Prognostic Index Normed (PIN) Model in HD

• Prognostic Index Normed (PIN) model is a semiparametric tool employing a Cox proportional‐hazards framework with motor, cognitive, and genetic predictors to estimate time to Huntington disease diagnosis.
• The model normalizes risk scores to zero mean and unit variance, offering a platform-agnostic approach that enables consistent risk thresholds and precise sample-size estimation.
• Empirical evaluations demonstrate robust predictive accuracy (C-index ≈0.83–0.84) with moderate operational complexity compared to higher-dimensional models.

The Prognostic Index Normed (PIN) model is a semiparametric risk stratification tool for modeling time to clinical diagnosis in Huntington disease (HD) settings, designed for use in highly censored datasets and applicable to preventative clinical trial design. The model leverages baseline clinical and genetic data to generate a normalized prognostic index, facilitating risk enrichment and sample-size estimation in multi-year HD studies. PIN is derived from a Cox proportional-hazards framework incorporating motor, cognitive, and genetic–age predictors, then rescales the risk scores for cohort generalizability. Recent comparative research has validated its discrimination ability and logistical benefits.

1. Mathematical Formulation

The PIN model fits a Cox proportional-hazards regression for the interval $X_i$ from paper entry to HD diagnosis, conditioned on three covariates:

• TMS$_i$: UHDRS Total Motor Score at enrollment,
• SDMT$_i$: Symbol-Digit Modalities Test at enrollment,
• CAP$_i$: CAG-Age Product, defined as $\text{Age}_i \times (\text{CAG}_i-34)$.

The conditional hazard at time $t$ is defined as: $$\lambda_X(t\mid \mathbf Z_i) = \lambda_0(t) \exp(\beta_1 \mathrm{TMS}_i + \beta_2 \mathrm{SDMT}_i + \beta_3 \mathrm{CAP}_i)$$ where $\lambda_0(t)$ is an unspecified baseline hazard and $$\boldsymbol\beta = (\beta_1, \beta_2, \beta_3)^\top$$ are the covariate-specific log-hazard ratios.

The unnormalized prognostic index for subject $i$ is: $$\mathrm{PI}_i = \hat\beta_1\,\mathrm{TMS}_i + \hat\beta_2\,\mathrm{SDMT}_i + \hat\beta_3\,\mathrm{CAP}_i$$

This specification reflects the model’s simplicity—three predictors capture motor, cognitive, and genetic–age risk aspects, with no additional penalization or adjustment for longitudinal changes.

2. Rationale and Mechanics of Normalization

PIN refers to a linear normalization of $\mathrm{PI}_i$ to have zero mean and unit variance in the training set. Given

$$\mu_{\rm train} = \frac{1}{N} \sum_{i=1}^N \mathrm{PI}_i^{(\rm train)}, \quad \sigma_{\rm train} = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (\mathrm{PI}_i^{(\rm train)} - \mu_{\rm train})^2}$$

the normalized index is: $$\mathrm{PIN}_i = \frac{\mathrm{PI}_i - \mu_{\rm train}}{\sigma_{\rm train}}$$

The normalization confers the following properties:

• A one-unit PIN increment corresponds to one standard deviation of risk in the original training cohort.
• PIN>0 partitions the cohort into the higher-risk half by construction.
• Risk thresholds and enrichment criteria become platform-agnostic, permitting consistent application across sites and future studies.

This approach enables PIN-based stratification for trial recruitment and risk grouping without site-specific refitting or calibration, which is particularly advantageous when merging legacy and contemporary cohorts.

3. Parameter Estimation Protocol

PIN is fit to pooled longitudinal cohorts (PREDICT-HD, COHORT, TRACK-HD), subject to minimum CAG repeat ($\mathrm{CAG}\ge36$) and diagnostic certainty level ($\mathrm{DCL}<4$) at baseline. The pooled sample comprises $N=1421$ participants, with right censoring in 77% of cases. Parameters are estimated via partial likelihood in a semiparametric Cox model, without lasso or ridge penalization due to the low dimensionality (three covariates). The baseline hazard $\lambda_0(t)$ is estimated via the Breslow method, yielding cohort-generalizable coefficients. This fitting approach ensures high transportability and interpretability across diverse paper populations.

4. Assessment of Predictive Accuracy under Heavy Censoring

PIN’s discrimination is externally validated on the ENROLL-HD dataset ($n=3113$, 77% censored) using metrics that correct for heavy censoring:

• Uno’s C-statistic (2011):

$$C_{\rm Uno}(\tau) = \frac{\sum_{i \neq j} \Delta_i\,\widehat G(W_i)^{-2}\,1\{W_i < W_j, W_i < \tau, \mathrm{PIN}_i > \mathrm{PIN}_j\}}{\sum_{i \neq j} \Delta_i\,\widehat G(W_i)^{-2}\,1\{W_i < W_j, W_i < \tau\}}$$

where $W_i = \min(X_i, C_i)$, $\Delta_i = 1\{X_i \leq C_i\}$, and $\widehat G$ is the Kaplan–Meier estimator of the censoring survival.

• Time-dependent ROC/AUC (Heagerty et al., 2000):

Time-dependent TPR($q,\tau$), FPR($q,\tau$), and integrated AUC($\tau$), all adjusted using Kaplan–Meier estimates to overcome censoring bias.

Empirical results indicate a PIN C-index of ≈0.83–0.84, closely trailing the MRS (≈0.86), and exceeding CAP and Langbehn models (≈0.80).

5. Comparative Performance and Operational Considerations

Model	Covariates	C-index (ENROLL-HD)	Logistical Burden
MRS	8–10	≈0.86	High
PIN	4	≈0.83–0.84	Moderate
CAP	2	≈0.80	Low
Langbehn	2 (parametric)	≈0.80	Low

PIN offers nearly equivalent risk ranking as MRS—with half the covariate input and generalizable coefficients trained over merged cohorts. CAP and Langbehn models remain defensible where only limited baseline data are available. PIN’s principal limitation is reliance on the proportional hazards assumption and static baseline predictors, rendering it less suited for scenarios with substantial time-dependent covariate variation.

6. PIN-based Sample Size Estimation for Preventative HD Trials

PIN facilitates trial sample size calculations via ROC-optimized threshold enrichment:

• Select trial duration $T$ (e.g., 2–5 years).
• From PIN-adjusted ROC at $T$, identify threshold $q^*$ maximizing Youden’s $$J(q,T) = \mathrm{TPR}(q,T) - (1 - \mathrm{FPR}(q,T))$$.
• Estimate event rate in untreated arm:

$$\hat p_0 = \mathrm{TPR}(q^*,T) \times \Pr(\mathrm{PIN} \geq q^*)$$

• For predicted risk reduction $\varepsilon$ (e.g., 30–50%):

$\hat p_1 = (1 - \varepsilon)\,\hat p_0$

• Compute sample size per arm:

$$n_{\rm per\,arm} = \frac{(Z_{1-\alpha/2} + Z_{1-\beta})^2 [\hat p_0 (1-\hat p_0) + \hat p_1 (1-\hat p_1)]}{(\hat p_0 - \hat p_1)^2}$$

for $\alpha = 0.05$, $1-\beta = 0.80$.

PIN-based enrichment for a 3-year trial requires approximately 140–250 patients per arm (contingent on hypothesized effect size), substantially fewer than unenriched recruitment and only modestly more than the MRS-based design. The benefit lies in reduced measurement effort and analysis generalizability, with only a small penalty in statistical power.

7. Significance and Prospects

PIN exemplifies a practical compromise between accuracy and operational simplicity in HD risk modeling. It provides a cohort-generalizable, interpretable, four-variable index that is readily normalized, enabling the calibration of enrichment thresholds and sample size estimates for multicenter preventative trials. The model’s external validation under extreme censoring and systematic comparison to competing approaches reinforce its utility, particularly where measurement resources are limited or harmonization constraints preclude high-dimensional input. A plausible implication is that the adoption of PIN may mitigate underpowered trial designs that have resulted from earlier event rate misestimates. However, scenarios with dynamic or multidimensional risk profiles may require alternative or more flexible modeling frameworks.