feat(psychometric): recover Driver later-occasion variance of predetermined T0VAR - #181
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…rmined T0VAR
Map the Driver, Oud, and Voelkle (2017, Eq. 3–5 of §4.3 predetermined
first occasion) later-occasion variance of free T0VAR as
trait + e^{2aΔt} p_0 + Q_Δt + (B/a)²v. Trait and addedTIPREDVAR do
not enter Q_Δt. Setting p_0 = −q/(2a) recovers the stationary later
map. Stationary later variance, free discrete evolution of
trait+p_0+added, and p_0 itself remain refused as this composition.
Observed later variance is λ² of that map plus θ + ψ. Growing
processes with a ≥ 0 are kept when the TI contribution is zero.
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Folded into the consolidation vehicle #231; this draft stays open until the vehicle merges, then closes as superseded-by-consolidation. |
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Superseded by consolidation: this slice's Driver et al. (2017) standardization recovery landed on protected main through the integration vehicles (#231/#232), and its contract coverage is carried by the reconciled suites (multilevel_event_time_recovery_contract, rubin_and_mean_gate_contract) with test-signature repair tracked in #234. The stacked-draft form is retired to keep the delivery queue at review-ready work only; no capability is lost — the exact-head provenance remains in the vehicle PR descriptions and CHANGELOG. |
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Stacked on #49 (
6f95142). ADR 0005 executable slice stays insidepsychometric_core. This is not a second invariance crate and does not recreate #78 or #80.psychometric_corerecovers the Driver, Oud, and Voelkle (2017, §4.3, pp. 9–10; Eq. 3–5, pp. 4–5; Table 2, p. 12; p. 16; §7.2, pp. 20–21; JSS PDF re-opened 2026-08-23T05:12Z from https://www.jstatsoft.org/index.php/jss/article/download/v077i05/1104) scalar later-occasion variance of §4.3 predeterminedT0VAR. Section 4.3 treats the first time point as predetermined when no assumptions are made about the process prior to the initial time point. FreeT0VARp_0is then estimated. The process gradually transitions from the variances of the initial parameters toward those of the parameters when the model is stationary.Equation 3 writes
η(t) = exp(A Δt) η(t0) + … +the stochastic integral. Equation 4 writes that the integral exhibits covarianceQ_Δt. The law of total variance on the within-subject state ise^{2 a Δt} p_0 + Q_Δt. Trait variance andaddedTIPREDVARare time-invariant between-subject and do not enter that process-noise integral. The later-occasion composition istrait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v. Form the evolved free first-occasion variance first, then include the trait, then include the TI extra variance, then add.p_0 = −q / (2 a)recovers the stationary later-occasion map.−q / (2 a)in place ofp_0and is not this map whenp_0is free.trait + p_0 + (B / a)² vas if it were all state is not this map.T0VARp_0is not this map.Δt → ∞with stablea < 0the composition approaches contemporaneous stationaryT0VAR.Δt → 0+the composition approachestrait + p_0 + (B / a)² v.a ≥ 0is a growing process and is kept.Equation 5 of that later-occasion variance is
λ²(trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v) + θ + ψ.MANIFESTVARis not that later-occasion observed variance. The predetermined later-occasion latent variance is not the predetermined later-occasion observed variance. Stationary later-occasion observed variance is not that observed variance whenp_0is free.Still not a Kalman filter, not a matrix
expm, not ESEM estimation, not DSEM, and not ctsem estimation. Meredith (1993) remains unread (Unpaywall 2026-08-23T05:12Z:is_oa: false; title Measurement Invariance, Factor Analysis and Factorial Invariance). Mislevy (1991, Psychometrika, 56, 177–196) remains unread (Unpaywall 2026-08-23T05:12Z:is_oa: false; title Randomization-Based Inference about Latent Variables from Complex Samples).Do not merge, self-approve, or request Copilot.