feat(psychometric): recover 2017-era commented asymTOTALVAR after addedTIPREDVAR on main

[seonghobae]

Summary

psychometric_core recovers the 2017-era commented-out asymTOTALVAR after addedTIPREDVAR as the three-term scalar -q / (2 a) + trait / a² + (B / a)² v on current main.

This is an independent successor of queued two-term asymTOTALVAR (#340). It does not depend on unpublished crates and is not stacked on #340. Table 2/3 do not name asymTOTALVAR. 2017-era ctsem summary.ctsemFit.R (cran/ctsem 2.5.0) comments asymTOTALVAR <- asymDIFFUSION + asymTRAITVAR with asymTRAITVAR <- solve(DRIFT) %*% TRAITVAR %*% t(solve(DRIFT)), then later comments asymTOTALVAR <- asymTOTALVAR + addedTIPREDVAR immediately after forming addedTIPREDVAR <- asymTIPREDEFFECT %*% TIPREDVAR %*% t(asymTIPREDEFFECT). Driver, Oud, and Voelkle (2017, Eq. 1, p. 4; Eq. 4, p. 5; Table 2, p. 12; §4.3, p. 9; §7.2, pp. 20–21; JSS PDF re-opened 2026-08-31T04:40Z) write dη = (Aη + ξ + Bz + Mx) dt + G dW. The Lyapunov within-subject variance is asymDIFFUSION -q / (2 a). At a stable equilibrium a random intercept in CINT units has process-mean variance trait / a². Section 7.2 names addedTIPREDVAR the stable between-subject variance accounted for by a time-independent predictor, (B / a)² v. Form the two-term total first (1 / a, then square, then multiply by trait, then add asymDIFFUSION). Then include addedTIPREDVAR (unit asymptotic effect, then square, then multiply by v). Then add. This crate does not currently export recover_asymptotic_total_variance or recover_asymptotic_trait_variance; form those terms inline.

A zero trait, a zero diffusion, and a zero TI extra is exactly zero even if a ≥ 0. a ≥ 0 with a nonzero contribution fails closed. A zero TI extra keeps the two-term total. A zero trait and a zero diffusion keep addedTIPREDVAR. p + trait / a² is the two-term commented total and is not this map when addedTIPREDVAR ≠ 0. Stationary T0VAR trait + p + (B / a)² v keeps TRAITVAR in process units and is not this map. (B / a)² v is addedTIPREDVAR and equals this total when q = 0 and trait = 0 and remains a distinct named quantity.

Numeric: a = -0.5, trait = 1, q = 0.4, B = 0.5, v = 1 → p = 0.4, trait / a² = 4, (B / a)² v = 1, total 5.4. Two-term 4.4 and stationary T0VAR 2.4 are the refuses.

Still not a Kalman filter, not a matrix expm, not ESEM estimation, not DSEM, and not ctsem estimation. Meredith (1993) and Mislevy (1991) remain unread (closed OA).

Claim boundary

• Do not treat the two-term commented p + trait / a² as this later three-term total.
• Do not treat stationary T0VAR trait + p + (B / a)² v as this map.
• Do not treat addedTIPREDVAR (B / a)² v as this total.
• Do not duplicate queued feat(psychometric): recover 2017-era commented asymTOTALVAR as p + trait/a² on main #340 two-term asymTOTALVAR.
• Do not add asymTOTALVARstd in this PR.

Test plan

• cargo fmt --all
• cargo clippy -p psychometric_core --all-targets -- -D warnings
• cargo test -p psychometric_core --all-targets
• RUSTDOCFLAGS='-D warnings' cargo doc -p psychometric_core --no-deps

Do not convert #310. Do not self-approve. Do not merge without independent non-author APPROVE and exact-head required-check success.

[coderabbitai]

Important

Draft PR not reviewed

Draft PRs are not automatically reviewed by default.

• Trigger a manual review

To automatically review draft PRs, update your CodeRabbit configuration:

reviews:
  auto_review:
    drafts: true

---

Thanks for using CodeRabbit! It's free for OSS, and your support helps us grow. If you like it, consider giving us a shout-out.

❤️ Share

• X
• Mastodon
• Reddit
• LinkedIn

_{Comment @coderabbitai help to get the list of available commands.}