# Causal progress with imperfect placebo treatments and outcomes
[Draft]

Adam Rohde, Chad Hazlett111Associate Professor, Departments of Statistics &
Political Science, UCLA<EMAIL_ADDRESS>PhD, Department of Statistics,
UCLA. Senior Decision Scientist, ZipRecruiter<EMAIL_ADDRESS>
(October 2023)

###### Abstract

In the quest to make defensible causal claims from observational data, it is
sometimes possible to leverage information from “placebo treatments” and
“placebo outcomes” (or “negative control outcomes”). Existing approaches
employing such information focus largely on point identification and assume
(i) “perfect placebos”, meaning placebo treatments have precisely zero effect
on the outcome and the real treatment has precisely zero effect on a placebo
outcome; and (ii) “equiconfounding”, meaning that the treatment-outcome
relationship where one is a placebo suffers the same amount of confounding as
does the real treatment-outcome relationship, on some scale. We instead
consider an omitted variable bias framework, in which users can postulate non-
zero effects of placebo treatment on real outcomes or of real treatments on
placebo outcomes, and the relative strengths of confounding suffered by a
placebo treatment/outcome compared to the true treatment-outcome relationship.
Once postulated, these assumptions identify or bound the linear estimates of
treatment effects. While applicable in many settings, one ubiquitious use-case
for this approach is to employ pre-treatment outcomes as (perfect) placebo
outcomes. In this setting, the parallel trends assumption of difference-in-
difference is in fact a strict equiconfounding assumption on a particular
scale, which can be relaxed in our framework. Finally, we demonstrate the use
of our framework with two applications, employing an R package that implements
these approaches.

##  1 Introduction

Unobserved confounding plagues attempts to draw causal inferences from
observational data. When we cannot rule out unobserved confounding, we may
look to additional sources of information and alternative strategies to help
fully or partially identify the range of defensible causal effects. One set of
approaches is to seek out observed variables that are “similar to” the
treatment or outcome, in the form of placebo treatments or placebo outcomes
(also known as “negative control outcomes” and “negative control exposures,”
respectively). In broad terms, if these variables are also subject to the
problematic confounding, they may offer investigators a way to measure the
strength of confounding. If this strength of confounding is assumed to be
“similar” (in some specified sense) to the confounding suffered by the main
treatment-outcome relationship it can be used to guess at and remove the bias
in the estimated treatment effect. Existing approaches that operationalize
this idea have focused on point identification, and rely on two strict
assumptions. First, the placebo treatments/outcomes involved must be “perfect
placebos”, meaning there must be precisely zero effect of the placebo
treatment on the real outcome, or real treatment on placebo outcome. Second,
these approaches must connect the amount of confounding suffered by the
placebo relationship to that in the relationship of interest, typically by
assuming they are equal on some scale. We call this the “equiconfounding”
assumption. As discussed below, such approaches include “proximal causal
inference” (Miao et al., 2020; Tchetgen et al., 2020), which in its original
form, requires a perfect placebo treatment, perfect placebo outcome, and a
“bridge function” assumption that invokes equiconfounding on some
transformation. We discuss this and a related recent development with a single
proxy variable (Tchetgen et al., 2023) below. Less obvious but central to our
presentation here, classical difference-in-difference (DID) also employs a
perfect placebo (outcome) and equiconfounding, which appears as the parallel
trends assumption.

Unfortunately, when seeking to defend causal claims in real world
circumstances, these assumptions are often not defensible. While some placebos
may be arguably “perfect” (such as pre-treatment outcome measures acting as
placebo outcomes), for many variables one might use as a placebo outcome or
treatment, this is not the case. More problematic still is the equiconfounding
assumption. On what basis can investigators hope to argue that unobserved
confounders have precisely the same impact on the treatment-outcome
relationship as in the relationship with the placebo treatment or outcome?

Is it nevertheless possible to make use of the information that a given
variable is a possibly imperfect placebo, and is exposed to some fraction of
the confounding that affect the true treatment-outcome relationship? In this
paper we examine what progress can be made by sidestepping the two strong
assumptions of placebo perfection and equiconfounding, determining how
arguments about limited violations of these assumptions lead to defensible
ranges of causal claims. Specifically, consider a set of assumptions about (i)
the relative confounding felt by the treatment-outcome relationship compared
to that in the placebo treatment to real outcome or real treatment to placebo
outcome relationship; and (ii) the degree of “imperfection” in each placebo.
For any range of assumptions on these that can(not) be ruled out by
argumentation, there is a set of effect estimates that consequently can(not)
be ruled out as possible. This approach simply and mechanically shows the
consequence of whatever assumption one would be willing to pose regarding
these placebo relationships. One can expect relatively uninformative results
when little is known about the extent to which these assumptions are violated,
whereas the approach will be more informative to the degree that investigators
are better able to defend narrower assumptions about the degree of placebo
imperfection or the relative strength of confounding.

In what follows, we begin by discussing causal structures in general that
involve placebo treatments and outcomes, including cases where these labels
become ambiguous. Our treatment employs linear models in an omitted variable
bias framework. We focus first on a single placebo outcome, then show the
corresponding analysis for placebo treatments.222We reserve our analysis of
“double placebos”—cases with a placebo treatment and outcome—for the appendix,
noting that this relates most closely to developments under the heading of
proximal causal inference. We give special attention to the connection of this
approach to DID, where (i) the pre-treatment outcome is a (perfect) placebo
outcome, and (ii) the parallel trends assumption is a sharp equiconfounding
assumption on the additive scale. This relationship is clarifying, but also
demonstrates the applicability of our approach as a relaxation of DID that
readily enables investigators to examine the credible causal interval obtained
over a set of assumptions that would be strictly more defensible than those of
conventional DID. Finally, we provide two applications. First, we look to
observational data designed to examine the impact of the National Supported
Work Demonstration, a job training program in the 1970s. A pre-treatment
measure of pariticpants’ income serves as a (perfect) placebo outcome.
Assumptions are then required on how strongly pre-treatment income is
confounded with treatment status, compared to how strongly subsequent income
is confounded with treatment status. Under arguably defensible assumptions,
the resulting range of estimates is informative and includes the benchmark
value obtained from a randomized trial with the same treatment group. Second,
we show that for a likely range of relative levels of confounding, the Zika
virus decreased birth rates in Brazil in 2016, using 2014 birth rates as a
placebo outcome. We conduct these analyses using the placeboLM package for R.
Vignettes for this package (will eventually!) include examples with imperfect
placebos, placebo outcomes other than pre-treatment outcomes, and placebo
treatments.

##  2 Leveraging placebos in linear models

###  2.1 Causal structures of interest

Placebo treatments and outcomes, perfect or imperfect, are defined by the role
they play in some causal model. We employ directed acyclic graphs (DAGs)
(Pearl, 1995) to commnunicate these roles. Figure 1 shows simple structures
with “perfect” placebo treatments ($P$) and placebo outcomes ($N$). It is
notable that placebo treatments and outcomes can look very similar in their
causal role; indeed Figure 1(a) and (b) are mirror images, replacing the role
of $P$ with $N$. We explore such ambiguities in the context of a broader set
of cases provided in Section 2.4. To call a placebo “imperfect” is to allow
for an additional causal effect, and so always implies an additional edge in
the causal structure. For example, Figure 1(c) adds the $D\to N$ edge to
Figure 1(a), while Figure 1(d) adds the $P\to Y$ edge to Figure 1(b).

Figure 1:  Simple DAGs with a placebo outcome, $N$, and with a placebo
treatment, $P$. $D$ is treatment, $Y$ is outcome, $\mathbf{Z}$ contains
unobserved confounders, and $\mathbf{X}$ contains observed covariates.

(a)$D$$Y$$N$$\mathbf{Z}$$\mathbf{X}$

(b)$D$$Y$$P$$\mathbf{Z}$$\mathbf{X}$

(c)$D$$Y$$N$$\mathbf{Z}$$\mathbf{X}$

(d)$D$$Y$$P$$\mathbf{Z}$$\mathbf{X}$

###  2.2 An omitted variable bias approach to placebos: Placebo outcomes

Our proposed framework is a product of the omitted variable bias analysis of
linear models (e.g. Angrist and Pischke, 2008; Cinelli and Hazlett, 2020;
Hansen, 2022), extended to make use of assumptions involving placebo
treatments or outcomes.333We also consider relaxations to this parametric
approach in Appendix D, in which we illustrate similar methods adapted to
partially linear and non-parametric estimation of treatment effects. In this
analysis we begin by discussing placebo outcomes, and build on this in the
subsequent analysis of placebo treatments and of “double placebos” in the
appendix. Suppose that we want to estimate the “long” regression of $Y$ on
$D,\mathbf{X}$, and $\mathbf{Z}$ in Equation 1. In particular, we are
interested in the coefficient $\beta_{Y\sim D|Z,X}$ as a linear approximation
to the causal effect of $D$ on $Y$. $\mathbf{X}$ and $\mathbf{Z}$ are vectors
of covariates.444Note that in this section and others, we will refer to
including $\mathbf{Z}$ in the long regression as “de-confounding”. Readers
should recognize that, in reality, we are only partialling out the linear
relationships with $\mathbf{Z}$. This is, therefore, a linear approximation to
full non-parametric elimination of confounding from $\mathbf{Z}$. On the other
hand, $\mathbf{Z}$ can be assumed to contain arbitrary functions of the
relevant variables, making this very flexible for de-confounding.

$\displaystyle Y$ $\displaystyle=\beta_{Y\sim D|Z,X}D+\mathbf{X}\beta_{Y\sim
X|D,Z}+\mathbf{Z}\beta_{Y\sim Z|D,X}+\epsilon_{l}$ (1)

In reality, however, the variables $\mathbf{Z}$ are not observed, and
henceforth are referred to as “unobserved confounders”. We are restricted to
estimating the “short” regression of $Y$ on $D$ and $\mathbf{X}$ in Equation
2.

$\displaystyle Y$ $\displaystyle=\beta_{Y\sim D|X}D+\mathbf{X}\beta_{Y\sim
X|D}+\epsilon_{s}$ (2)

The difference between $\beta_{Y\sim D|X}$ and $\beta_{Y\sim D|Z,X}$ is
referred to as omitted variable bias, which we label
$\text{bias}_{\text{(YD.X)}}$,

$\displaystyle\text{bias}_{\text{(YD.X)}}\overset{\Delta}{=}\beta_{Y\sim
D|X}-\beta_{Y\sim D|Z,X}$ (3)

Note that for any given sample and estimate, there would be a particular
difference between the estimate obtained from the short regression and the
long regression, $\widehat{\text{bias}}_{\text{(YD.X)}}$, representing the
difference between what we get in reality and what we would have gotten if
$\mathbf{Z}$ was available.

To incorporate placebo related information, suppose (momentarily) the
investigator argues that the relationship between the treatment and a placebo
outcome (the $D\to N$ relationship) suffers from a similar level of unobserved
confounding as does the relationship between treatment and real outcome (the
$D\to Y$ relationship). Now consider the “long” regression of $N$ on $D$,
$\mathbf{X}$ and $\mathbf{Z}$ (Equation 4), and the short regression of $N$ on
just $\mathbf{X}$ and $D$ (Equation 5). Note that we can consider $\mathbf{Z}$
to be a rich enough set of variables to de-confound both the $D\to Y$ and the
$D\to N$ relationships.

As with $Y$, we are only able to consider estimating the short regression for
$N$, as $\mathbf{Z}$ is unobserved. Here too the difference between
$\beta_{N\sim D|X}$ and $\beta_{N\sim D|Z,X}$ is omitted variable bias,
labelled $\text{bias}_{\text{(ND.X)}}$.

$\displaystyle N$ $\displaystyle=\beta_{N\sim D|Z,X}D+\mathbf{X}\beta_{N\sim
X|D,Z}+\mathbf{Z}\beta_{N\sim Z|D,X}+\xi_{l}$ (4) $\displaystyle N$
$\displaystyle=\beta_{N\sim D|X}D+\mathbf{X}\beta_{N\sim X|D}+\xi_{s}$ (5)
$\displaystyle\text{bias}_{\text{(ND.X)}}\overset{\Delta}{=}\beta_{N\sim
D|X}-\beta_{N\sim D|Z,X}$ (6)

In the simple case where one assumes $N$ is a perfect placebo outcome,
$\beta_{N\sim D|Z,X}=0$, so $\text{bias}_{\text{(ND.X)}}$ is revealed by the
observed $\beta_{N\sim D|X}$. The key maneuver is to then rewrite the bias we
are worried about ($\text{bias}_{\text{(YD.X)}}$) as it relates to the bias in
the placebo relationship we can, under these assumptions, observe
($\text{bias}_{\text{(ND.X)}}$). A first way to do so is simply to define
$m\in\mathbb{R}$ such that:

$\displaystyle\text{bias}_{\text{(YD.X)}}=m\times\text{bias}_{\text{(ND.X)}}$
(7)

Plugging this in allows us to solve for $\beta_{Y\sim D|X,Z}$ as simply the
naive estimate ($\beta_{Y\sim D|X}$), minus $m$ times $\beta_{N\sim D|X}$.
More generally, allowing an imperfect placebo and thus $\beta_{N\sim
D|Z,X}\neq 0$, we have

$\displaystyle\text{bias}_{\text{(YD.X)}}$ $\displaystyle=\beta_{Y\sim
D|X}-\beta_{Y\sim D|Z,X}$ (8) $\displaystyle=m\times\left(\beta_{N\sim
D|X}-\beta_{N\sim D|Z,X}\right)$ (9) $\displaystyle\beta_{Y\sim D|Z,X}$
$\displaystyle=\beta_{Y\sim D|X}-m\times\left(\beta_{N\sim D|X}-\beta_{N\sim
D|Z,X}\right)$ (10)

Equation 10 provides all we need for partial identification of $\beta_{Y\sim
D|Z,X}$. Sample estimates of $\beta_{Y\sim D|X}$ and $\beta_{N\sim D|X}$ are
provided by the data, while the remaining terms ($m$ and $\beta_{N\sim
D|Z,X}$) parameterize assumptions from outside the data, and can be postulated
for purposes of partial identification or sensitivity analysis. Specifically,
the value of $\beta_{N\sim D|Z,X}$ relates to violations of placebo
perfection; i.e. the $D\to N$ path in Figure 1(c). The value of $m$
parameterizes “relative confounding”, comparing the strength of the
$D\leftarrow Z\to Y$ relationship to the $D\leftarrow Z\to N$ relationship in
Figure 1(a). This comparison thus comes down to $Z\to Y$ versus $Z\to N$.

Only one special case assumes that (i) the placebo outcome $N$ is “perfect”
and so $\beta_{N\sim D|Z,X}$ is zero; and (ii) $m=1$, meaning
“equiconfounding” on the coeficient scale is assumed. As examined below, this
corresponds to classical DID (wherein $N$ would typically represent a pre-
treatment measure of the outcome). Outside of this narrow case, some range of
$m$ values may need to be tested based on what the researcher can plausibly
defend.

#### Re-expressing $m$.

One important concern with the parameterization above is that $N$ and $Y$ may
be dissimilar enough in conception or measurement that we do not understand
them to be on the same scale. This complicates the interpretation of $m$,
which is sensitive to any scale differences between $N$ and $Y$.

A similar problem arises in sensitivitiy analysis with omitted variables on
the “coefficient” scale (Cinelli and Hazlett, 2020), and we propose a similar
solution: reparameterization in terms of “(partial) variance explained”, i.e.
partial-correlation parameters. Specifically,555In Equations 11, 12, and 13,
$Z_{\text{(Y.DX)}}=\mathbf{Z}\beta_{Y\sim Z|D,X}$ is the linear combination of
$\mathbf{Z}$ that “de-confounds” the $D\to Y$ relationship in the same way
that including $\mathbf{Z}$ in Equation 1 does. Similarly,
$Z_{\text{(N.DX)}}=\mathbf{Z}\beta_{N\sim Z|D,X}$ is the linear combination of
$\mathbf{Z}$ that “de-confounds” the $D\to N$ relationship in the same way
that including $\mathbf{Z}$ in Equation 4 does.

$\displaystyle\text{bias}_{\text{(YD.X)}}=\beta_{Y\sim D|X}-\beta_{Y\sim
D|Z,X}=R_{Y\sim Z|D,X}f_{D\sim Z_{\text{(Y.DX)}}|X}(\text{SD}(Y^{\perp
D,X})/\text{SD}(D^{\perp X}))$ (11)
$\displaystyle\text{bias}_{\text{(ND.X)}}=\beta_{N\sim D|X}-\beta_{N\sim
D|Z,X}=R_{N\sim Z|D,X}f_{D\sim Z_{\text{(N.DX)}}|X}(\text{SD}(N^{\perp
D,X})/\text{SD}(D^{\perp X}))$

$\displaystyle
m=\frac{{\text{bias}_{\text{(YD.X)}}}}{{\text{bias}_{\text{(ND.X)}}}}=\frac{R_{Y\sim
Z|D,X}f_{D\sim Z_{\text{(Y.DX)}}|X}}{R_{N\sim Z|D,X}f_{D\sim
Z_{\text{(N.DX)}}|X}}\times\frac{\text{SD}(Y^{\perp D,X})}{\text{SD}(N^{\perp
D,X})}=k\times\frac{\text{SD}(Y^{\perp D,X})}{\text{SD}(N^{\perp D,X})}$ (12)

In this arrangement, $k$ is defined as

$k\overset{\Delta}{=}\frac{R_{Y\sim Z|D,X}f_{D\sim
Z_{\text{(Y.DX)}}|X}}{R_{N\sim Z|D,X}f_{D\sim Z_{\text{(N.DX)}}|X}}$ (13)

The first ratio, $\frac{R_{Y\sim Z|D,X}}{R_{N\sim Z|D,X}}$, is naturally
interpretable, as it is the ratio of partial correlations of $Z$ with $Y$
compared to $Z$ with $N$ (both after partialing out $D$ and $X$). This
interpretation of $k$ is complicated by the ratio $\frac{f_{D\sim
Z_{\text{(Y.DX)}}|X}}{f_{D\sim Z_{\text{(N.DX)}}|X}}$. In the case of a single
$Z$, this is unproblematic as it equals 1. In the more general case, it poses
a greater nuisance as the $Z$ variables may combine differently in a model
predicting $N$ as compared to one predicting $Y$. In this case, a more
practical interpretation is given by noting

$k=\frac{{\text{bias}_{\text{(YD.X)}}}}{{\text{bias}_{\text{(ND.X)}}}}\times\frac{1}{\text{SF}}=\frac{m}{SF}\text{,
where $\text{SF}=\frac{\text{SD}(Y^{\perp D,X})}{\text{SD}(N^{\perp
D,X})}$}\\\ $ (14)

Equation 14 provides an expression for $k$ in terms of $m$ and a “scale
factor”, $SF$, that makes up for scale differences between $Y$ and $N$, after
removing the influence of $D$ and $X$. Using $k$ rather than $m$, the bias and
adjusted coefficient can simply be rewritten as

$\displaystyle\text{bias}_{\text{(YD.X)}}=k\times\left(\beta_{N\sim
D|X}-\beta_{N\sim D|Z,X}\right)\times\frac{\text{SD}(Y^{\perp
D,X})}{\text{SD}(N^{\perp D,X})}$ (15) $\displaystyle\beta_{Y\sim D|Z,X}$
$\displaystyle=\beta_{Y\sim D|X}-k\times\left(\beta_{N\sim D|X}-\beta_{N\sim
D|Z,X}\right)\times\frac{\text{SD}(Y^{\perp D,X})}{\text{SD}(N^{\perp D,X})}$
(16)

#### Estimator and partial identification.

Equation 16 provides us with a moment estimator for $\beta_{Y\sim D|Z,X}$ in
Equation 17.

$\displaystyle\hat{\beta}_{Y\sim D|Z,X}$ $\displaystyle=\hat{\beta}_{Y\sim
D|X}-k\times\left(\hat{\beta}_{N\sim D|X}-\beta_{N\sim
D|Z,X}\right)\times\frac{\|Y^{\perp D,X}\|_{2}}{\|N^{\perp D,X}\|_{2}}$ (17)

Given $k$ and $\beta_{N\sim D|Z,X}$, the estimator $\hat{\beta}_{Y\sim D|Z,X}$
from Equation 17 is consistent for $\beta_{Y\sim D|Z,X}$ and asymptotically
normal. We will discuss estimating standard errors below. Using Equation 17 in
a partial identification framework, we can reason about plausible ranges of
values for $\beta_{N\sim D|Z,X}$ and $k$ to arrive at a range for plausible
estimates of $\beta_{Y\sim D|Z,X}$.

In this way, we can partially identify $\beta_{Y\sim D|Z,X}$ subject to any
particular assumption about $k$ (or $m$) and $\beta_{N\sim D|Z,X}$. We imagine
two main modes of inference or use-cases. The first approach is “postulative
partial identification”: one first reasons about the imperfection in the
placebo, i.e. decide how far $\beta_{N\sim D|Z,X}$ may reasonably be from
zero. Then, one considers $k$, the ratio of confounding strengths experienced
by $D\to N$ versus $D\to Y$. For any plausible range that can be put on these
values, adjusted coefficients for $\hat{\beta}_{Y\sim D|Z,X}$ can be computed.
The alternative approach, more akin to sensitivity analysis, is to invert this
process: Determine what must be assumed about these parameters in order to
reach a particular conclusion or set of conclusions that are of substantive
interest. Investigators can then question whether the required values of these
parameters are defensible.

###  2.3 Placebo treatments

Similar machinery can leverage imperfect placebo treatments rather than
imperfect placebo outcomes. Suppose that we want to run the long regression
for $Y$ on $D$ (Equation 1) but, again, $\mathbf{Z}$ is an unobserved vector
of variables that we would have liked to include in the regression but cannot.
Suppose also that we have a placebo treatment, $P$, for which we believe the
confounding of $P$ and $Y$ from $\mathbf{Z}$ can be compared to the
confounding of $D$ and $Y$ from $\mathbf{Z}$. We might consider running two
separate regressions as we did for placebo outcomes. However, that approach is
not easily adapted to imperfect placebo treatments, where there may be a
direct relationship between $P$ and $Y$.666Suppose we have a causal model like
Figure 1(d). $\beta_{Y\sim Z|P,X}$ (or $R_{Y\sim Z|P,X}$) will capture the
causal relationship $D\to Y$, since the path $Z\to D\to Y$ will remain open
conditional on $P$ and $X$ alone. Expressing the bias in the regression
coefficient on $D$ from the short regression using a two regression approach
like we used for placebo outcomes would, therefore, require us to reason about
the causal effect of interest itself, which is not useful. Instead, suppose
that $P$ is not a descendant of $D$, $D$ is not a descendant of $P$, $P$ is a
good or neutral control for the effect of $D$ on $Y$, and $P$ is a good or
neutral control for $D$ on $Y$, conditional on $Z$ and $X$.777We require that
$D$ and $P$ are good or neutral controls for each other in that they do not
increase the bias in our estimate of the coefficient we are targeting
($\beta_{Y\sim D|X,Z}$). This will simplify interpretation. If a user is
concerned that there could be a path opened or closed by conditioning on $D$
or $P$, we suggest drawing the DAG to check. See Section 2.4 for further
discussion. Further, suppose that we want to run the “long” regression of $Y$
on $D,P,\mathbf{X}$, and $\mathbf{Z}$ in Equation 18. In particular, we are
interested in the coefficient $\beta_{Y\sim D|P,Z,X}$. However, since
$\mathbf{Z}$ is unobserved, we must run the “short” regression of $Y$ on $D,P$
and $\mathbf{X}$ in Equation 19, rather than the desired long regression.

$\displaystyle Y$ $\displaystyle=\beta_{Y\sim D|P,Z,X}D+\beta_{Y\sim
P|D,Z,X}P+\mathbf{X}\beta_{Y\sim X|D,P,Z}+\mathbf{Z}\beta_{Y\sim
Z|D,P,X}+\epsilon_{l}$ (18) $\displaystyle Y$ $\displaystyle=\beta_{Y\sim
D|P,X}D+\beta_{Y\sim P|D,X}P+\mathbf{X}\beta_{Y\sim X|D,P}+\epsilon_{s}$ (19)

Following a similar approach as we took for placebo outcomes, we can arrive at
similar partial identification expressions for the placebo treatment case,

$\displaystyle\text{bias}_{\text{(YD.PX)}}\overset{\Delta}{=}\beta_{Y\sim
D|P,X}-\beta_{Y\sim D|P,Z,X}=k\times\left(\beta_{Y\sim P|D,X}-\beta_{Y\sim
P|D,Z,X}\right)\times\frac{\text{SD}(P^{\perp D,X})}{\text{SD}(D^{\perp
P,X})}$ (20) $\displaystyle\beta_{Y\sim D|P,Z,X}=\beta_{Y\sim
D|P,X}-k\times\left(\beta_{Y\sim P|D,X}-\beta_{Y\sim
P|D,Z,X}\right)\times\frac{\text{SD}(P^{\perp D,X})}{\text{SD}(D^{\perp
P,X})}$ (21)

All components of these expressions except for
$k\overset{\Delta}{=}\frac{R_{Y\sim Z|D,P,X}f_{D\sim
Z_{\text{(Y.DPX)}}|P,X}}{R_{Y\sim Z|D,P,X}f_{P\sim
Z_{\text{(Y.DPX)}}|D,X}}=\frac{\text{bias}_{\text{(YD.PX)}}}{\text{bias}_{\text{(YP.DX)}}\times\text{SF}}$
and $\beta_{Y\sim P|D,Z,X}$ can be estimated from the data. $\beta_{Y\sim
P|D,Z,X}$ captures any direct causal relationship between $P$ and $Y$. Since
both $P$ and $Y$ are observed variables, their relationship conditional on
$D,Z,X$ should be able to be specified as the regression coefficient
$\beta_{Y\sim P|D,Z,X}$ despite scale considerations. As with placebo
outcomes, $k$ captures the relative levels of confounding. Note that we can
compare the level of confounding of the $P\to Y$ relationship from
$\mathbf{Z}$ to the level of confounding of the $D\to Y$ relationship from
$\mathbf{Z}$ by using the same linear combination,
$Z_{\text{(Y.DPX)}}=\mathbf{Z}\beta_{Y\sim Z|D,P,X}$. This is because in the
placebo treatment approach we run a single regression with $Y$ as the outcome
that includes both $D$ and $P$ on the right-hand side. We assume that
$\mathbf{Z}$ contains a rich enough set of unobserved variables to de-confound
both $P\to Y$ and $D\to Y$. $k$ is the ratio of the level of total confounding
of $D\to Y$ from $D\leftarrow Z\to Y$ to the level of total confounding of
$P\to Y$ from $P\leftarrow Z\to Y$, after re-scaling one of these biases to
account for scale differences between $P$ and $D$.888Note that we do not
cancel terms in $k=\frac{R_{Y\sim Z|D,P,X}f_{D\sim
Z_{\text{(Y.DPX)}}|P,X}}{R_{Y\sim Z|D,P,X}f_{P\sim Z_{\text{(Y.DPX)}}|D,X}}$
since this would force us to reason about how $f_{D\sim
Z_{\text{(Y.DPX)}}|P,X}$ differs from $f_{D\sim\mathbf{Z}|P,X}$ and how
$f_{P\sim Z_{\text{(Y.DPX)}}|D,X}$ differs from $f_{P\sim\mathbf{Z}|D,X}$. It
is more intuitive to reason about relative levels of bias, which does not
require us to directly consider those differences. We can again use this for
“postulatory” partial identification by considering plausible ranges of values
for $k$ and $\beta_{Y\sim P|D,Z,X}$, or as sensitivity analysis by inverting
the analysis to determine what values of $k$ and $\beta_{Y\sim P|D,Z,X}$ would
have to be defended in order to defend a given substantive conclusion.

#### Estimator and partial identification

Equation 21 provides us with a moment estimator for $\beta_{Y\sim D|P,Z,X}$ in
Equation 22.999It is also possible to show that
$\frac{\widehat{\text{SD}}(P^{\perp D,X})}{\widehat{\text{SD}}(D^{\perp
P,X})}=\frac{\widehat{\text{SD}}(P^{\perp D,X})}{\widehat{\text{SD}}(D^{\perp
P,X})}\times\frac{\widehat{\text{SD}}(Y^{\perp
D,P,X})}{\widehat{\text{SD}}(Y^{\perp
D,P,X})}=\frac{\text{se}(\hat{\beta}_{Y\sim
D|P,X})}{\text{se}(\hat{\beta}_{Y\sim
P|D,X})}\times\sqrt{\frac{\text{df}_{D}}{\text{df}_{P}}}$, which is an
expression containing only commonly reported regression summary statistics
from the short regression that is estimable from the observed data.
$\text{se}(\hat{\beta}_{Y\sim D|P,X})$ and $\text{se}(\hat{\beta}_{Y\sim
P|D,X})$ are the estimated standard errors from standard regression estimation
of the short regression. $\text{df}_{Y}$ and $\text{df}_{N}$ are the degrees
of freedom from the estimated short regression, these will be equal in this
case.

$\displaystyle\hat{\beta}_{Y\sim D|P,Z,X}=\hat{\beta}_{Y\sim
D|P,X}-k\times\left(\hat{\beta}_{Y\sim P|D,X}-\beta_{Y\sim
P|D,Z,X}\right)\times\frac{\|P^{\perp D,X}\|_{2}}{\|D^{\perp P,X}\|_{2}}$ (22)

Given $k$ and $\beta_{Y\sim P|D,Z,X}$, the estimator $\hat{\beta}_{Y\sim
D|P,Z,X}$ from Equation 22 is consistent for $\beta_{Y\sim D|P,Z,X}$ and
asymptotically normal. We discuss computing standard errors below. Using
Equation 22 in a partial identification framework, we can reason about
plausible ranges of values for $\beta_{Y\sim P|D,Z,X}$ and $k$ to arrive at a
range of plausible estimates of $\beta_{Y\sim D|P,Z,X}$. In the special case
when there is only a single omitted variable, then $k$ can also be written as
$k=\frac{f_{D\sim Z|P,X}}{f_{P\sim Z|D,X}}$. This can be thought of as
comparing the strength of the $Z\to D$ “arm” of the confounding path
$D\leftarrow Z\to Y$ to the relative strength of the $Z\to P$ “arm” of the
confounding path $P\leftarrow Z\to Y$ in Figure 1(b).

#### Statistical inference.

For fixed values for the partial identification parameters, the moment
estimators in Equations 17, and 22 are consistent for the target regression
coefficient on the treatment from the “long” regression and are asymptotically
normal. Non-parametric bootstrap can be used to estimate standard errors.

###  2.4 Varieties of (single) placebos

The approaches presented above are properties of linear regression, and apply
to any (pair of) regression(s). However, an assumed causal structure (embodied
by a graph) is necessary to link the estimates to causal quantities of
interest. While Figure 1 represents four graphs in which a placebo variable
can be considered a placebo outcome or treatment, there are additional cases
representing placebos as mediators, observed confounders, or post-outcome
variables. Tables 2 and 3 illustrate 8 different causal structure with a
single placebo variable in different positions, inclusive of those in Figure
1. We assume that all relevant unobserved common causes are bundled into the
vector of unobserved variables $\mathbf{Z}$; and we refer to the placebo as
$P$ in every case as the $P$ versus $N$ distinction is not always appropriate
or clear.101010Observed covariates $\mathbf{X}$ are omitted from the graphs
for clarity. These tables detail the specific regressions one would run and
adjustment one would make for partial identification purposes in each case. In
some causal structures, a given placebo variable can be understood to have
different roles, allowing it to be handled in more than one way. In such a
case, we can partially identify effects in more than one way by employing
different types of assumptions. Different assumptions can lead to different
conclusions. When ranges of assumed plausible values for parameters include
the truth, different approaches will lead to similar conclusions.

Consider Table 2[c]. The causal structure includes the edge $P\to Y$. There
are two options for how we might approach this setting. First, we might
consider $P$ to be an imperfect placebo treatment. The machinery that we
detailed above (and outlined in the entry for Table 2[a] - Placebo Treatment)
is all we need. Second, we might consider $P$ to be a placebo outcome. Suppose
we simply applied the placebo outcome approach we’ve already seen. In this
case, $\text{bias}_{\text{(YD.X)}}$, the numerator of $k$, will capture both
$D\leftarrow Z\to Y$ and $D\leftarrow Z\to P\to Y$, while
$\text{bias}_{\text{(PD.X)}}$, the denominator of $k$, will capture
$D\leftarrow Z\to P$. $D\leftarrow Z\to P$ appears in both the numerator and
denominator of $k$, making the parameter more difficult to interpret. The
solution is simple: include $P$ in the regression where $Y$ is the outcome. We
see that the “Short Regressions” column of the table shows this update. $P$
plays a role like an observed confounder: Controlling for it blocks a non-
causal path between $D$ and $Y$. In Appendix A, we derive the expression for
$\beta_{Y\sim D|P,Z,X}$ that we see in the “Target Coefficient Expression”
column. This contains two coefficients from the short regressions and a scale
factor that are all estimable from observed data (i.e., data on $Y,D,P,X$). It
also contains a $k$ parameter that compares the level of unobserved
confounding from $D\leftarrow Z\to Y$ (i.e., $\text{bias}_{\text{(YD.PX)}}$)
to that from $D\leftarrow Z\to P$ (i.e., $\text{bias}_{\text{(PD.X)}}$), after
accounting for scale differences, and an imperfect placebo parameter capturing
the direct causal relationship between $D$ and $P$, which in this graph is
zero. These are listed in the “Parameters” column. The remaining entries in
Tables 2 and 3 can be read similarly. When the causal structure is that of a
mediator (i.e., there exists a path $D\to P\to Y$ in addition to $D\to Y$),
the placebo approach becomes very complicated, and we do not recommend our
approach.

##  3 Difference in difference as a special case

In this section, we illustrate how traditional DID can be seen as a special
case of the placebo outcome approach laid out above. We start by considering
difference in means (DIM) decompositions for binary treatments. For an outcome
$Y$ and binary treatment where the treated group is indicated by $G$,

$\displaystyle\text{DIM}_{Y}$
$\displaystyle=\mathbb{E}[Y|G=1]-\mathbb{E}[Y|G=0]=\mathbb{E}[Y_{1}|G=1]-\mathbb{E}[Y_{0}|G=0]$
(23)
$\displaystyle=\underbrace{\mathbb{E}[Y_{1}|G=1]-\mathbb{E}[Y_{0}|G=1]}_{\text{ATT}_{Y}}+\underbrace{\mathbb{E}[Y_{0}|G=1]-\mathbb{E}[Y_{0}|G=0]}_{\text{Bias}_{Y}}$

Similarly, for a placebo outcome $N$,

$\displaystyle\text{DIM}_{N}$
$\displaystyle=\mathbb{E}[N|G=1]-\mathbb{E}[N|G=0]=\mathbb{E}[Y_{1}|G=1]-\mathbb{E}[Y_{0}|G=0]$
(24)
$\displaystyle=\underbrace{\mathbb{E}[N_{1}|G=1]-\mathbb{E}[N_{0}|G=1]}_{\text{ATT}_{N}}+\underbrace{\mathbb{E}[N_{0}|G=1]-\mathbb{E}[N_{0}|G=0]}_{\text{Bias}_{N}}$

With these in hand, it is easy to see that there exists an $m\in\mathbb{R}$
such that we can obtain the expression for $\text{ATT}_{Y}$,

$\displaystyle\text{ATT}_{Y}$ $\displaystyle=\text{DIM}_{Y}-\text{Bias}_{Y}$
(25) $\displaystyle=\text{DIM}_{Y}-m\times\text{Bias}_{N}$
$\displaystyle=\text{DIM}_{Y}-m\times(\text{DIM}_{N}-\text{ATT}_{N}),\text{
where }m=\frac{\text{Bias}_{Y}}{\text{Bias}_{N}}$

But how does this expression for $\text{ATT}_{Y}$ relate to the traditional
DID identification strategy? DID identifies
$\text{ATT}_{Y}=\text{DIM}_{Y}-\text{DIM}_{N}$ by making two assumptions.
First, we must assume that $\text{ATT}_{N}=0$. That is, we have a perfect
placebo outcome. In DID, the placebo outcome is typically a pre-treatment
version of the outcome, meaning that the treatment cannot have a causal effect
on the placebo outcome. Thus, this first assumption may be satisfied by
design. The second identifying assumption required by DID is “parallel
trends”, which requires that the expected overtime change in non-treatment
outcome in the treated group and control group are identical. It is easy to
show that the parallel trends assumption is equivalent to an assumption of
equiconfounding ($m=1$) (Equation 26; Sofer et al. (2016)).

$\displaystyle\text{Parallel
Trends}\overset{\Delta}{=}\underbrace{\mathbb{E}[Y_{0}|G=1]-\mathbb{E}[N_{0}|G=1]}_{\text{Trend}_{G=1}}$
$\displaystyle=\underbrace{\mathbb{E}[Y_{0}|G=0]-\mathbb{E}[N_{0}|G=0]}_{\text{Trend}_{G=0}}$
(26)
$\displaystyle\iff\underbrace{\mathbb{E}[Y_{0}|G=1]-\mathbb{E}[Y_{0}|G=0]}_{\text{Bias}_{Y}}$
$\displaystyle=\underbrace{\mathbb{E}[N_{0}|G=1]-\mathbb{E}[N_{0}|G=0]}_{\text{Bias}_{N}}\overset{\Delta}{=}\text{equiconfounding}$

To connect this with the approach above: Since $G$ is binary, a regression of
$Y$ (or $N$) on $G$ will give the difference in means $\text{DIM}_{Y}$ (or
$\text{DIM}_{N}$), respectively, as the coefficient on $G$. Under the standard
DID assumptions, we would have $\text{ATT}_{Y}=\beta_{Y\sim G}-\beta_{N\sim
G}$. To relax parallel trends/ equiconfounding, using omitted variable bias,
we instead consider

$\displaystyle\text{ATT}_{Y}+\text{Bias}_{Y}=\text{DIM}_{Y}=\beta_{Y\sim
G}=\beta_{Y\sim G|Z}+\text{Bias}_{Y,\text{reg.}}$ (27)
$\displaystyle\text{ATT}_{N}+\text{Bias}_{N}=\text{DIM}_{N}=\beta_{N\sim
G}=\beta_{N\sim G|Z}+\text{Bias}_{N,\text{reg.}}$ (28)

where $\mathbf{Z}$ contains all the remaining confounders of the $G,Y$ and
$G,N$ relationships. Under assumptions of effect homogeneity, or using OLS
regression as an approximation, we have that $\text{ATT}_{Y}=\beta_{Y\sim
G|Z}$, which means that $\text{Bias}_{Y}=\text{Bias}_{Y,\text{reg.}}$.
Similarly, $\text{ATT}_{N}=\beta_{N\sim G|Z}$, which means that
$\text{Bias}_{N}=\text{Bias}_{N,\text{reg.}}$. Plugging into the expression
for $\text{ATT}_{Y}$, we get Equation 29, where $m$ is the ratio of biases for
the regression coefficients.

$\displaystyle\text{ATT}_{Y}$ $\displaystyle=\beta_{Y\sim
G}-m\times(\beta_{N\sim G}-\beta_{N\sim G|Z})$ (29)

As in previous sections, we could re-express $m$ to account for scale
differences. For a discussion of how assumptions on $m$ correspond to
assumptions on differential trends, see Appendix C.

##  4 Examples

[Note: As a draft, both of these examples are of placebo outcomes and look
like the DID-relaxing situation; in the future we would like to swap one of
these for a different sort of example.]

###  4.1 National Supported Work Demonstration

The National Supported Work Demonstration (NSW) was a job training program.
From 1975 to 1979, it aimed to help disadvantaged workers build basic skills.
The program randomly assigned participants to training positions. These
consisted of a group that received the benefits of the program (the treatment
group) and a group that did not receive any benefits (the control group)
(LaLonde, 1986). The data also include a non-experimental dataset in which
individuals not from the NSW program but from the Panel Study of Income
Dynamics (PSID) are added as additional control units. The data that we work
with is “a subset that consists of males with three (2 pre-training and 1
post-training) years of earnings data (the outcome of interest). This subset
has been considered in (Dehejia and Wahba 1999; Smith and Todd 2005; Firpo
2007)” (Callaway, 2019).111111Data can be found in the qte package (in R) by
running the command data(lalonde,package = "qte") We use the non-experimental
dataset (lalonde.psid) to illustrate the methods discussed in this paper and
then compare these results with the estimates from the experimental data
(called lalonde.exp).121212Since this data includes a randomized experiment,
we can also estimate the effect using the experimental data to get an unbiased
estimate of the true treatment effect. This experimental estimate can be used
to back out the true value for $k$, assuming that $\beta_{N\sim D|Z,X}=0$. The
difference in means estimate of the treatment effect from the experimental
data is 1,795.55 USD higher 1978 earnings in dollars for the treated group
relative to the control group. The estimate is 1,693.12 USD when we control
for observed covariates.

We use 1974 earnings (`re74`) as a placebo outcome for 1978 earnings (`re78`).
The treatment (`treat`) is experiencing the job training program. We also use
the set of covariates ($X$) in the data including age, education (years and
degree or not), marital status, and demographic indicators. We recognize that
there are unobserved covariates ($Z$) that we wish we could condition on but
cannot. Ideally, we would like to estimate the regression `lm(re78 ~ treat + X
+ Z)`. Due to the unobserved confounders, we will not be able to point
identify the coefficient on `treat` from this “long” regression. Instead, we
partially identify $\beta_{Y\sim D|X,Z}$ with the approach proposed above, and
leverage information on 1974 earnings as a placebo outcome.

Since the placebo outcome is pre-treatment earnings, it is reasonable to
assume that the treatment has no causal effect on the the placebo outcome.
Therefore, we consider the perfect placebo assumption, $\beta_{P\sim
D|Z,X}=0$, plausible here.131313We perform these analyses using the placeboLM
package the authors built for this purpose, currently under development. It
can be installed via devtools::install_github("Adam-Rohde/placeboLM").

This package can be used in a manner reminiscent of the `lm()` function in
`R`. Arguments to the `placeboLM` function include the dataset, the outcome
variable, the treatment variable, the placebo variable (and whether we want to
think of it as a placebo outcome or treatment), the direction or presence of
the edge from the treatment ($D$) to the placebo ($P$), the direction or
presence of the edge from the placebo ($P$) to the outcome ($Y$), the observed
covariates, and finally the ranges of the partial identification parameters
that we want to consider or potentially defend. The first partial
identification parameter is $k$, which captures the ratio of the level of
confounding of $D$ and $Y$, conditional on $P$, to the level of confounding of
$D$ and $P$, after accounting for scale differences. We can think of this as
the $D,Y$ confounding as a percentage of the $D,P$ confounding, where scale
differences between $P$ and $Y$ have been accounted for. The second partial
identification parameter is $\beta_{P\sim D|Z,X}$ (coded as
`coef_P_D_given_XZ`) which captures the direct causal relationship between $D$
and $P$ and we already noted should be zero in this case. For this example, we
set the partial identification parameter ranges to be conservative and assume
that the placebo (1974 earnings) can also directly cause the outcome (1978
earnings).

placeblm.out = placeboLM( data = "lalonde.psid", outcome = "re78", treatment =
"treat", placebo_outcome = "re74", observed_covariates = c("age", "education",
"black", "hispanic", "married", "nodegree"), partialIDparam_minmax = list(k =
c(-2,2), coef_P_D_given_XZ = c(-15000,15000)) )

One way to examine the results is to produce a table like Table 1, in which
each row corresponds to an assumption, such as the selection on observatblaes
(SOO) assumption (i.e. no unobserved confounding), standard DID, or DID on the
$k$-scale, with $k=1$.

Another way to view the result is on a contour plot, showing how the adjusted
estimate would vary across values of $k$ and $\beta_{P\sim D|X,Z}$, shown in
Figure 2. Finally, a third useful way to examine the results is with a line
plot of the estimates as we vary one of the two partial identification
parameters. We can specify percentiles of the other partial identification
parameter and have a line plot generated for each. Since we believe
$\beta_{P\sim D|X,Z}=0$ with confidence in this case, we can focus on only
this “slice”. The line plot also includes 95% confidence intervals. See Figure
3.141414The three figures above are produced by placeboLM with the helper
functions placeboLM_table(), placeboLM_contour_plot(), and
placeboLM_line_plot() respectively.

Table 1:  Partial identification table for National Supported Work example with 1974 earnings as placebo outcome. | `k` | `coef_P_D_given_XZ` | Estimate | Std. Error | CI Low | CI High
---|---|---|---|---|---|---
SOO | 0 | 0 | -5928.11 | 822.57 | -7563.28 | -4337.09
Standard DID | 0.845 | 0 | 1718.01 | 852.98 | 27.35 | 3393.96
`k=1` DID | 1 | 0 | 3115.31 | 882.42 | 1337.4 | 4783.3
Grid | -1 | -7500 | -6100.92 | 1375.32 | -8843.01 | -3539.27
Grid | -1 | 0 | -14971.53 | 1379.99 | -17615.87 | -12160.46
Grid | -1 | 7500 | -23842.13 | 1484.07 | -26566.69 | -20663
Grid | 0 | -7500 | -5928.11 | 809.4 | -7473.88 | -4304.65
Grid | 0 | 0 | -5928.11 | 842.29 | -7549.07 | -4215.56
Grid | 0 | 7500 | -5928.11 | 832.39 | -7523.88 | -4262.12
Grid | 1 | -7500 | -5755.3 | 871.06 | -7441.71 | -4051.86
Grid | 1 | 0 | 3115.31 | 926.64 | 1389.42 | 4907.9
Grid | 1 | 7500 | 11985.91 | 984.87 | 10176.39 | 13963.14
Figure 2:  Contour plot for National Supported Work example with 1974 earnings
as placebo outcome. Figure 3:  Line plot for National Supported Work example
with 1974 earnings as placebo outcome.

Beginning with Table 1, the SOO-based estimate is negative. While perhaps
unexpected, there is little to be said about this estimate as the underlying
assumption of no unobserved confounding is not defensible. Under standard DID
($m=1$), the effect of job training estimated under this assumption is an
increase of 1,718 USD—extremely close to the benchmark experimental estimate
(1,693 USD). The $k=1$ DID estimate is also reasonable, at 3,115 USD. Note
that the assumption that $\beta_{P\sim D|Z,X}=0$, the experimental estimate
would imply that the “true” value for $k$ is about 0.84, i.e. the bias for the
treatment and 1978 earnings is 84% of the bias for the treatment and 1974
earnings, after accounting for scale differences.

Without knowledge of the experimental estimate, an analyst might focus
principally on the line plot and make an argument that the bias for the
treatment and 1978 earnings and the bias for the treatment and 1974 earnings
have at least the same sign. This alone would put us on the right half of the
line plot. We might further guess that the levels of confounding or bias are
somewhat similar though perhaps not identical, since the two earnings measures
are separated in time by a few years. Hence we might consider values for $k$
that are relatively close to 1, say 2/3 to 1.5. This would lead us to defense
estimates of roughly $0$ to $8000$ USD as credible. In this way, it would be
possible to construct a reasonable and largely informative range of estimates
under more defensible assumptions than the strict and usually undefended ones
of equiconfounding (parallel trends) required for DID. We also note that while
the placebo outcome here was a pre-treatment version of the outcome variable,
similar analyses can be conducted using any placebo outcome variable.

###  4.2 Effects of Zika virus on birth rates

In our second application, we investigate the effect Zika virus had on birth
rates in Brazil. Zika virus can cause birth complications, thus potentially
discouraging or delaying planned pregnancies and births. We use data
previously analyzed by Taddeo et al. (2022) and Tchetgen et al. (2023), also
employing placebo-based techniques. In Brazil in 2015, the state of Pernambuco
saw high rates of Zika, while zero cases were reported in the state of Rio
Grande do Sul. As in previous works, we regard the former state as the treated
group and the later as the control group. We use municipality level overall
birth rates per 1000 people in 2016 as our primary outcome of interest, while
the same measure in 2014 is our placebo outcome. We make the reasonable
assumption that the treatment has no direct causal effect on the pre-treatment
outcome. Using data provided by Amorim (2022), we perform the analysis using
placeboLM, indicating that the 2014 birth rate is the placebo outcome, letting
$k$ range from 0 to 2, and letting $\beta_{P\sim D|X,Z}$ range from -10 to 10
to provide a very inclusive first look.

Figure 4:  Line plot for Zika virus example with 2014 birth rate as placebo
outcome.

The naive difference in means estimate finds that Zika is associated with 3.4
additional births per 1000 people. By contrast, the two DID estimates ($m=1$
and $k=1$) are negative, indicating 1.2-1.3 fewer births per thousand people
in Pernambuco (the Zika-treated state). Relaxing either of these parallel
trend assumptions, we see that if the level of confounding of 2016 birth rate
and treatment is approximately 70% of the level of confounding of 2014 birth
rate and treatment or more, then the effect estimates are negative. The
estimate reaches a 2.0 births per thousand reduction when one assumes the
relative level of confounding is roughly 120% ($k=1.2$).

These findings are consistent with the results in Taddeo et al. (2022) and
Tchetgen et al. (2023), but we note a key difference in the required
assumptions. Specifically, we could conceive of the state indicator, which is
used as a proxy for whether or not the population was exposed to Zika virus,
as another direct cause of 2014 birth rates, in that the two states likely
differ in important ways that lead to divergent birth rates in 2014. Such a
structure is readily accommodated in our framework by considering values for
`coef_P_D_given_XZ` other than zero, but as discussed further below, it would
violate a key assumption employed in Tchetgen et al. (2023).

##  5 Discussion

To begin with the connection to DID we draw out, prior work has also noted the
connection between placebo outcomes and DID (Sofer et al., 2016). There have
also been several recent attempts to relax the parallel trends assumption
within the context of DID or to improve on the current practice of testing for
parallel trends in the pre-treatment period (see e.g. Manski and Pepper, 2018;
Bilinski and Hatfield, 2018; Freyaldenhoven et al., 2019; Keele et al., 2019;
Ryan et al., 2019; Ye et al., 2020; Gibson and Zimmerman, 2021; Rambachan and
Roth, 2021. Gibson and Zimmerman (2021) points out that violations of parallel
trends can be considered using an application of the omitted variable bias
framework from Cinelli and Hazlett (2020) in a first difference regression;
see also Rambachan and Roth (2021). Unlike our approach, this strategy would
be limited to the DID setting in which the pre-treatment outcome is directly
comparable to the post-treatment outcome, and it requires that researchers
observe the pre-treatment outcome and the post-treatment outcome for the same
set of units; that is, it requires panel data. The principal differences
between such an approach and applying ours to DID, however, is in the way
unobserved confounding is reasoned about. If one constructs a first-difference
regression and submits it to OVB-based sensitivity analysis, then one would go
on to reason about how strongly unobserved confounders would explain residual
variance in the treatment assignment process, and in the change in (non-
treatment) outcomes over time. Using our approach, the investigator would
instead consider the pre-treatment outcome as the placebo outcome, and reason
about the relative confounding of this pre-treatment outcome with the
treatment, compared to the real outcome and the treatment. Which of these
approaches is most useful is a matter of specific application and domain
expertise.

Returning to the broader setting of placebo treatments and outcomes, beyond
the DID connection, efforts to employ placebos to gain causal leverage are not
new. Lipsitch et al. (2010) provide an early discussion of the potential of
using placebo treatments and outcomes. Arnold et al. (2016) and Arnold and
Ercumen (2016) discuss the potential to use placebos to address sample
selection and measurement bias as well as bias in randomized experiments. Shi
et al. (2020) review many established procedures and recent developments in
this growing literature. An important recent area of research has been dubbed
“proximal” causal inference (see Tchetgen et al., 2020 for an introduction).
This line of work leverages “double-negative control designs” (Shi et al.,
2020) in which both a placebo outcome and a placebo treatment are present.
Under certain assumptions and conditions, papers such as Miao et al. (2018,
2020) and Mastouri et al. (2021) show non-parametric identification using both
of these placebos. While this work is very promising, it requires two
placebos, certain null effect assumptions (i.e., “perfect” placebos), and
regularity conditions.151515In Appendix B, we show a similar point
identification result for regression when both a “perfect” placebo treatment
and a “perfect” placebo outcome are available. We also relax the “double
placebo” setting to allow for imperfect placebos. Most critically, it requires
a “bridge function” assumption, which is tantamount to an equiconfounding
assumption on a transformation of the placebo variable. Also notable in this
direction is recent work (Tchetgen et al., 2023) in the single-placebo case.
While a potentially valuable approach, it applies under different causal
structures than those studied here. Specifically, it applies where the outcome
variable and placebo are associated only due to the effect of the treatment.
This rules out cases where the placebo and true outcome share a common cause
confounder, as well as all other DAGs in Tables 2 and 3. It would, however, be
a good fit in contexts akin to classical measurement error, such as when the
placebo is a descendant of the outcome and not of any confounding variables.

More broadly, the novelty of our approach lies in taking the OVB approach as a
framework for relaxing assumption of perfect placebos and relaxing
equiconfounding assumptions. The main limitations are, first, our dependency
on linearity (though see Appendix D for partially linear and non-parametric
extensions). Second, like any sensitivity or partial identification approach,
the key question to consider in choosing among methods is whether the
quantities investigators must reason about can be understood and debated, so
that it is an aid rather than hinderance to transparency. Finally, and also in
common with other partial identifications, these tools may leave researchers
unable to defend informative ranges of estimates in some cases. This, in our
view, is a feature not a bug, as it reminds us that these tools simply reflect
what can (or cannot) be concluded given the assumptions we are (or are not)
prepared to defend.

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Table 2:  Taxonomy of Single Placebo Graphs |  Graph ($\mathbf{X}$ omitted for clarity) |  Placebo Type | Short Regression(s) | Parameters |  Target Coefficient Expression
---|---|---|---|---|---
[a] |  $D$$Y$$P$$\mathbf{Z}$ |  Placebo Outcome | $\begin{aligned} Y=&\beta_{Y\sim D|X}D+\\\ &\mathbf{X}\beta_{Y\sim X|D}+\epsilon_{s}\\\ P=&\beta_{P\sim D|X}D+\\\ &\mathbf{X}\beta_{P\sim X|D}+\xi_{s}\end{aligned}$ | $\begin{aligned} &k=\frac{\text{bias}_{\text{(YD.X)}}}{\text{bias}_{\text{(PD.X)}}\times\text{SF}}\\\ &\beta_{P\sim D|X,Z}(=0\text{ in this graph})\\\ &\text{SF}=\frac{\text{SD}(Y^{\perp D,X})}{\text{SD}(P^{\perp D,X})}\end{aligned}$ |  $\begin{aligned} &\beta_{Y\sim D|Z,X}=\beta_{Y\sim D|X}-\\\ &k\times\left(\beta_{P\sim D|X}-\beta_{P\sim D|Z,X}\right)\\\ &\times\text{SF}\end{aligned}$
|  Placebo Treatment | $\begin{aligned} Y=&\beta_{Y\sim D|P,X}D+\\\ &\beta_{Y\sim P|D,X}P+\\\ &\mathbf{X}\beta_{Y\sim X|D,P}+\epsilon_{s}\end{aligned}$ | $\begin{aligned} &k=\frac{\text{bias}_{\text{(YD.PX)}}}{\text{bias}_{\text{(YP.DX)}}\times\text{SF}}\\\ &\beta_{Y\sim P|D,Z,X}(=0\text{ in this graph})\\\ &\text{SF}=\frac{\text{SD}(P^{\perp D,X})}{\text{SD}(D^{\perp P,X})}\end{aligned}$ |  $\begin{aligned} &\beta_{Y\sim D|P,Z,X}=\beta_{Y\sim D|P,X}-\\\ &k\times\left(\beta_{Y\sim P|D,X}-\beta_{Y\sim P|D,Z,X}\right)\\\ &\times\text{SF}\end{aligned}$
[b] |  $D$$Y$$P$$\mathbf{Z}$ |  Placebo Outcome (see above row); $\beta_{P\sim D|X,Z}\neq 0$
[c] |  $D$$Y$$P$$\mathbf{Z}$ |  Placebo Treatment (see above row); $\beta_{Y\sim P|D,Z,X}\neq 0$
|  Observed Confounder 1 | $\begin{aligned} Y=&\beta_{Y\sim D|P,X}D+\\\ &\beta_{Y\sim P|D,X}P+\\\ &\mathbf{X}\beta_{Y\sim X|D,P}+\epsilon_{s}\\\ P=&\beta_{P\sim D|X}D+\\\ &\mathbf{X}\beta_{P\sim X|D}+\xi_{s}\end{aligned}$ | $\begin{aligned} &k=\frac{\text{bias}_{\text{(YD.PX)}}}{\text{bias}_{\text{(PD.X)}}\times\text{SF}}\\\ &\beta_{P\sim D|Z,X}(=0\text{ in this graph})\\\ &\text{SF}=\frac{\text{SD}(Y^{\perp D,P,X})}{\text{SD}(D^{\perp P,X})}\times\frac{\text{SD}(D^{\perp X})}{\text{SD}(P^{\perp D,X})}\end{aligned}$ |  $\begin{aligned} &\beta_{Y\sim D|P,Z,X}=\beta_{Y\sim D|P,X}-\\\ &k\times(\beta_{P\sim D|X}-\beta_{P\sim D|Z,X})\\\ &\times\text{SF}\end{aligned}$
[d] |  $D$$Y$$P$$\mathbf{Z}$ |  See Zhang and Ding (2022) for partial identification of indirect and direct effects
|  Could use Cinelli and Hazlett (2020) for partial identification of total
effects; $R^{2}_{Y\sim Z|D,X}$ includes $Z\to P\to Y$
|  Placebo Outcome (see above row); $\beta_{P\sim D|X,Z}$ is part of total
effect; $\text{bias}_{\text{(YD.X)}}$ includes $Z\to P\to Y$
|  Mediator | $\begin{aligned} Y=&\beta_{Y\sim D|X}D+\\\ &\mathbf{X}\beta_{Y\sim X|D}+\epsilon_{s}\\\ Y=&\beta_{Y\sim D|P,X}D+\\\ &\beta_{Y\sim P|D,X}P+\\\ &\mathbf{X}\beta_{Y\sim X|D,P}+\xi_{s}\end{aligned}$ | $\begin{aligned} &k=\frac{{\text{bias}_{\text{(YD.X)}}}}{{\text{bias}_{\text{(YP.DX)}}}\times\text{SF}}\\\ &(\text{bias}_{\text{(YD.X)}}\text{ includes $Z\to P\to Y$})\\\ &\beta_{Y\sim P|D,Z,X}\text{ is part of total effect}\\\ &\text{SF}=\frac{\text{SD}(P^{\perp D,X})}{\text{SD}(D^{\perp X})}\frac{\text{SD}(Y^{\perp D,X})}{\text{SD}(Y^{\perp D,P,X})}\end{aligned}$ |  $\begin{aligned} &\beta_{Y\sim D|Z,X}=\beta_{Y\sim D|X}-\\\ &k\times(\beta_{Y\sim P|D,X}-\beta_{Y\sim P|D,Z,X})\\\ &\times\text{SF}\end{aligned}$
Table 3:  Taxonomy of Single Placebo Graphs - Continued |  Graph ($\mathbf{X}$ omitted for clarity) |  Placebo Type | Short Regression(s) | Parameters |  Target Coefficient Expression
---|---|---|---|---|---
[e] |  $D$$Y$$P$$\mathbf{Z}$ |  Observed Confounder 2 | $\begin{aligned} Y=&\beta_{Y\sim D|P,X}D+\\\ &\beta_{Y\sim P|D,X}P+\\\ &\mathbf{X}\beta_{Y\sim X|D,P}+\epsilon_{s}\\\ D=&\beta_{D\sim P|X}P+\\\ &\mathbf{X}\beta_{D\sim X|P}+\xi_{s}\end{aligned}$ | $\begin{aligned} &k=\frac{{\text{bias}_{\text{(YD.PX)}}}}{{\text{bias}_{\text{(DP.X)}}}\times\text{SF}}\\\ &\beta_{D\sim P|Z,X}\neq 0\\\ &\text{SF}=\frac{\text{SD}(Y^{\perp D,P,X})}{\text{SD}(D^{\perp P,X})}\times\frac{\text{SD}(P^{\perp X})}{\text{SD}(D^{\perp P,X})}\end{aligned}$ |  $\begin{aligned} &\beta_{Y\sim D|P,Z,X}=\beta_{Y\sim D|P,X}-\\\ &k\times(\beta_{D\sim P|X}-\beta_{D\sim P|Z,X})\\\ &\times\text{SF}\end{aligned}$
[f] |  $D$$Y$$P$$\mathbf{Z}$ | Observed Confounder 2 (see above row)
[g] |  $D$$Y$$P$$\mathbf{Z}$ |  Post-Outcome | $\begin{aligned} Y=&\beta_{Y\sim D|X}D+\\\ &\mathbf{X}\beta_{Y\sim X|D}+\epsilon_{s}\\\ P=&\beta_{P\sim D|Y,X}D+\\\ &\beta_{P\sim Y|D,X}Y+\\\ &\mathbf{X}\beta_{P\sim X|D,Y}+\xi_{s}\end{aligned}$ | $\begin{aligned} &k=\frac{{\text{bias}_{\text{(YD.X)}}}}{{\text{bias}_{\text{(PY.DX)}}}\times\text{SF}}\\\ &\beta_{P\sim Y|D,Z,X}\neq 0\\\ &\text{SF}=\frac{\text{SD}(Y^{\perp D,X})}{\text{SD}(D^{\perp X})}\times\frac{\text{SD}(Y^{\perp D,X})}{\text{SD}(P^{\perp Y,D,X})}\end{aligned}$ |  $\begin{aligned} &\beta_{Y\sim D|Z,X}=\beta_{Y\sim D|X}-\\\ &k\times(\beta_{P\sim Y|D,X}-\beta_{P\sim Y|D,Z,X})\\\ &\times\text{SF}\end{aligned}$
[h] |  $D$$Y$$P$$\mathbf{Z}$ | Post-Outcome (see above row)

##  Appendix A Single Placebo Derivations

#### Placebo Treatment (Table 2[a])

$\displaystyle\text{bias}_{\text{(YD.PX)}}=\beta_{Y\sim D|P,X}-\beta_{Y\sim
D|P,Z,X}=R_{Y\sim Z|D,P,X}f_{D\sim Z_{\text{(Y.DPX)}}|P,X}(\text{SD}(Y^{\perp
D,P,X})/\text{SD}(D^{\perp P,X}))$ (30)
$\displaystyle\text{bias}_{\text{(YP.DX)}}=\beta_{Y\sim P|D,X}-\beta_{Y\sim
P|D,Z,X}=R_{Y\sim Z|D,P,X}f_{P\sim Z_{\text{(Y.DPX)}}|D,X}(\text{SD}(Y^{\perp
D,P,X})/\text{SD}(P^{\perp D,X}))$
$\displaystyle\text{bias}_{\text{(YD.PX)}}=m\times\text{bias}_{\text{(YP.DX)}}\implies\beta_{Y\sim
D|P,Z,X}=\beta_{Y\sim D|P,X}-m\times(\beta_{Y\sim P|D,X}-\beta_{Y\sim
P|D,Z,X})$ $\displaystyle
m=\frac{{\text{bias}_{\text{(YD.PX)}}}}{{\text{bias}_{\text{(YP.DX)}}}}=\frac{R_{Y\sim
Z|D,P,X}f_{D\sim Z_{\text{(Y.DPX)}}|P,X}}{R_{Y\sim Z|D,P,X}f_{P\sim
Z_{\text{(Y.DPX)}}|D,X}}\times\frac{\text{SD}(P^{\perp
D,X})}{\text{SD}(D^{\perp P,X})}=k\times\frac{\text{SD}(P^{\perp
D,X})}{\text{SD}(D^{\perp P,X})}$ $\displaystyle\implies
k=\frac{{\text{bias}_{\text{(YD.PX)}}}}{{\text{bias}_{\text{(YP.DX)}}}\times\frac{\text{SD}(P^{\perp
D,X})}{\text{SD}(D^{\perp P,X})}}$ $\displaystyle\therefore\beta_{Y\sim
D|P,Z,X}=\beta_{Y\sim D|P,X}-k\times\left(\beta_{Y\sim P|D,X}-\beta_{Y\sim
P|D,Z,X}\right)\times\frac{\text{SD}(P^{\perp D,X})}{\text{SD}(D^{\perp
P,X})}$

#### Observed Confounder 1 (Table 2[c])

$\displaystyle\text{bias}_{\text{(YD.PX)}}=\beta_{Y\sim D|P,X}-\beta_{Y\sim
D|P,Z,X}=R_{Y\sim Z|D,P,X}f_{D\sim Z_{\text{(Y.DPX)}}|P,X}(\text{SD}(Y^{\perp
D,P,X})/\text{SD}(D^{\perp P,X}))$ (31)
$\displaystyle\text{bias}_{\text{(PD.X)}}=\beta_{P\sim D|X}-\beta_{P\sim
D|Z,X}=R_{P\sim Z|D,X}f_{D\sim Z_{\text{(P.DX)}}|X}(\text{SD}(P^{\perp
D,X})/\text{SD}(D^{\perp X}))$
$\displaystyle\text{bias}_{\text{(YD.PX)}}=m\times\text{bias}_{\text{(PD.X)}}\implies\beta_{Y\sim
D|P,Z,X}=\beta_{Y\sim D|P,X}-m\times(\beta_{P\sim D|X}-\beta_{P\sim D|Z,X})$
$\displaystyle
m=\frac{{\text{bias}_{\text{(YD.PX)}}}}{{\text{bias}_{\text{(PD.X)}}}}=\frac{R_{Y\sim
Z|D,P,X}f_{D\sim Z_{\text{(Y.DPX)}}|P,X}}{R_{P\sim Z|D,X}f_{D\sim
Z_{\text{(P.DX)}}|X}}\times\frac{\text{SD}(Y^{\perp
D,P,X})}{\text{SD}(D^{\perp P,X})}\times\frac{\text{SD}(D^{\perp
X})}{\text{SD}(P^{\perp D,X})}=k\times\frac{\text{SD}(Y^{\perp
D,P,X})}{\text{SD}(D^{\perp P,X})}\times\frac{\text{SD}(D^{\perp
X})}{\text{SD}(P^{\perp D,X})}$ $\displaystyle\implies
k=\frac{{\text{bias}_{\text{(YD.PX)}}}}{{\text{bias}_{\text{(PD.X)}}}\times\frac{\text{SD}(Y^{\perp
D,P,X})}{\text{SD}(D^{\perp P,X})}\times\frac{\text{SD}(D^{\perp
X})}{\text{SD}(P^{\perp D,X})}}$ $\displaystyle\therefore\beta_{Y\sim
D|P,Z,X}=\beta_{Y\sim D|P,X}-k\times(\beta_{P\sim D|X}-\beta_{P\sim
D|Z,X})\times\frac{\text{SD}(Y^{\perp D,P,X})}{\text{SD}(D^{\perp
P,X})}\times\frac{\text{SD}(D^{\perp X})}{\text{SD}(P^{\perp D,X})}$

#### Mediator (Table 2[d])

$\displaystyle\text{bias}_{\text{(YD.X)}}=\beta_{Y\sim D|X}-\beta_{Y\sim
D|Z,X}=R_{Y\sim Z|D,X}f_{D\sim Z_{\text{(Y.DX)}}|X}(\text{SD}(Y^{\perp
D,X})/\text{SD}(D^{\perp X}))$ (32)
$\displaystyle\text{bias}_{\text{(YP.DX)}}=\beta_{Y\sim P|D,X}-\beta_{Y\sim
P|D,Z,X}=R_{Y\sim Z|D,P,X}f_{P\sim Z_{\text{(Y.DPX)}}|D,X}(\text{SD}(Y^{\perp
D,P,X})/\text{SD}(P^{\perp D,X}))$
$\displaystyle\text{bias}_{\text{(YD.X)}}=m\times\text{bias}_{\text{(YP.DX)}}\implies\beta_{Y\sim
D|Z,X}=\beta_{Y\sim D|X}-m\times(\beta_{Y\sim P|D,X}-\beta_{Y\sim P|D,Z,X})$
$\displaystyle
m=\frac{{\text{bias}_{\text{(YD.X)}}}}{{\text{bias}_{\text{(YP.DX)}}}}=\frac{R_{Y\sim
Z|D,X}f_{D\sim Z_{\text{(Y.DX)}}|X}}{R_{Y\sim Z|D,P,X}f_{P\sim
Z_{\text{(Y.DPX)}}|D,X}}\frac{\text{SD}(P^{\perp D,X})}{\text{SD}(D^{\perp
X})}\frac{\text{SD}(Y^{\perp D,X})}{\text{SD}(Y^{\perp
D,P,X})}=k\times\frac{\text{SD}(P^{\perp D,X})}{\text{SD}(D^{\perp
X})}\frac{\text{SD}(Y^{\perp D,X})}{\text{SD}(Y^{\perp D,P,X})}$
$\displaystyle\implies
k=\frac{{\text{bias}_{\text{(YD.X)}}}}{{\text{bias}_{\text{(YP.DX)}}}\frac{\text{SD}(P^{\perp
D,X})}{\text{SD}(D^{\perp X})}\frac{\text{SD}(Y^{\perp
D,X})}{\text{SD}(Y^{\perp D,P,X})}}$ $\displaystyle\therefore\beta_{Y\sim
D|Z,X}=\beta_{Y\sim D|X}-k\times(\beta_{Y\sim P|D,X}-\beta_{Y\sim
P|D,Z,X})\times\frac{\text{SD}(P^{\perp D,X})}{\text{SD}(D^{\perp
X})}\frac{\text{SD}(Y^{\perp D,X})}{\text{SD}(Y^{\perp D,P,X})}$

#### Observed Confounder 2 (Table 3[e])

$\displaystyle\text{bias}_{\text{(YD.PX)}}=\beta_{Y\sim D|P,X}-\beta_{Y\sim
D|P,Z,X}=R_{Y\sim Z|D,P,X}f_{D\sim Z_{\text{(Y.DPX)}}|P,X}(\text{SD}(Y^{\perp
D,P,X})/\text{SD}(D^{\perp P,X}))$ (33)
$\displaystyle\text{bias}_{\text{(DP.X)}}=\beta_{D\sim P|X}-\beta_{D\sim
P|Z,X}=R_{D\sim Z|P,X}f_{P\sim Z_{\text{(D.PX)}}|X}(\text{SD}(D^{\perp
P,X})/\text{SD}(P^{\perp X}))$
$\displaystyle\text{bias}_{\text{(YD.PX)}}=m\times\text{bias}_{\text{(DP.X)}}\implies\beta_{Y\sim
D|P,Z,X}=\beta_{Y\sim D|P,X}-m\times(\beta_{D\sim P|X}-\beta_{D\sim P|Z,X})$
$\displaystyle
m=\frac{{\text{bias}_{\text{(YD.PX)}}}}{{\text{bias}_{\text{(DP.X)}}}}=\frac{R_{Y\sim
Z|D,P,X}f_{D\sim Z_{\text{(Y.DPX)}}|P,X}}{R_{D\sim Z|P,X}f_{P\sim
Z_{\text{(D.PX)}}|X}}\times\frac{\text{SD}(Y^{\perp
D,P,X})}{\text{SD}(D^{\perp P,X})}\times\frac{\text{SD}(P^{\perp
X})}{\text{SD}(D^{\perp P,X})}=k\times\frac{\text{SD}(Y^{\perp
D,P,X})}{\text{SD}(D^{\perp P,X})}\times\frac{\text{SD}(P^{\perp
X})}{\text{SD}(D^{\perp P,X})}$ $\displaystyle\implies
k=\frac{{\text{bias}_{\text{(YD.PX)}}}}{{\text{bias}_{\text{(DP.X)}}}\times\frac{\text{SD}(Y^{\perp
D,P,X})}{\text{SD}(D^{\perp P,X})}\frac{\text{SD}(P^{\perp
X})}{\text{SD}(D^{\perp P,X})}}$ $\displaystyle\therefore\beta_{Y\sim
D|P,Z,X}=\beta_{Y\sim D|P,X}-k\times(\beta_{D\sim P|X}-\beta_{D\sim
P|Z,X})\times\frac{\text{SD}(Y^{\perp D,P,X})}{\text{SD}(D^{\perp
P,X})}\times\frac{\text{SD}(P^{\perp X})}{\text{SD}(D^{\perp P,X})}$

#### Post-outcome (Table 3[g])

$\displaystyle\text{bias}_{\text{(YD.X)}}=\beta_{Y\sim D|X}-\beta_{Y\sim
D|Z,X}=R_{Y\sim Z|D,X}f_{D\sim Z_{\text{(Y.DX)}}|X}(\text{SD}(Y^{\perp
D,X})/\text{SD}(D^{\perp X}))$ (34)
$\displaystyle\text{bias}_{\text{(PY.DX)}}=\beta_{P\sim Y|D,X}-\beta_{P\sim
Y|D,Z,X}=R_{P\sim Z|Y,D,X}f_{Y\sim Z_{\text{(P.YDX)}}|D,X}(\text{SD}(P^{\perp
Y,D,X})/\text{SD}(Y^{\perp D,X}))$
$\displaystyle\text{bias}_{\text{(YD.X)}}=m\times\text{bias}_{\text{(PY.DX)}}\implies\beta_{Y\sim
D|Z,X}=\beta_{Y\sim D|X}-m\times(\beta_{P\sim Y|D,X}-\beta_{P\sim Y|D,Z,X})$
$\displaystyle
m=\frac{{\text{bias}_{\text{(YD.X)}}}}{{\text{bias}_{\text{(PY.DX)}}}}=\frac{R_{Y\sim
Z|D,X}f_{D\sim Z_{\text{(Y.DX)}}|X}}{R_{P\sim Z|Y,D,X}f_{Y\sim
Z_{\text{(P.YDX)}}|D,X}}\times\frac{\text{SD}(Y^{\perp
D,P,X})}{\text{SD}(D^{\perp P,X})}\times\frac{\text{SD}(P^{\perp
X})}{\text{SD}(D^{\perp P,X})}=k\times\frac{\text{SD}(Y^{\perp
D,X})}{\text{SD}(D^{\perp X})}\times\frac{\text{SD}(Y^{\perp
D,X})}{\text{SD}(P^{\perp Y,D,X})}$ $\displaystyle\implies
k=\frac{{\text{bias}_{\text{(YD.X)}}}}{{\text{bias}_{\text{(PY.DX)}}}\times\frac{\text{SD}(Y^{\perp
D,X})}{\text{SD}(D^{\perp X})}\times\frac{\text{SD}(Y^{\perp
D,X})}{\text{SD}(P^{\perp Y,D,X})}}$ $\displaystyle\therefore\beta_{Y\sim
D|Z,X}=\beta_{Y\sim D|X}-k\times(\beta_{P\sim Y|D,X}-\beta_{P\sim
Y|D,Z,X})\times\frac{\text{SD}(Y^{\perp D,X})}{\text{SD}(D^{\perp
X})}\times\frac{\text{SD}(Y^{\perp D,X})}{\text{SD}(P^{\perp Y,D,X})}$

##  Appendix B Double placebos

Consider a setting like Figure 5 \- that is, we observe both a placebo outcome
($N$) and a placebo treatment ($P$). Suppose we are interested in
$\beta_{Y\sim D|Z,X}$ from the “long” regression in Equation 1. However, $Z$
is unobserved and so we must only consider estimating “short” regression in
Equation 2. We might consider the following two sets of short and long
regressions. Again, $\mathbf{Z}$ is a set of unobserved confounders. We can
only consider estimating the short regressions (i.e., Equations 35b and 35d).
$Z_{\text{(Y.DPX)}}=\mathbf{Z}\beta_{Y\sim Z|D,P,X}$ is the linear combination
that de-confounds the $Y,D$ and the $Y,P$ relationship.
$Z_{\text{(N.DPX)}}=\mathbf{Z}\beta_{N\sim Z|D,P,X}$ is the linear combination
that de-confounds the $N,D$ and the $N,P$ relationship. We choose $\mathbf{Z}$
to be a rich enough set of variables to allow for this interpretation.

$\displaystyle Y$ $\displaystyle=\beta_{Y\sim D|P,Z,X}D+\beta_{Y\sim
P|D,Z,X}P+\mathbf{X}\beta_{Y\sim X|D,P,Z}+\mathbf{Z}\beta_{Y\sim
Z|D,P,X}+\epsilon_{y,l}$ (35a) $\displaystyle Y$ $\displaystyle=\beta_{Y\sim
D|P,X}D+\beta_{Y\sim P|D,X}P+\mathbf{X}\beta_{Y\sim X|D,P}+\epsilon_{y,s}$
(35b) $\displaystyle N$ $\displaystyle=\beta_{N\sim D|P,Z,X}D+\beta_{N\sim
P|D,Z,X}P+\mathbf{X}\beta_{N\sim X|D,P,Z}+\mathbf{Z}\beta_{N\sim
Z|D,P,X}+\epsilon_{n,l}$ (35c) $\displaystyle N$ $\displaystyle=\beta_{N\sim
D|P,X}D+\beta_{N\sim P|D,X}P+\mathbf{X}\beta_{N\sim X|D,P}+\epsilon_{n,s}$
(35d)

Figure 5:  A single causal graph with both a placebo outcome, $N$, and with a
placebo treatment, $P$. $D$ is treatment, $Y$ is outcome, and $\mathbf{Z}$
contains unobserved confounders. $\mathbf{X}$ contains observed covariates.

(a) Perfect Double Placebo$D$$Y$$P$$N$$\mathbf{Z}$$\mathbf{X}$

(a) Imperfect Double Placebo$D$$Y$$P$$N$$\mathbf{Z}$$\mathbf{X}$

Consider the following OVB expressions, where we use the relevant linear
combination for each relationship.

$\displaystyle\beta_{Y\sim D|P,X}-\beta_{Y\sim
D|P,Z_{\text{(Y.DPX)}},X}=\beta_{Y\sim
Z_{\text{(Y.DPX)}}|D,P,X}\beta_{Z_{\text{(Y.DPX)}}\sim D|P,X}$ (36a)
$\displaystyle\beta_{Y\sim P|D,X}-\beta_{Y\sim
P|D,Z_{\text{(Y.DPX)}},X}=\beta_{Y\sim
Z_{\text{(Y.DPX)}}|D,P,X}\beta_{Z_{\text{(Y.DPX)}}\sim P|D,X}$ (36b)
$\displaystyle\beta_{N\sim D|P,X}-\beta_{N\sim
D|P,Z_{\text{(N.DPX)}},X}=\beta_{N\sim
Z_{\text{(N.DPX)}}|D,P,X}\beta_{Z_{\text{(N.DPX)}}\sim D|P,X}$ (36c)
$\displaystyle\beta_{N\sim P|D,X}-\beta_{N\sim
P|D,Z_{\text{(N.DPX)}},X}=\beta_{N\sim
Z_{\text{(N.DPX)}}|D,P,X}\beta_{Z_{\text{(N.DPX)}}\sim P|D,X}$ (36d)

We can re-write the middle two of these as follows.

$\displaystyle\beta_{Y\sim Z_{\text{(Y.DPX)}}|D,P,X}$
$\displaystyle=\frac{\beta_{Y\sim P|D,X}-\beta_{Y\sim
P|D,Z_{\text{(Y.DPX)}},X}}{\beta_{Z_{\text{(Y.DPX)}}\sim P|D,X}}$ (37a)
$\displaystyle\beta_{Z_{\text{(N.DPX)}}\sim D|P,X}$
$\displaystyle=\frac{\beta_{N\sim D|P,X}-\beta_{N\sim
D|P,Z_{\text{(N.DPX)}},X}}{\beta_{N\sim Z_{\text{(N.DPX)}}|D,P,X}}$ (37b)

Now we note that there are $m_{A}$ and $m_{B}$ that satisfy the following.

$\displaystyle\beta_{Z_{\text{(Y.DPX)}}\sim D|P,X}$
$\displaystyle=m_{\text{A}}\times\beta_{Z_{\text{(N.DPX)}}\sim D|P,X}$ (38a)
$\displaystyle\beta_{Z_{\text{(Y.DPX)}}\sim P|D,X}$
$\displaystyle=m_{\text{B}}\times\beta_{Z_{\text{(N.DPX)}}\sim P|D,X}$ (38b)

We can derive the following expression for $\beta_{Y\sim
D|P,Z_{\text{(Y.DPX)}},X}$.

$\displaystyle\beta_{Y\sim D|P,X}-\beta_{Y\sim D|P,Z_{\text{(Y.DPX)}},X}$
$\displaystyle=\beta_{Y\sim
Z_{\text{(Y.DPX)}}|D,P,X}\beta_{Z_{\text{(Y.DPX)}}\sim D|P,X}$ (39)
$\displaystyle=m_{\text{A}}\times\beta_{Y\sim
Z_{\text{(Y.DPX)}}|D,P,X}\beta_{Z_{\text{(N.DPX)}}\sim D|P,X}$
$\displaystyle=m_{\text{A}}\times\left[\frac{\beta_{Y\sim P|D,X}-\beta_{Y\sim
P|D,Z_{\text{(Y.DPX)}},X}}{\beta_{Z_{\text{(Y.DPX)}}\sim
P|D,X}}\right]\left[\frac{\beta_{N\sim D|P,X}-\beta_{N\sim
D|P,Z_{\text{(N.DPX)}},X}}{\beta_{N\sim Z_{\text{(N.DPX)}}|D,P,X}}\right]$
$\displaystyle=\frac{m_{\text{A}}}{m_{\text{B}}}\times\frac{(\beta_{Y\sim
P|D,X}-\beta_{Y\sim P|D,Z_{\text{(Y.DPX)}},X})(\beta_{N\sim
D|P,X}-\beta_{N\sim D|P,Z_{\text{(N.DPX)}},X})}{[\beta_{N\sim
P|D,X}-\beta_{N\sim P|D,Z_{\text{(N.DPX)}},X}]}$
$\displaystyle\therefore\beta_{Y\sim D|P,Z_{\text{(Y.DPX)}},X}$
$\displaystyle=\beta_{Y\sim
D|P,X}-\frac{m_{\text{A}}}{m_{\text{B}}}\times\frac{(\beta_{Y\sim
P|D,X}-\beta_{Y\sim P|D,Z_{\text{(Y.DPX)}},X})(\beta_{N\sim
D|P,X}-\beta_{N\sim D|P,Z_{\text{(N.DPX)}},X})}{(\beta_{N\sim
P|D,X}-\beta_{N\sim P|D,Z_{\text{(N.DPX)}},X})}$ $\displaystyle\text{Now,
}\frac{m_{\text{A}}}{m_{\text{B}}}$
$\displaystyle=\frac{\beta_{Z_{\text{(Y.DPX)}}\sim
D|P,X}}{\beta_{Z_{\text{(N.DPX)}}\sim
D|P,X}}\frac{\beta_{Z_{\text{(Y.DPX)}}\sim
P|D,X}}{\beta_{Z_{\text{(N.DPX)}}\sim P|D,X}}$ (40)
$\displaystyle=\frac{R_{Z_{\text{(Y.DPX)}}\sim D|P,X}R_{Z_{\text{(N.DPX)}}\sim
P|D,X}}{R_{Z_{\text{(N.DPX)}}\sim D|P,X}R_{Z_{\text{(Y.DPX)}}\sim
P|D,X}}\times\frac{\text{SD}(Z_{\text{(Y.DPX)}}^{\perp
P,X})}{\text{SD}(Z_{\text{(N.DPX)}}^{\perp
P,X})}\frac{\text{SD}(Z_{\text{(N.DPX)}}^{\perp
D,X})}{\text{SD}(Z_{\text{(Y.DPX)}}^{\perp D,X})}$
$\displaystyle=\frac{R_{Z_{\text{(Y.DPX)}}\sim D|P,X}R_{Z_{\text{(N.DPX)}}\sim
P|D,X}}{R_{Z_{\text{(N.DPX)}}\sim D|P,X}R_{Z_{\text{(Y.DPX)}}\sim
P|D,X}}\times\frac{\text{SD}(Z_{\text{(Y.DPX)}}^{\perp
P,X})}{\text{SD}(Z_{\text{(Y.DPX)}}^{\perp
D,P,X})}\frac{\text{SD}(Z_{\text{(N.DPX)}}^{\perp
D,P,X})}{\text{SD}(Z_{\text{(N.DPX)}}^{\perp
P,X})}\frac{\text{SD}(Z_{\text{(N.DPX)}}^{\perp
D,X})}{\text{SD}(Z_{\text{(N.DPX)}}^{\perp
D,P,X})}\frac{\text{SD}(Z_{\text{(Y.DPX)}}^{\perp
D,P,X})}{\text{SD}(Z_{\text{(Y.DPX)}}^{\perp D,X})}$
$\displaystyle=\frac{R_{Z_{\text{(Y.DPX)}}\sim D|P,X}R_{Z_{\text{(N.DPX)}}\sim
P|D,X}}{R_{Z_{\text{(N.DPX)}}\sim D|P,X}R_{Z_{\text{(Y.DPX)}}\sim
P|D,X}}\times\frac{\sqrt{1-R^{2}_{Z_{\text{(N.DPX)}}\sim
D|P,X}}\sqrt{1-R^{2}_{Z_{\text{(Y.DPX)}}\sim
P|D,X}}}{\sqrt{1-R^{2}_{Z_{\text{(Y.DPX)}}\sim
D|P,X}}\sqrt{1-R^{2}_{Z_{\text{(N.DPX)}}\sim P|D,X}}}$
$\displaystyle=\frac{f_{Z_{\text{(Y.DPX)}}\sim D|P,X}f_{Z_{\text{(N.DPX)}}\sim
P|D,X}}{f_{Z_{\text{(N.DPX)}}\sim D|P,X}f_{Z_{\text{(Y.DPX)}}\sim P|D,X}}$
$\displaystyle\therefore\frac{m_{\text{A}}}{m_{\text{B}}}$
$\displaystyle=\frac{R_{Y\sim
Z_{\text{(Y.DPX)}}|D,P,X}f_{Z_{\text{(Y.DPX)}}\sim D|P,X}}{R_{Y\sim
Z_{\text{(Y.DPX)}}|D,P,X}f_{Z_{\text{(Y.DPX)}}\sim P|D,X}}\times\frac{R_{N\sim
Z_{\text{(N.DPX)}}|D,P,X}f_{Z_{\text{(N.DPX)}}\sim P|D,X}}{R_{N\sim
Z_{\text{(N.DPX)}}|D,P,X}f_{Z_{\text{(N.DPX)}}\sim
D|P,X}}\overset{\Delta}{=}k_{\text{(YD / YP)}}\times k_{\text{(NP / ND)}}$

Thus, we arrive at a final expression for $\beta_{Y\sim
D|P,Z_{\text{(Y.DPX)}},X}$:

$\displaystyle\beta_{Y\sim D|P,Z_{\text{(Y.DPX)}},X}$ (41)
$\displaystyle=\beta_{Y\sim D|P,X}-k_{\text{(YD / YP)}}\times k_{\text{(NP /
ND)}}\times\frac{(\beta_{Y\sim P|D,X}-\beta_{Y\sim
P|D,Z_{\text{(Y.DPX)}},X})(\beta_{N\sim D|P,X}-\beta_{N\sim
D|P,Z_{\text{(N.DPX)}},X})}{(\beta_{N\sim P|D,X}-\beta_{N\sim
P|D,Z_{\text{(N.DPX)}},X})}$

This is a fairly complicated expression that involves five partial
identification or sensitivity parameters. We suspect this may be unwieldy in
many practical cases. However, it may be useful when certain assumptions are
defensible. But first a few notes on what this expression contains.

  * •

$k_{\text{(YD / YP)}}$ captures the scaled ratio of the level of confounding
of the $Y,D$ relationship (conditional on $P$) to the level of confounding of
the $Y,P$ relationship (conditional on $D$), after re-scaling one of these
biases, or the ratio of bias factors.

  * •

$k_{\text{(NP / ND)}}$ captures the scaled ratio of the level of confounding
of the $N,P$ relationship (conditional on $D$) to the level of confounding of
the $N,D$ relationship (conditional on $P$), after re-scaling one of these
biases, or the ratio of bias factors.

  * •

$\beta_{Y\sim P|D,Z_{\text{(Y.DPX)}},X}$ measures the causal of $P$ on $Y$
(conditional on $D$). This could further be broken down into component edges
with an additional application of OVB.

  * •

$\beta_{N\sim D|P,Z_{\text{(N.DPX)}},X}$ measures the causal of $D$ on $N$
(conditional on $P$).

  * •

$\beta_{N\sim P|D,Z_{\text{(N.DPX)}},X}$ measures the causal of $P$ on $N$
(conditional on $D$).

  * •

We could also use $k_{\text{(YD / ND)}}\times k_{\text{(NP /
YP)}}=\frac{R_{Y\sim Z_{\text{(Y.DPX)}}|D,P,X}f_{Z_{\text{(Y.DPX)}}\sim
D|P,X}}{R_{N\sim Z_{\text{(N.DPX)}}|D,P,X}f_{Z_{\text{(N.DPX)}}\sim
D|P,X}}\times\frac{R_{N\sim Z_{\text{(N.DPX)}}|D,P,X}f_{Z_{\text{(N.DPX)}}\sim
P|D,X}}{R_{Y\sim Z_{\text{(Y.DPX)}}|D,P,X}f_{Z_{\text{(Y.DPX)}}\sim P|D,X}}$
and have similar interpretation.

This should have a similar feel to what we saw in the main text. In certain
settings, this expression simplifies and may be more practically useful.

  1. 1.

If $Z_{\text{(Y.DPX)}}=Z_{\text{(N.DPX)}}$ or both $R_{Z_{\text{(Y.DPX)}}\sim
D|P,X}=R_{Z_{\text{(N.DPX)}}\sim D|P,X}$ and $R_{Z_{\text{(Y.DPX)}}\sim
P|D,X}=R_{Z_{\text{(N.DPX)}}\sim P|D,X}$, then $k_{\text{(YD / YP)}}\times
k_{\text{(NP / ND)}}=1$. A simple case of this is when there is a single
unobserved confounder $Z$. When we believe this may be the case, we actually
do not need to reason about the level of relative confounding at all.

  2. 2.

When we have perfect placebos (i.e., when some or all of $\beta_{Y\sim
P|D,Z_{\text{(Y.DPX)}},X}$, $\beta_{N\sim D|P,Z_{\text{(N.DPX)}},X}$,
$\beta_{N\sim P|D,Z_{\text{(N.DPX)}},X}$ equal zero), the expression also
simplifies.

  3. 3.

When both of the above hold (e.g., we have a single unobserved $Z$ and perfect
placebos), we can point identify $\beta_{Y\sim D|P,Z,X}$ without reasoning
about the relative level of confounding.

##  Appendix C Relative bias as difference in trends in DID

Any assumption for $m$ can be turned into an assumption about how the trends
in control potential outcomes compare. An assumption about a difference in
confounding might initially be stated as, for example,

$\displaystyle\underbrace{\mathbb{E}[Y_{0}|G=1]-\mathbb{E}[Y_{0}|G=0]}_{\text{Bias}_{Y}}$
$\displaystyle=m\times(\underbrace{\underbrace{\mathbb{E}[N|G=1]-\mathbb{E}[N|G=0]}_{\text{DIM}_{N}}-\beta_{N\sim
G|Z}}_{\text{Bias}_{N}})$ (42) $\displaystyle\iff\mathbb{E}[Y_{0}|G=1]$
$\displaystyle=\mathbb{E}[Y_{0}|G=0]+m\times(\mathbb{E}[N|G=1]-\mathbb{E}[N|G=0]-\beta_{N\sim
G|Z})$

An assumption about a difference in trend would be something like Equation 43.

$\displaystyle\underbrace{\mathbb{E}[Y_{0}|G=1]-\mathbb{E}[N|G=1]}_{\text{Trend}_{G=1}}$
$\displaystyle=w\times(\underbrace{\mathbb{E}[Y_{0}|G=0]-\mathbb{E}[N|G=0]}_{\text{Trend}_{G=0}})$
(43) $\displaystyle\iff\mathbb{E}[Y_{0}|G=1]$
$\displaystyle=\mathbb{E}[N|G=1]+w\times(\mathbb{E}[Y_{0}|G=0]-\mathbb{E}[N|G=0])$

For this exposition, we assume $\beta_{N\sim G|Z}=0$. Both Equations 42 and 43
provide an expression for $\mathbb{E}[Y_{0}|G=1]$, which, in the DID setting,
is unobserved while all the other components of the expressions are estimable
from the data. We can set these two expressions for $\mathbb{E}[Y_{0}|G=1]$
equal to each other and then solve for $w$ to get the trend assumption that
corresponds to our assumption on $m$.

$\displaystyle\mathbb{E}[Y_{0}|G=0]+m\times(\mathbb{E}[N|G=1]-\mathbb{E}[N|G=0])=\mathbb{E}[N|G=1]+w\times(\mathbb{E}[Y_{0}|G=0]-\mathbb{E}[N|G=0])$
(44)

$\displaystyle\iff w$
$\displaystyle=\frac{\mathbb{E}[Y_{0}|G=0]+m\times(\mathbb{E}[N|G=1]-\mathbb{E}[N|G=0])-\mathbb{E}[N|G=1]}{\mathbb{E}[Y_{0}|G=0]-\mathbb{E}[N|G=0]}$
(45) $\displaystyle\iff m$
$\displaystyle=\frac{\mathbb{E}[N|G=1]+w\times(\mathbb{E}[Y_{0}|G=0]-\mathbb{E}[N|G=0])-\mathbb{E}[Y_{0}|G=0]}{\mathbb{E}[N|G=1]-\mathbb{E}[N|G=0]}$

We could similarly write $m$ in terms of $w$, if we preferred to make an
assumption on $w$, the difference in the trend, and wanted to know what this
implies about the unequal confounding. Additionally, we know that
$m=k\times\frac{\text{SD}(Y^{\perp G})}{\text{SD}(N^{\perp G})}$ so that we
can make an assumption on $k$ and see what this implies about $w$, the trend,
or make a trend assumption on $w$ and see what this implies about $k$.

##  Appendix D Partially linear and non-parametric framework

If investigators are interested in partially linear of fully non-parametric
approaches to estimating treatment effects, we can use the omitted variables
frameworks from Chernozhukov et al. (2022) to arrive at partial identification
frameworks similar to those presented in the main text for settings in which
placebo outcomes are available. We do not demonstrate the partially linear or
non-parametric case for placebo treatments, but in principle this is also
possible.

###  D.1 Partially linear framework

An investigator may be interested in estimating a partially linear model. $Y$
is the outcome; $N$ is a placebo outcome; $D$ is the treatment; $\mathbf{X}$
is a vector of observed covariates; $\mathbf{Z}$ is a vector of unobserved
covariates. We consider the following short and long partially linear models.

$\displaystyle Y$
$\displaystyle=\theta_{l,Y}D+f_{l}(\mathbf{X},\mathbf{Z})+\epsilon_{l}$ (46a)
$\displaystyle Y$ $\displaystyle=\theta_{s,Y}D+f_{s}(\mathbf{X})+\epsilon_{s}$
(46b) $\displaystyle N$
$\displaystyle=\theta_{l,N}D+g_{l}(\mathbf{X},\mathbf{Z})+\xi_{l}$ (47a)
$\displaystyle N$ $\displaystyle=\theta_{s,N}D+g_{s}(\mathbf{X})+\xi_{s}$
(47b)

There is some $m$ such that
$\text{bias}_{\text{(YD.X)}}=m\times\text{bias}_{\text{(ND.X)}}$, where
$\text{bias}_{\text{(YD.X)}}\overset{\Delta}{=}\theta_{s,Y}-\theta_{l,Y}$ and
$\text{bias}_{\text{(ND.X)}}\overset{\Delta}{=}\theta_{s,N}-\theta_{l,N}$.
This implies that Equation 48 holds. Using Equation 48 for partial
identification will run into the issue that
$m\overset{\Delta}{=}\frac{\text{bias}_{\text{(YD.X)}}}{\text{bias}_{\text{(ND.X)}}}$
may be difficult to interpret when $Y$ and $N$ have differing scales. Can we
do better?

$\displaystyle\theta_{l,Y}$
$\displaystyle=\theta_{s,Y}-m\times(\theta_{s,N}-\theta_{l,N})$ (48)

Chernozhukov et al. (2022) show that omitted variable bias in partially linear
setting can be written in the following way.

$\displaystyle|\theta_{s,Y}-\theta_{l,Y}|^{2}=|{\text{bias}_{\text{(YD.X)}}}|^{2}$
$\displaystyle=\rho_{Y}^{2}\times S_{Y}^{2}\times\eta^{2}_{Y\sim
Z|D,X}\times\frac{\eta^{2}_{D\sim Z|X}}{1-\eta^{2}_{D\sim Z|X}}$ (49a)
$\displaystyle|\theta_{s,N}-\theta_{l,N}|^{2}=|{\text{bias}_{\text{(ND.X)}}}|^{2}$
$\displaystyle=\rho_{N}^{2}\times S_{N}^{2}\times\eta^{2}_{N\sim
Z|D,X}\times\frac{\eta^{2}_{D\sim Z|X}}{1-\eta^{2}_{D\sim Z|X}}$ (49b)

$\text{BF}_{\text{(YD.X)}}\overset{\Delta}{=}\eta^{2}_{Y\sim
Z|D,X}\frac{\eta^{2}_{D\sim Z|X}}{1-\eta^{2}_{D\sim Z|X}}$ and
$\text{BF}_{\text{(ND.X)}}\overset{\Delta}{=}\eta^{2}_{N\sim
Z|D,X}\frac{\eta^{2}_{D\sim Z|X}}{1-\eta^{2}_{D\sim Z|X}}$ are “bias factors”
that are scaled versions of the maximum level of unobserved confounding from
$\mathbf{Z}$. $S_{Y}^{2}$ and $S_{N}^{2}$ are identified and therefore
estimable from the observed data. $\eta^{2}_{Y\sim Z|D,X}$ and
$\eta^{2}_{D\sim Z|X}$ are Pearson’s correlation ratios (or the non-parametric
$R^{2}$s).161616$\eta^{2}_{D\sim
Z|X}=\frac{\text{Var}(\mathbb{E}[D|Z,X])-\text{Var}(\mathbb{E}[D|X])}{\text{Var}(D)-\text{Var}(\mathbb{E}[D|X])}=\frac{\eta^{2}_{D\sim
Z,X}-\eta^{2}_{D\sim X}}{1-\eta^{2}_{D\sim X}}$. $\eta^{2}_{Y\sim Z|D,X}$ can
be similarly interpreted. $\eta^{2}_{Y\sim Z|D,X}$ is the proportion of
residual variation in $Y$ explained by $Z$. $\eta^{2}_{D\sim Z|X}$ is the
proportion of residual variation in $D$ explained by $Z$. $\rho_{Y}^{2}$ and
$\rho_{N}^{2}$ are referred to as “degree of adversity” in Chernozhukov et al.
(2022) and reduce the actual level of bias from the maximum possible. We can
now see that we can re-express $m$ as in Equation 50.

$\displaystyle|m|^{2}=\frac{|\text{bias}_{\text{(YD.X)}}|^{2}}{|\text{bias}_{\text{(ND.X)}}|^{2}}$
$\displaystyle=\frac{\rho_{Y}^{2}\times S_{Y}^{2}\times\eta^{2}_{Y\sim
Z|D,X}\times\frac{\eta^{2}_{D\sim Z|X}}{1-\eta^{2}_{D\sim
Z|X}}}{\rho_{N}^{2}\times S_{N}^{2}\times\eta^{2}_{N\sim
Z|D,X}\times\frac{\eta^{2}_{D\sim Z|X}}{1-\eta^{2}_{D\sim Z|X}}}$ (50)
$\displaystyle=\frac{\rho_{Y}^{2}}{\rho_{N}^{2}}\times\frac{\eta^{2}_{Y\sim
Z|D,X}\times\frac{\eta^{2}_{D\sim Z|X}}{1-\eta^{2}_{D\sim
Z|X}}}{\eta^{2}_{N\sim Z|D,X}\times\frac{\eta^{2}_{D\sim
Z|X}}{1-\eta^{2}_{D\sim Z|X}}}\times\frac{S_{Y}^{2}}{S_{N}^{2}}=\gamma\times
k\times\frac{S_{Y}^{2}}{S_{N}^{2}}$

$\gamma\overset{\Delta}{=}\frac{\rho_{Y}^{2}}{\rho_{N}^{2}}$ and
$k\overset{\Delta}{=}\frac{\eta^{2}_{Y\sim Z|D,X}\times\frac{\eta^{2}_{D\sim
Z|X}}{1-\eta^{2}_{D\sim Z|X}}}{\eta^{2}_{N\sim
Z|D,X}\times\frac{\eta^{2}_{D\sim Z|X}}{1-\eta^{2}_{D\sim
Z|X}}}=\frac{\eta^{2}_{Y\sim Z|D,X}}{\eta^{2}_{N\sim Z|D,X}}$. Which means
that we can write Equation 51. We could use Equation 51 for partial
identification by reasoning about plausible ranges of values for $k$,
$\gamma$, and $\theta_{l,N}$, while estimating the remaining terms from
observed data. $k$ represents the relative level of confounding where
differences in scale have been accounted for. $\theta_{l,N}$ is the effect of
$D$ on $N$. $\gamma$ is the relative level of adversity. $k$ and
$\theta_{l,N}$ should be easy to reason about using external knowledge. It is
possible that $\gamma$ would be more difficult to consider. We leave this
discussion here for now, however. A reasonable simplifying assumption may be
that $\gamma$ is close to 1. See Chernozhukov et al. (2022) for further
discussion of $\eta^{2}$s, $S_{Y}^{2}$, $S_{N}^{2}$, $\rho_{Y}^{2}$, and
$\rho_{N}^{2}$.

$\displaystyle\theta_{l,Y}$
$\displaystyle=\theta_{s,Y}-\text{sign}(m)\times\sqrt{\gamma\times
k}\times(\theta_{s,N}-\theta_{l,N})\times\sqrt{\frac{S_{Y}^{2}}{S_{N}^{2}}}$
(51)

###  D.2 Non-parametric framework

An investigator may also be interested in estimating a linear functional of
the conditional expectation function of the outcome in a fully non-parametric
setting, like
$\theta_{l,Y}=\mathbb{E}[Y_{1}-Y_{0}]=\mathbb{E}[f_{Y}(1,\mathbf{X},\mathbf{Z})-f_{Y}(0,\mathbf{X},\mathbf{Z})]$
for a binary treatment $D$, where $Y_{d}=f_{Y}(d,\mathbf{X},\mathbf{Z},U_{Y})$
is the equation for $Y$ in the structural causal model under intervention to
set $D=d$. Again, the investigator is only able to estimate
$\theta_{s,Y}=\mathbb{E}[f_{Y}^{*}(1,\mathbf{X})-f_{Y}^{*}(0,\mathbf{X})]$,
where
$f_{Y}^{*}(D,X)\overset{\Delta}{=}\mathbb{E}[Y|D,\mathbf{X}]=\mathbb{E}[f_{Y}(D,X,\mathbf{Z})|D,\mathbf{X}]$.
We define $\theta_{l,N}$ and $\theta_{s,N}$ similarly for a placebo outcome.
We could again consider an approach like Equation 52. But again
$m\overset{\Delta}{=}\frac{\text{bias}_{\text{(YD.X)}}}{\text{bias}_{\text{(ND.X)}}}$
may be difficult to interpret when $Y$ and $N$ have differing scales.

$\displaystyle\theta_{l,Y}$
$\displaystyle=\theta_{s,Y}-m\times(\theta_{s,N}-\theta_{l,N})$ (52)

Chernozhukov et al. (2022) also show that omitted variable bias in fully non-
parametric setting can be written as an expression in terms of
$\eta^{2}_{Y\sim Z|D,X}$ and a second term that, in the case of targeting
$\theta_{l,Y}=\mathbb{E}[Y_{1}-Y_{0}]$ with a binary treatment $D$, is the
“average gain in the conditional precision with which we predict $D$ by using
$Z$ in addition to $X$,” which is somewhat similar to $\eta^{2}_{D\sim Z|X}$.
Specifically, we have the following.

$\displaystyle|\theta_{s,Y}-\theta_{l,Y}|^{2}=|{\text{bias}_{\text{(YD.X)}}}|^{2}$
$\displaystyle=\rho_{Y}^{2}\times S_{Y}^{2}\times\eta^{2}_{Y\sim
Z|D,X}\times\frac{1-R^{2}_{\alpha\sim\alpha_{s}}}{R^{2}_{\alpha\sim\alpha_{s}}}$
(53a)
$\displaystyle|\theta_{s,N}-\theta_{l,N}|^{2}=|{\text{bias}_{\text{(YD.X)}}}|^{2}$
$\displaystyle=\rho_{N}^{2}\times S_{N}^{2}\times\eta^{2}_{N\sim
Z|D,X}\times\frac{1-R^{2}_{\alpha\sim\alpha_{s}}}{R^{2}_{\alpha\sim\alpha_{s}}}$
(53b)

$\text{BF}_{\text{(YD.X)}}\overset{\Delta}{=}\eta^{2}_{Y\sim
Z|D,X}\frac{1-R^{2}_{\alpha\sim\alpha_{s}}}{R^{2}_{\alpha\sim\alpha_{s}}}$ and
$\text{BF}_{\text{(ND.X)}}\overset{\Delta}{=}\eta^{2}_{N\sim
Z|D,X}\frac{1-R^{2}_{\alpha\sim\alpha_{s}}}{R^{2}_{\alpha\sim\alpha_{s}}}$ are
“bias factors” that are scaled versions of the maximum level of unobserved
confounding from $\mathbf{Z}$. $S_{Y}^{2}$ and $S_{N}^{2}$ are identified and
therefore estimable from the data. See Chernozhukov et al. (2022) for
discussion of $S_{Y}^{2}$, $S_{N}^{2}$, and $R^{2}_{\alpha\sim\alpha_{s}}$.
Proceeding as in the last section, we can arrive at Equation 54. Here again
$k=\frac{\eta^{2}_{Y\sim Z|D,X}}{\eta^{2}_{N\sim Z|D,X}}$.

$\displaystyle\theta_{l,Y}$
$\displaystyle=\theta_{s,Y}-\text{sign}(m)\times\sqrt{\gamma\times
k}\times(\theta_{s,N}-\theta_{l,N})\times\sqrt{\frac{S_{Y}^{2}}{S_{N}^{2}}}$
(54)

##  Appendix E Re-expression of bias factors

$\displaystyle\beta_{Y\sim D|X}-\beta_{Y\sim D|Z,X}$ $\displaystyle=R_{Y\sim
Z|D,X}f_{D\sim Z|X}\frac{\text{SD}(Y^{\perp D,X})}{\text{SD}(D^{\perp X})}$
$\displaystyle\implies R_{Y\sim Z|D,X}f_{D\sim Z|X}$
$\displaystyle=(\beta_{Y\sim D|X}-\beta_{Y\sim D|Z,X})\frac{\text{SD}(D^{\perp
X})}{\text{SD}(Y^{\perp D,X})}$ $\displaystyle=R_{Y\sim
D|X}\frac{\text{SD}(Y^{\perp X})}{\text{SD}(D^{\perp
X})}\frac{\text{SD}(D^{\perp X})}{\text{SD}(Y^{\perp D,X})}-R_{Y\sim
D|Z,X}\frac{\text{SD}(Y^{\perp X,Z})}{\text{SD}(D^{\perp
X,Z})}\frac{\text{SD}(D^{\perp X})}{\text{SD}(Y^{\perp
D,X})}\frac{\text{SD}(Y^{\perp D,X,Z})}{\text{SD}(Y^{\perp D,X,Z})}$
$\displaystyle=f_{Y\sim D|X}-f_{Y\sim D|Z,X}\frac{\text{SD}(Y^{\perp
D,X,Z})}{\text{SD}(D^{\perp X,Z})}\frac{\text{SD}(D^{\perp
X})}{\text{SD}(Y^{\perp D,X})}=f_{Y\sim D|X}-f_{Y\sim
D|Z,X}\sqrt{\frac{1-R^{2}_{Y\sim Z|D,X}}{1-R^{2}_{D\sim Z|X}}}$
$\displaystyle\implies 1$ $\displaystyle=\frac{f_{Y\sim D|X}}{R_{Y\sim
Z|D,X}f_{D\sim Z|X}}-\frac{f_{Y\sim D|Z,X}}{f_{Y\sim Z|D,X}R_{D\sim Z|X}}$