CSEM in light of those assumptions rather than treating it as a universal property of the test.

Therefore, before reporting or interpreting a CSEM:

1. 1. Identify the precision question you want to answer.

2. 2. Specify the conditioning information that defines the estimand.

3. 3. Choose an estimator whose assumptions match that estimand.

4. 4. Interpret the resulting CSEM only within those assumptions.