feat(psychometric): recover Driver first-occasion variance of predetermined T0VAR - #183
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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.
… T0VAR
Map the Driver, Oud, and Voelkle (2017, Eq. 3–5 of §4.3 predetermined
first occasion) lagged covariance of free T0VAR as
trait + e^{aΔt} p_0 + (B/a)²v. Trait and addedTIPREDVAR do not
decay. Setting p_0 = −q/(2a) recovers the stationary lagged map.
Stationary lagged covariance, later-occasion variance, the decayed
total, and p_0 itself remain refused as this composition. Observed
lagged covariance is λ² of that map plus ψ. Independent ε_t does
not enter. A zero-diffusion carry with a ≥ 0 is kept.
…rmined T0VAR Driver, Oud, and Voelkle (2017, §4.3) treat the first time point as predetermined when no assumptions are made about the process prior to the initial time point. Free T0VAR p_0 is then estimated. Between-subject TRAITVAR and addedTIPREDVAR are inherently stationary. The first-occasion composition is trait + p_0 + (B/a)² v. Equation 5 maps that variance as λ²(trait + p_0 + (B/a)² v) + θ + ψ. Setting p_0 = −q/(2a) recovers the stationary first-occasion map. Lagged and later maps approach this composition as Δt → 0+. Still not a Kalman filter, not a matrix expm, not ESEM estimation, not DSEM, and not ctsem estimation.
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Folded into the consolidation vehicle #231; this draft stays open until the vehicle merges, then closes as superseded-by-consolidation.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Superseded by consolidation: this slice's Driver et al.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 #182 (
2d4d6bf) / #181 (542806b) / #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. 5, p. 5; Table 2, p. 12; p. 16; §7.2, pp. 20–21; JSS PDF re-opened 2026-08-23T10:03Z from https://www.jstatsoft.org/index.php/jss/article/download/v077i05/1104) scalar first-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. Between-subjectTRAITVARandaddedTIPREDVARare inherently stationary (p. 10).The first-occasion composition is
trait + p_0 + (B / a)² v. Form the free first-occasion state variance first, then include the trait, then include the TI extra variance, then add.p_0 = −q / (2 a)recovers the stationary first-occasion map.−q / (2 a)in place ofp_0and is not this map whenp_0is free.T0VARp_0is not this map.trait + e^{a Δt} p_0 + (B / a)² vdecays the state and is not this map.trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² vincludesQ_Δtand is not this map.Δt → 0+those maps approach this composition.a ≥ 0cannot hold a finite TI extra variance when that contribution is nonzero and fails closed.Equation 5 of that first-occasion variance is
λ²(trait + p_0 + (B / a)² v) + θ + ψ.MANIFESTVARis not that first-occasion observed variance. The predetermined first-occasion latent variance is not the predetermined first-occasion observed variance. Stationary first-occasion observed variance is not that observed variance whenp_0is free. Predetermined later observed variance includesQ_Δtand is not that first-occasion observed variance.Still not a Kalman filter, not a matrix
expm, not ESEM estimation, not DSEM, and not ctsem estimation. Meredith (1993) remains unread (Unpaywall 2026-08-23T10:03Z:is_oa: false; title Measurement Invariance, Factor Analysis and Factorial Invariance). Mislevy (1991, Psychometrika, 56, 177–196) remains unread (Unpaywall 2026-08-23T10:03Z:is_oa: false; title Randomization-Based Inference about Latent Variables from Complex Samples).Do not merge, self-approve, or request Copilot.