Effect-Wise Inference for Smoothing Spline ANOVA on Tensor-Product Sobolev Space

TL;DR

This work develops a unified framework for effect-wise inference in smoothing spline ANOVA on tensor-product Sobolev spaces, enabling independent inference for each main effect and interaction within a nonparametric model. It establishes a functional Bahadur representation, deriving per-effect convergence rates and asymptotic distributions for both local confidence intervals and global Wald-type tests, with minimax optimality up to logarithmic factors. A key theoretical device is the orthogonality of effect subspaces under both the data-driven inner product and the smoothing penalty, which allows effect-wise analysis even with interactions. The authors demonstrate superior finite-sample performance through simulations and apply the method to the Colorado temperature dataset, yielding interpretable local and global inferences that align with known geographic and seasonal patterns. The approach offers principled tools for effect selection and interpretation in flexible nonparametric models and lays groundwork for extensions to high-dimensional settings and function-on-function regression.

Abstract

Functional ANOVA provides a nonparametric modeling framework for multivariate covariates, enabling flexible estimation and interpretation of effect functions such as main effects and interaction effects. However, effect-wise inference in such models remains challenging. Existing methods focus primarily on inference for entire functions rather than individual effects. Methods addressing effect-wise inference face substantial limitations: the inability to accommodate interactions, a lack of rigorous theoretical foundations, or restriction to pointwise inference. To address these limitations, we develop a unified framework for effect-wise inference in smoothing spline ANOVA on a subspace of tensor product Sobolev space. For each effect function, we establish rates of convergence, pointwise confidence intervals, and a Wald-type test for whether the effect is zero, with power achieving the minimax distinguishable rate up to a logarithmic factor. Main effects achieve the optimal univariate rates, and interactions achieve optimal rates up to logarithmic factors. The theoretical foundation relies on an orthogonality decomposition of effect subspaces, which enables the extension of the functional Bahadur representation framework to effect-wise inference in smoothing spline ANOVA with interactions. Simulation studies and real-data application to the Colorado temperature dataset demonstrate superior performance compared to existing methods.

Figures (10)

• Figure 4.1: RMISE of effect-wise estimators. Each panel corresponds to a specific effect $S$.
• Figure 4.2: Interval length of effect-wise confidence intervals. Each panel corresponds to a specific effect $S$. Boxplots display the distribution of interval lengths across replicates for ssaec and ssaebc, and the empirical length is indicated by a dot.
• Figure 4.3: Coverage of effect-wise confidence intervals. Each panel corresponds to a specific effect $S$. Coverages for ssaec and ssaebc are averaged over replicates. The dashed horizontal line indicates the nominal level $1 - \alpha = 0.95$.
• Figure 4.4: Empirical size of effect-wise Wald-type tests. Each panel corresponds to a specific effect $S$, representing scenarios for testing $\mathrm{H}_{0,S}: f^\ast_S = 0$ with $\rho_S = 0$. The dashed horizontal line indicates the significance level $\alpha = 0.05$.
• Figure 4.5: Empirical power of effect-wise Wald-type tests for main effects. Each row corresponds to a specific main effect $S$, representing scenarios for testing $\mathrm{H}_{0,S}: f^\ast_S = 0$ with $\rho_S \in \{0.3, 0.4, 0.5\}$.

Theorems & Definitions (28)

• Lemma 2.1
• Corollary 1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Corollary 2
• Theorem 3.1
• Remark 3.1
• Theorem 3.2
• Theorem 3.3