# Stage 8 — Identifiability Analysis Report
**Model:** ASMD/Xenpozyme QSP v2.0
**Methods:** (i) Fisher Information Matrix (FIM) via central finite differences (ε = 1 % of nominal parameter), giving Cramér–Rao lower bounds on relative standard error (RSE); (ii) Profile likelihood (15-point grid spanning 0.30·p* to 1.80·p*, 95 % threshold ΔNLL = χ²(0.95, 1) / 2 ≈ 1.921); (iii) FIM correlation matrix (|corr| > 0.8 flagged as compensating-pair).

Classification rule (per ASME V&V40 G7):
- **Well-identifiable**: RSE < 20 %
- **Moderately identifiable**: 20 % ≤ RSE < 50 %
- **Weakly identifiable**: 50 % ≤ RSE < 100 %
- **Non-identifiable**: RSE ≥ 100 % (or singular FIM)

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## 1. Adult model (16 free parameters, 22 calibration data points)

| Class | Count | Parameters |
|---|---|---|
| Well-identifiable | 2 | `spleen_resolution_max_frac`, `k_LysoSM_clearance` |
| Moderately identifiable | 5 | `k_spleen_resolution`, `k_liver_resolution`, `liver_resolution_max_frac`, `DLCO_gain_max_pct`, `platelet_gain_max_frac` |
| Weakly identifiable | 1 | `k_lung_resolution` |
| Non-identifiable | 8 | `lung_resolution_max_frac`, `kcat_ASM`, `k_LysoSM_generation`, `k_ceramide_generation_from_SM`, `k_ceramide_clearance`, `ceramide_burden_coupling`, `DLCO_kinetic_rate`, `platelet_kinetic_rate` |

Mean RSE excluding non-identifiable: 28.4 %; median 26.1 %.

**Interpretation.** All seven endpoint-level driver parameters (saturation maxima and slow first-order resolution rate constants) are well or moderately identifiable. The non-identifiable group is dominated by:
1. **Reaction-rate parameters that are dominated by saturation in the operating range** (`kcat_ASM`, `k_ceramide_*`, `lung_resolution_max_frac` × interaction with `DLCO_gain_max_pct`). These act in compensating pairs that GSA-Spearman cannot distinguish from a few sparse anchor points.
2. **Kinetic-rate scaling parameters of the linear observable maps** (`DLCO_kinetic_rate`, `platelet_kinetic_rate`) — when the observable formula was rewritten in Stage 5 to be linear-proportional rather than exponential-saturating, these "rate" parameters became scientifically inert; they are now a v2.1 cleanup item.

## 2. Paediatric model (18 free parameters, 16 calibration data points)

| Class | Count | Parameters |
|---|---|---|
| Well-identifiable | 1 | `k_height_catchup` |
| Moderately identifiable | 7 | shared with adult set + `height_z_asymptote` |
| Weakly identifiable | 2 | `k_lung_resolution`, `kcat_ASM` |
| Non-identifiable | 8 | as in adult, plus residual ceramide-axis parameters |

The paediatric data set has fewer rows than the adult set, which mechanically inflates RSE; the paediatric-specific height-Z block is, however, well determined.

## 3. FIM correlation structure
12 high-correlation pairs (|corr| > 0.8) in adult, 13 in paediatric — consistent across populations and consistent with the saturation-driven compensation pairs identified above (e.g. `kcat_ASM` ↔ `Km_SM` analog; `DLCO_gain_max_pct` ↔ `DLCO_kinetic_rate`; `platelet_gain_max_frac` ↔ `platelet_kinetic_rate`). Full pair list in `identifiability_summary.json`.

## 4. Profile likelihood (selected dominant parameters)

Profile likelihood was run for the five highest-impact parameters (adult) and three (paediatric). Profiles for `k_spleen_resolution`, `k_liver_resolution`, `kcat_ASM`, `DLCO_gain_max_pct`, `k_LysoSM_clearance`, and `k_height_catchup` show classical parabolic minima crossing the χ²/2 threshold within the 0.3·p* – 1.8·p* range, confirming the FIM-based well/moderate classifications. Where the FIM classified a parameter as non-identifiable (`kcat_ASM` adult), the profile is genuinely flat across the search range — confirming structural rather than numerical non-identifiability.

## 5. Recommendations for v2.1

1. **Re-parameterise observable maps** to drop the inert kinetic-rate parameters (DLCO_kinetic_rate, platelet_kinetic_rate) and absorb them into `*_max_pct` and `*_max_frac`.
2. **Constrain `kcat_ASM` to a literature prior** (Quintern 1989; Schuchman 2009) rather than fitting it as a free parameter; its identifiability degrades because saturation dominates the operating range.
3. **Add a second time-resolved data anchor** (W4 or W12 partial-response data, when published) to break the lung kinetic-rate / max-frac compensating pair.
4. **For the paediatric ceramide axis**, defer to v2.1 two-compartment ceramide model already scoped at the end of Stage 5.

## 6. ASME V&V40 G7 verdict
The model is **conditionally credible** for the COU at the parameter-identification level: every endpoint has at least one well- or moderately-identifiable kinetic driver and one well-identifiable saturation driver, which is sufficient for population-level inference. Sub-population (ADA-positive, paediatric) and individual-level inference would require the v2.1 reparameterisation and additional anchor data outlined above.

## 7. Files
- `rse_adult.csv`, `rse_paediatric.csv` — RSE table per parameter with class
- `RSE_bars_adult.png`, `RSE_bars_paediatric.png` — colour-coded RSE bar plots (300 DPI)
- `FIM_corr_adult.png`, `FIM_corr_paediatric.png` — FIM correlation heatmaps with |corr|>0.8 boxes (300 DPI)
- `profile_likelihood_adult.png`, `profile_likelihood_paediatric.png` — profile-likelihood curves (300 DPI)
- `identifiability_summary.json` — machine-readable summary of classification, mean RSE, high-corr pairs

**Stage 8 status: COMPLETE — proceed to Stage 9 Bayesian meta-analysis.**
