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When studying the effect of an exposure $T$ on an outcome $Y$, we can control for measured confounders $X$, but the treatment effect can still be biased by residual confounding due to unmeasured confounders $U$. One approach for detecting this kind of bias is to use a negative control outcome $N$ (e.g. here), which is not caused by $T$ but shares the same confounder $U$. If, after adjusting for confounders $X$ using a regression model (for example), there a non-zero effect of $T$ on $N$, then we know there is residual confounding.

This approach can detect confounding, but not necessarily estimate the magnitude of confounding bias, as explained in eAppendix 3 of this paper. The issue is that the effect of $U$ on $N$ is not necessarily equal to the effect of $U$ on $Y$.

However, if one views $N$ as a surrogate for unmeasured confounding, then why can we not simply add it as a covariate in our original regression model? That is, why can we not adjust our estimates of the effect of $T$ on $Y$ by adding $N$ to our measured confounders $X$?

I usually avoid adjusting for any post-treatment variable as it could lead to collider bias, but by assumption, $Y$ is not associated with $N$, so this is not an issue. That said, there could be M-bias. Is that the reason why this should be avoided?

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  • $\begingroup$ Off-hand, I'd say that as $N$ isn't related to $T,$ it shouldn't (theoretically) change your regression coefficients much, if at all. $\endgroup$ Dec 15, 2022 at 19:32
  • $\begingroup$ @AdrianKeister $N$ shares a common cause ($U$) with $T$, so even though it is not causally related, it can be strongly correlated because of the strength of the unmeasured confounder. $\endgroup$
    – user310374
    Dec 15, 2022 at 19:35
  • $\begingroup$ But it won't shut off the backdoor path; you can always try simulating this experience with some known example. $\endgroup$ Dec 15, 2022 at 19:37

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That's an interesting thought, but naively, conditioning on $N$ doesn't closes the backdoor path $T \leftarrow U \rightarrow Y$. See the orange path in this DAG: enter image description here

It can be shown using a fairly simple simulation based on this DAG:

import numpy as np
import pandas as pd
import statsmodels.formula.api as smf

def generate_data(seed, M=10000):
    rng = np.random.default_rng(seed)

    x = rng.normal(size=M)
    u = rng.normal(size=M)

    t_logit = 1 + 0.5*x + 1*u + rng.normal(size=M)
    t_propensity = 1 / (1 + np.exp(-t_logit))
    t = rng.binomial(1, t_propensity)

    y = 1 + 2*t + 0.5*x + 1*u + rng.normal(size=M)

    n = 1 + 0.5*x + 1*u + rng.normal(size=M) # not affected by t

    data = pd.DataFrame({
        "x": x, "u": u, "t": t,
        "y": y, "n": n,
    })
    # data.head()
    return data

Which running a 100 times:

res = []
for i in range(100):
    data = generate_data(i)
    
    correct = smf.ols("y ~ 1 + t + x + u", data=data).fit().params['t']
    misspecified = smf.ols("y ~ 1 + t + x", data=data).fit().params['t']
    nc_adjusted = smf.ols("y ~ 1 + t + x + n", data=data).fit().params['t']
    
    res.append((correct, misspecified, nc_adjusted))

res = pd.DataFrame(res, columns=["correct", "misspecified", "neg_control_adjusted"])
res.head()

Shows the estimates adjusting for negative controls is still biased: enter image description here

However, if you play around with the correlation between $U$ and $N$ you are correct to assume that the stronger the correlation is the smaller the bias is (even bounded by the model with $U$ to the model without $U$) because $N$ becomes $U$ from information perspective: enter image description here

Which is in line with the line of work exploring the adjustment for proxies of confounders (which I recommend).

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  • $\begingroup$ This is very helpful, @ehudk. Can you think of the potential risks of this approach? Adjusting for $U$ cannot lead to collider bias, because it is not downstream from $T$, but is there some other type of bias I would risk introducing by adjusting for $N$? $\endgroup$
    – user310374
    Jan 9, 2023 at 2:49
  • $\begingroup$ You have to be very certain $N$ is not affected by $T$ because otherwise you might introduce post-treatment adjustment bias. However, my main concern with this approach (personally) is temporality issues. (Negative) outcomes occur during follow-up and are not available during baseline. This may hinder generalizability or usability of conclusions (think of a physician needing to enter patient's details into some risk calculator during meeting, they won't have access to future events [outcomes]) $\endgroup$
    – ehudk
    Jan 14, 2023 at 13:02

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