feat(psychometric): recover Eq. 5 of analog addedT0TIPREDVAR extra observed TD 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 Equation 5 of the scalar analog of Driver, Oud, and Voelkle (2017, Eq. 5, p. 5; Table 2, p. 12 `TDPREDVAR` / `T0TDPREDCOV`; Table 3, p. 13 `T0TIPREDEFFECT`; p. 16; §7.2, pp. 20–21; 2017-era ctsem `summary.ctsemFit.R`; JSS PDF re-opened 2026-08-23T22:26Z from https://www.jstatsoft.org/index.php/jss/article/download/v077i05/1104) extra first-occasion time-dependent predictor variance. 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`. That file comments out `TDPREDVAR` and does not form `addedT0TDPREDVAR`. The scalar analog of that quadratic form using the stack's first-occasion TD coefficient `t0_m` and Table 2 `TDPREDVAR` `v` is `t0_m² v`. Equation 5 of that analog extra, with `θ = 0` and `ψ = 0`, is `λ² t0_m² v`. Form the analog extra 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 `t0_m` does not require stable `a < 0`. `t0_m² v` is the latent extra, not this observed extra. `λ² p_0 + θ` is first-occasion observed variance, not this extra. `λ² t0_b² v` is Eq. 5 of `addedT0TIPREDVAR` and is not this extra even when `t0_m = t0_b`. `MANIFESTVAR` `θ` is measurement error, not this extra. Still not a Kalman filter, not a matrix `expm`, not ESEM estimation, not DSEM, and not ctsem estimation. Meredith (1993) remains unread (Unpaywall 2026-08-23T22:26Z: `is_oa: false`; title *Measurement Invariance, Factor Analysis and Factorial Invariance*). Mislevy (1991, *Psychometrika, 56*, 177–196) remains unread (Unpaywall 2026-08-23T22:26Z: `is_oa: false`; title *Randomization-Based Inference about Latent Variables from Complex Samples*).
 - `psychometric_core` recovers the scalar analog of Driver, Oud, and Voelkle (2017, Table 2, p. 12 `TDPREDVAR` / `T0TDPREDCOV`; 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 2026-08-23T22:13Z from https://www.jstatsoft.org/index.php/jss/article/download/v077i05/1104) extra first-occasion time-dependent predictor variance. The 2017-era `summary.ctsemFit.R` forms `addedT0TIPREDVAR` as `T0TIPREDEFFECT %*% TIPREDVAR %*% t(T0TIPREDEFFECT)` immediately after `T0TIPREDEFFECTstd`. That file comments out `TDPREDVAR` and does not form `addedT0TDPREDVAR`. Table 2 names `T0TDPREDCOV` the covariance between latents at `T0` and time-dependent predictors, not this extra variance. Table 3 names `T0TIPREDEFFECT`, not a TD first-occasion effect matrix. The scalar analog of that quadratic form using the stack's first-occasion TD coefficient `t0_m` and Table 2 `TDPREDVAR` `v` is `t0_m² v`. Form `t0_m` 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 `t0_m` does not require stable `a < 0`. `t0_b² v` is `addedT0TIPREDVAR` and is not this extra even when `t0_m = t0_b`. `t0_m · √v / √p_0` is `T0TDPREDEFFECTstd` and is not this variance. `T0TDPREDCOV` is the covariance, not `t0_m² v`. Free `T0VAR` `p_0` is the first-occasion state, not the extra TD variance. `TRAITVAR` is a zero-drift latent process, not `t0_m² v`. Still not a Kalman filter, not a matrix `expm`, not ESEM estimation, not DSEM, and not ctsem estimation. Meredith (1993) remains unread (Unpaywall 2026-08-23T21:34Z: `is_oa: false`; title *Measurement Invariance, Factor Analysis and Factorial Invariance*). Mislevy (1991, *Psychometrika, 56*, 177–196) remains unread (Unpaywall 2026-08-23T21:34Z: `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 `T0TDPREDEFFECTstd`; Table 2, p. 12; p. 16; footnote 4; Eq. 3, p. 5; 2017-era ctsem `summary.ctsemFit.R`; JSS PDF re-opened 2026-08-23T21:34Z from https://www.jstatsoft.org/index.php/jss/article/download/v077i05/1104) scalar standardised first-occasion time-dependent predictor effect. Table 3 names `T0TDPREDEFFECT` the effect of time-dependent predictors on latents at `T0`. Table 2 names `M` `TDPREDEFFECT` and names `T0TDPREDCOV` the first-occasion covariance, not this coefficient. 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 time-dependent predictor variance `v`, not `TIPREDVAR`. 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_m · √v / √p_0`. Unstandardised `t0_m` is defined for a zero coefficient and for zero predictor variance; standardised `T0TDPREDEFFECT` 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 `m · √v / √(-q / (2 a))` uses `asymDIFFUSION` and is not this first-occasion map. `T0TIPREDEFFECTstd` `t0_b · √v / √p_0` is a different named matrix even when `t0_m = t0_b` and the predictor variances match. Section 7.1 warns that omitting trait variance confounds between- and within-person information. `t0_m · √v / √(trait + p_0 + added)` uses the total, not free `T0VAR`, and is not `T0TDPREDEFFECTstd` 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 2026-08-23T21:34Z: `is_oa: false`; title *Measurement Invariance, Factor Analysis and Factorial Invariance*). Mislevy (1991, *Psychometrika, 56*, 177–196) remains unread (Unpaywall 2026-08-23T21:34Z: `is_oa: false`; title *Randomization-Based Inference about Latent Variables from Complex Samples*).
 - `psychometric_core` recovers the Driver, Oud, and Voelkle (2017, p. 16 `TDPREDEFFECTstd`; Table 2, p. 12; Eq. 3, p. 5; footnote 4; §7.1, pp. 18–19; JSS PDF re-opened 2026-08-23T21:10Z from https://www.jstatsoft.org/index.php/jss/article/download/v077i05/1104) scalar standardised continuous time-dependent predictor effect. Page 16 prints continuous-time parameters and, when appropriate, standardised matrices with the suffix `std`. Table 2 names `M` `TDPREDEFFECT`. Footnote 4: standardisations use only the relevant variance, not the total. The affecting variance is time-dependent predictor variance `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 `m · √v / √(-q / (2 a))`. Unstandardised `M` is defined for a zero coefficient and for zero predictor variance; standardised `TDPREDEFFECT` is not. Zero `asymDIFFUSION` or zero `v` has no positive SD and fails closed. `TIPREDEFFECTstd` `B · √v / √p` is a different named matrix even when `M = B` and the predictor variances match. The finite-interval intercept-style standardisation `A^{-1}[e^{A Δt} − I] M · √v / √p` depends on the event interval and is not this continuous Dirac coefficient. Section 7.1 warns that omitting trait variance confounds between- and within-person information. `m · √v / √(trait + p + added)` uses the total, not `asymDIFFUSION`, and is not `TDPREDEFFECTstd` when `TRAITVAR` is nonzero. `TRAITVAR` is not the standardisation variance. Predecessor `event_time.rs` asymptotic-std `process_sd == 0` / `predictor_sd == 0` gates after already-checked `within == 0` and `v == 0` were unreachable; this slice drops them so stacked line/branch coverage can close. Still not a Kalman filter, not a matrix `expm`, not ESEM estimation, not DSEM, and not ctsem estimation. Meredith (1993) remains unread (Unpaywall 2026-08-23T21:10Z: `is_oa: false`; title *Measurement Invariance, Factor Analysis and Factorial Invariance*). Mislevy (1991, *Psychometrika, 56*, 177–196) remains unread (Unpaywall 2026-08-23T21:10Z: `is_oa: false`; title *Randomization-Based Inference about Latent Variables from Complex Samples*).

crates/psychometric_core/src/error.rs

@@ -955,6 +955,24 @@ pub enum PsychometricError {
     /// `TRAITVAR`. Section 4.3 `TRAITVAR` is a zero-drift latent
     /// process, not first-occasion TD extra variance.
     InitialTimeDependentVarianceIsNotTraitVariance,
+    /// Driver Eq. 5 of the analog first-occasion TD extra
+    /// `λ² t0_m² v` was treated as the latent extra `t0_m² v`. The
+    /// observed extra is not the latent extra.
+    InitialTimeDependentObservedVarianceIsNotInitialTimeDependentVariance,
+    /// Driver Eq. 5 of the analog first-occasion TD extra
+    /// `λ² t0_m² v` was treated as first-occasion observed variance
+    /// `λ² p_0 + θ`. The extra is not the full first-occasion
+    /// `Var(y_0)`.
+    InitialTimeDependentObservedVarianceIsNotInitialObservedVariance,
+    /// Driver Eq. 5 of the analog first-occasion TD extra
+    /// `λ² t0_m² v` was treated as Eq. 5 of `addedT0TIPREDVAR`
+    /// `λ² t0_b² v`. Equal numbers when `t0_m = t0_b` are still
+    /// distinct named quantities.
+    InitialTimeDependentObservedVarianceIsNotInitialTimeIndependentObservedVariance,
+    /// Driver Eq. 5 of the analog first-occasion TD extra
+    /// `λ² t0_m² v` was treated as `MANIFESTVAR` `θ`. Measurement
+    /// error is not extra observed TD variance.
+    InitialTimeDependentObservedVarianceIsNotMeasurementError,
 }
 
 impl fmt::Display for PsychometricError {
@@ -1657,6 +1675,18 @@ impl fmt::Display for PsychometricError {
             Self::InitialTimeDependentVarianceIsNotTraitVariance => {
                 "initial time-dependent predictor variance is not trait variance"
             }
+            Self::InitialTimeDependentObservedVarianceIsNotInitialTimeDependentVariance => {
+                "initial time-dependent observed predictor variance is not initial time-dependent predictor variance"
+            }
+            Self::InitialTimeDependentObservedVarianceIsNotInitialObservedVariance => {
+                "initial time-dependent observed predictor variance is not initial observed variance"
+            }
+            Self::InitialTimeDependentObservedVarianceIsNotInitialTimeIndependentObservedVariance => {
+                "initial time-dependent observed predictor variance is not initial time-independent observed predictor variance"
+            }
+            Self::InitialTimeDependentObservedVarianceIsNotMeasurementError => {
+                "initial time-dependent observed predictor variance is not measurement error"
+            }
         };
         formatter.write_str(message)
     }
@@ -2850,4 +2880,28 @@ mod tests {
             "initial time-dependent predictor variance is not trait variance"
         );
     }
+
+    #[test]
+    fn initial_time_dependent_observed_variance_boundary_messages_are_stable() {
+        assert_eq!(
+            PsychometricError::InitialTimeDependentObservedVarianceIsNotInitialTimeDependentVariance
+                .to_string(),
+            "initial time-dependent observed predictor variance is not initial time-dependent predictor variance"
+        );
+        assert_eq!(
+            PsychometricError::InitialTimeDependentObservedVarianceIsNotInitialObservedVariance
+                .to_string(),
+            "initial time-dependent observed predictor variance is not initial observed variance"
+        );
+        assert_eq!(
+            PsychometricError::InitialTimeDependentObservedVarianceIsNotInitialTimeIndependentObservedVariance
+                .to_string(),
+            "initial time-dependent observed predictor variance is not initial time-independent observed predictor variance"
+        );
+        assert_eq!(
+            PsychometricError::InitialTimeDependentObservedVarianceIsNotMeasurementError
+                .to_string(),
+            "initial time-dependent observed predictor variance is not measurement error"
+        );
+    }
 }