# Negative Controls

Negative controls are variables that meant to help when the study design is suspected to be invalid because of unmeasured confounders that are correlated with both the treatment and the outcome.

## Background

Confounding is a critical issue in observational studies because it can lead to biased or misleading conclusions about relationships between variables. A confounder is an extraneous variable that is related to both the independent variable (treatment or exposure) and the dependent variable (outcome), potentially distorting the true association. If confounding is not properly accounted for, researchers might incorrectly attribute an effect to the exposure when it is actually due to another factor. This can result in incorrect policy recommendations, ineffective interventions, or flawed scientific understanding. For example, in a study examining the relationship between physical activity and heart disease, failure to control for diet, a potential confounder, could lead to an overestimation or underestimation of the true effect of exercise.
Falsification tests are a robustness-checking technique used in observational studies to assess whether observed associations are likely due to confounding, bias, or model misspecification rather than a true causal effect. These tests help validate findings by applying the same analytical approach to a scenario where no effect is expected. If an association still appears where none should exist, it raises concerns that the primary analysis may suffer from confounding or other biases.
Negative controls are one type of falsification tests. The need to use negative controls usually arise in observational studies, when the study design can be questioned because of a potential confounding mechanism. A Negative control test can reject study design, but it cannot validate them. Either because there might be another confounding mechanism, or because of low statistical power. Negative controls are increasingly used in the epidemiology literature, but they show promise in social sciences fields such as economics. Negative controls are divided into two main categories: Negative Control Exposures (NCEs) and Negative Control Outcomes (NCOs).
Lousdal et al. examined the effect of screening participation on death from breast cancer. They hypothesized that screening participants are healthier than non-participants and, therefore, already at baseline have a lower risk of breast-cancer death. Therefore, they used proxies for better health as negative-control outcomes (NCOs) and proxies for healthier behavior as negative-control exposures (NCEs). Death from causes other than breast cancer was taken as NCO, as it is an outcome of better health, not effected by breast cancer screening. Dental care participation was taken to be NCE, as it is assumed to be a good proxy of health attentive behavior.

## Negative Control Exposure (NCE)

NCE is a variable that should not causally affect the outcome, but may suffer from the same confounding as the exposure-outcome relationship in question. A priori, there should be no statistical association between the NCE and the outcome. If an association is found, then it through the unmeasured confounder, and since the NCE and treatment share the same confounding mechanism, there is an alternative path, apart from the direct path from the treatment to the outcome. In that case, the study design is invalid.
For example, Yerushalmy used husband's smoking as an NCE. The exposure was maternal smoking; the outcomes were various birth factors, such as incidence of low birth weight, length of pregnancy, and neonatal mortality rates. It is assumed that husband's smoking share common confounders, such household health lifestyle with the pregnant woman's smoking, but it does not causally affect the fetus development. Nonetheless, Yerushalmy found a statistical association, And as a result, it casts doubt on the proposition that cigarette smoking causally interferes with intrauterine development of the fetus.
In another example,

## Negative Control Outcome (NCO)

Negative Control Outcomes are the more popular type of negative controls. NCO is a variable that is not causally affected by the treatment, but suspected to have a similar confounding mechanism as the treatment-outcome relationship. If the study design is valid, there should be no statistical association between the NCO and the treatment. Thus, an association between them suggest that the design is invalid.
For example, Jackson et al. used mortality from all causes outside of influenza season an NCO in a study examining influenza vaccine's effect on influenza-related deaths. A possible confounding mechanism is health status and lifestyle, such as the people who are more healthy in general also tend to take the influenza vaccine. Jackson et al. found that a preferential receipt of vaccine by relatively healthy seniors, and that differences in health status between vaccinated and unvaccinated groups leads to bias in estimates of influenza vaccine effectiveness. In a similar example, when discussing the impact of air pollutants on asthma hospital admissions, Sheppard et al. et al. used non-elderly appendicitis hospital admissions as NCO.

### Formal Conditions

Given a treatment $A$ and an outcome $Y$, in the presence of a set of control variables $X$, and unmeasured confounder $U$ for the $A-Y$ relationship. Shi et al. presented formal conditions for a negative control outcome ${\tilde {Y}}$,
* Stable Unit Treatment Value Assumption (SUTVA): For both ${Y}$ and ${\tilde {Y}}$ with regard to $A=a$.
* Latent Exchangeability: $Y^{A=a}\perp A|\;X,U$ Given $X$ and $U$, the potential outcome $Y^{A=a}$ is independent of the treatment.
* Irrelevancy: Ensures the irrelevancy of the treatment on the NCO.
** ${\tilde {Y}}^{A=a}={\tilde {Y}}^{A=a'}={\tilde {Y}}|\;U,X$: There is no causal effect of $A$ on ${\tilde {Y}}$ given $X$ and $U$.
** ${\tilde {Y}}\perp A|\;U,X$: There is no causal effect of $A$ on ${\tilde {Y}}$ given $X$ and $U$. The NCO is independent of the treatment given $X$ and $U$.
* U-Comparability: ${\tilde {Y}}\not {\perp }U|\;X$ The unmeasured confounders $U$ of the association between $A$ and $Y$ are the same for the association between $A$ and ${\tilde {Y}}$.
Given assumption 1 - 4, a non-null association between $A$ and ${\tilde {Y}}$, can be explained by $U$, and not by another mechanism. A possible violation of Latent Exchangeability will be when only the people that are influenced by a medicine will take it, even if both $X$ and $U$ are the same. For example, we would expect that given age and medical history ($X$), general health awareness ($U$), the intake of $A$ influenza vaccine will be independent of potential influenza related deaths ${\tilde {Y}}^{A=a}$. Otherwise, the Latent Exchangeability assumption is violated, and no identification can be made.
A violation of Irrelevancy occurs when there is a causal effect of $A$ on ${\tilde {Y}}$. For example, we would expect that given $X$ and $U$, the influenza vaccine does not influence all-cause mortality. If, however, during the influenza vaccine medical visit, the physician also performs a general physical test, recommends good health habits, and prescribes vitamins and essential drugs. In this case, there is likely a causal effect of $A$ on ${\tilde {Y}}$ (conditional on $X$ and $U$). Therefore, ${\tilde {Y}}$ cannot be used as NCO, as the test might fail even if the causal design is valid.
U-Comparability is violated when ${\tilde {Y}}{\perp }U$ , and therefore the lack of association between $A$ and ${\tilde {Y}}$ does not provide us any evidence for the invalidity of $A$. This violation would occur when we choose a poor NCO, that is not or very weakly correlated with the unmeasured confounders