Spatially Robust Inference with Predicted and Missing at Random Labels

Stephen Salerno; Zhenke Wu; Tyler McCormick

stat.MLMar 11, 2026

Spatially Robust Inference with Predicted and Missing at Random Labels

Authors: Stephen Salerno, Zhenke Wu, Tyler McCormick

Organizations: Fred Hutch Cancer Center · University of Michigan · University of Washington

Abstract

When outcome data are expensive or onerous to collect, scientists increasingly substitute predictions from machine learning and AI models for unlabeled cases, a process which has consequences for downstream statistical inference. While recent methods provide valid uncertainty quantification under independent sampling, real-world applications involve missing at random (MAR) labeling and spatial dependence. For inference in this setting, we propose a doubly robust estimator with cross-fit nuisances. We show that cross-fitting induces fold-level correlation that distorts spatial variance estimators, producing unstable or overly conservative confidence intervals. To address this, we propose a jackknife spatial heteroscedasticity and autocorrelation consistent (HAC) variance correction that separates spatial dependence from fold-induced noise. Under standard identification and dependence conditions, the resulting intervals are asymptotically valid. Simulations and benchmark datasets show substantial improvement in finite-sample calibration, particularly under MAR labeling and clustered sampling.

Figures & tables

Figure 1: Synthetic spatial-field simulation (superpopulation target): boxplots of empirical 90% coverage versus dependence level $\sigma$ . Panels split by missingness arm (MCAR vs MAR) and sampling regime (iid vs soft-block). For each $\sigma$ , each box summarizes the distribution across 100 population replicates of empirical coverage computed from 200 repeated sample draws of size $n=600$ from a single population replicate. Methods shown are Cross-PPI, PPI++, Bootstrap-PPI, and Spatial DR-JK-HAC. The dashed line marks nominal coverage. Under MAR, baseline models under-cover even under iid sampling; under MCAR, these models under-cover when soft-block sampling makes spatial dependence visible. Spatial DR-JK-HAC stays near nominal in all four cells, at the cost of modestly wider intervals (see Appendix D ).

Panel: MAR
Dataset	Cross-PPI	PPI++ (IID)	Bootstrap-PPI	DR-IID (CF-m)	Spatial DR-HAC	Spatial DR-JK-HAC (K=5, $t$ )
Forest	0.613	0.619	0.575	0.825	0.825	0.925
Galaxies	0.775	0.875	0.825	0.825	0.825	0.875
Census income	0.631	0.831	0.613	0.725	0.744	0.806
Malaria	0.850	0.900	0.725	0.750	0.750	0.850
Health+	0.456	0.406	0.119	0.925	0.938	0.912

Table 1: Coverage for observed data auxiliary-split prediction design, averaged over IID and soft-block sampling.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Panel: MAR
Dataset	Spatial DR-HAC	Spatial DR-JK-HAC (K=5, $t$ )	Spatial DR-JK-HAC (K=10, $t$ )	Spatial DR-JK-HAC (K=10, $z$ )
Forest	0.825	0.925	0.838	0.819
Galaxies	0.825	0.875	0.838	0.825
Census income	0.744	0.806	0.787	0.769
Malaria	0.750	0.850	0.819	0.794
Health+	0.938	0.912	0.931	0.881

Appendix

Table 2: Observed-data auxiliary-split prediction design: superpopulation-style coverage, fold-count sensitivity for dependence-aware methods.

Panel: MAR
Dataset	Spatial DR-HAC	Spatial DR-JK-HAC (K=5, $t$ )	Spatial DR-JK-HAC (K=10, $t$ )	Spatial DR-JK-HAC (K=10, $z$ )
Forest	0.160	0.232	0.184	0.165
Galaxies	0.214	0.314	0.255	0.229
Census income	30273.567	41671.111	32924.730	29543.385
Malaria	0.823	1.140	0.931	0.847
Health+	11.337	15.016	11.098	9.984

Appendix

Table 3: Observed-data auxiliary-split prediction design: mean confidence-interval width by dataset and arm (nominal 90%).

Figure 2: Synthetic simulation (superpopulation target): confidence-interval length boxplots by missingness arm and sampling regime. The $x$ -axis is $\sigma\in\{0,40,80,120\}$ . Each box summarizes mean interval length over 200 repeated sample draws across 100 replicate populations. Purple shades denote DR-based methods (DR-iid, spatial DR-HAC, spatial DR-JK-HAC). The figure is self-contained: the dependence-aware methods inflate width relative to iid post-prediction baselines, and spatial DR-JK-HAC narrows intervals relative to spatial DR-HAC while retaining dependence-aware coverage.

Figure 3: Synthetic simulation: coverage boxplots for the full method menu. Methods include Cross-PPI, PPI++ (iid), Bootstrap-PPI (Efron 2025), DR-iid, spatial DR-HAC, and spatial DR-JK-HAC. The figure is self-contained: DR-iid and spatial DR-HAC improve over iid post-prediction baselines under dependence and MAR but do not match spatial DR-JK-HAC in the most challenging soft-block MAR cells, motivating the jackknife-HAC correction in the main text.

Figure 4: Replicate-level interval visualization under MAR and soft-block sampling. Each interval is centered by subtracting its replicate target mean, so the dashed vertical line at $0$ is the target in every panel. Cross-PPI (blue) shows systematic mis-centering in multiple panels, consistent with the MAR under-coverage seen in Figure 1 . Spatial DR-JK-HAC (purple) remains centered and attains near-nominal coverage.

Figure 5: Replicate-level interval visualization under MCAR and soft-block sampling, with the same centering convention as Figure 4 . Under MCAR, iid baselines are often closer to nominal than under MAR, but Cross-PPI still under-covers in the dependence-rich synthetic panel as dependence becomes visible under soft-block sampling. Spatial DR-JK-HAC remains near nominal across panels at the cost of wider intervals.

Dataset	Moran’s $I$	Moran p-value	NN residual corr.	NN p-value
Forest	0.146	0.003	0.134	0.003
Galaxies	-0.010	0.355	0.022	0.326
Census income	0.015	0.110	0.040	0.060
Malaria	0.293	0.003	0.352	0.003
Health+	0.051	0.003	0.026	0.252

Appendix

Table 4: Residual dependence diagnostics on analysis coordinates (5,000 permutation draws).

Figure 6: Residual dependence diagnostics by dataset under the exact coordinates used in inference. Left panel reports Moran’s $I$ with permutation intervals and right panel reports nearest-neighbor residual correlation with the same permutation reference. The figure is self-contained: forest and malaria are clearly dependence-rich under observed coordinates, while galaxies and census are weak under proxy embeddings.

Figure 7: Residual semivariograms for each empirical dataset under analysis coordinates. Rising semivariograms indicate stronger local dependence. Forest and malaria show the clearest distance-gradient pattern.

Figure 8: Overlap diagnostics under the auxiliary-split observed-data design: design and estimated propensity lower-tail quantiles ( $q05$ ) by dataset and arm, averaged over sampling regimes. The dashed line at $0.10$ marks the declared overlap floor.

Figure 9: Inverse-propensity effective-sample-size ratio (ESS / labeled count) under the auxiliary-split observed-data design, averaged over sampling regimes. MAR arms have lower ESS ratios, quantifying finite-sample information loss from nonuniform labeling probabilities.

Method	MCAR cov.	MCAR width	MAR cov.	MAR width
Cross-PPI	0.813	0.283	0.463	0.273
PPI++ (iid)	0.800	0.273	0.541	0.269
Bootstrap-PPI (Efron 2025)	0.652	0.215	0.345	0.288
DR-iid	0.805	0.274	0.823	0.315
Spatial DR-HAC	0.864	0.308	0.869	0.346
Spatial DR-JK-HAC	0.908	0.385	0.903	0.430

Appendix

Table 5: Mechanism attribution in the hardest synthetic cell (

\sigma=120

, soft-block sampling, 20% labels, nominal 90% superpopulation target). Each entry reports empirical coverage and mean interval length over 100 populations and 200 draws per population (20,000 intervals per arm).

Figure 10: Modular extension simulation with overlapping two-way dependence at nominal $90\%$ . Cross-PPI remains anti-conservative in MAR. The jackknife two-way DR interval is less conservative than the direct two-way plug-in at weak dependence, while both methods are highly conservative in strong-overlap regimes.

Figure 11: Interval width in the modular extension simulation at nominal $90\%$ . Jackknife and direct two-way plug-in widths are nearly identical throughout, and both increase sharply as overlap dependence strengthens.

Figure 12: Modular extension coverage at nominal $80\%$ .

Figure 13: Modular extension coverage at nominal $95\%$ .

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