Outcome-Calibrated Regression and Predicted Outcome-Based Inference

Technical Summary: Outcome-Calibrated Regression and Predicted Outcome-Based Inference

Problem Statement
Ordinary Least Squares (OLS) regression is the standard tool for scientific inference, prized for its property as the Best Linear Unbiased Estimator (BLUE) under the Gauss-Markov theorem. This property ensures that predictions are conditionally unbiased given the covariates ($X$), i.e., $E(\hat{Y} - Y | X = x) = 0$. However, the authors identify a critical, often-overlooked deficiency: OLS predictions are generally not conditionally unbiased given the outcome ($Y$).

Due to the minimization of mean squared error, OLS predictions systematically shrink toward the population mean of the outcome ($\mu_Y$). This results in a "regression-to-the-mean" (RTM) phenomenon where large outcome values are underpredicted and small outcome values are overpredicted. Mathematically, the conditional prediction bias is $E(\hat{Y}_{OLS} - Y | Y = y) = (\eta_{OLS} - 1)(y - \mu_Y)$, where $\eta_{OLS} < 1$ represents the population slope of regressing predictions on true outcomes (equivalent to the population $R^2$).

This bias creates a nonignorable problem for predicted outcome-based inference, a common practice in fields like brain-age analysis, polygenic risk score studies, and causal inference where the true outcome $Y$ is unobserved or expensive to measure, and inference is performed using the predicted value $\hat{Y}$ against a variable $W$. The authors demonstrate that this shrinkage propagates into downstream analysis, systematically attenuating the estimated association between $\hat{Y}$ and $W$, leading to biased effect estimates and potentially erroneous scientific conclusions.

Methodology: Outcome-Calibrated Regression (OCR)
To resolve this, the authors propose Outcome-Calibrated Regression (OCR), a new framework that enforces outcome-conditional unbiasedness directly during model fitting.

1. Calibration Constraints: The method imposes two linear equality constraints on the regression coefficients $\beta$:

   • Unit Slope: The slope of the regression of predicted values on observed outcomes must be 1 ($\hat{\eta} = 1$). This ensures the covariance between predictions and observations equals the variance of the observations.
   • Zero Intercept: The intercept of this calibration regression must be 0 ($\hat{\alpha} = 0$). This ensures the mean of the fitted values matches the mean of the observed outcomes.

2. Closed-Form Solution: OCR is formulated as a constrained least squares problem:
   $$\hat{\beta}_{OCR} = \arg\min_{\beta} \frac{1}{2n}\|y - X\beta\|_2^2 \quad \text{s.t.} \quad A\beta = c$$
   where $A\beta = c$ encodes the slope and intercept constraints. The authors derive a closed-form solution using Lagrange multipliers, showing that the OCR estimator is a linear adjustment of the OLS estimator:
   $$\hat{\beta}_{OCR} = \hat{\beta}_{OLS} - K(A\hat{\beta}_{OLS} - c)$$
   Here, $K$ acts as a calibration correction matrix that maps the constraint violation of the OLS estimator into a coefficient adjustment.

3. Theoretical Properties:

   • Constrained Gauss-Markov: Among all linear unbiased estimators satisfying the calibration constraints, $\hat{\beta}_{OCR}$ has the minimum variance.
   • Downstream Unbiasedness: Under the assumption that the residual $(\hat{Y}_{OCR} - Y)$ is independent of the external variable $W$ conditional on $Y$, the regression coefficient of $\hat{Y}_{OCR}$ on $W$ recovers the true population coefficient $\theta$ without attenuation.
   • Inference: The paper provides formulas for the variance-covariance matrix of $\hat{\beta}_{OCR}$, standard errors, and valid confidence/prediction intervals, noting that the variance is generally no larger than that of OLS due to the restriction of the parameter space.

Key Results
The authors validate OCR through simulation studies comparing it against OLS across various sample sizes ($n$), predictor dimensions ($p$), and noise levels ($\sigma$).

• Calibration: While OLS consistently exhibited calibration slopes ($\hat{\eta}$) significantly less than 1 (ranging from 0.335 to 0.837 depending on noise), OCR achieved exact calibration ($\hat{\eta} = 1.000, \hat{\alpha} = 0$) across all scenarios.
• Downstream Inference: When regressing predicted outcomes on an external variable $W$ to estimate the true association $\theta$:
  • OLS: Produced severely biased estimates (Bias $\approx -0.50$) with 0% empirical coverage of the 95% confidence interval. The bias-to-standard deviation ratio was extremely high, indicating systematic bias dominated sampling variability.
  • OCR: Produced approximately unbiased estimates ($|Bias| < 0.25$, often much lower) with empirical coverage rates close to the nominal 95% (ranging from 95.2% to 98.6%).
• Trade-off: The enforcement of calibration constraints results in a higher Mean Squared Error (MSE) for the predictions themselves compared to OLS, particularly under high noise conditions. However, this increase in prediction error is the necessary cost to eliminate the systematic bias in downstream inference.

Significance and Claims
The paper claims that the regression-to-the-mean bias in OLS is an inherent property of the method, not a finite-sample artifact, and that its consequences for downstream inference have been largely uncharacterized until now.

The significance of OCR lies in its ability to enable valid statistical inference when using predicted outcomes as surrogates for unobserved ground truths. The authors state that OCR is particularly relevant for:

• Causal effect estimation with predicted exposures or outcomes.
• Brain-age gap analysis and biological aging clocks.
• Polygenic risk score association studies.
• Two-sample Mendelian randomization.
• Federated or transfer learning settings where only model predictions are shared.

By replacing standard OLS predictions with OCR-corrected predictions, researchers can eliminate the systematic attenuation of effect sizes caused by regression-to-the-mean, thereby reducing bias in downstream scientific conclusions. The authors present OCR not as a replacement for OLS in all contexts, but as a necessary framework specifically when the goal involves inference based on predicted outcomes.