Quantile Residual Diagnostics

• Quantile residual approach is a technique that transforms model-based cumulative probabilities into standard normal variates, providing a unified diagnostic tool for various models.
• It facilitates model assessment in regression, classification, and probabilistic settings, particularly when responses are non-normal, bounded, or discrete.
• The method supports effective outlier detection and model checking with adjustments for discrete data and leverage, ensuring a robust evaluation process.

The quantile residual approach is a foundational and increasingly dominant paradigm for diagnostic evaluation of regression, classification, and other probabilistic models, particularly when the conditional distribution of the response deviates from normality or possesses boundaries, discreteness, or other complexities that challenge classical residual analyses. Rather than relying on raw or standardized residuals with complex or non-normal distributions, quantile residuals transform model-based cumulative probabilities to the standard normal scale, providing a theoretically grounded, broadly applicable mechanism for model assessment, outlier detection, and the unification of diagnostics across diverse statistical settings.

1. Formal Definition and Foundation

The quantile residual, introduced in the context of generalized linear and related models, is mathematically defined for a fitted model in terms of the model's cumulative distribution function (CDF) and the standard normal quantile function. For a continuous response variable $Y_i$ with fitted cumulative distribution $F_{Y_i}(y_i;\hat\theta_i)$, the quantile residual is given by

$$r_i^q = \Phi^{-1}\bigl( F_{Y_i}(y_i; \hat\theta_i) \bigr)$$

where $\Phi^{-1}$ is the quantile function (inverse CDF) of the standard normal distribution, and $\hat\theta_i$ are fitted model parameters (Pereira, 2017, Scudilio et al., 2017). In the discrete response case, to account for CDF discontinuities, one employs a randomized version:

$$r_i^q = \Phi^{-1}\bigl( U_i \bigr), \qquad U_i \sim \mathrm{Uniform}[F_{Y^*}(y_i^-;\hat\theta_i), F_{Y^*}(y_i;\hat\theta_i)]$$

where $F_{Y^*}(y_i^-)$ is the left-limit of the discrete CDF at $y_i$ (Araripe et al., 2023, Padellini et al., 2018).

The validity of this approach rests on the probability integral transform: under model correctness and continuity, $F_{Y_i}(Y_i)$ is uniform, and $\Phi^{-1}$ then yields standard normal variates. For fitted models, this normality is approximate but improves as sample size increases and model specification is accurate.

2. Construction in Specific Model Classes

Beta Regression

For $Y_i\sim\mathrm{Beta}(\mu_i,\phi)$, with link $g(\mu_i)=x_i^\top\beta$, the fitted CDF $F_{\mathrm{Beta}}(y_i; \hat\mu_i, \hat\phi)$ is computed using the regularized incomplete beta function. The quantile residual formula becomes

$$r_i^q = \Phi^{-1}\!\bigl[ I_{y_i}(\hat\mu_i\hat\phi,\,(1-\hat\mu_i)\hat\phi) \bigr]$$

where $I_x(a,b)$ is the regularized incomplete beta function (Pereira, 2017).

Generalized Linear Models (GLMs)

In the GLM context, the quantile residual is

$r_i^{qu} = \Phi^{-1}\{F_{Y_i}(y_i;\hat\theta_i)\}$

where $F_{Y_i}$ is the CDF under the fitted GLM. To address heteroskedasticity and leverages, an adjusted quantile residual is recommended:

$r_i^{*qu} = r_i^{qu}/\sqrt{1-\hat{h}_{ii}}$

where $\hat{h}_{ii}$ is the i-th diagonal entry of the hat matrix (Scudilio et al., 2017).

Categorical and Multinomial Models

For polytomous response $y_i\in\{0,1\}^J$, the randomized quantile residual is constructed using model CDF and a uniform randomization over the mass at the observed outcome:

$$r_i^Q = \Phi^{-1}\bigl( F^*(y_i,u_i; \hat\pi_i) \bigr)$$

with $$F^*(y_i,u_i; \hat\pi_i) = F(y_i-1; \hat\pi_i) + u_i f(y_i; \hat\pi_i)$$ and $u_i\sim\mathrm{Uniform}(0,1)$ (Araripe et al., 2023).

Quantile Residual Lifetime (QRL)

In survival analysis, the quantile residual lifetime at landmark $s$ and quantile $q$ is

$$QRL_q(s) = \inf\{ t : P(T-s \leq t\,|\,T>s) \geq q \}$$

which is used as a regression target for residual life modeling in censored/multivariate event data (Yu et al., 2 Mar 2025).

3. Distributional Properties and Diagnostics

Under correct specification, quantile residuals are asymptotically standard normal irrespective of the response distribution or link function (subject to regularity and sample size) (Pereira, 2017, Scudilio et al., 2017, Araripe et al., 2023). This property underpins universal diagnostic tools: QQ-plots versus $\mathcal{N}(0,1)$, normality tests (e.g., Anderson–Darling, Shapiro–Wilk), standardized residual-fitted plots, and simulation-based envelope visualizations.

For common alternatives:

• Classical standardized or deviance residuals in GLMs may have substantially non-normal distributions in finite samples, especially for non-Gaussian responses or small $n$ (Scudilio et al., 2017).
• Weighted or adjusted residuals in beta regression typically show skewness, variance distortion, or improper kurtosis, particularly near boundaries or low $\phi$ (Pereira, 2017).
• Pearson and deviance residuals in multinomial models yield vectors with intractable, non-universal distributions (Araripe et al., 2023).

Quantile residuals therefore enable interpretable and statistically well-calibrated assessment, powerful for detecting misspecification and outliers.

4. Comparative Simulation and Empirical Evidence

Monte Carlo simulations across contexts consistently demonstrate that quantile residuals:

• Exhibit means ≈ 0, variances ≈ 1, skewness ≈ 0, kurtosis ≈ 3 (i.e., close to standard normal);
• Yield the lowest values of normality-distance metrics (e.g., Anderson–Darling statistic) compared to classical alternatives (Pereira, 2017, Scudilio et al., 2017);
• Retain normal approximation even for modest $n$ (e.g., $n=16$ in beta regression), and perform well across a range of precision, link, and covariate scenarios (Pereira, 2017);
• Facilitate normality-based outlier detection without manual re-centering or moment correction (Scudilio et al., 2017);
• Are effective in group or vector-valued discrete models when coupled with scalar distance summaries (Euclidean or Mahalanobis distances) (Araripe et al., 2023).

These findings are corroborated by applications: e.g., in beta regression, quantile residuals correctly flag lack of fit in model-misspecified applications, remain well-behaved under correct models, and offer interpretable diagnostics even when classical residuals suggest spurious lack of fit (Pereira, 2017).

5. Methodological and Computational Aspects

Computation of quantile residuals requires:

• Evaluation of the fitted model CDF at observed responses for continuous and continuous-interpolated models;
• (For discrete outcomes) Randomization within the CDF jump or "model-aware" continuous interpolations to allow valid normal transformations (Padellini et al., 2018, Araripe et al., 2023);
• Leverage-based adjustments for models with non-constant variance across predictions (Scudilio et al., 2017).

In high-dimensional or large-sample problems, quantile residuals can be paired with scalable algorithms—for example, using quantile-based cutoffs for robust randomized solvers in linear systems with sparse corruption (Haddock et al., 2023).

Hybrid approaches such as the residual-quantile adjustment (RQA) for physics-informed neural networks use residual quantiles for reweighting loss contributions, providing a robust mechanism to suppress the influence of extreme error regions and stabilize training (Han et al., 2022).

6. Extensions and Related Methodologies

The quantile residual framework generalizes to:

• Survival models (as in quantile residual lifetime regression), where quantiles of residual life are regressed against covariates with robust large-sample properties and effective variance estimation (Yu et al., 2 Mar 2025);
• Endogenous models, where quantile residuals derived from a first-stage quantile regression serve as control functions for consistent estimation in quantile-specific settings, including censored and heteroskedastic models (Kobayashi, 2015);
• Categorical and grouped data, where randomization and univariate distance functions allow deployment of residual-based assessments in settings where alternative residuals are inherently multivariate or not well-calibrated for diagnostics (Araripe et al., 2023).

Tables summarizing key domains and advantages:

Model Class	Quantile Residual Construction	Diagnostic Strengths
Beta regression	$\Phi^{-1}(F_\mathrm{Beta}(y;\hat\mu,\hat\phi))$	Closest to $\mathcal{N}(0,1)$, robust to mean/precision
GLMs	$\Phi^{-1}(F(y;\hat\theta))$, leverage adjustment	Uniform applicability, normality in small/large $n$
Categorical/multinomial	Randomized CDF, univariate transformations	Scalar diagnostics from multivariate models
Survival/QRL	Inversion of survival function, empirical weights	Estimation under censoring, interpretability

7. Practical Recommendations and Limitations

Empirical and theoretical evidence converge on the following guidance:

• Plot and test quantile residuals as the primary diagnostic for model checking in non-normal, bounded, or discrete outcome regression;
• Employ adjusted (leverage-corrected) quantile residuals in GLMs, especially for small $n$ or non-canonical links (Scudilio et al., 2017);
• For grouped or multivariate responses, use quantile residuals plus scalar distance reductions for practical diagnostics (Araripe et al., 2023);
• In high-dimensional or adversarial settings, use quantile-threshold-based robust algorithms (e.g., in linear solvers or PINNs) for computational stability (Haddock et al., 2023, Han et al., 2022).

Quantile residuals remain sensitive to inaccuracies in model CDF estimation; in very small samples or for complex data structures (e.g., longitudinal, hierarchical), further validation is needed to calibrate normality approximations and the impact of randomization (Araripe et al., 2023).

• Pereira, G.H.A., "On quantile residuals in beta regression" (Pereira, 2017)
• S. X. Yu, Y. Xiang, J.H. Jeong, "Quantile Residual Lifetime Regression for Multivariate Failure Time Data" (Yu et al., 2 Mar 2025)
• Nascimento et al., "Diagnostics for categorical response models based on quantile residuals and distance measures" (Araripe et al., 2023)
• Mansour et al., "Adjusted quantile residual for generalized linear models" (Scudilio et al., 2017)
• Kobayashi, G., "Bayesian Endogenous Tobit Quantile Regression" (Kobayashi, 2015)
• Martins et al., "Model-aware Quantile Regression for Discrete Data" (Padellini et al., 2018)
• Wang et al., "Residual-Quantile Adjustment for Adaptive Training of Physics-informed Neural Network" (Han et al., 2022)
• Fattahi et al., "On Subsampled Quantile Randomized Kaczmarz" (Haddock et al., 2023)