# Bounds on causal effects in $2^{K}$ factorial experiments with non-
compliance††thanks: Working paper. Comments welcome.

Matthew Blackwell Department of Government and Institute for Quantitative
Social Science, Harvard University. web: http://www.mattblackwell.org email:
<EMAIL_ADDRESS>Nicole E. Pashley Department of Statistics, Rutgers
University. email<EMAIL_ADDRESS>
###### Abstract

Factorial experiments are ubiquitous in the social and biomedical sciences,
but when units fail to comply with each assigned factors, identification and
estimation of the average treatment effects become impossible. Leveraging an
instrumental variables approach, previous studies have shown how to identify
and estimate the causal effect of treatment uptake among respondents who
comply with treatment. A major caveat is that these identification results
rely on strong assumptions on the effect of randomization on treatment uptake.
This paper shows how to bound these complier average treatment effects under
more mild assumptions on non-compliance.

##  1 Introduction

Factorial experiments are widespread in the social and biomedical sciences for
assessing how different interventions (or different features of an
intervention) affect a unit’s response. Unfortunately, in many of these
experiments, units fail to comply with one or more of the factors, making it
difficult to identify and estimate the actual effect of treatment uptake
rather than the effect of treatment randomization. Instrumental variables (IV)
has been a common approach to estimating causal effects in randomized
experiments with non-compliance (Angrist et al.,, 1996). The key intuition of
this approach is that while the average effect of treatment uptake is
unidentified, with some additional assumptions we may be able to identify
those effects among _compliers_ , or those who would always comply with the
treatment assignment they receive.

With a binary treatment, these additional assumptions (often called the
instrumental variables IV assumptions) can be benign or be ensured to hold
through the design of the experiment. In factorial settings, however, point
identification of complier average effects has required a treatment exclusion
restriction that restricts the assignment of each factor to only affect that
factor (Blackwell,, 2017; Blackwell and Pashley,, 2023). This assumption rules
out plausible interactions or spillovers between factors. In this paper, we
derive partial identification bounds on complier average factorial effects
under weaker assumptions about non-compliance. We focus on two assumptions:
for the factor of interest, we assume (1) the existence of a “least compliant
profile” of other factors; and (2) a weak exclusion restriction that allows
for spillovers between factors on treatment uptake. We use these assumption to
bound the main effects of factors among units that always comply with
treatment, whom we call the _constant compliers_. These bounds can be
informative of the direction of the constant complier main effects.

Several studies have previously investigated partial identification bounds in
the context of instrumental variables. Balke and Pearl, (1997) used linear
programming to obtain bounds on the overall average treatment effects under
the IV assumptions with a binary treatment. Cheng and Small, (2006) derived
bounds on the complier average treatment effect in a three-arm trial under an
additional “compliance monotonicity” assumption that we generalize to the
“least compliant profile” assumption for the factorial setting. More recently,
Song et al., (2024) derive bounds on the joint counterfactual distribution of
the outcome when the treatment and outcome are discrete with a finite number
of states. Our analysis leverages the factorial structure to generate
plausible assumptions and narrower bounds.

##  2 Setting

Consider an experiment on $n$ units with $K$ two-level factors, with levels
labeled as $-1$ and $+1$, and the potential for noncompliance on any of the
factors. There are $J=2^{K}$ possible treatment assignments, the number of
combinations of the $K$ factors, which we label
$\mathcal{Z}=\\{\bm{z}_{1},\dots,\bm{z}_{J}\\}$ where each
$\bm{z}=(z_{1},\dots,z_{K})\in\\{-1,1\\}^{K}$. Let $\bm{Z}_{i}$ be the
treatment assigned to unit $i$. Further, let $\bm{z}_{-k}\in\\{-1,1\\}^{K-1}$
denote a treatment assignment less the $k$th factor and similarly
$\mathcal{Z}_{-k}$ be the set of possible assignments on $K-1$ factors.

######  Assumption 1 (Complete randomization).

We assume assignment to the treatment combinations is completely at random,
fixing the number of units assigned to treatment $\bm{z}$ as $N_{\bm{z}}$ such
that $N_{\bm{z}}\geq 2$ for all $\bm{z}\in\mathcal{Z}$ and
$\sum_{\bm{z}\in\mathcal{Z}}N_{\bm{z}}=N$.

See Dasgupta et al., (2015) for more on standard factorial designs. Because of
the noncompliance, we define
$\bm{D}_{i}(\bm{z})=(D_{i1}(\bm{z}),D_{i2}(\bm{z}),\dots,D_{iK}(\bm{z}))$ to
be the treatment uptake for unit $i$ on each factor 1 through $K$ if assigned
to treatment $\bm{z}$ where each $D_{ik}\in\\{-1,1\\}$.

Assuming the Stable Unit Treatment Value Assumption (SUTVA) that there are no
hidden forms of treatment and no interference between units, each unit has a
potential outcome for each combination of assignment and uptake,
$Y_{i}(\bm{z},\bm{d})$. We assume that $Y_{i}(\bm{z},\bm{d})\in[0,1]$,
although extensions of our results to other bounded outcomes is
straightforward.

######  Assumption 2 (Exclusion restriction).

For $\bm{z},\bm{z}^{\prime}$ such that $D_{i}(\bm{z})=D_{i}(\bm{z}^{\prime})$,
$Y_{i}(\bm{z},D_{i}(\bm{z}))=Y_{i}(\bm{z}^{\prime},D_{i}(\bm{z}^{\prime}))$.

Under Assumption 2, each unit has only $2^{K}$ potential outcomes for this
experiment, one for each possible treatment assignment. Define the average of
potential outcomes under treatment $\bm{z}$ as
$\bar{Y}(\bm{z})=N^{-1}\sum_{i=1}^{N}Y_{i}(\bm{z})$ and the average uptake
value for factor $k$ under treatment $\bm{z}$ as
$\bar{D}_{k}(\bm{z})=N^{-1}\sum_{i=1}^{N}D_{ik}(\bm{z})$.

###  2.1 Compliance types in factorial experiments

A compliance type is a complete description of a unit’s uptake for every
possible treatment assignment, or the set
$\\{\bm{D}_{i}(\bm{z})\mid\mathbb{P}(\bm{Z}_{i}=\bm{z})>0\\}$. Without any
restrictions on compliance, there are $(2^{K})^{2^{K}}$ compliance types,
typically making it difficult to learn anything about the effect of treatment
uptake. A common solution to this problem in the binary IV literature is to
restrict the compliance types in such a way that we can identify the effect of
treatment uptake for some subset of compliance types. One common assumption is
monotonicity, which assumes that the effect of treatment assignment on
treatment uptake can never be negative. Blackwell and Pashley, (2023)
generalized this assumption to the factorial setting by having this hold
conditionally for each factor. Let $\bm{z}_{-k}$ be the vector $\bm{z}$
without the $k$th entry omitted, so we can write $(\bm{z}_{-k},z_{k}^{+})$ as
the $\bm{z}$ vector with the $k$th entry as $+1$ and $(\bm{z}_{-k},z_{k}^{-})$
as $\bm{z}$ with the $k$th entry as $-1$. Finally, let
$\mathcal{Z}_{-k}=\\{-1,+1\\}^{K-1}$ be the set of all possible treatment
assignment vectors without entry $k$.

######  Assumption 3 (Conditional monotonicity).

For each $\bm{z}_{-k}\in\mathcal{Z}_{-k}$ and $k\in\\{1,\ldots,K\\}$,
$D_{ik}(\bm{z}_{-k},z_{k}^{+})\geq D_{ik}(\bm{z}_{-k},z_{k}^{-})$ for all $i$.

With conditional monotonicity, we can characterize the compliance types for a
particular factor $k$ as they vary by treatment assignment on the other
factors $\bm{z}_{-k}$ as follows:

  1. 1.

Compliers for $\bm{z}_{-k}$: $D_{ik}(\bm{z}_{-k},z_{k}^{+})=+1$ and
$D_{ik}(\bm{z}_{-k},z_{k}^{-})=-1$,

  2. 2.

Always takers for $\bm{z}_{-k}$:
$D_{ik}(\bm{z}_{-k},z_{k}^{+})=D_{ik}(\bm{z}_{-k},z_{k}^{-})=+1$,

  3. 3.

Never takers for $\bm{z}_{-k}$:
$D_{ik}(\bm{z}_{-k},z_{k}^{+})=D_{ik}(\bm{z}_{-k},z_{k}^{-})=-1$.

Of course, a unit’s compliance might vary as a function of assignment to the
other factors if those other factors can change compliance behavior. For
example, in a $2^{2}$ experiment, there are nine compliance types for factor 1
denoted by their type when assigned to $+1$ for factor 2 and $-1$ for factor
2: {cc, ca, cn, ac, aa, an, nc, na, nn}.

We focus our attention on units that would comply with treatment assignment on
factor $k$ no matter the treatment assignment on other factors. We call these
units the _constant compliers_ for factor $k$ and define them as the units
with

$D_{ik}(\bm{z}_{-k},z_{k}^{+})-D_{ik}(\bm{z}_{-k},z_{k}^{-})=2,\quad\forall\;\bm{z}_{-k}\in\mathcal{Z}_{-k}.$

Let $C_{ik}=1$ if unit $i$ is a constant complier for factor $k$. We define
$\rho_{c_{k}}=\frac{1}{N}\sum_{i=1}^{N}C_{{ik}}$ as the proportion of units
that are constant compliers for factor $k$ and

$\bar{Y}_{c_{k}}(\bm{z})=\frac{1}{\rho_{c_{k}}N}\sum_{i=1}^{N}C_{ik}Y_{i}(\bm{z})$

as the average potential outcome under treatment assignment $\bm{z}$ for the
constant compliers for factor $k$.

We will group other compliance groups together by whether they ever comply
with assignment on factor $k$. For any particular assignment, $\bm{z}_{-k}$,
we can partition units into constant compliers, those who comply for
$\bm{z}_{-k}$ but fail to comply for other assignments, and those who do not
comply for $\bm{z}_{-k}$. We call the latter two groups the _conditional
compliers_ for $\bm{z}_{-k}$ and the _conditional noncompliers_ for
$\bm{z}_{-k}$, and denote their shares in the population as
$\rho_{cc(\bm{z}_{-k})}$ and $\rho_{cn(\bm{z}_{-k})}$, respectively. We also
define their average potential outcomes similarly to the constant compliers as
$\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z})$ and $\bar{Y}_{cn(\bm{z}_{-k})}(\bm{z})$.

Our identification results will focus on the factorial effects for the
constant compliers. For example, the main effect of factor $k$ would be

$\delta_{k}=\frac{1}{2^{K-1}}\sum_{\bm{z}_{{-k}}\in\mathcal{Z}_{-k}}\left[\bar{Y}_{c_{k}}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{c_{k}}(\bm{z}_{-k},z_{k}^{-})\right].$

Note that this is a linear combination of conditional effects of the form
$\bar{Y}_{c_{k}}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{c_{k}}(\bm{z}_{-k},z_{k}^{-})$.

##  3 Identification of main effects for constant compliers

What we can learn about the average factorial effects for constant compliers
will depend on the assumptions about compliance that we can maintain.
Blackwell and Pashley, (2023) assumed that compliance for one factor does not
vary with the assignment of other factors and obtained point identification.
Given the strength of that assumption, here we consider weaker assumptions
that might be applicable in a wide variety of empirical settings.

We begin with a useful decomposition of the intent-to-treat (ITT) effect of
factor $k$ conditional on assignment on the other factors, $\bm{z}_{-k}$.
Recall that the ITT is the effect of assignment on the outcome, ignoring
uptake entirely, which we can decompose as

$\displaystyle\gamma_{k}(\bm{z}_{-k})$
$\displaystyle\equiv\bar{Y}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}(\bm{z}_{-k},z_{k}^{-})=\rho_{c_{k}}\left[\bar{Y}_{c_{k}}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{c_{k}}(\bm{z}_{-k},z_{k}^{-})\right]$
(1)
$\displaystyle+\rho_{cc(\bm{z}_{-k})}\left[\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right]$
$\displaystyle+\rho_{cn(\bm{z}_{-k})}\left[\bar{Y}_{cn(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cn(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right].$

We can see that the ITT is a mixture of three effects: (1) the effect for
constant compliers, (2) the effect for conditional compliers, and (3) the
effect for conditional noncompliers. It may seem odd to have the effect for
conditional noncompliers, but recall that without further assumptions $Z_{ik}$
may affect compliance on other factors ($D_{ij}$ for $j\neq k$) even when it
has no effect on $D_{ik}$.

To bring a more compact notation to this framework, we now define a series of
vectors that encode what treatment vectors we will contrast for each factorial
effect (Dasgupta et al.,, 2015; Blackwell and Pashley,, 2023). We begin by
defining $\bm{Y}=(\bar{Y}(\bm{z}_{1}),\dots,\bar{Y}(\bm{z}_{J}))^{T}$, which
is the vector of average potential outcomes for different treatment
assignments. We then defined a series of $J$-length vectors, $\bm{g}_{j}$,
each of which has one half of its entries as $+1$ and the other half as $-1$
to select the appropriate entries of $\bm{Y}$ when we use the product
$\bm{g}^{T}_{j}\bm{Y}$. We index these contrast vectors by the interaction
level, so that the first $K$ vectors, $\bm{g}_{1},\ldots,\bm{g}_{K}$,
correspond to the main factorial effects of the $K$ factors, the next
$\binom{K}{2}$ vectors correspond to the two-ways interactive effects, and so
on. As one example, $\bm{g}_{k}$ where $k\leq K$ encodes the $k$th main effect
so any treatment assignment vector $(\bm{z}_{-k},z_{k}^{+})$ has a
corresponding $+1$ entry in $\bm{g}_{k}$ and any assignment vector
$(\bm{z}_{-k},z_{k}^{-})$ has a corresponding $-1$ entry in $\bm{g}$.

With this notation in hand, we can write the main factorial ITT for factor $k$
as

$\frac{1}{2^{K-1}}\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}\gamma_{k}(\bm{z}_{-k})=\frac{1}{2^{K-1}}\bm{g}_{k}^{T}\bm{Y},$

which we can combine with Equation (1), to obtain an expression for the
constant complier average factorial effect,

$\displaystyle\delta_{k}=\frac{1}{2^{K-1}\rho_{c_{k}}}\Bigg{[}$
$\displaystyle\bm{g}_{k}^{T}\bm{Y}-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}\rho_{cc(\bm{z}_{-k})}\left[\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right]$
$\displaystyle-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}\rho_{cn(\bm{z}_{-k})}\left[\bar{Y}_{cn(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cn(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right]\Bigg{]}.$

It is possible to identify the ITT effect on treatment uptake as well. Let
$\nu_{k}^{+}(\bm{z}_{-k})=\frac{1}{2}(\overline{D}_{k}(\bm{z}_{-k},z_{k}^{+})+1)$
be the proportion of units receiving $+1$ for factor $k$ under assignment
$(\bm{z}_{-k},z_{k}^{+})$ and
$\nu_{k}^{-}(\bm{z}_{-k})=\frac{1}{2}(\overline{D}_{k}(\bm{z}_{-k},z_{k}^{-})+1)$
be the proportion of units receiving $+1$ for factor $k$ under assignment
$(\bm{z}_{-k},z_{k}^{-})$. Then we can write the ITT for treatment uptake as

$\nu_{k}(\bm{z}_{-k})=\nu_{k}^{+}(\bm{z}_{-k})-\nu_{k}^{-}(\bm{z}_{-k})=\frac{1}{2}\left[\bar{D}_{k}(\bm{z}_{-k},z_{k}^{+})-\bar{D}_{k}(\bm{z}_{-k},z_{k}^{-})\right]=\rho_{k}+\rho_{cc(\bm{z}_{-k})},$

which tells us the proportion of respondents that are either conditional or
constant compliers. Without further assumptions, we cannot break apart these
two groups, so we introduce an assumption on the compliance behavior.

###  3.1 Bounds under a least compliant profile

Without further assumptions, the bounds for the constant complier effects will
be from $-1$ to $1$ because there is insufficient information about the size
of the constant complier group. To put structure on this problem, we first
assume there is a profile of the other factors that is the worst in terms of
compliance for factor $k$ for all units.

######  Assumption 4 (Least compliant profile for factor $k$).

There is a $\tilde{\bm{z}}_{-k}\in\mathcal{Z}_{-k}$, such that
$D_{ik}(\bm{z}_{j},z_{k}^{+})-D_{ik}(\bm{z}_{j},z_{k}^{-})\geq
D_{ik}(\tilde{\bm{z}}_{-k},z_{k}^{+})-D_{ik}(\tilde{\bm{z}}_{-k},z_{k}^{-})$
for all other $\bm{z}_{-k}\in\mathcal{Z}_{-k}$ and for all $i$.

Assumption 4 implies that there is a “most difficult” profile of the other
factors, $\tilde{\bm{z}}_{-k}$, in terms of compliance. In other words, if a
unit complies with assignment on factor $k$ under
$Z_{i,-k}=\tilde{\bm{z}}_{-k}$, then we know it would comply with all other
possible assignments. This implies that we can identify the proportion of
constant compliers as
$\nu_{k}(\tilde{\bm{z}}_{-k})=\frac{1}{2}\left[\bar{D}_{k}(\tilde{\bm{z}}_{-k},z_{k}^{+})-\bar{D}_{k}(\tilde{\bm{z}}_{-k},z_{k}^{-})\right]=\rho_{k}.$
Then we have
$\rho_{cc(\bm{z}_{-k})}=\nu_{k}(\bm{z}_{-k})-\nu_{k}(\tilde{\bm{z}}_{-k})$ and
$\rho_{cn(\bm{z}_{-k})}=1-\nu_{k}(\bm{z}_{-k})$. In Supplemental Materials C,
we discuss this and other assumptions in the context of a data example.

Our bounding strategy will rely on average effects for conditional compliers
being bounded between $-1$ and $1$. Furthermore, noncompliers actually provide
additional information because we can observe certain average outcomes for
those groups. For instance, suppose that a unit is an always-taker for factor
$k$ with assignment $\bm{z}_{-k}$, so that
$D_{ik}(\bm{z}_{-k},z_{k}^{+})=D_{ik}(\bm{z}_{-k},z_{k}^{-})=+1$. Under
conditional monotonicity, we can actually distinguish one set of always-
takers: those that take treatment when assigned control,
$D_{i}(\bm{z}_{-k},z_{k}^{-})=+1$. Let
$\overline{Y}_{a(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})$ and
$\overline{Y}_{n(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})$ be the average
potential outcomes for conditional always- and never-takers, respectively,
which we can identify under conditional monotonicity.

######  Proposition 1.

Suppose Assumptions 1, 2, and 3 hold, and Assumption 4 holds with respect to
factor $k$. Then,
$\delta_{k}\in[\widetilde{\delta}_{k}-b^{-}_{k},\widetilde{\delta}_{k}+b^{+}_{k}]\cap[-1,1]$,
where

$\displaystyle b^{-}_{k}$
$\displaystyle=\frac{1}{2^{{K-1}}\nu_{k}(\tilde{\bm{z}}_{-k})}\Big{[}\sum_{\bm{z}_{-k}}\nu_{k}(\bm{z}_{-k})-\nu_{k}(\tilde{\bm{z}}_{-k})+\nu_{k}^{-}(\bm{z}_{-k})\Big{]},$
(2) $\displaystyle b^{+}_{k}$
$\displaystyle=\frac{1}{2^{{K-1}}\nu_{k}(\tilde{\bm{z}}_{-k})}\Big{[}\sum_{\bm{z}_{-k}}\nu_{k}(\bm{z}_{-k})-\nu_{k}(\tilde{\bm{z}}_{-k})+(1-\nu_{k}^{+}(\bm{z}_{-k}))\Big{]},$
$\displaystyle\widetilde{\delta}_{k}$
$\displaystyle=\frac{1}{2^{{K-1}}\nu_{k}(\tilde{\bm{z}}_{-k})}\Big{[}\bm{g}_{k}^{T}\bm{Y}-\sum_{\bm{z}_{-k}}(1-\nu_{k}^{+}(\bm{z}_{-k}))\overline{Y}_{n(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\sum_{\bm{z}_{-k}}\nu_{k}^{-}(\bm{z}_{-k})\overline{Y}_{a(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\Big{]}.$

######  Remark 1.

There is a set of bounds that are larger if more interpretable than those in
Proposition 1. In particular, we can show that

$\delta_{k}\in\frac{\frac{1}{2^{K-1}}\bm{g}_{k}^{T}\bm{Y}}{\nu_{k}(\tilde{\bm{z}}_{-k})}\pm\frac{1-\nu_{k}(\tilde{\bm{z}}_{-k})}{\nu_{k}(\tilde{\bm{z}}_{-k})}.$

These bounds have a simple interpretation because the first term in this bound
is similar to the Wald estimator in that it is a ratio of two ITT estimators.
The second term, which determines the width of the bounds, captures how much
information we lose to conditional compliance. The width of the bounds,
$2(1-\nu_{k}(\tilde{\bm{z}}_{-k}))/\nu_{k}(\tilde{\bm{z}}_{-k})$, will only be
informative when constant compliance is very high.

###  3.2 Bounds under weak treatment exclusion

We now derive bounds for the constant complier effects when we place
restrictions on the first stage relationship between treatment assignment and
treatment take-up. In particular, we investigate an assumption that changing
the assignment on a factor that a respondent fails to conditionally comply
with cannot affect treatment take-up on other factors.

######  Assumption 5 (Weak treatment exclusion for factor $k$).

For any $\bm{z}_{-k}\in\mathcal{Z}_{-k}$, if
$D_{ik}(\bm{z}_{-k},z_{k}^{+})=D_{ik}(\bm{z}_{-k},z_{k}^{-})$, then
$\bm{D}_{i}(\bm{z}_{-k},z_{k}^{+})=\bm{D}_{i}(\bm{z}_{-k},z_{k}^{-})$ for all
$i$.

Assumption 5 is inspired by a similar assumption in the context of
noncompliance and interference in Imai et al., (2020) and is a weaker version
of the treatment exclusion assumption from Blackwell and Pashley, (2023),
which held that $D_{ik}(z_{k},\bm{z}_{-k})=D_{ik}(z_{k},\bm{z}^{\prime}_{-k})$
for all $z_{k},\bm{z}_{-k}$, $\bm{z}^{\prime}_{-k}$, and $k$. Blackwell and
Pashley, (2023) used this assumption to point identify the effects of various
complier average factorial effects. One downside to the stronger treatment
exclusion assumption, however, is that it eliminates any possibility of
interactions between factors on the assignment to uptake relationships.
Assumption 5 allows for a structured amount of interaction, where assignment
to factor $k$ can only affect uptake on other factors if unit $i$ is a
complier for $k$. Note that if all units are constant compliers on all factors
except for factor $k$, then Assumption 5 is automatically satisfied.

$D_{1}$$Z_{1}$$Z_{2}$$D_{2}$$U_{3}$$Y$ (a)

$D_{1}$$Z_{1}$$Z_{2}$$D_{2}$$U_{3}$$Y$ (b)

Figure 1:  Directed acyclic graphs of causal structures that would satisfy
weak treatment exclusion for both factors (a) and for just factor 2 (b).

There are many situations where Assumption 5 can hold by design. In
particular, Figure 1 shows two directed acyclic graphs (DAGs) with causal
structures where the first factor is assigned first, allowing for it to affect
the second factor but not vice versa. In both DAGs, assignment to the second
factor cannot affect uptake on the first factor, so weak exclusion for factor
2 holds. In Figure 1a, weak exclusion also holds with respect to factor 1
because assignment to factor 1 only affects uptake on factor 2 through uptake
on factor 1. Thus, there is no effect of $Z_{i1}$ on $D_{i2}$ for noncompliers
on factor 1. This DAG may be plausible in situations where noncompliers do not
know their assignment, making it impossible for it to affect their compliance
on factor 2. In Figure 1b, the direct arrow from $Z_{i1}$ to $D_{i2}$ means
weak exclusion is violated for factor 1, but the assumption does continue to
hold for factor 2. In these settings we still allow for arbitrary joint
dependence between treatment uptake on each factor and the outcome. The
important part of Assumption 5 is the restrictions on the effects of
assignment on uptake.

Assumptions 2 and 5 give us that if unit $i$ does not comply on factor $k$
given assignment $\bm{z}_{-k}$ on other factors, then
$Y_{i}(\bm{z}_{-k},z_{k}^{+})=Y_{i}(\bm{z}_{k},z_{k}^{-})$, which implies that

$\bar{Y}_{cn(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cn(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})=0$

in (1). Thus, under Assumption 5, we can write the constant complier factorial
effect as

$\displaystyle\delta_{k}=\frac{1}{2^{K-1}\nu_{k}(\tilde{\bm{z}}_{-k})}\Bigg{[}$
$\displaystyle\bm{g}_{k}^{T}\bm{Y}-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}(\nu_{k}(\bm{z}_{-k})-\nu_{k}(\tilde{\bm{z}}_{-k}))\left[\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right]\Bigg{]}.$

Under Assumption 4, we have
$\nu_{k}(\bm{z}_{-k})-\nu_{k}(\tilde{\bm{z}}_{-k})\geq 0$, which is the key to
the following bounding result.

######  Proposition 2.

Suppose Assumptions 1, 2, and 3 hold, and Assumptions 4 and 5 hold with
respect to factor $k$. Then,
$\delta_{k}\in[\delta^{\prime}_{k}-\tilde{b}_{k},\delta^{\prime}_{k}+\tilde{b}_{k}]\cap[-1,1]$,
where

$\displaystyle\tilde{b}_{k}$
$\displaystyle=\frac{1}{2^{K-1}\nu_{k}(\tilde{\bm{z}}_{-k})}\left(\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}(\nu_{k}(\bm{z}_{-k})-\nu_{k}(\tilde{\bm{z}}_{-k}))\right)$
(3) $\displaystyle\delta^{\prime}_{k}$
$\displaystyle=\frac{1}{2^{K-1}\nu_{k}(\tilde{\bm{z}}_{-k})}\bm{g}_{k}^{T}\bm{Y}.$

######  Remark 2.

We can also write
$\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}\nu_{k}(\bm{z}_{-k})=\frac{1}{2}\bm{g}_{k}^{T}\bm{D}_{k}$
where $\bm{D}_{k}(\bar{D}_{k}(\bm{z}_{1}),\dots,\bar{D}_{k}(\bm{z}_{J}))^{T}$.

######  Remark 3.

Consider the case where we believe Assumption 4 holds but we do not know which
combination of other factors, $\tilde{\bm{z}}_{-k}$ leads to the “least”
compliance. A seemingly reasonable strategy might be to estimate $\rho_{k}$
based on the assignment of other factors with the smallest observed
compliance. That is,
$\widehat{\rho}_{k}=\min_{\bm{z}_{-k}}\widehat{\bar{D}}_{k}(\bm{z}_{-k},z_{k}^{+})-\widehat{\bar{D}}_{k}(\bm{z}_{-k},z_{k}^{-})$.
This is not unbiased for $\rho_{k}$, but rather will equal in expectation some
value $t_{k}\leq\rho_{k}$. The question is: Will this be conservative, in the
sense that the bound resulting from using $t_{k}$ will cover the bound using
$\rho_{k}$? In fact, it is conservative (and will be wider than using the true
value $\rho_{k}$). Two further comments on this results are warranted: 1\.
Although the bounds using $t_{k}$ are conservative in the way described, they
will be centered on an estimate strictly greater in magnitude due to dividing
by a smaller proportion. 2\. This conservativeness only holds if Assumption 4
holds. Importantly, this assumption is for every unit in the experiment and
thus not directly testable. Arguably, if researchers do not have a substantive
reason to believe a particular treatment combination $\bm{z}_{-k}$ will yield
lower compliance on factor $k$ for all units (or if the observed data provides
strong evidence against the hypothesized combination), then the assumption in
general might be called into question.

##  4 Identification of interaction effects for marginal-perfect compliers

Note that any factorial effect involving factor $k$ can be written as a linear
combination of the conditional effects
$\bar{Y}(+1,\bm{z}_{-k})-\bar{Y}(-1,\bm{z}_{-k})$. Therefore, we can use the
same identification strategy as above to estimate the interaction effect among
marginal-perfect compliers for factor $k$, for any interaction involving
factor $k$.

In particular, consider estimating some interaction with factor $k$, called
$\delta_{f(k)}$ which has contrast vector $\bm{g}_{f(k)}$, among marginal-
perfect compliers. Following the logic above under assumptions 1, 2, 3, 4, and
5 for factor $k$, we have
$\delta_{f(k)}\in[\delta^{\prime}_{f(k)}-\tilde{b}_{k},\delta^{\prime}_{f(k)}+\tilde{b}_{k}]\cap[-1,1]$,
where

$\displaystyle\delta^{\prime}_{f(k)}$
$\displaystyle=\frac{1}{2^{K-1}\nu_{k}(\tilde{\bm{z}}_{-k})}\bm{g}_{f(k)}^{T}\bm{Y}.$

Here, we would end up with a different estimand for the same interaction for
the marginal-perfect compliers on factors involved which may not be desirable.
Under the treatment exclusion restriction assumption of Blackwell and Pashley,
(2023), one can point identify $\delta_{f(k)}$ and can in fact identify the
effect among individuals who are constant compliers for all factors involved
in the interaction or constant compliers across all factors in the experiment.
Partial identification of two-factor interactions among individuals who comply
with both factors under weaker assumptions is discussed in Supplementary
Materials B.

##  5 Estimation and inference

The bounds in the previous sections are unobserveable because they are
functions of the potential outcomes. We can derive estimators for the bounds
with a plug-in estimation approach that uses observed means.

$\overline{Y}^{\text{obs}}(\bm{z})=\frac{1}{N_{\bm{z}}}\sum_{i=1}^{N}\mathbb{I}(\bm{Z}_{i}=\bm{z})Y_{i},\qquad\overline{D}_{k}^{\text{obs}}(\bm{z})=\frac{1}{N_{\bm{z}}}\sum_{i=1}^{N}\mathbb{I}(\bm{Z}_{i}=\bm{z})D_{ik}.$

Let
$\widehat{\bm{Y}}=(\overline{Y}^{\text{obs}}(\bm{z}_{1}),\ldots,\overline{Y}^{\text{obs}}(\bm{z}_{J}))$
and let
$\widehat{\nu}_{k}(\bm{z}_{-k})=\overline{D}_{k}^{\text{obs}}(\bm{z}_{-k},z_{k}^{+})-\overline{D}_{k}^{\text{obs}}(\bm{z}_{-k},z_{k}^{-})$.
Then, a natural estimator for the bounds from Proposition 2 would be

$\displaystyle\widehat{\tilde{L}}_{k}$
$\displaystyle=\frac{1}{2^{K-1}\widehat{\nu}_{k}(\tilde{\bm{z}}_{-k})}\left(\bm{g}_{k}^{T}\widehat{\bm{Y}}-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}(\widehat{\nu}_{k}(\bm{z}_{-k})-\widehat{\nu}_{k}(\tilde{\bm{z}}_{-k}))\right),$
$\displaystyle\widehat{\tilde{U}}_{k}$
$\displaystyle=\frac{1}{2^{K-1}\widehat{\nu}_{k}(\tilde{\bm{z}}_{-k})}\left(\bm{g}_{k}^{T}\bm{Y}+\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}(\widehat{\nu}_{k}(\bm{z}_{-k})-\widehat{\nu}_{k}(\tilde{\bm{z}}_{-k}))\right).$

It is straightforward to obtain variance estimators for these quantities using
the delta method, based on either the superpopulation or finite-population
perspectives (see, for example Pashley,, 2019; Blackwell and Pashley,, 2023).
With these variance estimators, we can use the usual constructions to obtain
confidence intervals for each end of the interval, though we can also obtain
confidence intervals for the parameter of interest, $\delta_{k}$, rather than
the intervals, as described by Imbens and Manski, (2004). This latter
procedure involves altering the critical values from the confidence intervals
to account for how the true parameter cannot simultaneously be close to the
upper and lower bound when they are far apart. Thus, when the bounds are
sufficiently far apart, this allows us to construct $1-\alpha$ confidence
intervals for $\delta_{k}$ using $\alpha$-level critical values on both the
upper and lower bound, rather than the usual $\alpha/2$ values.

## References

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These supplementary materials are organized as follows: Section A provides
proofs of the results in the main paper. Section B gives additional results
for bounds on two-way interaction effects. Section C discusses a data
illustration with the methods discussed in the main paper.

##  Appendix A Proofs

###  A.1 Proof of Proposition 1

We prove the bounds under Assumptions 1, 2, 3, and 4.

###### Proof.

Given the results shown in Section 3, we can rewrite $\delta_{k}$ as

$\displaystyle\delta_{k}=$
$\displaystyle\frac{1}{2^{K-1}\nu_{k}(\tilde{\bm{z}}_{-k})}\Bigg{[}\bm{g}_{k}^{T}\bm{Y}-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}(\nu_{k}(\bm{z}_{-k})-\nu_{k}(\tilde{\bm{z}}_{-k}))\left[\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right]$
$\displaystyle\hskip
79.6678pt-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}(1-\nu_{k}(\bm{z}_{-k}))\left[\bar{Y}_{cn(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cn(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right]\Bigg{]}$
$\displaystyle=$
$\displaystyle\frac{1}{2^{K-1}\nu_{k}(\tilde{\bm{z}}_{-k})}\Bigg{[}\bm{g}_{k}^{T}\bm{Y}-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}(\nu_{k}(\bm{z}_{-k})-\nu_{k}(\tilde{\bm{z}}_{-k}))\left[\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right]$
$\displaystyle\hskip
79.6678pt-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}\rho_{n(\bm{z}_{-k})}\left[\bar{Y}_{n(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{n(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right]$
$\displaystyle\hskip
79.6678pt-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}\rho_{a(\bm{z}_{-k})}\left[\bar{Y}_{a(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{a(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right]\Bigg{]}.$

Rearrange to put the identified components first as follows:

$\displaystyle\delta_{k}=$
$\displaystyle\frac{1}{2^{K-1}\nu_{k}(\tilde{\bm{z}}_{-k})}\Bigg{[}\bm{g}_{k}^{T}\bm{Y}-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}\left(\rho_{n(\bm{z}_{-k})}\bar{Y}_{n(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\rho_{a(\bm{z}_{-k})}\bar{Y}_{a(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right)$
$\displaystyle\hskip
79.6678pt-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}(\nu_{k}(\bm{z}_{-k})-\nu_{k}(\tilde{\bm{z}}_{-k}))\left[\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right]$
$\displaystyle\hskip
79.6678pt-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}\left[\rho_{a(\bm{z}_{-k})}\bar{Y}_{a(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\rho_{n(\bm{z}_{-k})}\bar{Y}_{n(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right]\Bigg{]}.$

We can bound the unidentified components,

$\displaystyle-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}(\nu_{k}(\bm{z}_{-k})-\nu_{k}(\tilde{\bm{z}}_{-k}))\left[\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right]$
$\displaystyle\qquad-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}\left[\rho_{a(\bm{z}_{-k})}\bar{Y}_{a(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\rho_{n(\bm{z}_{-k})}\bar{Y}_{n(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right]$
$\displaystyle\in\Bigg{(}-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}(\nu_{k}(\bm{z}_{-k})-\nu_{k}(\tilde{\bm{z}}_{-k}))-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}\rho_{a(\bm{z}_{-k})},$
$\displaystyle\qquad\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}(\nu_{k}(\bm{z}_{-k})-\nu_{k}(\tilde{\bm{z}}_{-k}))+\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}\rho_{n(\bm{z}_{-k})}\Bigg{)}.$

We can finish the proof by noting that
$\rho_{a(\bm{z}_{-k})}=\nu_{k}^{-}(\bm{z}_{-k})$ and
$\rho_{n(\bm{z}_{-k})}=1-\nu_{k}^{+}(\bm{z}_{-k})$.

∎

###  A.2 Proof of Remark 1

Remark 1 is proved by bounding the first line of the proof in Section A.1.

###  A.3 Proof of Proposition 2

We prove the bounds under Assumptions 1, 2, 3, 4, and 5.

###### Proof.

We start with $\delta_{k}$ as

$\displaystyle\delta_{k}=$
$\displaystyle\frac{1}{2^{K-1}\nu_{k}(\tilde{\bm{z}}_{-k})}\Bigg{[}\bm{g}_{k}^{T}\bm{Y}-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}(\nu_{k}(\bm{z}_{-k})-\nu_{k}(\tilde{\bm{z}}_{-k}))\left[\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right]$
$\displaystyle\hskip
79.6678pt-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}(1-\nu_{k}(\bm{z}_{-k}))\left[\bar{Y}_{cn(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cn(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right]\Bigg{]}$
$\displaystyle=$
$\displaystyle\frac{1}{2^{K-1}\nu_{k}(\tilde{\bm{z}}_{-k})}\Bigg{[}\bm{g}_{k}^{T}\bm{Y}-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}(\nu_{k}(\bm{z}_{-k})-\nu_{k}(\tilde{\bm{z}}_{-k}))\left[\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right].$

The last line holds because under the assumptions, in particular Assumption 5,
the effect among conditional noncompliers is 0. What remains is to simply
bound the effect among conditional compliers, which must be in the interval
$[-1,1]$. ∎

###  A.4 Proof of Remark 3

###### Proof.

$\displaystyle\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}\nu_{k}(\bm{z}_{-k})$
$\displaystyle=\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}\frac{1}{2}\left[\bar{D}_{k}(\bm{z}_{-k},z_{k}^{+})-\bar{D}_{k}(\bm{z}_{-k},z_{k}^{-})\right]=\frac{1}{2}\bm{g}_{k}^{T}\bm{D}_{k}.$

∎

###  A.5 Proof of Remark 3

###### Proof.

First, we note that
$\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}\nu(\bm{z}_{-k})=\frac{1}{2}\bm{g}_{k}^{T}\bm{D}_{k}$.
Then, we can write the lower bound from Proposition 2 as

$\displaystyle\frac{1}{\rho_{k}}\left[\frac{1}{2^{K-1}}\bm{g}_{k}^{T}\bm{Y}-\frac{1}{2^{K}}\bm{g}_{k}^{T}\bm{D}_{k}+\rho_{k}\right]-\frac{1}{t_{k}}\left[\frac{1}{2^{K-1}}\bm{g}_{k}^{T}\bm{Y}-\frac{1}{2^{K}}\bm{g}_{k}^{T}\bm{D}_{k}+t_{k}\right]$
$\displaystyle=\frac{1}{\rho_{k}}\left[\frac{1}{2^{K-1}}\bm{g}_{k}^{T}\bm{Y}-\frac{1}{2^{K}}\bm{g}_{k}^{T}\bm{D}_{k}\right]-\frac{1}{t_{k}}\left[\frac{1}{2^{K-1}}\bm{g}_{k}^{T}\bm{Y}-\frac{1}{2^{K}}\bm{g}_{k}^{T}\bm{D}_{k}\right]$
$\displaystyle=\frac{t_{k}-\rho_{k}}{\rho_{k}t_{k}}\left[\frac{1}{2^{K-1}}\bm{g}_{k}^{T}\bm{Y}-\frac{1}{2^{K}}\bm{g}_{k}^{T}\bm{D}_{k}\right]$
$\displaystyle=\frac{t_{k}-\rho_{k}}{\rho_{k}t_{k}}\left[\rho_{k}\delta_{k}+\frac{1}{2^{K-1}}\sum_{\bm{z}_{-k}}\rho_{cc(\bm{z}_{-k})}\left(\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right)-\left(\rho_{k}+\frac{1}{2^{K-1}}\sum_{\bm{z}_{-k}}\rho_{cc(\bm{z}_{-k})}\right)\right]$
$\displaystyle=\underbrace{\frac{t_{k}-\rho_{k}}{\rho_{k}t_{k}}}_{\leq
0}\underbrace{\left[\rho_{k}\left(\delta_{k}-1\right)+\frac{1}{2^{K-1}}\sum_{\bm{z}_{-k}}\rho_{cc(\bm{z}_{-k})}\left(\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})-1\right)\right]}_{\leq
0}$ $\displaystyle\geq 0$

So the lower bound using $t_{k}$ is less than the lower bound using
$\rho_{k}$.

Now consider the upper bound.

$\displaystyle\frac{1}{\rho_{k}}\left[\frac{1}{2^{K-1}}\bm{g}_{k}^{T}\bm{Y}+\frac{1}{2^{K}}\bm{g}_{k}^{T}\bm{D}_{k}-\rho_{k}\right]-\frac{1}{t_{k}}\left[\frac{1}{2^{K-1}}\bm{g}_{k}^{T}\bm{Y}+\frac{1}{2^{K}}\bm{g}_{k}^{T}\bm{D}_{k}-t_{k}\right]$
$\displaystyle=\frac{1}{\rho_{k}}\left[\frac{1}{2^{K-1}}\bm{g}_{k}^{T}\bm{Y}+\frac{1}{2^{K}}\bm{g}_{k}^{T}\bm{D}_{k}\right]-\frac{1}{t_{k}}\left[\frac{1}{2^{K-1}}\bm{g}_{k}^{T}\bm{Y}+\frac{1}{2^{K}}\bm{g}_{k}^{T}\bm{D}_{k}\right]$
$\displaystyle=\frac{t_{k}-\rho_{k}}{\rho_{k}t_{k}}\left[\frac{1}{2^{K-1}}\bm{g}_{k}^{T}\bm{Y}+\frac{1}{2^{K}}\bm{g}_{k}^{T}\bm{D}_{k}\right]$
$\displaystyle=\frac{t_{k}-\rho_{k}}{\rho_{k}t_{k}}\left[\rho_{k}\delta_{k}+\frac{1}{2^{K-1}}\sum_{\bm{z}_{-k}}\rho_{cc(\bm{z}_{-k})}\left(\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-1})\right)+\left(\rho_{k}+\frac{1}{2^{K-1}}\sum_{\bm{z}_{-k}}\rho_{cc(\bm{z}_{-k})}\right)\right]$
$\displaystyle=\underbrace{\frac{t_{k}-\rho_{k}}{\rho_{k}t_{k}}}_{\leq
0}\underbrace{\left[\rho_{k}\left(\delta_{k}+1\right)+\frac{1}{2^{K-1}}\sum_{\bm{z}_{-k}}\rho_{cc(\bm{z}_{-k})}\left(\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})+1\right)\right]}_{\geq
0}$ $\displaystyle\leq 0$

So the upper bound using $t_{k}$ is greater than the upper bound using
$\rho_{k}$. ∎

###  A.6 Proof of bounds on interaction

We prove the bounds for interactions involving factor $k$ under Assumptions 1,
2, 3, 4, and 5.

###### Proof.

Define $\tilde{\bm{g}}_{f(k)}(\bm{z})\in\\{-1,1\\}$ give the sign of
$\bm{g}_{f(k)}$ for each treatment $\bm{z}_{k}$. We have $\delta_{f(k)}$ as

$\displaystyle\delta_{f(k)}=$
$\displaystyle\frac{1}{2^{K-1}\nu_{k}(\tilde{\bm{z}}_{-k})}\Bigg{[}\bm{g}_{f(k)}^{T}\bm{Y}$
$\displaystyle\hskip
79.6678pt-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}\tilde{\bm{g}}_{f(k)}(z^{+}_{k},\bm{z}_{-k})(\nu_{k}(\bm{z}_{-k})-\nu_{k}(\tilde{\bm{z}}_{-k}))\left[\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right]$
$\displaystyle\hskip
79.6678pt-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}\tilde{\bm{g}}_{f(k)}(z^{+}_{k},\bm{z}_{-k})(1-\nu_{k}(\bm{z}_{-k}))\left[\bar{Y}_{cn(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cn(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right]\Bigg{]}$
$\displaystyle=$
$\displaystyle\frac{1}{2^{K-1}\nu_{k}(\tilde{\bm{z}}_{-k})}\Bigg{[}\bm{g}_{f(k)}^{T}\bm{Y}$
$\displaystyle\hskip
79.6678pt-\sum_{\bm{z}_{-k}\in\mathcal{Z}_{-k}}\tilde{\bm{g}}_{f(k)}(z^{+}_{k},\bm{z}_{-k})(\nu_{k}(\bm{z}_{-k})-\nu_{k}(\tilde{\bm{z}}_{-k}))\left[\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{+})-\bar{Y}_{cc(\bm{z}_{-k})}(\bm{z}_{-k},z_{k}^{-})\right].$

As in the proof of Proposition 2, the last line holds because under the
assumptions, in particular Assumption 5, the effect among conditional
noncompliers is 0. We then bound the effect of conditional compliers in the
interval $[-1,1]$. ∎

##  Appendix B Additional interaction results

In Section 4, we consider estimating interactive effects for constant
compliers on a single factor $k$. However, as interactions inherently involve
multiple factors, we may instead be interested in estimating the effect among
units who are constant compliers on multiple factors. We focus on two-factor
interactions. Thus, if we are estimating the interaction between factor $k$
and $k^{\prime}$, we are interested in identifying the effect among
individuals who are constant compliers on both $k$ and $k^{\prime}$. This was
addressed by Blackwell and Pashley, (2023) under the treatment exclusion
restriction assumption on all factors. We introduce here two weaker
assumptions that can be used to obtain bounds.

Throughout this section, we will use notation $\bm{z}_{-(k,k^{\prime})}$ to
indicate the assignment vector $\bm{z}$ with the assignments for factor $k$
and $k^{\prime}$ removed. We define $D_{i,k\circ
k^{\prime}}(\bm{z})=D_{i,k}(\bm{z})D_{i,k^{\prime}}(\bm{z})$. Similarly,
$\bm{g}_{k\circ k^{\prime}}$ is the contrast vector for the two-factor
interaction between factors $k$ and $k^{\prime}$, which is defined by the
elemnt-wise (Hadamard) product between $\bm{g}_{k}$ and $\bm{g}_{k^{\prime}}$.
Also let $c_{k,k^{\prime}}$ be the subscript denoting constant compliers on
factors $k$ and $k^{\prime}$ so that, e.g., $\rho_{c_{k,k^{\prime}}}$ is the
proportion of units who are constant compliers on both $k$ and $k^{\prime}$.
The effect we wish to estimate is then

$\displaystyle\delta_{kk^{\prime}}$
$\displaystyle\equiv\frac{1}{2^{K-1}}\sum_{\bm{z}_{-(k,k^{\prime})}}\Big{(}\bar{Y}_{c_{kk^{\prime}}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-\bar{Y}_{c_{kk^{\prime}}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})$
$\displaystyle\qquad\qquad\qquad\qquad-\left[\bar{Y}_{c_{kk^{\prime}}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})-\bar{Y}_{c_{kk^{\prime}}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{-})\right]\Big{)}.$

To provide identification results for this effect, first, we need the
following assumption which is stronger than Assumption 4.

######  Assumption SM.6 (Least compliant profile for factors $k$ and
$k^{\prime}$).

There is a $\tilde{\bm{z}}_{-(k,k^{\prime})}\in\mathcal{Z}_{-(k,k^{\prime})}$,
such that

$\displaystyle D_{i,k\circ
k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-D_{i,k\circ
k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})-\left[D_{i,k\circ
k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})-D_{i,k\circ
k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{-})\right]$
$\displaystyle\geq D_{i,k\circ
k^{\prime}}(\tilde{\bm{z}}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-D_{i,k\circ
k^{\prime}}(\tilde{\bm{z}}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})-\left[D_{i,k\circ
k^{\prime}}(\tilde{\bm{z}}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})-D_{i,k\circ
k^{\prime}}(\tilde{\bm{z}}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{-})\right]$

and for all $i$.

Second, we need the following assumption that controls the interactive effect
of assignment on compliance between the two factors under consideration.

######  Assumption SM.7 (Conditional treatment exclusion restriction between
factors $k$ and $k^{\prime}$).

There is a conditional treatment exclusion between factors $k$ and
$k^{\prime}$ if

$D_{i,k}(\bm{z}_{-(k,k^{\prime})},z_{k}=z,z_{k^{\prime}}^{+})=D_{i,k}(\bm{z}_{-(k,k^{\prime})},z_{k}=z,z_{k^{\prime}}^{-})$

and

$D_{i,k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+}z,z_{k^{\prime}}=z)=D_{i,k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}=z)$

for all $z\in\\{-1,+1\\}$, $\bm{z}_{-(k,k^{\prime})}$, and $i$. In other
words, conditional on the assignment of other factors, the uptake on factor
$k$ does not depend on the assignment of factor $k^{\prime}$ and vice versa.

Assumption SM.7 is weaker than the treatment exclusion restriction assumption
of Blackwell and Pashley, (2023) because it is (i) only required on the two
factors under consideration (ii) conditional and thus allows compliance to
change across assignments for the other $K-2$ factors.

######  Proposition SM.3.

Under Assumptions 2, 3, 5, SM.6, and SM.7,
$\delta_{kk^{\prime}}\in[\widetilde{\delta}_{k,k^{\prime}}-b_{k,k^{\prime}},\widetilde{\delta}_{k,k^{\prime}}+b_{k,k^{\prime}}]\cap[-1,1]$
where

$\widetilde{\delta}_{k,k^{\prime}}=\frac{\frac{1}{2^{K-1}}\bm{g}_{k\circ
k^{\prime}}^{T}\bm{Y}}{\nu_{k,k^{\prime}}(\tilde{\bm{z}}_{-(k,k^{\prime})})}$

and

$b_{k,k^{\prime}}=\frac{1}{\nu_{k,k^{\prime}}(\tilde{\bm{z}}_{-(k,k^{\prime})})}\left[\frac{1}{2^{K}}\bm{g}_{k\circ
k^{\prime}}^{T}\bm{D}_{kk^{\prime}}-\nu_{k,k^{\prime}}(\tilde{\bm{z}}_{-(k,k^{\prime})})\right]$

###### Proof.

We start by showing identification of the proportion of constant compliers for
factors $k$ and $k^{\prime}$ using Assumptions 3, SM.6, and SM.7. We have the
conditional unit-level interactive effect for factor $k$ and $k^{\prime}$ on
uptake is

$\displaystyle D_{i,k\circ
k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-D_{i,k\circ
k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})-\left[D_{i,k\circ
k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})-D_{i,k\circ
k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{-})\right]$
$\displaystyle=D_{i,k}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})D_{i,k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-D_{i,k}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})D_{i,kk^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})$
$\displaystyle\quad-\left[D_{i,k}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})D_{i,k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})-D_{i,k}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{-})D_{i,k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{-})\right]$
$\displaystyle=D_{i,k}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})\left[D_{i,k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-D_{i,k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})\right]$
$\displaystyle\quad-
D_{i,k}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})\left[D_{i,k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})-D_{i,k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{-})\right]$
$\displaystyle=\left[D_{i,k}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-D_{i,k}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})\right]\left[D_{i,k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-D_{i,k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})\right]$
$\displaystyle=4C_{i,k,k^{\prime}}+4CC_{i,k,k^{\prime}},$

where $C_{i,k,k^{\prime}}$ $CC_{i,k,k^{\prime}}$ are indicators that unit $i$
is a constant complier and conditional complier for factors $k$ and
$k^{\prime}$, respectively. The second equality comes from the definition of
$D_{i,k\circ k^{\prime}}(\bm{z})$, the third and fourth equalities come from
Assumption SM.7, and the last follows from Assumption 3, and the fact that
only conditional compliers and constant compliers on both factors will have

$\left[D_{i,k}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-D_{i,k}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})\right]\left[D_{i,k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-D_{i,k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})\right]=4$

(and conditional noncompliers will have it equal to 0).

Thus, we must have the average conditional interactive effect for factor $k$
and $k^{\prime}$ on uptake,
$\nu_{k,k^{\prime}}(\tilde{\bm{z}}_{-(k,k^{\prime})})$, is

$\displaystyle 4\nu_{k,k^{\prime}}(\tilde{\bm{z}}_{-(k,k^{\prime})})=$
$\displaystyle\bar{D}_{k\circ
k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-\bar{D}_{k\circ
k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})-\left[\bar{D}_{k\circ
k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})-\bar{D}_{k\circ
k^{\prime}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{-})\right]$
$\displaystyle=4\rho_{c_{k,k^{\prime}}}+4\rho_{cc(\bm{z}_{-(k,k^{\prime})})}.$

Further, from Assumption SM.6, we must have
$\nu_{k,k^{\prime}}(\tilde{\bm{z}}_{-(k,k^{\prime})})=\rho_{c_{k,k^{\prime}}}$.

Next we show the bounds. We have the conditional intent-to-treat interactive
effect between factors $k$ and $k^{\prime}$ can be decomposed as

$\displaystyle\gamma_{k,k^{\prime}}(\bm{z}_{-(k,k^{\prime})})$
$\displaystyle\equiv\bar{Y}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-\bar{Y}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})-\left[\bar{Y}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})-\bar{Y}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{-})\right]$
$\displaystyle=\rho_{c_{k,k^{\prime}}}\left[\bar{Y}_{c_{k,k^{\prime}}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-\bar{Y}_{c_{k,k^{\prime}}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})-\left[\bar{Y}_{c_{k,k^{\prime}}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})-\bar{Y}_{c_{k,k^{\prime}}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{-})\right]\right]$
$\displaystyle\quad+\rho_{cc(\bm{z}_{-(k,k^{\prime})})}\Big{[}\bar{Y}_{cc(\bm{z}_{-(k,k^{\prime})})}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-\bar{Y}_{cc(\bm{z}_{-(k,k^{\prime})})}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})$
$\displaystyle\qquad\qquad\qquad\qquad-\left[\bar{Y}_{cc(\bm{z}_{-(k,k^{\prime})})}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})-\bar{Y}_{cc(\bm{z}_{-(k,k^{\prime})})}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{-})\right]\Big{]}$
$\displaystyle\quad+\rho_{cn(\bm{z}_{-(k,k^{\prime})})}\Big{[}\bar{Y}_{cn(\bm{z}_{-(k,k^{\prime})})}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-\bar{Y}_{cn(\bm{z}_{-(k,k^{\prime})})}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})$
$\displaystyle\qquad\qquad\qquad\qquad-\left[\bar{Y}_{cn(\bm{z}_{-(k,k^{\prime})})}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})-\bar{Y}_{cn(\bm{z}_{-(k,k^{\prime})})}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{-})\right]\Big{]}.$

Consider a unit $i$ who is a conditional noncomplier on either $k$ or
$k^{\prime}$, conditional on $\bm{z}_{-(k,k^{\prime})}$. Without loss of
generality, take unit $i$ to be a noncomplier on factor $k$. Then under
Assumptions 2, and 3, and SM.7,

$\displaystyle
Y_{i}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-Y_{i}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})-\left[Y_{i}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})-Y_{i}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{-})\right]$
$\displaystyle=0.$

This follows because under Assumptions 3 and SM.7, for this unit
$D_{ik}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}})=D_{ik}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}})$
for $z_{k^{\prime}}\in\\{z_{k^{\prime}}^{+},z_{k^{\prime}}^{-}\\}$. Then under
Assumption 5, we must have
$Y_{i}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-Y_{i}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})=0$
and
$Y_{i}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})-Y_{i}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{-})=0$.
Thus, we have

$\displaystyle\gamma_{k,k^{\prime}}(\bm{z}_{-(k,k^{\prime})})$
$\displaystyle=\rho_{c_{k,k^{\prime}}}\left[\bar{Y}_{c_{k,k^{\prime}}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-\bar{Y}_{c_{k,k^{\prime}}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})-\left[\bar{Y}_{c_{k,k^{\prime}}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})-\bar{Y}_{c_{k,k^{\prime}}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{-})\right]\right]$
$\displaystyle\quad+\rho_{cc(\bm{z}_{-(k,k^{\prime})})}\Big{[}\bar{Y}_{cc(\bm{z}_{-(k,k^{\prime})})}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-\bar{Y}_{cc(\bm{z}_{-(k,k^{\prime})})}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})$
$\displaystyle\qquad\qquad\qquad\qquad-\left[\bar{Y}_{cc(\bm{z}_{-(k,k^{\prime})})}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})-\bar{Y}_{cc(\bm{z}_{-(k,k^{\prime})})}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{-})\right]\Big{]}.$

Rearranging and bounding the conditional complier term gives us

$\displaystyle\rho_{c_{k,k^{\prime}}}\left[\bar{Y}_{c_{k,k^{\prime}}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-\bar{Y}_{c_{k,k^{\prime}}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})-\left[\bar{Y}_{c_{k,k^{\prime}}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})-\bar{Y}_{c_{k,k^{\prime}}}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{-})\right]\right]$
$\displaystyle=\gamma_{k,k^{\prime}}(\bm{z}_{-(k,k^{\prime})})-\rho_{cc(\bm{z}_{-(k,k^{\prime})})}\Big{[}\bar{Y}_{cc(\bm{z}_{-(k,k^{\prime})})}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{+})-\bar{Y}_{cc(\bm{z}_{-(k,k^{\prime})})}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{+})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-\left[\bar{Y}_{cc(\bm{z}_{-(k,k^{\prime})})}(\bm{z}_{-(k,k^{\prime})},z_{k}^{+},z_{k^{\prime}}^{-})-\bar{Y}_{cc(\bm{z}_{-(k,k^{\prime})})}(\bm{z}_{-(k,k^{\prime})},z_{k}^{-},z_{k^{\prime}}^{-})\right]\Big{]}$
$\displaystyle\in\left(\gamma_{k,k^{\prime}}(\bm{z}_{-(k,k^{\prime})})-2\rho_{cc(\bm{z}_{-(k,k^{\prime})})},\gamma_{k,k^{\prime}}(\bm{z}_{-(k,k^{\prime})})+2\rho_{cc(\bm{z}_{-(k,k^{\prime})})}\right)$
$\displaystyle=\left(\gamma_{k,k^{\prime}}(\bm{z}_{-(k,k^{\prime})})-2[\nu_{k,k^{\prime}}(\bm{z}_{-(k,k^{\prime})})-\nu_{k,k^{\prime}}(\tilde{\bm{z}}_{-(k,k^{\prime})})],\gamma_{k,k^{\prime}}(\bm{z}_{-(k,k^{\prime})})+2[\nu_{k,k^{\prime}}(\bm{z}_{-(k,k^{\prime})})-\nu_{k,k^{\prime}}(\tilde{\bm{z}}_{-(k,k^{\prime})})]\right).$

Aggregating across $\bm{z}_{-k,k^{\prime}}$, note that

$\frac{1}{2^{K-1}}\bm{g}_{k\circ
k^{\prime}}^{T}\bm{Y}=\frac{1}{2^{K-1}}\sum_{\bm{z}_{-(k,k^{\prime})}}\gamma_{k,k^{\prime}}(\bm{z}_{-(k,k^{\prime})})$

and

$\frac{1}{2^{K}}\bm{g}_{k\circ
k^{\prime}}^{T}\bm{D}_{kk^{\prime}}=\frac{1}{2^{K-2}}\sum_{\bm{z}_{-(k,k^{\prime})}}\nu_{k,k^{\prime}}(\bm{z}_{-(k,k^{\prime})}).$

So then we have the result

$\displaystyle\delta_{kk^{\prime}}$
$\displaystyle\in\Bigg{(}\max\left\\{0,\frac{1}{\rho_{c_{k,k^{\prime}}}}\left[\frac{1}{2^{K-1}}\bm{g}_{k\circ
k^{\prime}}^{T}\bm{Y}-\frac{1}{2^{K}}\bm{g}_{k\circ
k^{\prime}}^{T}\bm{D}_{kk^{\prime}}+\nu_{k,k^{\prime}}(\tilde{\bm{z}}_{-(k,k^{\prime})})\right]\right\\},$
$\displaystyle\qquad\min\left\\{1,\frac{1}{\rho_{c_{k,k^{\prime}}}}\left[\frac{1}{2^{K-1}}\bm{g}_{k\circ
k^{\prime}}^{T}\bm{Y}+\frac{1}{2^{K}}\bm{g}_{k\circ
k^{\prime}}^{T}\bm{D}_{kk^{\prime}}-\nu_{k,k^{\prime}}(\tilde{\bm{z}}_{-(k,k^{\prime})})\right]\right\\}\Bigg{)}.$

∎

##  Appendix C Data illustration

We reanalyze data from Blattman et al., (2017), an experiment on the use of
cognitive behavioral therapy (CBT) and cash infusions to change behavioral and
criminal behaviors among at-risk young men in Liberia, to illustrate our
methods. We focus on the outcome of whether individuals reported selling drugs
following the experimental intervention assigned, which was a binary outcome
(yes or no). The experiment was a $2^{2}$ factorial design, with factors CBT
and cash each having levels receive or do not receive. Compliance was one-
sided in this experiment in that individuals assigned not to receive an
intervention (CBT or cash) would not be able to receive that intervention.
There was some noncompliance on both factors with, unsurprisingly, lower
compliance on CBT (63% compliance on CBT and 98% compliance on cash).

Assumption 3 is assured due to the one-sided nature of the compliance.
Moreover, because of one-sided noncompliance, there are 5 compliance types for
CBT (or cash): {cc, ca, cn, nc, nn}, where the first type refers to CBT (or
cash) compliance when assigned to receive cash (or CBT) (complier is c, never
taker is n) and the second type is CBT (or cash) compliance when not assigned
to receive cash (or CBT).

Consider the weak-treatment exclusion restriction assumption (Assumption 5).
Being assigned to receive a cash transfer might change compliance with the CBT
assignment. A unit might be a conditional never-taker for CBT if not assigned
the cash transfer, but a complier if they were assigned the cash transfer.
Under this situation, Assumption 5 would hold for CBT but not for cash. These
assumptions must always be assessed based on subject-matter knowledge.

Figure SM.2 shows the point estimates of the bounds from Proposition 2 in red
with the black points and lines showing the point estimates and confidence
intervals under the stronger treatment exclusion assumption of Blackwell and
Pashley, (2023) for comparison. We quickly see that the bounds on both effects
do not include 0, though the confidence interval for cash based on the strong
assumption does. The point identified estimate based on the stronger
assumption falls within the bounds for each effect. Under Assumption 5 for CBT
but not cash, we would not trust the red bounds (or the stronger point
identification) on the cash estimate. However, it provides a useful
illustration here in comparison to CBT: as the compliance rate decreases, the
bounds and confidence intervals will both become narrower. Further, because
the bounds depend on estimates of
$(\nu_{k}(\bm{z}_{-k})-\nu_{k}(\tilde{\bm{z}}_{-k})$, the width of the bounds
will also depend on how much variability there is in observed compliance
across the treatment arms.

Figure SM.2: Red dots give bounds from Proposition 2. Black dots give point
estimates under treatment exclusion assumption of Blackwell and Pashley,
(2023) with black lines giving the 95% confidence intervals on those points.


# Quantifying polarization changes induced by rotating Dove prisms and
K-mirrors: Supplementary Information

Suman Karan||<EMAIL_ADDRESS>Ruchi||<EMAIL_ADDRESS>
$||$ SK and Ruchi contributed equally to this work  Pranay Mohta  Anand K. Jha
Department of Physics, Indian Institute of Technology Kanpur, Kanpur, Uttar
Pradesh, 208016, India

###### Abstract

Here we present the detailed calculations to obtain the polarization transfer
matrix for a Dove prism and a K-mirror. We also show the calculations of mean
polarization change D, which we use as a quantifier for the the polarization
changes of the transmitted field due to rotation of a Dove prism and a
K-mirror.

††preprint: arXiv

##  I Polarization transfer matrix

As shown in Fig. 1, $(\hat{x},\hat{y},\hat{z})$ is a laboratory coordinate
system, $(\hat{s}_{0},\hat{p}_{0},\hat{k}_{0})$ is the polarization coordinate
system for the incident field and $(\hat{s}_{1},\hat{p}_{1},\hat{k}_{1})$ is
the polarization coordinate system for the reflected field. Therefore, the
incident electric field $\vec{E}_{0}$ can be written as

$\displaystyle\vec{E}_{0}=E_{s_{0}}~{}\hat{s}_{0}+E_{p_{0}}~{}\hat{p}_{0}.$
(1)

Figure 1:  Schematic representation of the polarization coordinate systems for
the incident and the reflected fields.

where, $E_{s_{0}}$ and $E_{p_{0}}$ are components of electric field vector,
perpendicular and parallel to the plane of incidence, respectively. We note
that $\hat{s}_{1}$ is perpendicular to both $\hat{k}_{0}$ and $\hat{k}_{1}$.
Hence the directions of $\hat{p}_{1}$ is along $\hat{k}_{0}\times\hat{s}_{1}$.
The reflected electric field $\vec{E}^{{}^{\prime}}_{1}$ can be written as

$\displaystyle\vec{E}^{{}^{\prime}}_{1}=E^{{}^{\prime}}_{s_{1}}~{}\hat{s}_{1}+E^{{}^{\prime}}_{p_{1}}~{}\hat{p}_{1},$
(2)

where, $E^{{}^{\prime}}_{s_{1}}$ and $E^{{}^{\prime}}_{p_{1}}$ are components
of electric field vector, perpendicular and parallel to the plane of
incidence, respectively, and using Eq. (1), we write them as [1]

$\displaystyle
E^{{}^{\prime}}_{s_{1}}=E_{s_{0}}(\hat{s}_{0}\cdot\hat{s}_{1})+E_{p_{0}}(\hat{n}_{0}\cdot\hat{s}_{1}),$
(3) $\displaystyle
E^{{}^{\prime}}_{p_{1}}=-E_{s_{0}}(\hat{s}_{1}\cdot\hat{n}_{0})+E_{p_{0}}(\hat{s}_{1}\cdot\hat{s}_{0}),$
(4)

where $\hat{n}_{0}$ is normal to the surface$~{}0$ (see Fig. 1 ) and
$\hat{p}_{0}$ is parallel to $\hat{n}_{0}$. In the matrix notation, we express
Eqs. (3) and (4) as

$\displaystyle\begin{pmatrix}E^{{}^{\prime}}_{s_{1}}\\\
E^{{}^{\prime}}_{p_{1}}\end{pmatrix}=\begin{pmatrix}\hat{s}_{1}\cdot\hat{s}_{0}&\hat{s}_{1}\cdot\hat{n}_{0}\\\
-\hat{s}_{1}\cdot\hat{n}_{0}&\hat{s}_{1}\cdot\hat{s}_{0}\end{pmatrix}\begin{pmatrix}E_{s_{0}}\\\
E_{p_{0}}\end{pmatrix}.$ (5)

The reflected electric field components $E_{s_{1}}$ and $E_{p_{1}}$ can be
written as

$\displaystyle\begin{pmatrix}E_{s_{1}}\\\
E_{p_{1}}\end{pmatrix}=\begin{pmatrix}r_{s_{1}}&0\\\
0&r_{p_{1}}\end{pmatrix}\begin{pmatrix}E^{{}^{\prime}}_{s_{1}}\\\
E^{{}^{\prime}}_{p_{1}}\end{pmatrix},$ (6)

where $r_{s_{1}}$ and $r_{p_{1}}$ are the Fresnel’s reflection coefficients
[2]. From Eqs. (5) and (6), the reflected field with respect to the incident
electric field can be written as [1]

$\displaystyle\begin{pmatrix}E_{s_{1}}\\\
E_{p_{1}}\end{pmatrix}=\begin{pmatrix}r_{s_{1}}(\hat{s}_{1}\cdot\hat{s}_{0})&r_{s_{1}}(\hat{s}_{1}\cdot\hat{n}_{0})\\\
-r_{p_{1}}(\hat{s}_{1}\cdot\hat{n}_{0})&r_{p_{1}}(\hat{s}_{1}\cdot\hat{s}_{0})\end{pmatrix}\begin{pmatrix}E_{s_{0}}\\\
E_{p_{0}}\end{pmatrix}.$ (7)

For normal to the surface$~{}1$ along $\hat{n}_{1}$ direction, using relations
$\hat{s}_{1}=\frac{\hat{n}_{1}\times\hat{k}_{1}}{|\hat{n}_{1}\times\hat{k}_{1}|}$
and
$\hat{s}_{0}=\frac{\hat{n}_{0}\times\hat{k}_{0}}{|\hat{n}_{0}\times\hat{k}_{0}|}$
in Eq. 7, the generalized form of the transfer matrix for reflection from the
$q^{th}$ surface can be written as

$\displaystyle
M_{q}=\begin{bmatrix}r_{s_{q}}\frac{(\hat{n}_{q}\times\hat{k}_{q})\cdot(\hat{n}_{q-1}\times\hat{k}_{q-1})}{|\hat{n}_{q}\times\hat{k}_{q}||\hat{n}_{q-1}\times\hat{k}_{q-1}|}&r_{s_{q}}\frac{(\hat{n}_{q}\times\hat{k}_{q})\cdot\hat{n}_{q-1}}{|\hat{n}_{q}\times\hat{k}_{q}|}\\\
-r_{p_{q}}\frac{(\hat{n}_{q}\times\hat{k}_{q})\cdot\hat{n}_{q-1}}{|\hat{n}_{q}\times\hat{k}_{q}|}&r_{p_{q}}\frac{(\hat{n}_{q}\times\hat{k}_{q})\cdot(\hat{n}_{q-1}\times\hat{k}_{q-1})}{|\hat{n}_{q}\times\hat{k}_{q}||\hat{n}_{q-1}\times\hat{k}_{q-1}|}\end{bmatrix}.$
(8)

where $r_{s_{q}}$ and $r_{p_{q}}$ are the Fresnel’s reflection coefficients
[2] for $q^{th}$ surface and are defined as follows:

$\begin{split}r_{s_{q}}&=\frac{n_{i}\cos\theta_{i_{q}}-n_{t}\cos\theta_{t_{q}}}{n_{i}\cos\theta_{i_{q}}+n_{t}\cos\theta_{t_{q}}}\\\
r_{p_{q}}&=\frac{n_{t}\cos\theta_{i_{q}}-n_{i}\cos\theta_{t_{q}}}{n_{i}\cos\theta_{t_{q}}+n_{t}\cos\theta_{i_{q}}}\end{split}$
(9)

Similarly, in the case of transmission from the $q^{th}$ surface, we can write
the transfer matrix as

$\displaystyle
T_{q}=\begin{bmatrix}t_{s_{q}}\frac{(\hat{n}_{q}\times\hat{k}_{q})\cdot(\hat{n}_{q-1}\times\hat{k}_{q-1})}{|\hat{n}_{q}\times\hat{k}_{q}||\hat{n}_{q-1}\times\hat{k}_{q-1}|}&t_{s_{q}}\frac{(\hat{n}_{q}\times\hat{k}_{q})\cdot\hat{n}_{q-1}}{|\hat{n}_{q}\times\hat{k}_{q}|}\\\
-t_{p_{q}}\frac{(\hat{n}_{q}\times\hat{k}_{q})\cdot\hat{n}_{q-1}}{|\hat{n}_{q}\times\hat{k}_{q}|}&t_{p_{q}}\frac{(\hat{n}_{q}\times\hat{k}_{q})\cdot(\hat{n}_{q-1}\times\hat{k}_{q-1})}{|\hat{n}_{q}\times\hat{k}_{q}||\hat{n}_{q-1}\times\hat{k}_{q-1}|}\end{bmatrix},$
(10)

where $t_{s_{q}}$ and $t_{p_{q}}$ are the Fresnel’s transmission coefficients
[2] for $q^{th}$ surface and are given as:

$\begin{split}t_{s_{q}}=\frac{2n_{i}\cos\theta_{i_{q}}}{n_{i}\cos\theta_{i_{q}}+n_{t}\cos\theta_{t_{q}}}\quad\text{and}\quad
t_{p_{q}}=\frac{2n_{i}\cos\theta_{i_{q}}}{n_{i}\cos\theta_{t_{q}}+n_{t}\cos\theta_{i_{q}}}\end{split}.$
(11)

##  II Transfer matrix for dove prism

Figure 2: Schematic of a Dove prism characterized by a base angle $\alpha$.

Consider a Dove prism characterized by base angle, $\alpha$ as shown in Fig.
2. The transmitted electric field from a Dove prism in Jones matrix form can
be written as, $[E^{{}^{\prime}}_{x}\quad
E^{{}^{\prime}}_{y}]^{T}=T_{DP}\left(\phi\right)\times[E_{x}\quad E_{y}]^{T}$,
where $T_{DP}\left(\phi\right)$ is the polarization transfer matrix of a Dove
prism, rotated by an angle $\phi$. Figure 2 illustrates that when an optical
field passes through a Dove prism, it undergoes refraction at surface$~{}1$,
total internal reflection at surface$~{}2$, and again refraction at
surface$~{}3$. Therefore, we can write $T_{DP}\left(\phi\right)$ as [3]

$\displaystyle T_{DP}\left(\phi\right)=R(\phi)\times T_{3}\times TIR_{2}\times
T_{1}\times R(\phi)^{T},$ (12)

where $R(\phi)$ is the rotation matrix given by Eq. $1$ in the main text. The
quantities $T_{1}$, $TIR_{2}$ and $T_{3}$ are the transfer matrices
corresponding to transmission from surface $1$, total internal reflection from
the surface $2$ and transmission from the surface $3$, respectively. Using Eq.
(10) we can write $T_{1}$ as

$\displaystyle
T_{1}=\begin{bmatrix}-\frac{2\cos\theta_{i_{1}}}{\cos\theta_{i_{1}}+n\cos\theta_{t_{1}}}&0\\\
0&-\frac{2\cos\theta_{i_{1}}}{\cos\theta_{t_{1}}+n\cos\theta_{i_{1}}}\end{bmatrix},$
(13)

where $\theta_{i_{1}}=\pi/2-(\alpha+\theta_{in})$ is the angle of incidence of
the input beam with respect to normal to the surface $1$. $\theta_{in}$ is the
angle of the incident field with respect to $z$ direction, $n$ is the
refractive index of the Dove prism and for our experiment $n=1.51680$.
$\theta_{t_{1}}$ is the transmission angle from the surface $1$ with
$n\sin\theta_{t_{1}}=\sin\theta_{i_{1}}$, that is
$n\sin\theta_{t_{1}}=\cos~{}(\alpha+\theta_{in})$. Similarly, using Eq. (8),
$TIR_{2}$ can be expressed as

$\displaystyle
TIR_{2}=\begin{bmatrix}-\left(\frac{n\cos\theta_{i_{2}}-\cos\theta_{t_{2}}}{n\cos\theta_{i_{2}}+\cos\theta_{t_{2}}}\right)&0\\\
0&-\left(\frac{\cos\theta_{i_{2}}-n\cos\theta_{t_{2}}}{n\cos\theta_{t_{2}}+\cos\theta_{i_{2}}}\right)\end{bmatrix},$
(14)

where, $\theta_{i_{2}}=\alpha+\theta_{t_{1}}$ and
$\sin\theta_{t_{2}}=n\sin\theta_{i_{2}}$. Now we can write $T_{3}$ from Eq.
(10) as

$\displaystyle
T_{3}=\begin{bmatrix}\frac{2n\cos\theta_{t_{1}}}{n\cos\theta_{t_{1}}+\cos\theta_{i_{1}}}&0\\\
0&\frac{2n\cos\theta_{t_{1}}}{\cos\theta_{t_{1}}+n\cos\theta_{i_{1}}}\end{bmatrix}.$
(15)

where, $\theta_{i_{3}}=\theta_{t_{1}}$ and $\theta_{t_{3}}=\theta_{i_{1}}$.
Now, using Eqs. (13), (14) and (15) in Eq. (12) we write
$T_{DP}\left(\phi\right)$ as

$\displaystyle
T_{DP}\left(\phi\right)=\begin{bmatrix}T^{s}_{DP}\cos^{2}\phi+T^{p}_{DP}\sin^{2}\phi&\left(T^{s}_{DP}-T^{p}_{DP}\right)\sin\phi\cos\phi\\\
\left(T^{s}_{DP}-T^{p}_{DP}\right)\sin\phi\cos\phi&T^{p}_{DP}\cos^{2}\phi+T^{s}_{DP}\sin^{2}\phi\end{bmatrix},$
(16)

where, $T^{s}_{DP}$ and $T^{p}_{DP}$ can be written as

$\displaystyle
T^{s}_{DP}=\frac{2\cos\theta_{i_{1}}}{\cos\theta_{i_{1}}+n\cos\theta_{t_{1}}}\times\frac{n\cos\theta_{i_{2}}-\cos\theta_{t_{2}}}{n\cos\theta_{i_{2}}+\cos\theta_{t_{2}}}\times\frac{2n\cos~{}\theta_{t_{1}}}{n\cos~{}\theta_{t_{1}}+\cos\theta_{i_{1}}},$
(17)

and

$\displaystyle
T^{p}_{DP}=\frac{2\cos\theta_{i_{1}}}{n\cos\theta_{i_{1}}+\cos\theta_{t_{1}}}\times\frac{\cos\theta_{i_{2}}-n\cos\theta_{t_{2}}}{n\cos\theta_{t_{2}}+\cos\theta_{i_{2}}}\times\frac{2n\cos~{}\theta_{t_{1}}}{\cos~{}\theta_{t_{1}}+n\cos\theta_{i_{1}}}.$
(18)

##  III Transfer matrix for K-mirror

Figure 3: (a) Schematic of a K-mirror characterized by base angle $\beta$.

Consider a K-mirror with base angle $\beta$, as shown in Fig. 3. A K-mirror is
a monolithic arrangement of three mirrors that can be controlled
independently. The incident light undergoes three reflections from these
mirrors. Therefore, we can write the polarization transfer matrix of K-mirror
at a rotation angle $\phi$ as

$\displaystyle
T_{KM}\left(\phi\right)=R(\phi)\times(-M_{3})\times(-M_{2})\times M_{1}\times
R(\phi)^{T},$ (19)

where $M_{1}$,$M_{2}$ and $M_{3}$ are the reflection transfer matrices for
mirror $1$, mirror $2$ and mirror $3$, respectively (see Fig. 3 ). Using Eqs.
(9) and (8), we can write $M_{1}$ as

$\displaystyle
M_{1}=\begin{bmatrix}-\frac{\sin(\theta_{i_{M1}}-~{}\theta_{t_{M1}})}{\sin(\theta_{i_{M1}}+~{}\theta_{t_{M1}})}&0\\\
0&\frac{\tan(\theta_{i_{M1}}-~{}\theta_{t_{M1}})}{\tan(\theta_{i_{M1}}+~{}\theta_{t_{M1}})}\end{bmatrix},$
(20)

where $\theta_{i_{M1}}$ is the angle of incidence with respect to normal to
mirror $1$ and from the geometry, we calculate it as
$\theta_{i_{M1}}=\pi/2-(\beta+\theta_{in})$. $\beta$ is the base angle of the
K-mirror and $\theta_{in}$ is the angle of incidence of the incident field
with respect to the $z-$axis with
$n_{M}\sin\theta_{t_{M1}}=\sin\theta_{i_{M1}}$, where $n_{M}$ is the
refractive index of the metallic coating of the mirror $1$. Considering
$M_{1}$, $M_{2}$ and $M_{3}$ as identical mirrors, and using Eq. (8), we write
$M_{2}$ as

$\displaystyle
M_{2}=\begin{bmatrix}-\frac{\sin(\theta_{i_{M2}}-~{}\theta_{t_{M2}})}{\sin(\theta_{i_{M2}}+~{}\theta_{t_{M2}})}&0\\\
0&\frac{\tan(\theta_{i_{M2}}-~{}\theta_{t_{M2}})}{\tan(\theta_{i_{M2}}+~{}\theta_{t_{M2}})}\end{bmatrix},$
(21)

where $\theta_{i_{M2}}=\pi/2-(2\beta+\theta_{in})$ and $\theta_{t_{M2}}$
satisfies the following relation:
$n_{M}\sin\theta_{t_{M2}}=sin\theta_{i_{M2}}$. Similarly, $M_{3}$ can be
expressed as

$\displaystyle
M_{3}=\begin{bmatrix}-\frac{\sin(\theta_{i_{M3}}-~{}\theta_{t_{M3}})}{\sin(\theta_{i_{M3}}+~{}\theta_{t_{M3}})}&0\\\
0&\frac{\tan(\theta_{i_{M3}}-~{}\theta_{t_{M3}})}{\tan(\theta_{i_{M3}}+~{}\theta_{t_{M3}})}\end{bmatrix},$
(22)

where $\theta_{i_{M1}}=\pi/2-(\beta+\theta_{in})$ and
$n_{M}\sin\theta_{t_{M3}}=\sin\theta_{i_{M3}}$. Now, using Eqs. (20), (21) and
(22) in Eq. 19 we write $T_{KM}\left(\phi\right)$ as

$\displaystyle
T_{KM}\left(\phi\right)=\begin{bmatrix}T^{s}_{KM}\cos^{2}\phi+T^{p}_{KM}\sin^{2}\phi&\left(T^{s}_{KM}-T^{p}_{KM}\right)\sin\phi\cos\phi\\\
\left(T^{s}_{KM}-T^{p}_{KM}\right)\sin\phi\cos\phi&T^{p}_{KM}\cos^{2}\phi+T^{s}_{KM}\sin^{2}\phi\end{bmatrix}.$
(23)

Here, $T^{s}_{KM}$ and $T^{p}_{KM}$ have the following form

$\displaystyle
T^{s}_{KM}=-\frac{\sin(\theta_{i_{M1}}-~{}\theta_{t_{M1}})}{\sin(\theta_{i_{M1}}+~{}\theta_{t_{M1}})}\times\frac{\sin(\theta_{i_{M2}}-~{}\theta_{t_{M2}})}{\sin(\theta_{i_{M2}}+~{}\theta_{t_{M2}})}\times\frac{\sin(\theta_{i_{M3}}-~{}\theta_{t_{M3}})}{\sin(\theta_{i_{M3}}+~{}\theta_{t_{M3}})},$
(24)

and

$\displaystyle
T^{p}_{KM}=\frac{\tan(\theta_{i_{M1}}-~{}\theta_{t_{M1}})}{\tan(\theta_{i_{M1}}+~{}\theta_{t_{M1}})}\times\frac{\tan(\theta_{i_{M2}}-~{}\theta_{t_{M2}})}{\tan(\theta_{i_{M2}}+~{}\theta_{t_{M2}})}\times\frac{\tan(\theta_{i_{M3}}-~{}\theta_{t_{M3}})}{\tan(\theta_{i_{M3}}+~{}\theta_{t_{M3}})}.$
(25)

##  IV Calculation of mean polarization change $D$

In order to quantify the change of polarization of the transmitted field from
the Dove prism or the K-mirror, we calculate the mean polarization change $D$
between the transmitted and the incident field. When a Dove prism or a
K-mirror is rotated from $\phi=0$ to $\phi=\pi$, the transmitted state of
polarizations form a closed loop on the surface of the Poincaré sphere, as
shown in Fig. 4. The closed loop can be discretize into large number of points
where, each point represents the transmitted state of polarization at the
respective rotation angle $\phi$. The geodesic distance of each point from the
incident field as depicted through a black dot in Fig. 4, represents change of
polarization at every angle of rotation with respect to the incident state of
polarization. The mean value of these geodesic lengths is the mean
polarization change $D$, which we propose as a quantifier for minimizing the
polarization changes induced by a rotating Dove prism and a K-mirror.

Figure 4: Schematic representation of a continuous closed loop on the surface
of the Poincaré sphere in terms of discrete points P. $P_{in}$ represents the
incident state of polarization on the surface of the Poincaré sphere. $ds$
depicts the arclength element and the connecting black line between points $P$
and $P_{in}$ depicts the geodesic distance $d$ between them.

Now, to establish the mathematical result that relates the polarization
changes to the rotation angles and the base angles of the Dove prism and a
K-Mirror, we begin with the expression of the transmitted electric field in
Jones matrix form at any rotation angle $\phi$ of a Dove prism or a K-mirror
for any arbitrary incident state of polarization as

$\displaystyle\begin{bmatrix}E^{\prime}_{x}\\\
E^{\prime}_{y}\end{bmatrix}=\begin{bmatrix}\left(T^{s}_{j}\cos^{2}\phi+T^{p}_{j}\sin^{2}\phi\right)\cos\psi_{in}+\left(T^{s}_{j}-T^{p}_{j}\right)\sin\phi\cos\phi\sin\psi_{in}~{}e^{i\delta_{in}}\\\
\left(T^{s}_{j}-T^{p}_{j}\right)\sin\phi\cos\phi\cos\psi_{in}+\left(T^{p}_{j}\cos^{2}\phi+T^{s}_{j}\sin^{2}\phi\right)\sin\psi_{in}~{}e^{i\delta_{in}}\end{bmatrix}$
(26)

where $j=\left\\{DP,KM\right\\}$ for Dove prism and the K-mirror respectively
and the incident state of polarization is defined here as

$\begin{bmatrix}E_{x}\\\ E_{y}\end{bmatrix}=\begin{bmatrix}\cos\psi_{in}\\\
\sin\psi_{in}~{}e^{i\delta_{in}}\end{bmatrix}.$ (27)

Next, from Eqn. (26), we calculate the Stokes parameters for the transmitted
field at the rotation angle $\phi$. These Stokes parameters can be expressed
as

$S_{0}=|E^{\prime}_{x}|^{2}+|E^{\prime}_{y}|^{2}\\\
=\frac{1}{2}\left[|T^{p}_{j}|^{2}+|T^{s}_{j}|^{2}-\left(|T^{p}_{j}|^{2}-|T^{s}_{j}|^{2}\right)\left(\cos
2\psi_{in}\cos 2\phi+\cos\delta_{in}\sin 2\psi_{in}\sin 2\phi\right)\right],$
(28) $S_{1}=\frac{|E^{\prime}_{x}|^{2}-|E^{\prime}_{y}|^{2}}{S_{0}}\\\
=\frac{1}{4S_{0}}\left[|T^{s}_{j}|^{2}\left[2\cos 2\phi+\cos
2\psi_{in}\left(1+\cos 4\phi\right)+\cos\delta_{in}\sin 2\psi_{in}\sin
4\phi\right]\right.\\\ \left.+|T^{p}_{j}|^{2}\left[-2\cos 2\phi+\cos
2\psi_{in}\left(1+\cos 4\phi\right)+\cos\delta_{in}\sin 2\psi_{in}\sin
4\phi\right]\right.\\\
\left.+\mbox{Re}\left[T^{s*}_{j}T^{p}_{j}\right]\left(4\cos 2\psi_{in}\sin
2\phi^{2}-2\cos\delta_{in}\sin 2\psi_{in}\sin
4\phi\right)+4~{}\mbox{Im}\left[T^{s*}_{j}T^{p}_{j}\right]\sin\delta_{in}\sin
2\psi_{in}\sin 2\phi\right],$ (29)
$S_{2}=\frac{2\mbox{Re}\left[E^{\prime*}_{x}E^{\prime}_{y}\right]}{S_{0}}\\\
=\frac{1}{4S_{0}}\left[4\cos 2\phi\sin
2\psi_{in}\left(\mbox{Re}\left[T^{s*}_{j}T^{p}_{j}\right]\cos\delta_{in}\cos
2\phi-\mbox{Im}\left[T^{s*}_{j}T^{p}_{j}\right]\sin\delta_{in}\right)\right.\\\
\left.+2\sin 2\phi\left[|T^{s}_{j}|^{2}-|T^{p}_{j}|^{2}+|T^{s}_{j}|^{2}\cos
2\psi_{in}\cos
2\phi+\left(|T^{s}_{j}|^{2}+|T^{p}_{j}|^{2}\right)\cos\delta_{in}\sin
2\psi_{in}\sin 2\phi\right]\right.\\\ \left.+\cos 2\psi_{in}\sin
4\phi\left(|T^{p}_{j}|^{2}-2\mbox{Re}\left[T^{s*}_{j}T^{p}_{j}\right]\right)\right],$
(30)
$S_{3}=\frac{2\mbox{Im}\left[E^{\prime*}_{x}E^{\prime}_{y}\right]}{S_{0}}\\\
=\frac{1}{S_{0}}\left[\mbox{Im}\left[T^{s*}_{j}T^{p}_{j}\right]\left(\cos\delta_{in}\cos
2\phi\sin 2\psi_{in}-\cos 2\psi_{in}\sin
2\phi\right)+\mbox{Re}\left[T^{s*}_{j}T^{p}_{j}\right]\sin\delta_{in}\sin
2\psi_{in}\right],$ (31)

where $\mbox{Re}\left[z\right]$ and $\mbox{Im}\left[z\right]$ represent the
real and imaginary part of the complex number $z$ respectively.

Now, from the parametric Eqs. (28),(29) and (31) of the Stokes parameters
$S_{1},S_{2}~{}\rm{and}~{}S_{3}$, respectively, we write the small arclength
element of the closed loop $dS$ as

$ds=\sqrt{\left(\frac{dS_{1}}{d\phi}\right)^{2}+\left(\frac{dS_{2}}{d\phi}\right)^{2}+\left(\frac{dS_{3}}{d\phi}\right)^{2}}d\phi,$
(32)

where

$\left(\frac{dS_{1}}{d\phi}\right)^{2}+\left(\frac{dS_{2}}{d\phi}\right)^{2}+\left(\frac{dS_{3}}{d\phi}\right)^{2}\\\
=|T^{p}_{j}-T^{s}_{j}|^{2}\left[22\left(T^{p}_{j}|^{2}+T^{s}_{j}|^{2}\right)-4\mbox{Re}\left[T^{s*}_{j}T^{p}_{j}\right]+|T^{p}_{j}+T^{s}_{j}|^{2}\left\\{2\left(3+\cos
2\delta_{in}\right)\cos 4\psi_{in}\right.\right.\\\
\left.\left.+4\sin^{2}\delta\right\\}\cos
4\phi+\left(-12\mbox{Re}\left[T^{s*}_{j}T^{p}_{j}\right]+2|T^{p}_{j}|^{2}+2|T^{s}_{j}|^{2}\right)\left(\cos
2\delta_{in}+2\cos 4\psi_{in}\sin^{2}\delta_{in}\right)\right.\\\
\left.-32\cos 2\phi\left\\{\left(|T^{p}_{j}|^{2}-|T^{s}_{j}|^{2}\right)\cos
2\psi_{in}-\mbox{Im}\left[T^{s*}_{j}T^{p}_{j}\right]\sin
2\delta_{in}\sin^{2}2\psi_{in}\right\\}\right.\\\
\left.-32\left\\{\left(|T^{p}_{j}|^{2}-|T^{s}_{j}|^{2}\right)\cos\delta_{in}+2\mbox{Im}\left[T^{s*}_{j}T^{p}_{j}\right]\sin\delta_{in}\cos
2\psi_{in}\right\\}+8|T^{p}_{j}+T^{s}_{j}|^{2}\cos\delta_{in}\sin
4\psi_{in}\sin 4\phi\right]/\\\
4\left[|T^{p}_{j}|^{2}+|T^{s}_{j}|^{2}+\left(|T^{s}_{j}|^{2}-|T^{p}_{j}|^{2}\right)\left(\cos
2\psi_{in}\cos 2\phi+\cos\delta_{in}\sin 2\psi_{in}\sin
2\phi\right)\right]^{2}.$ (33)

Now, the geodesic distance between incident state of polarization point
$P_{in}\left(S^{in}_{1},S^{in}_{2},S^{in}_{3}\right)$ and an arbitrary point
on the closed surface $P\left(S_{1},S_{2},S_{3}\right)$, as illustrated
through a black curved line in Fig. 4 can be written as

$d=R\cos^{-1}\left[S_{1}S^{in}_{1}+S_{2}S^{in}_{2}+S_{3}S^{in}_{3}\right],$
(34)

where $R=1$ is the radius of the Poincaré sphere. $S^{in}_{1}=\cos
2\psi_{in}$, $S^{in}_{2}=\sin 2\psi_{in}\cos\delta_{in}$ and
$S^{in}_{3}=sin2\psi_{in}\sin\delta_{in}$ are for the incident state of
polarization given by Eqn. (27). Therefore, the mean polarization change $D$
due to rotation of the Dove prism and the K-mirror from $\phi=0$ to $\phi=\pi$
can be expressed as

$\qquad\qquad\qquad D=\frac{\int d~{}ds}{\int ds},\\\
=\frac{\int^{\pi}_{\phi=0}\cos^{-1}\left[S_{1}S^{in}_{1}+S_{2}S^{in}_{2}+S_{3}S^{in}_{3}\right]\sqrt{\left(\frac{dS_{1}}{d\phi}\right)^{2}+\left(\frac{dS_{2}}{d\phi}\right)^{2}+\left(\frac{dS_{3}}{d\phi}\right)^{2}}d\phi}{\int^{\pi}_{\phi=0}\sqrt{\left(\frac{dS_{1}}{d\phi}\right)^{2}+\left(\frac{dS_{2}}{d\phi}\right)^{2}+\left(\frac{dS_{3}}{d\phi}\right)^{2}}d\phi}.$
(35)

We use this mean polarization change $D$ as a quantifier for polarization
changes induced due to the rotation of the Dove prism or the K-mirror in the
transmitted field with respect to the incident field.

##  V Designing K-mirror with minimum polarization change

From the plots of mean polarization change $D$ as a function of base angles
$\alpha$ and $\beta$ as shown in Fig. $6$ in the main text, we find that, $D$
is minimum at a specific $\beta$ for a given reflection coating of the mirror.
Now, in order to design such K-mirror for experimental implications where
minimum change of polarization is required, we provide the dimension of the
device in terms of length $L$ and height $H$ as a function of clear aperture
size $h$ and the base angle $\beta$. From the ray diagram as shown in the Fig.
5, we can write $L=2\left(L_{1}+L_{2}\right)$ and $H=h+h^{{}^{\prime}}$. Next,
from the right angle triangle $\Delta AGF$ we write the length of the K-mirror
for clear aperture size $h$ as

$L=2h\cot~{}\beta.$ (36)

Figure 5: Ray diagram for a K-mirror at base angle $\beta$.

Similarly, from $\Delta ABC$ and $\Delta BEF$ one can show that
$L_{1}=\left(h+h^{{}^{\prime}}\right)\cot~{}2\beta$ and
$L_{2}=h^{{}^{\prime}}\cot~{}2\beta$, respectively. Therefore, by adding
$L_{1}$ and $L_{2}$, we can write as

$\left(2H-h\right)\cot~{}2\beta=\frac{L}{2},$ (37)

where we substitute $H=h+h^{{}^{\prime}}$, which is the height of the
K-mirror. Next, using Eqn. 36 in Eqn. 37 one can calculate $H$ as

$H=\frac{h}{2}\left[1+\frac{\tan~{}2\beta}{\tan~{}\beta}\right].$ (38)

We also calculate the minimum required length of the top mirror $L_{M}$ for
the clear aperture size $h$ . It can be inferred from the Fig. 5 that
$L_{M}=2L_{2}$. Next, thorough simple and straight forward trigonometric
calculations $L_{M}$ can be expressed as

$L_{M}=h\left[\cot~{}\beta-\cot~{}2\beta\right].$ (39)

Here we observe that for any allowed values of $h$ and $\beta$, $L_{M}$ is
less than $L$, which verifies the practical feasibility of the design. Now,
using the expression of length $L$ and height $H$ given by Eqs. (36) and (38)
one can estimate the dimension of the K-mirror at base angle $\beta$ where the
mean polarization change is minimum.

## References

  * Waluschka [1988] E. Waluschka, Polarization ray tracing, in  _SPIE Proceedings_, edited by R. A. Chipman (SPIE, 1988).
  * Hecht [1974] E. Hecht,  _Optics_, 4th ed. (Addison-Wesley, Reading, Mass., 1974).
  * Moreno [2004] I. Moreno, Jones matrix for image-rotation prisms, Applied Optics 43, 3373 (2004).