{"doi":"10.1093/biomtc/ujaf095","title":"Valid and efficient inference for nonparametric variable importance in two-phase studies","abstract":"We consider a common nonparametric regression setting, where the data consist of a response variable Y, some easily obtainable covariates $\\mathbf {X}$, and a set of costly covariates $\\mathbf {Z}$. Before establishing predictive models for Y, a natural question arises: Is it worthwhile to include $\\mathbf {Z}$ as predictors, given the additional cost of collecting data on $\\mathbf {Z}$ for both training the models and predicting Y for future individuals? Therefore, we aim to conduct preliminary investigations to infer importance of $\\mathbf {Z}$ in predicting Y in the presence of $\\mathbf {X}$. To achieve this goal, we propose a nonparametric variable importance measure for $\\mathbf {Z}$. It is defined as a parameter that aggregates maximum potential contributions of $\\mathbf {Z}$ in single or multiple predictive models, with contributions quantified by general loss functions. Considering two-phase data that provide a large number of observations for $(Y,\\mathbf {X})$ with the expensive $\\mathbf {Z}$ measured only in a small subsample, we develop a novel approach to infer the proposed importance measure, accommodating missingness of $\\mathbf {Z}$ in the sample by substituting functions of $(Y,\\mathbf {X})$ for each individual's contribution to the predictive loss of models involving $\\mathbf {Z}$. Our approach attains unified and efficient inference regardless of whether $\\mathbf {Z}$ makes zero or positive contribution to predicting Y, a desirable yet surprising property owing to data incompleteness. As intermediate steps of our theoretical development, we establish novel results in two relevant research areas, semi-supervised inference and two-phase nonparametric estimation. Numerical results from both simulated and real data demonstrate superior performance of our approach.","journal":"Biometrics","year":2025,"id":536445,"datarank":0.10397207708399181,"base_score":0.6931471805599453,"endowment":0.6931471805599453,"self_citation_contribution":0.10397207708399181,"citation_network_contribution":0.0,"self_endowment_contribution":0.10397207708399181,"citer_contribution":0.0,"corpus_percentile":null,"corpus_rank":null,"citation_count":1,"citer_count":0,"citers_with_citation_signal":0,"citers_with_endowment":0,"datacite_reuse_total":0,"is_dataset":false,"is_dataset_confidence":0.9528,"is_data_producer":false,"deposit_databanks":null,"is_oa":false,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":"2025-01-01","fair_score":null,"fair_percentile":null,"algorithm_id":"datarank_citation_only_1hop_v6","ranking_scope":"data_only","authors":[{"id":251335,"name":"Raymond J. Carroll","orcid":"0000-0002-5465-9682","position":1,"is_corresponding":false},{"id":285723,"name":"Jinbo Chen","orcid":"0009-0009-2049-6342","position":2,"is_corresponding":false},{"id":976677,"name":"Guorong Dai","orcid":"0000-0003-3179-2541","position":0,"is_corresponding":true}],"reference_count":21,"raw_metadata":{"citation_network_status":"fetched"},"created_at":"2026-07-19T02:52:05.227140Z","pmid":"40742446","pmcid":null,"fwci":null,"citation_percentile":null,"influential_citations":0,"oa_status":null,"license":null,"views":0,"total_file_size_bytes":0,"version_count":0,"fair_f":null,"fair_a":null,"fair_i":null,"fair_r":null,"fair_zscore":null,"fair_rationale":null,"fair_model":null,"fair_agent_version":null,"fair_fulltext_source":null,"fair_has_llm":null,"fair_computed_at":null,"clinical_trials":[],"software_tools":[],"db_accessions":[],"linked_datasets":[],"topics":[]}