{"doi":"10.1093/aje/kwad133","title":"NO UNMEASURED CONFOUNDING: KNOWN UNKNOWNS OR… NOT?","abstract":"A large majority of methods described in the causal literature rely on the assumption of no unmeasured confounding (NUC). When estimating treatment effects, the NUC assumption requires the measurement of all variables related to both treatment exposure and the outcome(s) of interest. In this short letter, we discuss 2 approaches which one might think could provide validation of the NUC assumption and show that neither is appropriate for this purpose. We close with a reminder to readers of the optimistic view of the complexity of data and how correlation between observed variables and unmeasured ones can reduce any bias associated with unmeasured information. We begin with some notation. Suppose we are interested in estimating an average treatment effect with a regression-based framework. Let |$Y$| denote a continuous outcome, |$X$| a measured confounder (possibly a vector), |$U$| an unmeasured confounder, and |$A$| a binary treatment. We assume the structural model>? where |${\\mathrm{\\varepsilon}}_i$| is a mean zero error term (see Web Appendix 1, available at https://doi.org/10.1093/aje/kwad133, for further details on the structural model). The function |$\\mathrm{\\gamma} \\left(A,X;\\mathrm{\\psi} \\right)$| represents the treatment effect, with |$\\mathrm{\\gamma} \\left(A=0,X;\\mathrm{\\psi} \\right)=0$| and |$\\mathrm{\\psi}$| the treatment effect estimand of interest. While |$\\mathrm{\\gamma} \\left(A,X;\\mathrm{\\psi} \\right)$|allows for the estimation of conditional treatment effects (i.e., conditional on |$X$|⁠), if |$\\mathrm{\\gamma} \\left(A,X;\\mathrm{\\psi} \\right)$| = |$\\mathrm{\\gamma} \\left(A;\\mathrm{\\psi} \\right)$| then we assume an unconditional treatment effect. The function |$f({X},{U};\\mathrm{\\beta} )$| represents the effect of confounders in the absence of treatment (i.e., when |$A=0$|⁠). In this letter, with some minor loss of generality, we assume that the effect of treatment, |$A$|⁠, may be modified by |$X$| but not |$U$|⁠. For simplicity, we assume linear relationships throughout numerical examples; however, the theory and conclusions are not constrained to that case. Assuming the structural model above permits the counterfactual formulation |${Y}_i(0)\\!={Y}_i(a)-\\mathrm{\\gamma} \\left(a,{x}_i;\\mathrm{\\psi} \\right)$| = |$f({x}_i,{u}_i;\\mathrm{\\beta} )+{\\mathrm{\\varepsilon}}_i$|⁠, where |${Y}_i(a)$| is the counterfactual outcome if individual |$i$| had received (potentially counter to fact) treatment |$a$|⁠. Letting the vector of all confounders be denoted by |$\\mathbf{L}=\\left(X,U\\right)$|⁠, the NUC assumption entails conditional independence between the counterfactual |$Y(A)$| and treatment |$A$|⁠, given the confounders |$\\mathbf{L}$|⁠, that is, as |$Y(A)\\perp A\\mid \\mathbf{L}$|⁠. Suppose, then, we had a candidate value of the true treatment effect, |$\\mathrm{\\psi}$|⁠, which we denote by |${\\mathrm{\\psi}}^{\\dagger }$|⁠, and posited structural model (which we shall assume is linear for ease of notation). Following Hernán and Robins (1) (see section 14.5), define |$H({\\mathrm{\\psi}}^{\\dagger})=Y-\\mathrm{\\gamma} \\left(A,X;{\\mathrm{\\psi}}^{\\dagger}\\right)$| as a surrogate for the counterfactual |$Y(0)$| based on an assumed value for the treatment effect |${\\mathrm{\\psi}}^{\\dagger }$|⁠. Then |$Y(0)\\perp A\\mid \\mathbf{L} \\;$|—that is, the NUC assumption—implies that the assignment of treatment does not depend on how an individual might react to treatment, or in its absence. Therefore 1) |${\\zeta}_1\\!=\\!0$| in |$E\\left[H\\left(\\mathrm{\\psi} \\right)|A,\\mathbf{L};\\zeta \\right]={\\zeta}_0+{\\zeta}_1A+{\\zeta}_2^{\\top}\\mathbf{L}$| and 2) |${\\mathrm{\\alpha}}_1\\!=\\!0$| in |$\\mathrm{logit}(\\Pr [A=1|H\\left(\\mathrm{\\psi} \\right)\\!,\\mathbf{L};\\mathrm{\\alpha} ])={\\mathrm{\\alpha}}_0+{\\mathrm{\\alpha}}_1H\\left(\\mathrm{\\psi} \\right)+{\\alpha}_2^\\top \\mathbf{L}$|when |$\\mathrm{\\psi}$| is the true causal treatment effect. It is tempting to use these facts as a means of testing the NUC assumption. However, in Web Appendix 2 (data generation according t","journal":"American Journal of Epidemiology","year":2023,"id":385014,"datarank":0.0,"base_score":0.0,"endowment":0.0,"self_citation_contribution":0.0,"citation_network_contribution":0.0,"self_endowment_contribution":0.0,"citer_contribution":0.0,"corpus_percentile":null,"corpus_rank":null,"citation_count":2,"citer_count":0,"citers_with_citation_signal":0,"citers_with_endowment":0,"datacite_reuse_total":0,"is_dataset":false,"is_dataset_confidence":0.9572,"is_data_producer":false,"deposit_databanks":null,"is_oa":true,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":"2023-01-01","fair_score":null,"fair_percentile":null,"algorithm_id":"datarank_citation_only_1hop_v6","ranking_scope":"data_only","authors":[{"id":422125,"name":"Erica E. M. Moodie","orcid":"0000-0002-7225-3977","position":1,"is_corresponding":false},{"id":422126,"name":"Susan M. Shortreed","orcid":"0000-0001-7918-601X","position":2,"is_corresponding":false},{"id":1152782,"name":"Juliana Schulz","orcid":"0000-0001-7312-0167","position":0,"is_corresponding":true}],"reference_count":8,"raw_metadata":null,"created_at":"2026-07-19T01:17:48.629718Z","pmid":"37280737","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":[]}