Table 2.

Summary of published methodologies using negative controls for detection (D), reduction (R), and correction (C) of confounding bias

	Reference and Setting	Main Assumptions Besides Assumptions 2–5	Methods
D	[32]: Time-series study. Z = future air pollution At+1.	(1) At+1 ⟂ Yt | At, Ut, Xt. (2) log[E(Yt)] = α + βAt + γXt + βfAt+1.	Bias detection by Wald-test on βf.
	[60, 61]: invalid NCE Z.	(1) Violation of exclusion restriction Y(a, z) ≠ Y(a). (2) Z is U-comparable with $A:Z\cancel{⫫}U|A,X$.	No evidence of Z-Y association adjusting for A implies no residual confounding of A-Y association.
R	[33, 49]: Time-series study. Z = future air pollution At+1.	(1) At+1 ⟂ Yt | At, Ut, Xt; $A_{t+1}\cancel{⫫}(A_t,U_t)|X_t$. (2) Yt(at, xt, ut) = β0 + β1αt + β2xt + β3ut + ϵt; E[ϵt | At = at, Ut = ut, Xt = xt] = 0. (3) $E[U_t|A_t=a_t,A_{t+1}=a_{t+1},X_{t=x_t}]={\alpha }_0+{\alpha }_1{a}_t+{\alpha }_2{x}_t+{\alpha }_3{a}_{t+1}$; sign(α1) = sign(α3). (4) E[At+1 | At = at, Xt = xt] = γ0 + γ1at + γ2xt; γ1 > 0.	Bias reduction by fitting E[Yt | At, Xt, At+1] instead of fitting E[Yt | At, Xt]. Further bias reduction considered in [49] by incorporating Xt+1 or At−1. Identification of β1 is possible with multiple future exposures under autoregressive model for exposure time series.
	[62]: Standardized mortality ratio in occupational cohort study.	(1) E[Y(1) | X = k]/E[Yref | X = k] = exp(αk − δk) E[W | X = k]/E[Wref | X = k] = exp(−ϵk). (2) sign(ϵk) = sign(δk) and 0 < |ϵk| < 2|δk|.	Adjust for bias δk via E[Y(1) | X = k]E[Wref | X = k]/E[Yref | X = k]E[W | X = k].
	[38, 40]: Define negative controls as drug–outcome pairs where one believes no causal effect exists.	(1) For a negative control drug-outcome pair, the effect estimate ${\beta }_i\sim N({\theta }_i,{\tau }_i^2)$, i = 1,..., n, where θi ∼ N(μ, σ2) is the true bias. (2) Under the null of no treatment effect, the effect estimate ${\beta }_{n+1}\stackrel{H_0}{\sim }N(\mu ,{\sigma }^2+{\tau }_{n+1}^2)$.	Estimate μ, σ by MLE with $L(\mu ,\sigma |\theta ,\tau )={\prod }_{i=1}^n\int p({\beta }_i|{\theta }_i,{\tau }_i)p({\theta }_i|\mu ,\sigma )d{\theta }_i$. Calibrated p-value computed via Wald-test of βn+1. Confidence interval calibrated similarly using distribution generated by positive controls.
C	[63, 64]: W, Y = Time-to-event outcome.	(1) There exist monotonic functions that describe U-Y and U-W associations: Y(0) = hy(U, X), W = hw(U, X). (2) Cox models for Y and W w/ hazard ratio $e^{{\beta }_y}$ and $e^{{\beta }_w}$.	The hazard ratio measuring the causal effect of treatment is $e^{{\beta }_y-{\beta }_w}$.
	[13, 65]: Generalized difference-in-differences using NCO.	(1) There exist monotonic functions that describe U-Y and U-W associations: Y(0) = hy(U, X), W = hw(U, X). (2) Positivity: if 0 < fW|A=1,X(W*) then 0 < fW|A=0,X(W*) < 1, where W* = (W | A = 1, X) is distributed as W in the exposed group.	The average treatment effect on the treated is $E[Y(1)-Y(0)|A=1]=E[Y|A=1]-E[F_{Y|A=0,X}^{-1})\cdot F_{W|A=0,X}(W*)]$. Generalized the difference-in-differences approach to the broader context of NCO.
	[66]: Calibration using NCO.	(1) W ⟂ A | X, Y(1), Y(0). (2) Rank preservation: Y = Y(0) + ΨA, and hence W ⟂ A | X, Y(0) by (1). (3) E[W | A, Y(0) = Y − ΨA, X] = β1 + β2X + β3Y(Ψ) + β4A, where β4 = 0 by (1).	The 95% CI for any Ψ0 consists of all Ψ for which ${\widehat{\beta }}_4(\mathrm{\Psi })\pm 1.96\mathrm{s}.\mathrm{e}.[{\widehat{\beta }}_4(\mathrm{\Psi })]$ contains 0; Under (1)–(3), fit E[W | A, Y, X] = β1 + β2X + β3Y + βΨA, then the causal effect Ψ = −βΨ/β3.
	[67–69]: Removing unwanted variation in gene-expression analysis.	(1) Y1×p = X1×qβq×p + U1×rΓr×p + ϵ1×p, p ≥ r + 1. (2) $W_{1\times s}=U_{1\times r}{\mathrm{\Gamma }}_{r\times s}^W+{ϵ}_{1\times s}^W$, s ≥ r, $\mathrm{Rank}({\mathrm{\Gamma }}_{r\times s}^W)=r$. (3) $(ϵ,{ϵ}^W)\sim N(0,\mathrm{diag}({\sigma }_1^2,\dots ,{\sigma }_{p+s}^2)),(ϵ,{ϵ}^W)⫫(X,U)$. (4) $U_{1\times r}=X_q{\alpha }_{q\times r}+{ϵ}_{1\times r}^U,{ϵ}^U\sim N(0,I_r),{ϵ}^U⫫X$.	[67, 68]: Estimate U by factor analysis of (2), then estimate β from (1). [69]: Estimate ΓW and Γ by factor analysis of Y = X(β + αΓ) + (ϵUΓ + ϵ) (5) and W = XαΓW + (ϵUΓW + ϵW) (6). Then estimate α from (6), and estimate β from (5).
	[12, 17, 36]: Nonparametric identification.	Assumption 7	Identify h in E[Y | A, Z, X] = E[h(W, A, X) | A, Z, X], then ATE = E[h(W, A = 1, X)] − E[h(W, A = 0, X)].