feat(psychometric): recover Driver Eq. 5 of addedT0TIPREDVAR extra observed TI variance

CHANGELOG.md

@@ -4,6 +4,7 @@ All notable changes to TEPP are documented here. The format follows Keep a Chang
 
 ## [Unreleased]
 
+- `psychometric_core` recovers the Driver, Oud, and Voelkle (2017, Eq. 5, p. 5; Table 3, p. 13 `T0TIPREDEFFECT`; Table 2, p. 12; p. 16; §7.2, pp. 20–21; 2017-era ctsem `summary.ctsemFit.R`; JSS PDF re-opened 2026-08-23T19:10Z from https://www.jstatsoft.org/index.php/jss/article/download/v077i05/1104) scalar extra observed-indicator time-independent predictor variance of 2017-era `addedT0TIPREDVAR`. Equation 5 writes `y_i(t) = Γ + Λ η_i(t) + ζ_i(t)` with `ζ ~ N(0, Θ)` and `Γ ~ N(τ, Ψ)`. The 2017-era `summary.ctsemFit.R` forms the latent extra `addedT0TIPREDVAR` as `T0TIPREDEFFECT %*% TIPREDVAR %*% t(T0TIPREDEFFECT)` immediately after `T0TIPREDEFFECTstd`. The scalar latent extra is `t0_b² v`. Equation 5 of that extra, with `θ = 0` and `ψ = 0`, is `λ² t0_b² v`. Form `addedT0TIPREDVAR` first, then `(λ extra) λ`. Do not form `λ²` first. A zero loading or zero extra is exactly zero. `v < 0` fails closed. `T0` is an event-time occasion, so a non-event clock fails closed. Free `T0TIPREDEFFECT` does not require stable `a < 0`. `t0_b² v` is the latent extra, not this observed extra. `λ² p_0 + θ` is first-occasion observed variance, not this extra. `λ² (B / a)² v` is Eq. 5 of `addedTIPREDVAR`, not this first-occasion observed extra. `MANIFESTVAR` `θ` is measurement error, not this extra. `Ψ` is intercept variance and is not extra TI. Still not a Kalman filter, not a matrix `expm`, not ESEM estimation, not DSEM, and not ctsem estimation. Meredith (1993) remains unread (Unpaywall 2026-08-23T19:10Z: `is_oa: false`; title *Measurement Invariance, Factor Analysis and Factorial Invariance*). Mislevy (1991, *Psychometrika, 56*, 177–196) remains unread (Unpaywall 2026-08-23T19:10Z: `is_oa: false`; title *Randomization-Based Inference about Latent Variables from Complex Samples*).
 - `psychometric_core` recovers the Driver, Oud, and Voelkle (2017, Table 3, p. 13 `T0TIPREDEFFECT`; p. 16; §7.2, pp. 20–21; Eq. 3, p. 5; 2017-era ctsem `summary.ctsemFit.R`; JSS PDF re-opened 2026-08-23T18:20Z from https://www.jstatsoft.org/index.php/jss/article/download/v077i05/1104) scalar first-occasion extra time-independent predictor variance `addedT0TIPREDVAR`. Table 3 names `T0TIPREDEFFECT` the effect of time-independent predictors on latents at `T0`. Page 16 prints extra summary matrices when `verbose = TRUE`. The 2017-era `summary.ctsemFit.R` forms `addedT0TIPREDVAR` as `T0TIPREDEFFECT %*% TIPREDVAR %*% t(T0TIPREDEFFECT)` immediately after `T0TIPREDEFFECTstd`. Section 7.2 names `addedTIPREDVAR` the stable between-subject variance accounted for by time-independent predictors at the process asymptote, `(B / a)² v`. The first-occasion analogue uses free `T0TIPREDEFFECT`, not `-B / a`. The scalar map is `t0_b² v`. Form `t0_b` first, then square, then multiply by `v`. A zero coefficient or zero predictor variance is exactly zero. `v < 0` fails closed. `T0` is an event-time occasion, so a non-event clock fails closed. Free `T0TIPREDEFFECT` does not require stable `a < 0`. `(B / a)² v` is `addedTIPREDVAR` and is not this first-occasion map. `t0_b · √v / √p_0` is `T0TIPREDEFFECTstd` and is not this variance. Free `T0VAR` `p_0` is the first-occasion state, not the extra TI variance. `TRAITVAR` is a zero-drift latent process, not `t0_b² v`. Still not a Kalman filter, not a matrix `expm`, not ESEM estimation, not DSEM, and not ctsem estimation. Meredith (1993) remains unread (Unpaywall 2026-08-23T18:20Z: `is_oa: false`; title *Measurement Invariance, Factor Analysis and Factorial Invariance*). Mislevy (1991, *Psychometrika, 56*, 177–196) remains unread (Unpaywall 2026-08-23T18:20Z: `is_oa: false`; title *Randomization-Based Inference about Latent Variables from Complex Samples*).
 - `psychometric_core` recovers the Driver, Oud, and Voelkle (2017, Table 3, p. 13 `T0TIPREDEFFECTstd`; p. 16; footnote 4; Eq. 3, p. 5; 2017-era ctsem `summary.ctsemFit.R`; JSS PDF re-opened 2026-08-23T17:20Z from https://www.jstatsoft.org/index.php/jss/article/download/v077i05/1104) scalar standardised first-occasion time-independent predictor effect. Table 3 names `T0TIPREDEFFECT` the effect of time-independent predictors on latents at `T0`. Page 16 prints standardised matrices with the suffix `std` when appropriate. Footnote 4: standardisations use only the relevant variance, not the total. The affecting variance is predictor variance `TIPREDVAR` `v`. The affected variance is free first-occasion `T0VAR` `p_0`, not within-subject `asymDIFFUSION` `-q / (2 a)`, because Table 3 is the first occasion, not the process dynamics. Form strictly positive `p_0` first, then strictly positive `v`, then `t0_b · √v / √p_0`. Unstandardised `t0_b` is defined for a zero coefficient and for zero predictor variance; standardised `T0TIPREDEFFECT` is not. Zero `p_0` or zero `v` has no positive SD and fails closed. `T0` is an event-time occasion, so a non-event clock fails closed. Free `T0VAR` does not require stable `a < 0`. The continuous standardisation `B · √v / √(-q / (2 a))` uses `asymDIFFUSION` and is not this first-occasion map. The asymptotic standardisation `(-B / a) · √v / √p` is the total change, not this first-occasion coefficient. Section 7.1 warns that omitting trait variance confounds between- and within-person information. `t0_b · √v / √(trait + p_0 + added)` uses the total, not free `T0VAR`, and is not `T0TIPREDEFFECTstd` when `TRAITVAR` is nonzero. `TRAITVAR` is not the standardisation variance. Still not a Kalman filter, not a matrix `expm`, not ESEM estimation, not DSEM, and not ctsem estimation. Meredith (1993) remains unread (Unpaywall 2026-08-23T17:20Z: `is_oa: false`; title *Measurement Invariance, Factor Analysis and Factorial Invariance*). Mislevy (1991, *Psychometrika, 56*, 177–196) remains unread (Unpaywall 2026-08-23T17:20Z: `is_oa: false`; title *Randomization-Based Inference about Latent Variables from Complex Samples*).
 - `psychometric_core` recovers the Driver, Oud, and Voelkle (2017, p. 16 `TIPREDEFFECTstd`; §7.2, pp. 20–21; Eq. 3, p. 5; Table 2, p. 12; footnote 4; JSS PDF re-opened 2026-08-23T16:21Z from https://www.jstatsoft.org/index.php/jss/article/download/v077i05/1104) scalar standardised continuous time-independent predictor effect. Page 16 prints continuous-time parameters and, when appropriate, standardised matrices with the suffix `std`. Table 2 names `B` `TIPREDEFFECT`. Footnote 4: standardisations use only the relevant variance, not the total. The affecting variance is predictor variance `TIPREDVAR` `v`. The affected variance is within-subject `asymDIFFUSION` `-q / (2 a)`, because the process dynamics are individual, or average individual, temporal dynamics. Form strictly positive `asymDIFFUSION` first, then strictly positive `v`, then `B · √v / √(-q / (2 a))`. Unstandardised `B` is defined for a zero coefficient and for zero predictor variance; standardised `TIPREDEFFECT` is not. Zero `asymDIFFUSION` or zero `v` has no positive SD and fails closed. The asymptotic standardisation `(-B / a) · √v / √p` is the total change, not this continuous coefficient. The finite-interval standardisation `A^{-1}[e^{A Δt} − I] B · √v / √p` depends on the event interval and is not this continuous map. Section 7.1 warns that omitting trait variance confounds between- and within-person information. `B · √v / √(trait + p + added)` uses the total, not `asymDIFFUSION`, and is not `TIPREDEFFECTstd` when `TRAITVAR` is nonzero. `TRAITVAR` is not the standardisation variance. Still not a Kalman filter, not a matrix `expm`, not ESEM estimation, not DSEM, and not ctsem estimation. Meredith (1993) remains unread (Unpaywall 2026-08-23T13:19Z: `is_oa: false`; title *Measurement Invariance, Factor Analysis and Factorial Invariance*). Mislevy (1991, *Psychometrika, 56*, 177–196) remains unread (Unpaywall 2026-08-23T13:19Z: `is_oa: false`; title *Randomization-Based Inference about Latent Variables from Complex Samples*).

crates/psychometric_core/src/error.rs

@@ -831,6 +831,23 @@ pub enum PsychometricError {
     /// `TRAITVAR`. Section 4.3 `TRAITVAR` is a zero-drift latent
     /// process, not first-occasion TI extra variance.
     InitialTimeIndependentVarianceIsNotTraitVariance,
+    /// Driver Eq. 5 of 2017-era `addedT0TIPREDVAR` `λ² t0_b² v` was
+    /// treated as the latent extra `t0_b² v`. The observed extra is
+    /// not the latent extra.
+    InitialTimeIndependentObservedVarianceIsNotInitialTimeIndependentVariance,
+    /// Driver Eq. 5 of 2017-era `addedT0TIPREDVAR` `λ² t0_b² v` was
+    /// treated as first-occasion observed variance `λ² p_0 + θ`.
+    /// The extra is not the full first-occasion `Var(y_0)`.
+    InitialTimeIndependentObservedVarianceIsNotInitialObservedVariance,
+    /// Driver Eq. 5 of 2017-era `addedT0TIPREDVAR` `λ² t0_b² v` was
+    /// treated as Eq. 5 of `addedTIPREDVAR` `λ² (B / a)² v`. The
+    /// first-occasion observed extra uses free `T0TIPREDEFFECT`,
+    /// not `-B / a`.
+    InitialTimeIndependentObservedVarianceIsNotAsymptoticTimeIndependentObservedVariance,
+    /// Driver Eq. 5 of 2017-era `addedT0TIPREDVAR` `λ² t0_b² v` was
+    /// treated as `MANIFESTVAR` `θ`. Measurement error is not extra
+    /// observed TI variance.
+    InitialTimeIndependentObservedVarianceIsNotMeasurementError,
 }
 
 impl fmt::Display for PsychometricError {
@@ -1458,6 +1475,18 @@ impl fmt::Display for PsychometricError {
             Self::InitialTimeIndependentVarianceIsNotTraitVariance => {
                 "initial time-independent predictor variance is not trait variance"
             }
+            Self::InitialTimeIndependentObservedVarianceIsNotInitialTimeIndependentVariance => {
+                "initial time-independent observed variance is not initial time-independent predictor variance"
+            }
+            Self::InitialTimeIndependentObservedVarianceIsNotInitialObservedVariance => {
+                "initial time-independent observed variance is not initial observed variance"
+            }
+            Self::InitialTimeIndependentObservedVarianceIsNotAsymptoticTimeIndependentObservedVariance => {
+                "initial time-independent observed variance is not asymptotic time-independent observed variance"
+            }
+            Self::InitialTimeIndependentObservedVarianceIsNotMeasurementError => {
+                "initial time-independent observed variance is not measurement-error variance"
+            }
         };
         formatter.write_str(message)
     }
@@ -2508,4 +2537,28 @@ mod tests {
             "initial time-independent predictor variance is not trait variance"
         );
     }
+
+    #[test]
+    fn initial_time_independent_observed_variance_boundary_messages_are_stable() {
+        assert_eq!(
+            PsychometricError::InitialTimeIndependentObservedVarianceIsNotInitialTimeIndependentVariance
+                .to_string(),
+            "initial time-independent observed variance is not initial time-independent predictor variance"
+        );
+        assert_eq!(
+            PsychometricError::InitialTimeIndependentObservedVarianceIsNotInitialObservedVariance
+                .to_string(),
+            "initial time-independent observed variance is not initial observed variance"
+        );
+        assert_eq!(
+            PsychometricError::InitialTimeIndependentObservedVarianceIsNotAsymptoticTimeIndependentObservedVariance
+                .to_string(),
+            "initial time-independent observed variance is not asymptotic time-independent observed variance"
+        );
+        assert_eq!(
+            PsychometricError::InitialTimeIndependentObservedVarianceIsNotMeasurementError
+                .to_string(),
+            "initial time-independent observed variance is not measurement-error variance"
+        );
+    }
 }