Extended Data Fig. 4: Directed acyclic graphs showing community-acquired SARS-CoV-2 (CA-SARS-CoV-2) infection as a negative control outcome for use in evaluating the relationship between an exposure, A, and hospital-acquired SARS-CoV-2 (HA-SARS-CoV-2).

Measured confounders, L, are assumed to be adjusted for in the analysis, while unmeasured variables, U, may distort the estimated measure of association between exposure and hospital-acquired SARS-CoV-2 infection, generating a non-causal association. (a) Suppose that exposure, A, is a cause of HA-SARS-CoV-2 but not of CA-SARS-CoV-2, while unmeasured variables, U, are causes of both HA-SARS-CoV-2 and CA-SARS-CoV-2 but not of A (for example, factors affecting susceptibility to infection). In this case, in an analysis that adjusts for L, the association between A and HA-SARS-CoV-2 is a consequence of the causal link between A and HA-SARS-CoV-2, and no such association would be seen between A and the control outcome, CA-SARS-CoV-2. b) Conversely, if U is a cause of A, HA-SARS-CoV-2 and CA-SARS-CoV-2, but A is neither a cause of HA-SARS-CoV-2 nor of CA-SARS-CoV-2 then in an analysis adjusting for L associations between A and HA-SARS-CoV-2 and between A and CA-SARS-CoV-2 are expected as a consequence of the confounding factors, U. If a) and b) were the only possible causal relationships to be considered, an association between A and HA-SARS-CoV-2 but not between A and CA-SARS-CoV-2 after adjusting for L would provide evidence in support of a), where A is a cause of HA-SARS-CoV-2, while an association between A and CA-SARS-CoV-2 (after adjusting for L), would support b) as the backdoor path through U is open. c) If A is both a cause of HA-SARS-CoV-2 and there are unmeasured confounders, U, an association between A and HA-SARS-CoV-2 after adjusting for L is a consequence of both the direct causal link and confounding; in this case we would also expect an association between A and CA-SARS-CoV-2 after adjusting for L arising entirely as a result of confounding.