secReferences

# Regression-based causal inference with factorial experiments: estimands,
model specifications, and design-based properties

Anqi Zhao and Peng Ding 111Anqi Zhao, Department of Statistics and Data
Science, National University of Singapore, 117546, Singapore (E-mail:
staza@nus.edu.sg). Peng Ding, Department of Statistics, University of
California, Berkeley, CA 94720 (E-mail: pengdingpku@berkeley.edu).

###### Abstract

Factorial designs are widely used due to their ability to accommodate multiple
factors simultaneously. Factor-based regression with main effects and some
interactions is the dominant strategy for downstream analysis, delivering
point estimators and standard errors simultaneously via one least-squares fit.
Justification of these convenient estimators from the design-based perspective
requires quantifying their sampling properties under the assignment mechanism
whilst conditioning on the potential outcomes. To this end, we derive the
sampling properties of the regression estimators under a wide range of
specifications, and establish the appropriateness of the corresponding robust
standard errors for the Wald-type inference. The result clarifies the causal
interpretation of the coefficients in these factor-based regressions, and
inspires the definition of general factorial effects to unify the standard
definitions of factorial effects in various literatures. We also quantify the
bias-variance trade-off between the saturated and unsaturated regressions from
the design-based perspective.

Keywords: Factorial effect; potential outcome; randomization inference; robust
standard error

##  Introduction

Factorial designs are increasingly popular in field experiments in social
sciences (e.g., Duflo et al. 2007; Dasgupta et al. 2015; Branson et al. 2016;
Egami and Imai 2019) in addition to the traditional agricultural, industrial,
and biomedical applications (e.g., Wu and Hamada 2009). Factor-based
regression remains the dominant strategy for downstream analysis (e.g., Karlan
and List 2007; Eriksson and Rooth 2014; Torres et al. 2021), enabling not only
direct estimation of the factorial effects as regression coefficients but also
flexible unsaturated specifications to reduce model complexity. A formal
justification of its role in causal inference, however, requires both clearly
defining the estimands of interest and deriving the sampling properties of the
resulting estimators under the potential outcomes framework.

This article makes several contributions. First, we clarify the causal
interpretation of the coefficients in factor-based linear regressions and
propose a location-shift strategy to reproduce the design-based inference of
various factorial effects via least squares. Importantly, we show that the
robust covariance affords an asymptotically conservative estimator of the true
sampling covariance from the design-based perspective, justifying its use for
large-sample Wald-type inference. Second, we review and clarify the standard
definitions of factorial effects in the causal inference, experimental design,
epidemiology, and social sciences literatures, and extend them to allow for
arbitrary weighting schemes to accommodate external validity concerns. Third,
we derive the design-based properties of estimators from unsaturated factor-
based regressions for the first time, and quantify the bias-variance trade-off
between the saturated and unsaturated regressions from the design-based
perspective.

We use $Y_{i}\sim x_{i}$ to denote the least-squares regression of $Y_{i}$ on
$x_{i}$ and focus on not only the causal interpretation of the regression
coefficients for estimating the general factorial effects but also the design-
based properties of the robust covariance, also known as the
Eicker–Huber–White covariance, for large-sample Wald-type inference. The terms
“regression”, “coefficients”, and “robust covariance” refer to the numeric
outputs of least squares free of any modeling assumptions; we evaluate their
sampling properties from the design-based perspective. We omit the discussion
of the ordinary covariance derived under homoskedasticity due to its lack of
design-based guarantees even with the simple treatment-control experiment
(Freedman 2008).

Let $1_{N}$ and $1_{Q}$ be the $N\times 1$ and $Q\times 1$ vectors of ones,
respectively. Let $\mathcal{I}(\cdot)$ be the indicator function. Let
$[m]=\\{1,\ldots,m\\}$ be the set of $1$ to $m$ for positive integer $m$. For
two symmetric matrices $M_{1}$ and $M_{2}$, write $M_{1}\geq 0$ if $M_{1}$ is
positive semi-definite and write $M_{1}\leq M_{2}$ or $M_{1}\geq M_{2}$ if
$(M_{2}-M_{1})$ is positive or negative semi-definite, respectively.

##  Framework, causal effects, and treatment-based regression

Consider an experiment with $N$ units, $i=1,\ldots,N$, and $Q$ treatment
levels, $z\in\mathcal{T}=\\{1,\ldots,Q\\}$. Let $Y_{i}(z)$ be the potential
outcome of unit $i$ if assigned to level $z$, and let
$\bar{Y}(z)=N^{-1}\sum_{i=1}^{N}Y_{i}(z)$ be the average, vectorized as
${\bar{Y}}=({\bar{Y}}(1),\ldots,{\bar{Y}}(Q))^{\mathrm{\scriptscriptstyle
T}}$. Let $S=(S(z,z^{\prime}))_{z,z^{\prime}\in\mathcal{T}}$ be the finite-
population covariance matrix of the potential outcomes with
$S(z,z^{\prime})=(N-1)^{-1}\sum_{i=1}^{N}\\{Y_{i}(z)-{\bar{Y}}(z)\\}\\{Y_{i}(z^{\prime})-{\bar{Y}}(z^{\prime})\\}$.
The goal is to estimate $\tau=G{\bar{Y}}$ for some contrast matrix $G$ with
rows orthogonal to $1_{Q}$. Complete randomization assigns completely at
random $N_{z}\geq 2$ units to level $z$ with $\sum_{z\in\mathcal{T}}N_{z}=N$
and $e_{z}=N_{z}/N$. For unit $i$, let $Z_{i}\in\mathcal{T}$ denote the
treatment level and $Y_{i}=\sum_{z\in\mathcal{T}}\mathcal{I}(Z_{i}=z)Y_{i}(z)$
denote the observed outcome. Let $\hat{Y}(z)=N_{z}^{-1}\sum_{i:Z_{i}=z}Y_{i}$
be the average observed outcome under level $z$, vectorized as
$\hat{Y}=(\hat{Y}(1),\ldots,\hat{Y}(Q))^{\mathrm{\scriptscriptstyle T}}$. Then
$\hat{\tau}=G\hat{Y}$ affords an intuitive choice for estimating $\tau$.

Design-based inference, also known as the randomization inference, concerns
the sampling properties of estimators over the distribution of the treatment
indicators, conditioning on the potential outcomes. Throughout the paper, we
focus on complete randomization and invoke Condition 1 below for asymptotic
properties (Li and Ding 2017).

######  Condition 1.

As $N$ goes to infinity, for all $z\in\mathcal{T}$, (i) $N_{z}\geq 2$ and
$e_{z}$ has a limit between $(0,1)$, (ii) ${\bar{Y}}$ and $S$ have finite
limits, and (iii) $\max_{1\leq i\leq
N}\\{Y_{i}(z)-\bar{Y}(z)\\}^{2}/N\rightarrow 0$.

Under complete randomization, $\hat{Y}$ is unbiased for ${\bar{Y}}$ with
covariance
$\textup{cov}(\hat{Y})=\operatorname{diag}\\{S(z,z)/N_{z}\\}_{z\in\mathcal{T}}-N^{-1}S$.
Define
$\hat{V}=\operatorname{diag}\\{\hat{S}(z,z)/N_{z}\\}_{z\in\mathcal{T}}$, where
$\hat{S}(z,z)=(N_{z}-1)^{-1}\sum_{i:Z_{i}=z}\\{Y_{i}-\hat{Y}(z)\\}^{2}$, as a
moment estimator of $\textup{cov}(\hat{Y})$. It is conservative in the sense
of $E(\hat{V})-\textup{cov}(\hat{Y})=N^{-1}S\geq 0$. Condition 1 further
ensures $\hat{Y}$ is asymptotically Normal with
$N\\{\hat{V}-\textup{cov}(\hat{Y})\\}=S+o_{\text{p}}(1)$ (Li and Ding 2017).
The Wald-type inference of $\tau$ can thus be conducted using
$\hat{\tau}=G\hat{Y}$ and
$\hat{\textup{cov}}(\hat{\tau})=G\hat{V}G^{\mathrm{\scriptscriptstyle T}}$ as
the point estimator and estimated covariance, respectively. It is in general
conservative due to the over-estimation of the covariance; one exception is
when the treatment effects are constant across all units as specified by
Condition 2 below.

######  Condition 2.

For all $z,z^{\prime}\in\mathcal{T}$,
$Y_{i}(z)-Y_{i}(z^{\prime})=c(z,z^{\prime})$ are constant across
$i=1,\ldots,N$. This ensures $S(z,z^{\prime})$ are identical for all
$z,z^{\prime}\in\mathcal{T}$, denoted by $S(z,z^{\prime})=s_{0}$.

Treatment-based regression affords a convenient tool for computing $\hat{Y}$
and $\hat{V}$ from least squares. The regression
$Y_{i}\sim\mathcal{I}(Z_{i}=1)+\cdots+\mathcal{I}(Z_{i}=Q)$ without an
intercept yields coefficient vector $\hat{\beta}$ and robust covariance
$\hat{V}_{0}$ that satisfy $\hat{\beta}=\hat{Y}$ and
$\hat{V}_{0}=\operatorname{diag}(1-N_{z}^{-1})_{z\in\mathcal{T}}\hat{V}=\hat{V}+o_{\text{p}}(1)$
(Wu and Ding 2020, Section 3.3). The Wald-type inference of $\tau$ can thus
also be conducted using $G\hat{\beta}$ and
$G\hat{V}_{0}G^{\mathrm{\scriptscriptstyle T}}$ as the point estimator and
estimated covariance, respectively.

As a special case, this setup encompasses the $Q_{1}\times\cdots\times Q_{K}$
factorial experiment, which involves $Q=\prod_{k=1}^{K}Q_{k}$ treatment levels
as the combinations of $K\geq 2$ factors with $Q_{k}\ (k=1,\ldots,K)$ levels,
respectively. Treatment-based regression accordingly affords a principled way
to study general factorial experiments. It is nevertheless not the dominant
strategy in practice when the estimands of interest take some special forms.
In the case where the goal is to estimate the main effects or interactions of
the factors under study, a more prevalent practice is to regress the outcome
on the factors themselves and interpret the coefficients as the corresponding
factorial effects of interest. This seemingly straightforward approach has
several variants across different fields, which turn out to target factorial
effects under distinct weighting schemes. Our first contribution unifies these
variants under a class of location-shifted factor-based regressions, and
establishes the design-based properties of the resulting coefficients and
robust covariances.

More importantly, treatment-based regression is saturated and requires the
estimation of $Q=\prod_{k=1}^{K}Q_{k}\geq 2^{K}$ parameters. This could be
demanding in terms of sample size even with a moderate number of factors.
Factor-based regression, on the other hand, enables flexible unsaturated
specifications that include only the main effects and possibly some lower-
order interactions corresponding to the factorial effects of interest. Despite
the intuitiveness of such an approach and its dominance in practice, the
existing literature on the design-based properties of factor-based regression
focuses on saturated specifications (Dasgupta et al. 2015; Lu 2016), and
leaves the theory of their unsaturated counterparts an open question. Our
second contribution fills this gap and establishes the design-based properties
of unsaturated factor-based regressions.

Due to the notational burden involved in the general setting, we start with
the $2^{2}$ and $2^{3}$ experiments to illustrate the main ideas, and then
unify the results under the $2^{K}$ experiment. The results convey all key
points for the theory of the general $Q_{1}\times\cdots\times Q_{K}$
experiment. We give the formal theory on the general case in the Supplementary
Material.

##  The $2^{2}$ factorial experiment

###  3.1 A review of existing strategies

The $2^{2}$ factorial experiment is the simplest factorial experiment with two
binary factors, A and B. The $Q=2^{2}=4$ treatment combinations consist of
$\mathcal{T}=\\{(00),(01),(10),(11)\\}$, indexed by $z=(ab)$ for $a,b=0,1$.
Let $A_{i},B_{i}\in\\{0,1\\}$ indicate the levels of the factors received by
unit $i$. We first review five factor-based regression strategies commonly
used for analyzing $2^{2}$ experiments, and then clarify their respective
causal interpretations.

The canonical factor-based regression takes the form $Y_{i}\sim
1+A_{i}+B_{i}+A_{i}B_{i}$. Strategy (i) directly uses the coefficients of
$(A_{i},B_{i},A_{i}B_{i})$, denoted by
$\hat{\gamma}_{0}=(\hat{\gamma}_{0,\textsc{a}},\hat{\gamma}_{0,\textsc{b}},\hat{\gamma}_{0,\textsc{ab}})^{\mathrm{\scriptscriptstyle
T}}$, to estimate the main effects of factors A and B and their interaction,
respectively. Strategy (ii) uses
$(\hat{\gamma}_{0,\textsc{a}}+B_{i}\hat{\gamma}_{0,\textsc{ab}},\hat{\gamma}_{0,\textsc{b}}+A_{i}\hat{\gamma}_{0,\textsc{ab}},\hat{\gamma}_{0,\textsc{ab}})$
to estimate the main effects and interaction at the unit level, respectively,
and then takes their respective averages to estimate the factorial effects at
the population level. Define $e_{\textsc{a}=1}=N^{-1}\sum_{i=1}^{N}A_{i}$ and
$e_{\textsc{b}=1}=N^{-1}\sum_{i=1}^{N}B_{i}$ as the empirical probabilities of
factors A and B, respectively. The final estimators equal
$\hat{\gamma}_{\textup{e}}=(\hat{\gamma}_{\textup{e},\textsc{a}},\hat{\gamma}_{\textup{e},\textsc{b}},\hat{\gamma}_{\textup{e},\textsc{ab}})^{\mathrm{\scriptscriptstyle
T}}$ where
$\hat{\gamma}_{\textup{e},\textsc{a}}=\hat{\gamma}_{0,\textsc{a}}+e_{\textsc{b}=1}\hat{\gamma}_{0,\textsc{ab}}$,
$\hat{\gamma}_{\textup{e},\textsc{b}}=\hat{\gamma}_{0,\textsc{b}}+e_{\textsc{a}=1}\hat{\gamma}_{0,\textsc{ab}}$,
and $\hat{\gamma}_{\textup{e},\textsc{ab}}=\hat{\gamma}_{0,\textsc{ab}}$.
Strategy (ii) is popular in econometrics, with the estimators of the main
effects, namely $\hat{\gamma}_{\textup{e},\textsc{a}}$ and
$\hat{\gamma}_{\textup{e},\textsc{b}}$, also known as the average partial or
marginal effects (Greene 2018). Strategy (iii) codes the factors by their
signs as $A_{i}^{\scaleto{\textup{s}}{4pt}}=2A_{i}-1$ and
$B_{i}^{\scaleto{\textup{s}}{4pt}}=2B_{i}-1\in\\{+1,-1\\}$, and uses the
coefficients from $Y_{i}\sim
1+A_{i}^{\scaleto{\textup{s}}{4pt}}+B_{i}^{\scaleto{\textup{s}}{4pt}}+A_{i}^{\scaleto{\textup{s}}{4pt}}B_{i}^{\scaleto{\textup{s}}{4pt}}$,
after multiplied by two, to estimate the main effects and interaction,
respectively (Wu and Hamada 2009; Lu 2016). Let
$\hat{\gamma}_{\textup{s}}=(\hat{\gamma}_{\textup{s},\textsc{a}},\hat{\gamma}_{\textup{s},\textsc{b}},\hat{\gamma}_{\textup{s},\textsc{ab}})^{\mathrm{\scriptscriptstyle
T}}$ denote the estimators under strategy (iii). This gives three strategies
for simultaneously estimating the main effects and interaction via one least-
squares fit.

Strategies (iv) and (v), on the other hand, focus on only the two main
effects. Strategy (iv) considers two separate regressions, $Y_{i}\sim 1+A_{i}$
and $Y_{i}\sim 1+B_{i}$, and estimates the two main effects by the
coefficients of $A_{i}$ and $B_{i}$, respectively (e.g., Bertrand and
Mullainathan 2004; Eriksson and Rooth 2014). Strategy (v) considers the
additive regression $Y_{i}\sim 1+A_{i}+B_{i}$, and estimates the two effects
via one least-squares fit.

Refer to a factor-based regression as saturated if it contains all possible
interactions between the factors in addition to the constant term and main
effects. The regressions under strategies (i)–(iii) are saturated whereas
those under strategies (iv) and (v) are unsaturated.

###  3.2 Unifying the saturated regressions and introducing the general
factorial effects

We now unify strategies (i)–(iii) under a class of location-shifted factor-
based regressions that turn out to target factorial effects under different
weighting schemes. The result highlights the correspondence between the
location shifts in specifying the models and the weighting schemes in defining
the factorial effects.

To this end, we first formalize the notion of general factorial effects, which
are central to clarifying the effective estimands under strategies (i)–(iii).
Define
$\tau_{\textsc{a}|b}=\tau_{\textsc{a}|{\textsc{b}=b}}=\bar{Y}(1b)-\bar{Y}(0b)$
and
$\tau_{\textsc{b}|a}=\tau_{\textsc{b}|{\textsc{a}=a}}=\bar{Y}(a1)-\bar{Y}(a0)$
as the conditional effects of factors A and B when the level of the other
factor is fixed at $b\in\\{0,1\\}$ and $a\in\\{0,1\\}$, respectively. As a
convention, we abbreviate the “${\textsc{a}=a}$” and “${\textsc{b}=b}$” in the
subscripts as “$a$” and “$b$”, respectively, when no confusion would arise.
Define

$\displaystyle\tau_{\textsc{a}}(\pi_{\textsc{b}})=\pi_{{\textsc{b}=0}}\cdot\tau_{\textsc{a}|{\textsc{b}=0}}+\pi_{{\textsc{b}=1}}\cdot\tau_{\textsc{a}|{\textsc{b}=1}},\qquad\tau_{\textsc{b}}(\pi_{\textsc{a}})=\pi_{\textsc{a}=0}\cdot\tau_{\textsc{b}|\textsc{a}=0}+\pi_{\textsc{a}=1}\cdot\tau_{\textsc{b}|{\textsc{a}=1}}$

as the main effects of factors A and B under weighting schemes
$\pi_{\textsc{b}}=(\pi_{{\textsc{b}=0}},\pi_{{\textsc{b}=1}})$ and
$\pi_{\textsc{a}}=(\pi_{\textsc{a}=0},\pi_{\textsc{a}=1})$, respectively, with
$0\leq\pi_{\textsc{a}=a},\pi_{\textsc{b}=b}\leq 1$ for $a,b=0,1$ and
$\pi_{\textsc{a}=0}+\pi_{\textsc{a}=1}=\pi_{{\textsc{b}=0}}+\pi_{{\textsc{b}=1}}=1$.
As a convention, the subscript of the weighting scheme indicates the factor
that is being marginalized out. The standard main effects correspond to
$\pi_{\textsc{a}}=\pi_{\textsc{b}}=(1/2,1/2)$, weighting all conditional
effects equally (Dasgupta et al. 2015).

Define
$\tau_{\textsc{ab}}={\bar{Y}}(11)-{\bar{Y}}(10)-{\bar{Y}}(01)+{\bar{Y}}(00)$
as the interaction between A and B. It satisfies
$\tau_{\textsc{ab}}=\tau_{\textsc{a}|{\textsc{b}=1}}-\tau_{\textsc{a}|{\textsc{b}=0}}=\tau_{\textsc{b}|{\textsc{a}=1}}-\tau_{\textsc{b}|{\textsc{a}=0}}$
and characterizes the difference in the conditional effects of one factor at
the two levels of the other factor. Note that
$\tau_{\textsc{a}}(\pi_{\textsc{b}}^{\prime})-\tau_{\textsc{a}}(\pi_{\textsc{b}})=(\pi_{{\textsc{b}=1}}^{\prime}-\pi_{{\textsc{b}=1}})\tau_{\textsc{ab}}$
and
$\tau_{\textsc{b}}(\pi_{\textsc{a}}^{\prime})-\tau_{\textsc{b}}(\pi_{\textsc{a}})=(\pi_{\textsc{a}=1}^{\prime}-\pi_{\textsc{a}=1})\tau_{\textsc{ab}}$
such that $\tau_{\textsc{ab}}$ also quantifies the difference in causal
estimands between different weighting schemes. The absence of the interaction,
namely $\tau_{\textsc{a}\textsc{b}}=0$, ensures that
$\tau_{\textsc{a}}(\pi_{\textsc{b}})=\tau_{{\textsc{a}|{\textsc{b}=0}}}$ and
$\tau_{\textsc{b}}(\pi_{\textsc{a}})=\tau_{{\textsc{b}|\textsc{a}=0}}$ are
constant across all possible weighting schemes.

Recall
${\bar{Y}}=({\bar{Y}}(00),{\bar{Y}}(01),{\bar{Y}}(10),{\bar{Y}}(11))^{\mathrm{\scriptscriptstyle
T}}$. Vectorize the main effects and interaction as
$\tau_{\pi}=(\tau_{\textsc{a}}(\pi_{\textsc{b}}),\tau_{\textsc{b}}(\pi_{\textsc{a}}),\tau_{\textsc{ab}})^{\mathrm{\scriptscriptstyle
T}}=G_{\pi}{\bar{Y}}$ with $\pi=(\pi_{\textsc{a}},\pi_{\textsc{b}})$ and the
contrast matrix $G_{\pi}$ consisting of row vectors
$(-\pi_{{\textsc{b}=0}},-\pi_{{\textsc{b}=1}},\pi_{{\textsc{b}=0}},\pi_{{\textsc{b}=1}})$,
$(-\pi_{\textsc{a}=0},\pi_{\textsc{a}=0},-\pi_{\textsc{a}=1},\pi_{\textsc{a}=1})$,
and $(1,-1,-1,1)$. An unbiased estimator for $\tau_{\pi}$ is
$\hat{\tau}_{\pi}=G_{\pi}\hat{Y}=(\hat{\tau}_{\textsc{a}}(\pi_{\textsc{b}}),\hat{\tau}_{\textsc{b}}(\pi_{\textsc{a}}),\hat{\tau}_{\textsc{ab}})^{\mathrm{\scriptscriptstyle
T}}$.

Let $e_{\textsc{a}=0}=1-e_{\textsc{a}=1}$ and
$e_{\textsc{b}=0}=1-e_{\textsc{b}=1}$ be the proportions of units that receive
level 0 of factors A and B in the experiment, respectively. Proposition 1
below is numeric and clarifies the causal interpretations of the regression
estimators from strategies (i)–(iii).

######  Proposition 1.

Under the $2^{2}$ experiment, the coefficients from strategies (i)–(iii)
satisfy

  1. (i)

$\hat{\gamma}_{0}=(\hat{\tau}_{\textsc{a}}(1,0),\hat{\tau}_{\textsc{b}}(1,0),\hat{\tau}_{\textsc{ab}})^{\mathrm{\scriptscriptstyle
T}}$ with $\pi_{\textsc{a}}=\pi_{\textsc{b}}=(1,0)$;

  2. (ii)

$\hat{\gamma}_{\textup{e}}=(\hat{\tau}_{\textsc{a}}(e_{\textsc{b}=0},e_{\textsc{b}=1}),\hat{\tau}_{\textsc{b}}(e_{\textsc{a}=0},e_{\textsc{a}=1}),\hat{\tau}_{\textsc{ab}})^{\mathrm{\scriptscriptstyle
T}}$ with $\pi_{f}=(e_{f=0},e_{f=1})$ for $f=\textsc{a},\textsc{b}$;

  3. (iii)

$\hat{\gamma}_{\textup{s}}=(\hat{\tau}_{\textsc{a}}(1/2,1/2),\hat{\tau}_{\textsc{b}}(1/2,1/2),\hat{\tau}_{\textsc{ab}}/2)^{\mathrm{\scriptscriptstyle
T}}$ with $\pi_{\textsc{a}}=\pi_{\textsc{b}}=(1/2,1/2)$.

Strategies (i)–(iii) thus yield identical estimators of $\tau_{\textsc{ab}}$
up to a scaling factor yet target at distinct main effects under different
weighting schemes. Strategy (i) is unbiased for estimating
$\tau_{\textsc{a}}(1,0)=\tau_{\textsc{a}|{\textsc{b}=0}}$ and
$\tau_{\textsc{b}}(1,0)=\tau_{\textsc{b}|\textsc{a}=0}$ as the conditional
effects when the other factor is at the baseline level. Strategy (ii) is
unbiased for estimating $\tau_{\textsc{a}}(e_{\textsc{b}=0},e_{\textsc{b}=1})$
and $\tau_{\textsc{b}}(e_{\textsc{a}=0},e_{\textsc{a}=1})$; the average
partial effects in econometrics thus weight the conditional effects by the
empirical treatment probabilities. Strategy (iii) is unbiased for estimating
the standard effects $\tau_{\textsc{a}}=\tau_{\textsc{a}}(1/2,1/2)$ and
$\tau_{\textsc{b}}=\tau_{\textsc{b}}(1/2,1/2)$ that weight all conditional
effects equally. This clarifies the causal interpretations of
$\hat{\gamma}_{0}$, $\hat{\gamma}_{\textup{e}}$, and
$\hat{\gamma}_{\textup{s}}$ from strategies (i)–(iii), respectively. In
particular, $\hat{\gamma}_{\textup{s}}$ targets the standard factorial effects
regardless of whether the experiment is balanced or not.

Inspired by how transformation on factors allows us to obtain the moment
estimators of the standard main effects directly as regression coefficients
under strategy (iii), we now propose a location-shift strategy to generalize
strategies (i)–(iii) and estimate $\tau_{\pi}$ with arbitrary weights
$\pi=(\pi_{\textsc{a}},\pi_{\textsc{b}})$ via least squares. For
$A_{i}^{\prime}=A_{i}-\delta_{\textsc{a}}$ and
$B_{i}^{\prime}=B_{i}-\delta_{\textsc{b}}$ with prespecified
$0\leq\delta_{\textsc{a}},\delta_{\textsc{b}}\leq 1$, define the location-
shifted regression

$\displaystyle Y_{i}\sim
1+A_{i}^{\prime}+B_{i}^{\prime}+A_{i}^{\prime}B_{i}^{\prime}$ (1)

with coefficients
$\hat{\gamma}=(\hat{\gamma}_{\textsc{a}},\hat{\gamma}_{\textsc{b}},\hat{\gamma}_{{\textsc{a}\textsc{b}}})^{\mathrm{\scriptscriptstyle
T}}$ and robust covariance $\hat{\Psi}$ for the three non-intercept terms.
Strategies (i)–(iii) are special cases: setting
$(\delta_{\textsc{a}},\delta_{\textsc{b}})=(0,0)$ equals strategy (i); setting
$(\delta_{\textsc{a}},\delta_{\textsc{b}})=(e_{\textsc{a}=1},e_{\textsc{b}=1})$
equals strategy (ii) in the sense of $\hat{\gamma}=\hat{\gamma}_{\textup{e}}$
by Proposition 2 below; setting
$(\delta_{\textsc{a}},\delta_{\textsc{b}})=(1/2,1/2)$ equals strategy (iii) up
to scaling factors of two or four.

Recall $\hat{\tau}_{\pi}=G_{\pi}\hat{Y}$ as an unbiased estimator of
$\tau_{\pi}=G_{\pi}\bar{Y}$. Let
$\hat{\textup{cov}}(\hat{\tau}_{\pi})=G_{\pi}\hat{V}G_{\pi}^{\mathrm{\scriptscriptstyle
T}}$ be the corresponding estimated covariance, recalling $\hat{V}$ as a
conservative estimator of $\textup{cov}(\hat{Y})$. Proposition 2 below states
the numeric correspondence between $\\{\hat{\gamma},\hat{\Psi}\\}$ and
$\\{\hat{\tau}_{\pi},\hat{\textup{cov}}(\hat{\tau}_{\pi})\\}$, elucidating the
design-based properties of $\hat{\gamma}$ and $\hat{\Psi}$ for general
$(\delta_{\textsc{a}},\delta_{\textsc{b}})$.

######  Proposition 2.

Under the $2^{2}$ experiment, the outputs of (1) satisfy
$\hat{\gamma}=\hat{\tau}_{\pi}$ and
$\hat{\Psi}=\hat{\textup{cov}}(\hat{\tau}_{\pi})-G_{\pi}\operatorname{diag}(N_{z}^{-1})\hat{V}G_{\pi}^{\mathrm{\scriptscriptstyle
T}}$ for $\pi=(\pi_{\textsc{a}},\pi_{\textsc{b}})$ with
$\pi_{\textsc{a}}=(1-\delta_{\textsc{a}},\delta_{\textsc{a}})$ and
$\pi_{\textsc{b}}=(1-\delta_{\textsc{b}},\delta_{\textsc{b}})$.

Proposition 2 ensures that $\hat{\gamma}$ from (1) is unbiased for estimating
$\tau_{\pi}$ with
$\pi_{\textsc{a}}=(1-\delta_{\textsc{a}},\delta_{\textsc{a}})$ and
$\pi_{\textsc{b}}=(1-\delta_{\textsc{b}},\delta_{\textsc{b}})$. Location
shifts of $A_{i}$ and $B_{i}$ by
$(\delta_{\textsc{a}},\delta_{\textsc{b}})=(\pi_{\textsc{a}=1},\pi_{{\textsc{b}=1}})$
thus enable direct estimation of $\tau_{\pi}$ from (1) for arbitrary $\pi$.
This gives the intuition for requiring
$0\leq\delta_{\textsc{a}},\delta_{\textsc{b}}\leq 1$ introduced before.
Moreover, the difference between $\hat{\Psi}$ and
$\hat{\textup{cov}}(\hat{\tau}_{\pi})$ diminishes as $N$ goes to infinity.
This enables the large-sample Wald-type inference of $\tau_{\pi}$ by using
$\hat{\gamma}$ and $\hat{\Psi}$ as the point estimator and estimated
covariance, respectively.

######  Remark 1.

The classical experimental design literature focuses mostly on the standard
main effects (Wu and Hamada 2009), with equal weights on all conditional
effects:
$\tau_{\textsc{a}}=2^{-1}(\tau_{\textsc{a}|{\textsc{b}=0}}+\tau_{\textsc{a}|{\textsc{b}=1}})$
and
$\tau_{\textsc{b}}=2^{-1}(\tau_{\textsc{b}|\textsc{a}=0}+\tau_{\textsc{b}|{\textsc{a}=1}})$.
The standard main effects, together with balanced experiments with $N_{z}=N/Q$
for all $z\in\mathcal{T}$, have many advantages in practice. Corollary 1 later
states a result for the $2^{K}$ experiment with a general $K$.

Applications in practice, however, may not always value
$\tau_{\textsc{a}|{\textsc{b}=0}}$ and $\tau_{\textsc{a}|{\textsc{b}=1}}$, and
likewise $\tau_{\textsc{b}|\textsc{a}=0}$ and
$\tau_{\textsc{b}|{\textsc{a}=1}}$, equally. Alternative weighting schemes
based on perceived importance could thus also merit attention and afford
possibly more relevant summary of the marginal effects (Finney 1948). We give
an example based on consideration of external validity of the experimental
results.

Assume the experiment in question is a pilot study for a large-scale
implementation that intends $1/3$ of the population to receive level $1$ of
factor B marginally. Now that we know $2/3$ of the population will be
experiencing the effect of factor A at the baseline level of factor B, the
general effect
$\tau_{\textsc{a}}(2/3,1/3)=2/3\cdot\tau_{\textsc{a}|{\textsc{b}=0}}+1/3\cdot\tau_{\textsc{a}|{\textsc{b}=1}}$
can be a better summary of the effect of factor A compared with the standard
effect with equal weights. This illustrates the connection between the general
weighting schemes and external validity.

When $\tau_{\textsc{ab}}\neq 0$, we are also interested in finding the optimal
level of factor B to maximize the effect of factor A. This requires us to
compare $\tau_{\textsc{a}|{\textsc{b}=1}}$ and
$\tau_{\textsc{a}|{\textsc{b}=0}}$, which correspond to two special estimands
$\tau_{\textsc{a}}(0,1)$ and $\tau_{\textsc{a}}(1,0)$.

In summary, the choice of estimand depends on the scientific question of
interest. We provide the theory for the general estimand which includes the
above examples as special cases.

###  3.3 Factor-based regression with unsaturated models

Strategies (iv) and (v) concern only the main effects of factors A and B. To
this end, strategy (iv) fits two separate regressions for estimating the main
effects of factors A and B, respectively. The resulting estimators equal the
differences in means between $\\{Y_{i}:f_{i}=1\\}$ and $\\{Y_{i}:f_{i}=0\\}$
for $f=A,B$, respectively, and are biased for estimating factorial effects of
the form $\tau_{\textsc{a}}(\pi_{\textsc{b}})$ and
$\tau_{\textsc{b}}(\pi_{\textsc{a}})$ in general. We thus exclude it from the
ensuing discussion.

Strategy (v), on the other hand, estimates the two main effects together via
one additive regression. Consider a generalized version, incorporating the
location-shift transformation:

$\displaystyle Y_{i}\sim 1+A^{\prime}_{i}+B^{\prime}_{i}.$ (2)

We first derive the effective estimands of (2) as a pair of general factorial
effects, and then state the bias-variance trade-off between (1) and (2). The
result establishes the optimality of (2) for estimating arbitrary $\tau_{\pi}$
when the nuisance effect $\tau_{\textsc{a}\textsc{b}}$ indeed does not exist.

Let $\tilde{\gamma}_{\textsc{a}}$ and $\tilde{\gamma}_{\textsc{b}}$ be the
coefficients of $A_{i}^{\prime}$ and $B_{i}^{\prime}$ from (2), respectively.
Let $\hat{\tau}_{\textsc{a}|{\textsc{b}=b}}$ and
$\hat{\tau}_{\textsc{b}|{\textsc{a}=a}}$ be the moment estimators of
$\tau_{\textsc{a}|{\textsc{b}=b}}$ and $\tau_{\textsc{b}|{\textsc{a}=a}}$ for
$a,b=0,1$, respectively.

######  Proposition 3.

Under the $2^{2}$ experiment, the coefficients from (2) satisfy

$\displaystyle\tilde{\gamma}_{\textsc{a}}=\tilde{\pi}_{{\textsc{b}=0}}\cdot\hat{\tau}_{\textsc{a}|{\textsc{b}=0}}+\tilde{\pi}_{{\textsc{b}=1}}\cdot\hat{\tau}_{\textsc{a}|{\textsc{b}=1}},\qquad\tilde{\gamma}_{\textsc{b}}=\tilde{\pi}_{{\textsc{a}=0}}\cdot\hat{\tau}_{\textsc{b}|{\textsc{a}=0}}+\tilde{\pi}_{{\textsc{a}=1}}\cdot\hat{\tau}_{\textsc{b}|{\textsc{a}=1}}$

with
$\tilde{\pi}_{\textsc{b}}=(\tilde{\pi}_{\textsc{b}=0},\tilde{\pi}_{\textsc{b}=1})=\sigma^{-1}(e_{01}^{-1}+e_{11}^{-1},e_{00}^{-1}+e_{10}^{-1})$
and
$\tilde{\pi}_{\textsc{a}}=(\tilde{\pi}_{\textsc{a}=0},\tilde{\pi}_{\textsc{a}=1})=\sigma^{-1}(e_{10}^{-1}+e_{11}^{-1},e_{00}^{-1}+e_{01}^{-1})$,
where $\sigma=\sum_{z\in\mathcal{T}}e_{z}^{-1}$.

Proposition 3 shows $\tilde{\gamma}_{\textsc{a}}$ and
$\tilde{\gamma}_{\textsc{b}}$ as the moment estimators of
$\tau_{\textsc{a}}(\tilde{\pi}_{\textsc{b}})$ and
$\tau_{\textsc{b}}(\tilde{\pi}_{\textsc{a}})$ under a specific weighting
scheme that is fully determined by $(e_{z})_{z\in\mathcal{T}}$ and independent
of $(\delta_{\textsc{a}},\delta_{\textsc{b}})$. Therefore, the unsaturated
regression (2) no longer accommodates flexible weighting schemes even with
location-shifted factors. Under balanced designs with equal treatment sizes
$N_{z}=N/4$ for all $z\in\mathcal{T}$, we have
$\tilde{\gamma}_{\textsc{a}}=\hat{\tau}_{\textsc{a}}(1/2,1/2)$ and
$\tilde{\gamma}_{\textsc{b}}=\hat{\tau}_{\textsc{b}}(1/2,1/2)$ give the moment
estimators of the standard main effects, and thus equal the coefficients of
$A_{i}^{\prime}$ and $B_{i}^{\prime}$ from the saturated regression (1) with
$\delta_{\textsc{a}}=\delta_{\textsc{b}}=1/2$. This is no coincidence but due
to the fact that the columns of the design matrix of (1) with
$\delta_{\textsc{a}}=\delta_{\textsc{b}}=1/2$ are mutually orthogonal such
that the deletion of $A_{i}^{\prime}B_{i}^{\prime}$ has no effect on the
estimation of the remaining coefficients. This highlights the connection
between standard effects and balanced designs from a different angle, echoing
the classical principle that recommends the use of balanced designs whenever
possible.

In general, $\tilde{\gamma}_{\textsc{a}}$ and $\tilde{\gamma}_{\textsc{b}}$
are biased for $\tau_{\textsc{a}}(\pi_{\textsc{b}})$ and
$\tau_{\textsc{b}}(\pi_{\textsc{a}})$ unless
$(\pi_{\textsc{a}},\pi_{\textsc{b}})=(\tilde{\pi}_{\textsc{a}},\tilde{\pi}_{\textsc{b}})$
or the interaction $\tau_{\textsc{a}\textsc{b}}$ does not exist. Nevertheless,
under Condition 2, they minimize the sampling variances of
$\hat{\tau}_{\textsc{a}}(\pi_{\textsc{b}})={\pi}_{\textsc{b}=0}\cdot\hat{\tau}_{\textsc{a}|{\textsc{b}=0}}+{\pi}_{\textsc{b}=1}\cdot\hat{\tau}_{\textsc{a}|{\textsc{b}=1}}$
and
$\hat{\tau}_{\textsc{b}}(\pi_{\textsc{a}})={\pi}_{\textsc{a}=0}\cdot\hat{\tau}_{\textsc{b}|{\textsc{a}=0}}+{\pi}_{\textsc{a}=1}\cdot\hat{\tau}_{\textsc{b}|{\textsc{a}=1}}$
over all possible $\pi_{\textsc{b}}$ and $\pi_{\textsc{a}}$, respectively. In
particular, the constant treatment effects ensure
$\textup{var}(\hat{\tau}_{\textsc{a}|{\textsc{b}=0}})=s_{0}(N_{00}^{-1}+N_{10}^{-1})$
and
$\textup{var}(\hat{\tau}_{\textsc{a}|{\textsc{b}=1}})=s_{0}(N_{01}^{-1}+N_{11}^{-1})$.
To minimize the variance of $\hat{\tau}_{\textsc{a}}(\pi_{\textsc{b}})$ is
thus equivalent to having the weights proportional to the inverses of
$\textup{var}(\hat{\tau}_{\textsc{a}|{\textsc{b}=0}})$ and
$\textup{var}(\hat{\tau}_{\textsc{a}|{\textsc{b}=1}})$, respectively,
resulting in $(\tilde{\pi}_{\textsc{b}=0},\tilde{\pi}_{\textsc{b}=1})$ as
defined in Proposition 3. Similar discussion extends to
$\tilde{\gamma}_{\textsc{b}}$. This demonstrates the bias-variance trade-off
between (1) and (2).

This concludes our discussion on the $2^{2}$ experiment. We next extend the
results to the $2^{3}$ experiment to illustrate one additional point: with
more than two factors, the factor-based regression is capable of estimating
only a subset of all causally-meaningful factorial effects in general, yet
regains generality in the absence of three-way interactions.

##  The $2^{3}$ factorial experiment

###  4.1 Notation and definition of the general factorial effects

The $2^{3}$ factorial experiment features $Q=2^{3}=8$ treatment combinations
arising from three binary factors, A, B, and C. Let $A_{i}$, $B_{i}$, and
$C_{i}\in\\{0,1\\}$ indicate the levels of the factors received by unit $i$.
The eight treatment combinations consist of
$\mathcal{T}=\\{(abc):a,b,c=0,1\\}$. Let $\bar{Y}(abc)$ be the average
potential outcome under treatment combination $(abc)\in\mathcal{T}$. Define
the conditional effects of factors A, B, and C as

$\tau_{\textsc{a}|bc}=\bar{Y}(1bc)-\bar{Y}(0bc),\quad\tau_{\textsc{b}|ac}=\bar{Y}(a1c)-\bar{Y}(a0c),\quad\tau_{{\textsc{c}}|ab}=\bar{Y}(ab1)-\bar{Y}(ab0),$

respectively, with the rest two factors fixed at $bc,ac,ab\in\\{0,1\\}^{2}$.
Define the conditional two-way interactions between factors A and B, factors A
and C, and factors B and C as

$\displaystyle\tau_{{\textsc{a}\textsc{b}}|c}$ $\displaystyle=$
$\displaystyle\bar{Y}(11c)-\bar{Y}(10c)-\bar{Y}(01c)+\bar{Y}(00c),$
$\displaystyle\tau_{{\textsc{a}{\textsc{c}}}|b}$ $\displaystyle=$
$\displaystyle\bar{Y}(1b1)-\bar{Y}(1b0)-\bar{Y}(0b1)+\bar{Y}(0b0),$
$\displaystyle\tau_{{\textsc{b}{\textsc{c}}}|a}$ $\displaystyle=$
$\displaystyle\bar{Y}(a11)-\bar{Y}(a10)-\bar{Y}(a01)+\bar{Y}(a00),$

respectively, with the third factor fixed at $c,b,a\in\\{0,1\\}$. When
potential confusion arises, we write out “${\textsc{a}=a}$”,
“${\textsc{b}=b}$”, and “${{\textsc{c}}=c}$” for $a$, $b$, and $c$ in the
subscripts to emphasize both the factors and their respective levels; for
example,
$\tau_{\textsc{a}|bc}=\tau_{\textsc{a}|{\textsc{b}=b},{{\textsc{c}}=c}}$ and
$\tau_{{\textsc{a}\textsc{b}}|c}=\tau_{{\textsc{a}\textsc{b}}|{{\textsc{c}}=c}}$.
These conditional effects afford the building blocks for defining the general
factorial effects.

To simplify the presentation, we call a set of $W$ numbers,
$(\pi_{1},\ldots,\pi_{W})$, a $W$-dimensional weighting vector if
$\sum_{w=1}^{W}\pi_{w}=1$ and $\pi_{w}\geq 0$; a weighting scheme is then a
collection of weighting vectors with composition clear from the context.
Throughout this section, assume
$\pi_{\textsc{a}\textsc{b}}=(\pi_{ab})_{a,b=0,1},\pi_{\textsc{a}{\textsc{c}}}=(\pi_{ac})_{a,c=0,1},\pi_{\textsc{b}{\textsc{c}}}=(\pi_{bc})_{b,c=0,1}$
are some prespecified four-dimensional weighting vectors, and
$\pi_{\textsc{a}}=(\pi_{a})_{a=0,1},\pi_{\textsc{b}}=(\pi_{b})_{b=0,1},\pi_{\textsc{c}}=(\pi_{c})_{c=0,1}$
are some prespecified two-dimensional weighting vectors. Summarize them as
$\pi=\\{\pi_{\textsc{a}\textsc{b}},\pi_{\textsc{a}{\textsc{c}}},\pi_{\textsc{b}{\textsc{c}}},\pi_{\textsc{a}},\pi_{\textsc{b}},\pi_{\textsc{c}}\\}=\\{\pi_{ab},\pi_{ac},\pi_{bc},\pi_{a},\pi_{b},\pi_{c}:a,b,c=0,1\\}$.

######  Definition 1.

Under the $2^{3}$ experiment, define

$\tau_{\textsc{a}}(\pi_{\textsc{b}{\textsc{c}}})=\sum_{b,c}\pi_{bc}\cdot\tau_{\textsc{a}|bc},\quad\tau_{\textsc{b}}(\pi_{\textsc{a}{\textsc{c}}})=\sum_{a,c}\pi_{ac}\cdot\tau_{\textsc{b}|ac},\quad\tau_{{\textsc{c}}}(\pi_{\textsc{a}\textsc{b}})=\sum_{a,b}\pi_{ab}\cdot\tau_{{\textsc{c}}|ab}$

as the main effects of factors A, B, and C under weighting vectors
$\pi_{\textsc{b}{\textsc{c}}}$, $\pi_{\textsc{a}{\textsc{c}}}$, and
$\pi_{\textsc{a}\textsc{b}}$, respectively; define

$\tau_{\textsc{a}\textsc{b}}(\pi_{\textsc{c}})=\sum_{c=0,1}\pi_{c}\cdot\tau_{{\textsc{a}\textsc{b}}|c},\quad\tau_{\textsc{a}{\textsc{c}}}(\pi_{\textsc{b}})=\sum_{b=0,1}\pi_{b}\cdot\tau_{{\textsc{a}{\textsc{c}}}|b},\quad\tau_{\textsc{b}{\textsc{c}}}(\pi_{\textsc{a}})=\sum_{a=0,1}\pi_{a}\cdot\tau_{{\textsc{b}{\textsc{c}}}|a}$

as the two-way interactions between factors A and B, factors A and C, and
factors B and C under weighting vectors $\pi_{\textsc{c}}$,
$\pi_{\textsc{b}}$, and $\pi_{\textsc{a}}$, respectively; define

$\tau_{{\textsc{a}\textsc{b}{\textsc{c}}}}=\tau_{{\textsc{a}\textsc{b}}|{{\textsc{c}}=1}}-\tau_{{\textsc{a}\textsc{b}}|{{\textsc{c}}=0}}=\tau_{{\textsc{a}{\textsc{c}}}|{\textsc{b}=1}}-\tau_{{\textsc{a}{\textsc{c}}|{\textsc{b}=0}}}=\tau_{\textsc{b}{\textsc{c}}|\textsc{a}=1}-\tau_{\textsc{b}{\textsc{c}}|\textsc{a}=0}=\sum_{a,b,c}(-1)^{a+b+c+1}{\bar{Y}}(abc)$

as the three-way interaction between factors A, B, and C.

Definition 1 gives a total of $2^{3}-1=7$ general factorial effects,
vectorized as
$\tau_{\pi}=(\tau_{\textsc{a}}(\pi_{\textsc{b}{\textsc{c}}}),\tau_{\textsc{b}}(\pi_{\textsc{a}{\textsc{c}}}),\tau_{\textsc{c}}(\pi_{\textsc{a}\textsc{b}}),\tau_{{\textsc{a}\textsc{b}}}(\pi_{\textsc{c}}),\tau_{{\textsc{a}{\textsc{c}}}}(\pi_{\textsc{b}}),\tau_{{\textsc{b}{\textsc{c}}}}(\pi_{\textsc{a}}),\tau_{{\textsc{a}\textsc{b}{\textsc{c}}}})^{\mathrm{\scriptscriptstyle
T}}=G_{\pi}{\bar{Y}}.$ Following the convention from the $2^{2}$ experiment,
the subscripts of the weighting vectors indicate the factors that are being
marginalized out. Refer to $\pi$ as the equal weighting scheme if
$\pi_{ab}=\pi_{bc}=\pi_{ac}=1/4$ and $\pi_{a}=\pi_{b}=\pi_{c}=1/2$ for all
$a,b,c=0,1$; refer to $\pi$ as the empirical weighting scheme if
$\pi_{a}=N^{-1}\sum_{i=1}^{N}\mathcal{I}(A_{i}=a)$,
$\pi_{ab}=N^{-1}\sum_{i=1}^{N}\mathcal{I}(A_{i}=a,B_{i}=b)$, etc., equaling
the empirical treatment proportions in the experiment. Although Definition 1
can be general, we focus on the following coherent weighting scheme throughout
the paper.

######  Definition 2.

A weighting scheme $\pi$ is coherent if there exists a probability
distribution over $\mathcal{T}$, represented by $\pi_{abc}={\rm
pr}(A=a,B=b,C=c)$ for $a,b,c=0,1$, such that

$\displaystyle\begin{array}[]{lll}\pi_{a}={\rm pr}(A=a),&\pi_{b}={\rm
pr}(B=b),&\pi_{c}={\rm pr}(C=c),\\\ \pi_{ab}={\rm pr}(A=a,B=b),&\pi_{ac}={\rm
pr}(A=a,C=c),&\pi_{bc}={\rm pr}(B=b,C=c).\end{array}$

Coherence imposes mild restrictions on the elements in $\pi$ and, building on
the intuition from Remark 1, provides the causal interpretation of the general
factorial effects from a thought experiment perspective. Consider a target
thought experiment in which we assign unit $i$ to combination
$(abc)\in\mathcal{T}$ with probability ${\rm pr}\\{Z_{i}=(abc)\\}={\rm
pr}(A_{i}=a,B_{i}=b,C_{i}=c)=\pi_{abc}$. The weighting vector
$\pi_{\textsc{b}{\textsc{c}}}=(\pi_{bc})_{b,c=0,1}$ gives the marginal
distribution of $(B_{i},C_{i})$ and renders the weighted average
$\tau_{\textsc{a},i}(\pi_{\textsc{b}{\textsc{c}}})=\sum_{b,c}\pi_{bc}\cdot\tau_{\textsc{a}|bc,i}$,
where $\tau_{\textsc{a}|bc,i}=Y_{i}(1bc)-Y_{i}(0bc)$, an intuitive summary of
the main effect of factor A on unit $i$, accounting for the target treatment
probabilities of factors B and C (see also Hainmueller et al. 2014; Egami and
Imai 2019; de la Cuesta et al. 2021). Averaging
$\tau_{\textsc{a},i}(\pi_{\textsc{b}{\textsc{c}}})$ over $i=1,\ldots,N$ yields
$N^{-1}\sum_{i=1}^{N}\tau_{\textsc{a},i}(\pi_{\textsc{b}{\textsc{c}}})=\tau_{\textsc{a}}(\pi_{\textsc{b}{\textsc{c}}})$
as the average effect at the population level. The general weights as such
allow for external validity beyond the actual experiment being conducted. The
equal weighting scheme is coherent with $\pi_{abc}=1/8$, implying balanced
design in the thought experiment. The empirical weighting scheme is also
coherent with $\pi_{abc}=e_{abc}=N_{abc}/N$.

###  4.2 Factor-based regression with the saturated model

Define $A_{i}^{\prime}=A_{i}-\delta_{\textsc{a}}$,
$B_{i}^{\prime}=B_{i}-\delta_{\textsc{b}}$, and
$C_{i}^{\prime}=C_{i}-\delta_{\textsc{c}}$ for prespecified
$\delta=(\delta_{\textsc{a}},\delta_{\textsc{b}},\delta_{\textsc{c}})$ with
$0\leq\delta_{\textsc{a}},\delta_{\textsc{b}},\delta_{\textsc{c}}\leq 1$, and
extend (1) to the $2^{3}$ experiment to define

$\displaystyle Y_{i}\sim
1+A^{\prime}_{i}+B^{\prime}_{i}+C^{\prime}_{i}+A^{\prime}_{i}B^{\prime}_{i}+A^{\prime}_{i}C^{\prime}_{i}+B^{\prime}_{i}C^{\prime}_{i}+A^{\prime}_{i}B^{\prime}_{i}C^{\prime}_{i}.$
(4)

Let $\hat{\gamma}$ and $\hat{\Psi}$ be the coefficient vector and robust
covariance of the $2^{3}-1=7$ non-intercept terms in (4), respectively. We
study in this subsection their design-based properties, illustrating two
important characteristics of factor-based regressions with more than two
factors. First, saturated regressions like (4) can only recover a subset of
the coherent factorial effects with weighting schemes featuring a product
structure in Definition 3 below. Second, the absence of the three-way
interaction restores the generality of (4) for estimating all coherent
factorial effects.

######  Definition 3.

A coherent weighting scheme $\pi$ is a product weighting scheme if
$\pi_{abc}=\pi_{a}\pi_{b}\pi_{c}$ for $a,b,c=0,1$.

A product weighting scheme $\pi$ is fully determined by the values of
$(\pi_{\textsc{a}=1},\pi_{\textsc{b}=1},\pi_{{\textsc{c}}=1})$ and implies
independent factors in the corresponding thought experiment. The equal
weighting scheme satisfies Definition 3 with
$\pi_{\textsc{a}=1}=\pi_{{\textsc{b}=1}}=\pi_{{\textsc{c}}=1}=1/2$; the
empirical weighting scheme, on the other hand, in general does not.

Let $\delta_{\scaleto{\times}{4pt}}$ be the product weighting scheme with
${\rm pr}(A_{i}=1)=\delta_{\textsc{a}}$, ${\rm
pr}(B_{i}=1)=\delta_{\textsc{b}}$, and ${\rm pr}(C_{i}=1)=\delta_{\textsc{c}}$
in the corresponding thought experiment. As a convention, we use “$\times$” in
the subscript to indicate product weighting schemes. Let
$\tau_{\delta{\scaleto{\times}{4pt}}}=G_{\delta{\scaleto{\times}{4pt}}}{\bar{Y}}$
be the corresponding vector of general factorial effects,
$\hat{\tau}_{\delta{\scaleto{\times}{4pt}}}=G_{\delta{\scaleto{\times}{4pt}}}\hat{Y}$
be its moment estimator, and
$\hat{\textup{cov}}(\hat{\tau}_{{\delta{\scaleto{\times}{4pt}}}})=G_{\delta{\scaleto{\times}{4pt}}}\hat{V}G_{\delta{\scaleto{\times}{4pt}}}^{\mathrm{\scriptscriptstyle
T}}$ be the estimated covariance of
$\hat{\tau}_{\delta{\scaleto{\times}{4pt}}}$, respectively. Proposition 4
below gives the numeric correspondence between $\\{\hat{\gamma},\hat{\Psi}\\}$
and
$\\{\hat{\tau}_{\delta{\scaleto{\times}{4pt}}},\hat{\textup{cov}}(\hat{\tau}_{{\delta{\scaleto{\times}{4pt}}}})\\}$,
elucidating the utility of (4) for inferring
$\tau_{\delta{\scaleto{\times}{4pt}}}$.

######  Proposition 4.

Under the $2^{3}$ experiment, the outputs of (4) satisfy
$\hat{\gamma}=\hat{\tau}_{\delta{\scaleto{\times}{4pt}}}$ and
$\hat{\Psi}=\hat{\textup{cov}}(\hat{\tau}_{{\delta{\scaleto{\times}{4pt}}}})-G_{\delta{\scaleto{\times}{4pt}}}\operatorname{diag}(N_{z}^{-1})\hat{V}G_{\delta{\scaleto{\times}{4pt}}}^{\mathrm{\scriptscriptstyle
T}}.$

Proposition 4 highlights the commonality and difference between the $2^{2}$
and $2^{3}$ experiments. On the one hand, it ensures the asymptotic
equivalence between $\\{\hat{\gamma},\hat{\Psi}\\}$ and
$\\{\hat{\tau}_{\delta{\scaleto{\times}{4pt}}},\hat{\textup{cov}}(\hat{\tau}_{{\delta{\scaleto{\times}{4pt}}}})\\}$
as $N$ goes to infinity, and thereby allows for the large-sample Wald-type
inference of $\tau_{\delta{\scaleto{\times}{4pt}}}$ based on (4). On the other
hand, the product structure of $\delta_{\scaleto{\times}{4pt}}$ constrains the
generality of (4), and suggests that it recovers the full vector of
$\tau_{\pi}$ simultaneously if and only if
$(\delta_{\textsc{a}},\delta_{\textsc{b}},\delta_{\textsc{c}})=(\pi_{{\textsc{a}=1}},\pi_{{\textsc{b}=1}},\pi_{{{\textsc{c}}=1}})$
and $\pi$ is a product weighting scheme. The standard effects satisfy the
product structure with
$\pi_{\textsc{a}=1}=\pi_{{\textsc{b}=1}}=\pi_{{\textsc{c}}=1}=1/2$ and thus
admit of direct estimation with
$\delta_{\textsc{a}}=\delta_{\textsc{b}}=\delta_{\textsc{c}}=1/2$.

The resulting specification is equivalent to that under the $\\{+1,-1\\}$
coding system up to a constant scaling factor on each regressor, suggesting
the specificity of the $\\{+1,-1\\}$ coding system to the standard effects (Wu
and Hamada 2009; Lu 2016). The partial effects, on the contrary, may or may
not satisfy the product structure, and are thus not necessarily directly
estimable from (4); see Remark C1 in the Supplementary Material. This affords
a useful guideline for designing and analyzing factorial experiments.

One exception, however, is when the three-way interaction does not exist. The
absence of $\tau_{{\textsc{a}\textsc{b}{\textsc{c}}}}$ leaves the class of
product weighting schemes equivalent to the class of coherent weighting
schemes in defining the general factorial effects. We formalize the idea in
Proposition 5 below. For an arbitrary weighting scheme $\pi$, let
$\pi_{\scaleto{\times}{4pt}}$ be the product weighting scheme with ${\rm
pr}(A_{i}=1)=\pi_{\textsc{a}=1}$, ${\rm pr}(B_{i}=1)=\pi_{{\textsc{b}=1}}$,
and ${\rm pr}(C_{i}=1)=\pi_{{\textsc{c}}=1}$ in the corresponding thought
experiment. By definition, $\pi_{\scaleto{\times}{4pt}}$ and $\pi$ share the
same marginal treatment probabilities in the underlying thought experiments,
and satisfy $\pi_{\scaleto{\times}{4pt}}=\pi$ if $\pi$ is already a product
weighting scheme.

######  Proposition 5.

Under the $2^{3}$ experiment, if
$\tau_{{\textsc{a}\textsc{b}{\textsc{c}}}}=0$, then
$\tau_{\pi}=\tau_{\pi{\scaleto{\times}{4pt}}}$ for all coherent $\pi$, where
$\tau_{\pi}=G_{\pi}{\bar{Y}}$ and
$\tau_{\pi{\scaleto{\times}{4pt}}}=G_{\pi{\scaleto{\times}{4pt}}}{\bar{Y}}$
are the vectors of general factorial effects under weighting schemes $\pi$ and
$\pi_{\scaleto{\times}{4pt}}$, respectively.

Propositions 4 and 5 together justify the inference of $\tau_{\pi}$ from (4)
with
$(\delta_{\textsc{a}},\delta_{\textsc{b}},\delta_{\textsc{c}})=(\pi_{{\textsc{a}=1}},\pi_{{\textsc{b}=1}},\pi_{{{\textsc{c}}=1}})$
for all coherent $\pi$ when $\tau_{\textsc{a}\textsc{b}{\textsc{c}}}=0$. The
absence of the three-way interaction restores the generality of factor-based
regressions for all coherent weighting schemes.

###  4.3 Factor-based regression with an unsaturated model

Consider an extension to (2),

$\displaystyle Y_{i}\sim
1+A^{\prime}_{i}+B^{\prime}_{i}+C^{\prime}_{i}+A^{\prime}_{i}B^{\prime}_{i}+A^{\prime}_{i}C^{\prime}_{i}+B^{\prime}_{i}C^{\prime}_{i},$
(5)

when only the main effects and two-way interactions are of interest,
vectorized as

$\tau_{\pi,{\scaleto{+}{4pt}}}=(\tau_{\textsc{a}}(\pi_{\textsc{b}{\textsc{c}}}),\tau_{\textsc{b}}(\pi_{\textsc{a}{\textsc{c}}}),\tau_{\textsc{c}}(\pi_{\textsc{a}\textsc{b}}),\tau_{\textsc{ab}}(\pi_{\textsc{c}}),\tau_{\textsc{a}{\textsc{c}}}(\pi_{\textsc{b}}),\tau_{\textsc{b}{\textsc{c}}}(\pi_{\textsc{a}}))^{\mathrm{\scriptscriptstyle
T}}=\tau_{\pi}\backslash\\{\tau_{\textsc{a}\textsc{b}{\textsc{c}}}\\}.$

Let $\tilde{\gamma}_{\scaleto{+}{4pt}}$ and
$\tilde{\Psi}_{{\scaleto{+}{4pt}}}$ be the coefficient vector and robust
covariance of the six non-intercept terms from (5). We use
“$\tilde{\color[rgb]{1,1,1}{a}}$” to signify outputs from unsaturated
regressions, and “+” to signify quantities associated with the effects of
interest throughout the paper. Let $\hat{\gamma}_{\scaleto{+}{4pt}}$ and
$\hat{\gamma}_{{\textsc{a}\textsc{b}{\textsc{c}}}}$ be the coefficients of
$(A^{\prime}_{i},B^{\prime}_{i},C^{\prime}_{i},A^{\prime}_{i}B^{\prime}_{i},A^{\prime}_{i}C^{\prime}_{i},B^{\prime}_{i}C^{\prime}_{i})$
and $A_{i}^{\prime}B_{i}^{\prime}C_{i}^{\prime}$ from (4), respectively, with
$\hat{\gamma}=(\hat{\gamma}_{\scaleto{+}{4pt}}^{\mathrm{\scriptscriptstyle
T}},\hat{\gamma}_{{\textsc{a}\textsc{b}{\textsc{c}}}})^{\mathrm{\scriptscriptstyle
T}}$. Proposition 6 below extends the result in Proposition 3 to the $2^{3}$
experiment, elucidating the design-based properties of
$\tilde{\gamma}_{\scaleto{+}{4pt}}$ via its link with $\hat{\gamma}$.

######  Proposition 6.

Under the $2^{3}$ experiment, we have
$\tilde{\gamma}_{\scaleto{+}{4pt}}=\hat{\gamma}_{\scaleto{+}{4pt}}+D\hat{\gamma}_{{\textsc{a}\textsc{b}{\textsc{c}}}}$
with

$\displaystyle
D=\left(\sum_{z\in\mathcal{T}}e_{z}^{-1}\right)^{-1}\left(\begin{array}[]{ccc|ccc}&&\color[rgb]{1,1,1}1&\delta_{\textsc{b}}&\delta_{\textsc{c}}&0\\\
\color[rgb]{1,1,1}1&I_{3}&&\delta_{\textsc{a}}&0&\delta_{\textsc{c}}\\\
&&&0&\delta_{\textsc{a}}&\delta_{\textsc{b}}\\\ \hline\cr&&&\\\ &0_{3\times
3}&&&I_{3}\\\
&&&&&\end{array}\right)\left(\begin{array}[]{r}-\sum_{a}e_{a00}^{-1}\\\
-\sum_{b}e_{0b0}^{-1}\\\ -\sum_{c}e_{00c}^{-1}\\\
\hline\cr\sum_{ab}e_{ab0}^{-1}\\\ \sum_{ac}e_{a0c}^{-1}\\\
\sum_{bc}e_{0bc}^{-1}\end{array}\right)-\left(\begin{array}[]{c}\delta_{\textsc{b}}\delta_{\textsc{c}}\\\
\delta_{\textsc{a}}\delta_{\textsc{c}}\\\
\delta_{\textsc{a}}\delta_{\textsc{b}}\\\ \hline\cr\delta_{\textsc{c}}\\\
\delta_{\textsc{b}}\\\ \delta_{\textsc{a}}\end{array}\right).$

Recall that $\hat{\gamma}_{{\scaleto{+}{4pt}}}$ and
$\hat{\gamma}_{{\textsc{a}\textsc{b}{\textsc{c}}}}$ equal the moment
estimators of $\tau_{\delta{\scaleto{\times}{4pt}},{\scaleto{+}{4pt}}}$ and
$\tau_{\textsc{a}\textsc{b}{\textsc{c}}}$, respectively, denoted by
$\hat{\tau}_{\delta{\scaleto{\times}{4pt}},{\scaleto{+}{4pt}}}$ and
$\hat{\tau}_{\textsc{a}\textsc{b}{\textsc{c}}}$. The coefficients from (5)
thus recover the exact moment estimator
$\hat{\tau}_{\delta{\scaleto{\times}{4pt}},{\scaleto{+}{4pt}}}$ if and only if
$D=0_{6}$ or $\hat{\tau}_{\textsc{a}\textsc{b}{\textsc{c}}}=0$. The former in
general entails $e_{z}=1/8$ for all $z\in\mathcal{T}$ and
$\delta_{\textsc{a}}=\delta_{\textsc{b}}=\delta_{\textsc{c}}=1/2$, implying
both balanced design and standard effects as the estimands. In particular,
$e_{z}=1/8\ (z\in\mathcal{T})$ and
$\delta_{\textsc{a}}=\delta_{\textsc{b}}=\delta_{\textsc{c}}=1/2$ ensure that
the columns of the design matrix of the saturated regression (4) are mutually
orthogonal such that deletion of any subset of the columns has no effect on
the estimation of the remaining coefficients, with (5) being a special case.
This is in line with the intuition from the $2^{2}$ case and echos the
classical principle that advocates the use of balanced designs whenever
possible.

On the other hand, Proposition 6 implies
$E(\tilde{\gamma}_{\scaleto{+}{4pt}})-\tau_{\delta{\scaleto{\times}{4pt}},{\scaleto{+}{4pt}}}=D\tau_{\textsc{a}\textsc{b}{\textsc{c}}}$
such that $\tilde{\gamma}_{\scaleto{+}{4pt}}$ is unbiased for
$\tau_{\delta{\scaleto{\times}{4pt}},{\scaleto{+}{4pt}}}$ as long as the
nuisance effect $\tau_{\textsc{a}\textsc{b}{\textsc{c}}}$ indeed does not
exist. This, together with the equivalence between $\tau_{\pi}$ and
$\tau_{\pi{\scaleto{\times}{4pt}}}$ in the absence of
$\tau_{\textsc{a}\textsc{b}{\textsc{c}}}$, ensures the generality of (5) for
estimating $\tau_{\pi,{\scaleto{+}{4pt}}}$. More precisely, under
$\tau_{{\textsc{a}\textsc{b}{\textsc{c}}}}=0$, the coefficient
$\tilde{\gamma}_{\scaleto{+}{4pt}}$ from (5) with
$(\delta_{\textsc{a}},\delta_{\textsc{b}},\delta_{\textsc{c}})=(\pi_{{\textsc{a}=1}},\pi_{{\textsc{b}=1}},\pi_{{{\textsc{c}}=1}})$
is unbiased for $\tau_{\pi,{\scaleto{+}{4pt}}}$ for all coherent weighting
schemes $\pi$.

The violation of the no three-way interaction condition, on the other hand,
subjects $\tilde{\gamma}_{\scaleto{+}{4pt}}$ to non-diminishing bias, namely
$D\tau_{\textsc{a}\textsc{b}{\textsc{c}}}$. The intuition on the bias-variance
trade-off from the $2^{2}$ case extends here and ensures that
$\tilde{\gamma}_{\scaleto{+}{4pt}}$ is more precise than
$\hat{\gamma}_{{\scaleto{+}{4pt}}}$ under Condition 2 regardless of whether
$\tau_{{\textsc{a}\textsc{b}{\textsc{c}}}}=0$ or not.

##  A general theory for the $2^{K}$ factorial experiment

###  5.1  Overview and notation

The $2^{K}$ factorial experiment features $Q=2^{K}$ treatment combinations
arising from $K$ binary factors, indexed by $k=1,\ldots,K$. Of interest is the
utility of the corresponding factor-based regressions for inferring the
factorial effects of interest from the design-based perspective. To this end,
we first extend the definitions of general factorial effects, coherent
weighting scheme, and product weighting scheme to the $2^{K}$ design, and
demonstrate the utility of location-shifted regressions for recovering general
effects under product weighting schemes. We then show the equivalence between
the coherent and product weighting schemes under the no three-way interactions
condition. We finally quantify the bias-variance trade-off between the
saturated and unsaturated specifications.

We use the following notation to facilitate the discussion. Let
$Z_{ik}\in\\{0,1\\}$ denote the level of factor $k$ received by unit $i$ for
$\ i=1,\ldots,N$ and $k=1,\ldots,K$. Let
${\mathcal{F}}_{k}=\\{0,1\\}=\\{0_{k},1_{k}\\}$ be the set of possible levels
of factor $k$, where we use the subscript $k$ to differentiate the factors.
Let
$\mathcal{T}=\prod_{k=1}^{K}{\mathcal{F}}_{k}=\\{z=(z_{1},\ldots,z_{K}):z_{k}\in{\mathcal{F}}_{k};\
k=1,\ldots,K\\}$ be the set of the $2^{K}$ treatment combinations. Let
$\mathcal{P}_{K}=\\{{{\mathcal{K}}}:\emptyset\neq{{\mathcal{K}}}\subseteq[K]\\}$
be the set of all non-empty subsets of $[K]$. For
${{\mathcal{K}}}\in\mathcal{P}_{K}$, let
${z_{\scaleto{\mathcal{K}}{4.5pt}}}=(z_{k})_{\scaleto{k\in{{\mathcal{K}}}}{6pt}}$
and $z_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}}=(z_{k})_{k\not\in{{\mathcal{K}}}}$
index the combinations of factors in ${{\mathcal{K}}}$ and
${\overline{{\mathcal{K}}}}=[K]\backslash{{\mathcal{K}}}$, respectively,
taking values from
${\mathcal{F}}_{\scaleto{\mathcal{K}}{4.5pt}}=\prod_{\scaleto{k\in{{\mathcal{K}}}}{6pt}}{\mathcal{F}}_{k}=\\{0,1\\}^{|{{\mathcal{K}}}|}$
and ${\mathcal{F}}_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}}=\prod_{k\not\in{\scaleto{\mathcal{K}}{4.5pt}}}{\mathcal{F}}_{k}=\\{0,1\\}^{K-|{{\mathcal{K}}}|}$.
In particular, $z_{\scaleto{[K]}{6pt}}=z\in\mathcal{T}$ and
${\mathcal{F}}_{{\scaleto{[K]}{6pt}}}=\mathcal{T}$.

###  5.2 Definition of the conditional factorial effects

Consider $K$ types of factorial effects, quantifying the main effect of a
factor when applied alone and the two- to $K$-way interactions when multiple
factors are applied together. Refer to them interchangeably as the first- to
$K$th-order factorial effects, respectively. Building on the intuition from
the $2^{2}$ and $2^{3}$ experiments, we first define the conditional factorial
effects in this subsection, and then define the general factorial effects as
their respective weighted averages in the next subsection.

As a general rule, we define by induction the $m$th-order conditional
factorial effect as the difference between two $(m-1)$th-order conditional
effects for $m=2,\ldots,K$ (Wu and Hamada 2009). For notational simplicity, we
illustrate the definition of the $m$th-order effects using the first $m$
factors with ${{\mathcal{K}}}=[m]$,
$z_{{\scaleto{(m+1):K}{7pt}}}=(z_{k})_{k=m+1}^{K}$, and
${\mathcal{F}}_{{\scaleto{(m+1):K}{7pt}}}=\prod_{k=m+1}^{K}{\mathcal{F}}_{k}=\\{0,1\\}^{K-m}$.

######  Definition 4.

Let ${\bar{Y}}(z_{1},z_{\scaleto{2:K}{5pt}})$ be the average potential outcome
under $z=(z_{1},z_{\scaleto{2:K}{5pt}})\in\mathcal{T}$, and define
$\tau_{1}(z_{\scaleto{2:K}{5pt}})={\bar{Y}}(1_{1},z_{\scaleto{2:K}{5pt}})-{\bar{Y}}(0_{1},z_{\scaleto{2:K}{5pt}})$
as the conditional main effect of factor 1 when factors $2$ to $K$ are fixed
at $z_{\scaleto{2:K}{5pt}}\in{\mathcal{F}}_{{\scaleto{2:K}{5pt}}}$.

Given $\tau_{\scaleto{[m-1]}{7pt}}(z_{{\scaleto{m:K}{5pt}}})$ as the
conditional $(m-1)$th-order factorial effect of factors 1 to $(m-1)$ when the
rest of the factors are fixed at
$z_{\scaleto{m:K}{5pt}}\in{\mathcal{F}}_{\scaleto{m:K}{5pt}}$ for
$m=2,\ldots,K-1$, define

$\displaystyle\tau_{\scaleto{[m]}{7pt}}(z_{{\scaleto{(m+1):K}{7pt}}})=\tau_{\scaleto{[m-1]}{7pt}}(1_{m},z_{{\scaleto{(m+1):K}{7pt}}})-\tau_{\scaleto{[m-1]}{7pt}}(0_{m},z_{{\scaleto{(m+1):K}{7pt}}})$

as the conditional $m$th-order factorial effect of factors 1 to $m$ when the
rest of the factors are fixed at
$z_{{\scaleto{(m+1):K}{7pt}}}\in{\mathcal{F}}_{\scaleto{(m+1):K}{7pt}}$.

For $m=K$, define
$\tau_{\scaleto{[K]}{6pt}}=\tau_{{\scaleto{[K-1]}{6pt}}}(1_{K})-\tau_{{\scaleto{[K-1]}{6pt}}}(0_{K})$
as the $K$-way interaction of factors 1 to $K$.

Based on Definition 4, we can obtain the explicit form of
$\tau_{\scaleto{[m]}{7pt}}(z_{{\scaleto{(m+1):K}{7pt}}})$ in terms of the
$\bar{Y}(z)$’s, and show that the order in which new factors are added to the
combination in the induction does not matter. Definition 4 extends to general
${{\mathcal{K}}}\in\mathcal{P}_{K}$ by symmetry. Denote by
$\tau_{{\scaleto{\mathcal{K}}{4.5pt}}}(z_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}})$ the conditional
$|{{\mathcal{K}}}|$-th order factorial effect of factors in ${{\mathcal{K}}}$
when the rest of the factors are fixed at $z_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}}\in{\mathcal{F}}_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}}$. This gives a total of
$|{\mathcal{F}}_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}}|=2^{K-|{{\mathcal{K}}}|}$
conditional factorial effects for the $|{{\mathcal{K}}}|$ factors in a fixed
${{\mathcal{K}}}$. The notation from the $2^{2}$ case is a special case with
$\tau_{\textsc{a}|b}=\tau_{\textsc{a}}(b)$ and
$\tau_{\textsc{b}|a}=\tau_{\textsc{b}}(a)$; likewise for
$\tau_{\textsc{a}|bc}=\tau_{\textsc{a}}(bc)$,
$\tau_{\textsc{ab}|c}=\tau_{\textsc{ab}}(c)$, etc. from the $2^{3}$ case.

###  5.3 Definition of the general factorial effects

We next define the general factorial effects as weighted averages of their
respective conditional counterparts. Consider $\pi(z)={\rm
pr}(Z_{i1}=z_{1},\ldots,Z_{iK}=z_{K})$ for
$z=(z_{1},\ldots,z_{K})\in\mathcal{T}$ as the treatment probabilities under
some target thought experiment. The marginal distribution of
$Z_{i,{\scaleto{\mathcal{K}}{4.5pt}}}=(Z_{ik})_{{\scaleto{k\in{{\mathcal{K}}}}{6pt}}}$
equals
$\pi_{\scaleto{\mathcal{K}}{4.5pt}}=\\{\pi({z_{\scaleto{\mathcal{K}}{4.5pt}}}):{z_{\scaleto{\mathcal{K}}{4.5pt}}}\in{\mathcal{F}}_{\scaleto{\mathcal{K}}{4.5pt}}\\}$
with $\pi({z_{\scaleto{\mathcal{K}}{4.5pt}}})={\rm
pr}(Z_{i,{\scaleto{\mathcal{K}}{4.5pt}}}=z_{\scaleto{\mathcal{K}}{4.5pt}})=\sum_{{z_{\scaleto{\overline{{{\mathcal{K}}}}}{5pt}}}\in{\mathcal{F}}_{\scaleto{\overline{{{\mathcal{K}}}}}{5pt}}}\pi({z_{\scaleto{\mathcal{K}}{4.5pt}}},z_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}})$. It induces an
intuitive weighting scheme for averaging over factors in ${{\mathcal{K}}}$
when defining the general factorial effect of factors in
${\overline{{\mathcal{K}}}}$. The
$\pi_{\textsc{a}}=(\pi_{\textsc{a}=0},\pi_{\textsc{a}=1})$ and
$\pi_{\textsc{a}\textsc{b}}=(\pi_{ab})_{a,b=0,1}$ from the $2^{2}$ and $2^{3}$
experiments are both special cases of $\pi_{\scaleto{\mathcal{K}}{4.5pt}}$
with ${{\mathcal{K}}}=\\{\text{A}\\}$ and $\\{\text{A},\text{B}\\}$,
respectively. Building on the intuition from Definition 2, we call
$\pi=\\{\pi_{\scaleto{\mathcal{K}}{4.5pt}}:{{\mathcal{K}}}\in\mathcal{P}_{K}\\}$
the coherent weighting scheme induced by the joint distribution
$\\{\pi(z):z\in\mathcal{T}\\}$.

######  Definition 5.

Given a coherent weighting scheme $\pi$ and conditional factorial effects
$\tau_{{\scaleto{\mathcal{K}}{4.5pt}}}(z_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}})$ from Definition 4 for
all $\mathcal{K}\in\mathcal{P}_{K}$ and $z_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}}\in{\mathcal{F}}_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}}$, define
$\tau_{{\scaleto{\mathcal{K}}{4.5pt}},\pi}=\sum_{{z_{\scaleto{\overline{{{\mathcal{K}}}}}{5pt}}}\in{\mathcal{F}}_{\scaleto{\overline{{{\mathcal{K}}}}}{5pt}}}\pi(z_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}})\cdot\tau_{{\scaleto{\mathcal{K}}{4.5pt}}}(z_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}})$ as the general
factorial effect of factors in ${{\mathcal{K}}}$ under $\pi$, vectorized as
$\tau_{\pi}=\\{\tau_{{\scaleto{\mathcal{K}}{4.5pt}},\pi}:{{\mathcal{K}}}\in\mathcal{P}_{K}\\}=G_{\pi}{\bar{Y}}$.

Definitions 4 and 5 together define the $(2^{K}-1)$ general factorial effects
under the coherent weighting scheme $\pi$. Refer to
$\tau_{{\scaleto{\mathcal{K}}{4.5pt}},\pi}$ as the standard effect if
$\pi(z_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}})=|{\mathcal{F}}_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}}|^{-1}=2^{-|{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}}|}$ are identical for all
$z_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}}\in{\mathcal{F}}_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}}$. Refer to
$\tau_{{\scaleto{\mathcal{K}}{4.5pt}},\pi}$ as the empirical effect if
$\pi(z_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}})=N^{-1}\sum_{i=1}^{N}\mathcal{I}(Z_{i,{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}}}=z_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}})$ equals the empirical
proportion in the actual experiment.

###  5.4 Factor-based regression with the saturated model

Motivated by (1) for the $2^{2}$ experiment and (4) for the $2^{3}$
experiment, we define $Z^{\prime}_{ik}=Z_{ik}-\delta_{k}$ and
$Z^{\prime}_{i,{\scaleto{\mathcal{K}}{4.5pt}}}=\prod_{{\scaleto{k\in{{\mathcal{K}}}}{6pt}}}Z^{\prime}_{ik}$
as a location-shifted generalization for some prespecified
$(\delta_{k})_{k=1}^{K}$ with $0\leq\delta_{k}\leq 1$, and consider the
saturated factor-based regression

$\displaystyle Y_{i}\sim 1+\sum_{k=1}^{K}Z^{\prime}_{ik}+\sum_{1\leq k\neq
k^{\prime}\leq
K}Z^{\prime}_{ik}Z^{\prime}_{ik^{\prime}}+\dots+\prod_{k=1}^{K}Z^{\prime}_{ik}\sim
1+\sum_{{{\mathcal{K}}}\in\mathcal{P}_{K}}Z^{\prime}_{i,{\scaleto{\mathcal{K}}{4.5pt}}}.$
(7)

Let $\hat{\gamma}$ and $\hat{\Psi}$ be the coefficient vector and robust
covariance of the $(Q-1)$ non-intercept terms in (7), respectively, with
elements arranged in the same order of ${{\mathcal{K}}}$’s as those in
$\tau_{\pi}$. We derive below their utility for the Wald-type inference of
$\tau_{\pi}$.

To begin with, the notion of product weighting scheme extends naturally to the
current setting as $\pi(z)=\prod_{k=1}^{K}\pi(z_{k}),$ and is fully determined
by the values of $\\{\pi(1_{k})\\}_{k=1}^{K}$. The equal weighting scheme for
the standard effects satisfies the product structure with $\pi(1_{k})=1/2$.
The empirical weighting scheme, on the other hand, may not. Building on the
intuition from the $2^{3}$ experiment, Definition 6 below introduces two
product weighting schemes of particular importance, arising from the estimand
of interest and the location-shift parameters, respectively.

######  Definition 6.

For an arbitrary coherent weighting scheme $\pi$, let
$\pi_{\scaleto{\times}{4pt}}$ be the product weighting scheme with
$\pi_{\scaleto{\times}{4pt}}(1_{k})=\pi(1_{k})$ for $k=1,\ldots,K$.

For arbitrary location-shift parameters $(\delta_{k})_{k=1}^{K}$ with
$0\leq\delta_{k}\leq 1$, let $\delta_{\scaleto{\times}{4pt}}$ be the product
weighting scheme with $\delta_{\scaleto{\times}{4pt}}(1_{k})=\delta_{k}$ for
$k=1,\ldots,K$.

The product weighting scheme $\pi_{\scaleto{\times}{4pt}}$ satisfies
$\pi_{\scaleto{\times}{4pt}}=\pi$ if $\pi$ is already a product weighting
scheme. The product weighting scheme $\delta_{\scaleto{\times}{4pt}}$ features
$\delta_{\scaleto{\times}{4pt}}(z)=\prod_{k=1}^{K}\delta_{k}^{z_{k}}(1-\delta_{k})^{1-z_{k}}$
for all $z\in\mathcal{T}$. Let
$\tau_{\delta{\scaleto{\times}{4pt}}}=G_{\delta{\scaleto{\times}{4pt}}}{\bar{Y}}$
be the corresponding vector of general factorial effects,
$\hat{\tau}_{\delta{\scaleto{\times}{4pt}}}=G_{\delta{\scaleto{\times}{4pt}}}\hat{Y}$
be its moment estimator, and
$\hat{\textup{cov}}(\hat{\tau}_{{\delta{\scaleto{\times}{4pt}}}})=G_{\delta{\scaleto{\times}{4pt}}}\hat{V}G^{\mathrm{\scriptscriptstyle
T}}_{\delta{\scaleto{\times}{4pt}}}$ be the estimated covariance,
respectively. Theorem 1 below gives the numeric correspondence between
$\\{\hat{\gamma},\hat{\Psi}\\}$ and
$\\{\hat{\tau}_{\delta{\scaleto{\times}{4pt}}},\hat{\textup{cov}}(\hat{\tau}_{{\delta{\scaleto{\times}{4pt}}}})\\}$
for inferring $\tau_{\delta{\scaleto{\times}{4pt}}}$.

######  Theorem 1.

Under the $2^{K}$ experiment, the outputs of (7) satisfy
$\hat{\gamma}=\hat{\tau}_{\delta{\scaleto{\times}{4pt}}}$ and
$\hat{\Psi}=\hat{\textup{cov}}(\hat{\tau}_{{\delta{\scaleto{\times}{4pt}}}})-G_{\delta{\scaleto{\times}{4pt}}}\operatorname{diag}(N_{z}^{-1})\hat{V}G_{\delta{\scaleto{\times}{4pt}}}^{\mathrm{\scriptscriptstyle
T}}$.

Theorem 1 unifies the results from the $2^{2}$ and $2^{3}$ experiments and
justifies the utility of $\hat{\gamma}$ and $\hat{\Psi}$ from (7) for
inferring $\tau_{\pi}$ when $\pi$ is a product weighting scheme with
$\pi(1_{k})=\delta_{k}$ for $k=1,\ldots,K$. Despite the constrained
applicability in general, the intuition from Proposition 5 extends here and
ensures the generality of (7) in the absence of three-way interactions.

######  Condition 3.

Assume $\tau_{{\scaleto{\mathcal{K}}{4.5pt}}}(z_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}})=0$ for all $z_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}}$ with
$|{{\mathcal{K}}}|=3$.

Condition 3 rules out the existence of three-way interactions and thus that of
all $m$-way interactions for $3<m\leq K$ by Definition 4. We will refer to it
as the no three-way interactions condition hence for simplicity.

######  Theorem 2.

Under the $2^{K}$ experiment and Condition 3, we have
$\tau_{\pi}=\tau_{\pi{\scaleto{\times}{4pt}}}$ for all coherent $\pi$, where
$\tau_{\pi}$ and $\tau_{\pi{\scaleto{\times}{4pt}}}$ are the vectors of
general factorial effects under $\pi$ and $\pi_{\scaleto{\times}{4pt}}$,
respectively.

Theorems 1 and 2 together allow us to use (7) with $\delta_{k}=\pi(1_{k})$ for
the Wald-type inference of all $\tau_{\pi}$ with coherent $\pi$ in the absence
of three-way interactions. The proof of Theorem 2 further shows that the
requirement of $\tau_{{\scaleto{\mathcal{K}}{4.5pt}}}(z_{\kern
0.35002pt\overline{\scaleto{{{\mathcal{K}}}}{4.5pt}}})=0$ for all
$|{{\mathcal{K}}}|=3$ is not only sufficient but also necessary for
$\tau_{\pi}=\tau_{\pi{\scaleto{\times}{4pt}}}$ to hold if $\pi$ is coherent
but not a product weighting scheme. Thus, we cannot relax the
$|{{\mathcal{K}}}|=3$ in Condition 3 to $|{{\mathcal{K}}}|=m$ for some $m>3$
for Theorem 2 to hold.

###  5.5 Factor-based regression with unsaturated models

Motivated by (2) for the $2^{2}$ experiment and (5) for the $2^{3}$
experiment, we next consider

$\displaystyle Y_{i}\sim
1+\sum_{{{\mathcal{K}}}\in{{\mathcal{F}}_{\scaleto{+}{4pt}}}}Z^{\prime}_{i,{\scaleto{\mathcal{K}}{4.5pt}}},$
(8)

where ${{\mathcal{F}}_{\scaleto{+}{4pt}}}\subset\mathcal{P}_{K}$, as an
unsaturated variant of (7) when only a subset of the $(Q-1)$ factorial effects
are of interest, vectorized as
$\tau_{\pi,{\scaleto{+}{4pt}}}=\\{\tau_{{\scaleto{\mathcal{K}}{4.5pt}},\pi}:{{\mathcal{K}}}\in{\mathcal{F}}_{\scaleto{+}{4pt}}\\}$.
A commonly-used special case is $Y_{i}\sim
1+Z_{i1}^{\prime}+\dots+Z_{iK}^{\prime}$ with only the first-order terms and
${{\mathcal{F}}_{\scaleto{+}{4pt}}}=\\{\\{k\\}:k=1,\ldots,K\\}$. The additive
form ensures that the location-shift transformation has no effect on the
estimation of the non-intercept coefficients. Another commonly-used special
case is $Y_{i}\sim 1+\sum_{k=1}^{K}Z^{\prime}_{ik}+\sum_{1\leq k\neq
k^{\prime}\leq K}Z^{\prime}_{ik}Z^{\prime}_{ik^{\prime}}$ with only the main
effects and two-way interactions and
${{\mathcal{F}}_{\scaleto{+}{4pt}}}=\\{\\{k\\},\\{k,k^{\prime}\\}:k,k^{\prime}=1,\ldots,K\text{
with }\ k\neq k^{\prime}\\}$.

Let $\tilde{\gamma}_{{\scaleto{+}{4pt}}}$ and
$\tilde{\Psi}_{{\scaleto{+}{4pt}}}$ be the coefficient vector and robust
covariance of the $|{\mathcal{F}}_{\scaleto{+}{4pt}}|$ non-intercept terms in
(8), respectively. We establish in this subsection their utility for inferring
$\tau_{\pi,{\scaleto{+}{4pt}}}$. Recall $\hat{\gamma}$ as the coefficient
vector of the non-intercept terms from (7). Partition it into
$\hat{\gamma}_{{\scaleto{+}{4pt}}}$ and $\hat{\gamma}_{{\scaleto{-}{4pt}}}$,
corresponding to the coefficients of
$(Z^{\prime}_{i,{\scaleto{\mathcal{K}}{4.5pt}}})_{{{\mathcal{K}}}\in{{\mathcal{F}}_{\scaleto{+}{4pt}}}}$
and
$(Z^{\prime}_{i,{\scaleto{\mathcal{K}}{4.5pt}}})_{{{\mathcal{K}}}\not\in{{\mathcal{F}}_{\scaleto{+}{4pt}}}}$,
respectively. As a convention, we use “$+$” and “$-$” in the subscripts to
signify effects included in and omitted from the unsaturated regression (8),
respectively.

Let $F$ be the $N\times Q$ design matrix of (7), concatenating columns of
$1_{N}$ and $(Z^{\prime}_{i,{\scaleto{\mathcal{K}}{4.5pt}}})_{i=1}^{N}$ for
all ${{\mathcal{K}}}\in\mathcal{P}_{K}$. Let $F_{\scaleto{+}{4pt}}$ be the
$N\times(1+|{{\mathcal{F}}_{\scaleto{+}{4pt}}}|)$ design matrix of (8), and
$F_{\scaleto{-}{4pt}}=F\backslash F_{\scaleto{+}{4pt}}$ be the submatrix of
$F$ omitted from (8), concatenating columns of
$(Z^{\prime}_{i,{\scaleto{\mathcal{K}}{4.5pt}}})_{i=1}^{N}$ for
${{\mathcal{K}}}\not\in{{\mathcal{F}}_{\scaleto{+}{4pt}}}$. Assume throughout
that the elements in $\tilde{\gamma}_{{\scaleto{+}{4pt}}}$,
$\hat{\gamma}_{{\scaleto{+}{4pt}}}$, and $\hat{\gamma}_{{\scaleto{-}{4pt}}}$
are arranged in the same relative order of ${{\mathcal{K}}}$ as those in
$\tau_{\pi}$; likewise for the columns in $F_{\scaleto{+}{4pt}}$ and
$F_{\scaleto{-}{4pt}}$.

Let $\Phi={(F_{\scaleto{+}{4pt}}^{\mathrm{\scriptscriptstyle
T}}F_{\scaleto{+}{4pt}})^{-1}F_{\scaleto{+}{4pt}}^{\mathrm{\scriptscriptstyle
T}}F_{\scaleto{-}{4pt}}}$ be the coefficient matrix from the column-wise
regression of $F_{\scaleto{-}{4pt}}$ on $F_{\scaleto{+}{4pt}}$, which is a
deterministic function of $(e_{z})_{z\in\mathcal{T}}$ by Lemma B4 in the
Supplementary Material. Let $R=F_{\scaleto{-}{4pt}}-F_{\scaleto{+}{4pt}}\Phi$
be the corresponding residual matrix, $D$ be the submatrix of $\Phi$ without
the first row, and ${F_{{\scaleto{+}{4pt}}[,-1]}}$ be the submatrix of
$F_{\scaleto{+}{4pt}}$ without the first column. Let $P_{N}$ be the projection
matrix orthogonal to $1_{N}$, and
$Y=(Y_{1},\ldots,Y_{N})^{\mathrm{\scriptscriptstyle T}}$ be the vector of the
observed outcomes. Theorem 3 below states the numeric correspondence between
$\tilde{\gamma}_{\scaleto{+}{4pt}}$ and $\hat{\gamma}$ under the $2^{K}$
factorial experiment, generalizing Propositions 3 and 6.

######  Theorem 3.

Under the $2^{K}$ experiment, the coefficients from (7) and (8) satisfy
$\tilde{\gamma}_{{\scaleto{+}{4pt}}}=\hat{\gamma}_{{\scaleto{+}{4pt}}}+D\hat{\gamma}_{{\scaleto{-}{4pt}}}$,
where $D\hat{\gamma}_{{\scaleto{-}{4pt}}}=0$ if and only if
${F_{{\scaleto{+}{4pt}}[,-1]}^{\mathrm{\scriptscriptstyle
T}}}P_{N}F_{\scaleto{-}{4pt}}(R^{\mathrm{\scriptscriptstyle
T}}R)^{-1}R^{\mathrm{\scriptscriptstyle T}}Y=0$. Two sufficient conditions for
$D\hat{\gamma}_{{\scaleto{-}{4pt}}}=0$ for all $Y$ are
$F_{\scaleto{+}{4pt}}^{\mathrm{\scriptscriptstyle T}}F_{\scaleto{-}{4pt}}=0$
or $F_{\scaleto{+}{4pt}}^{\mathrm{\scriptscriptstyle
T}}P_{N}F_{\scaleto{-}{4pt}}=0$.

Recall that $\hat{\gamma}_{{\scaleto{+}{4pt}}}$ and
$\hat{\gamma}_{\scaleto{-}{4pt}}$ coincide with the moment estimators of
$\tau_{\delta{\scaleto{\times}{4pt}},{\scaleto{+}{4pt}}}=\\{\tau_{{\scaleto{\mathcal{K}}{4.5pt}},{\delta{\scaleto{\times}{4pt}}}}:{{\mathcal{K}}}\in{\mathcal{F}}_{\scaleto{+}{4pt}}\\}$
and
$\tau_{{\delta{\scaleto{\times}{4pt}}},{\scaleto{-}{4pt}}}=\\{\tau_{{\scaleto{\mathcal{K}}{4.5pt}},{\delta{\scaleto{\times}{4pt}}}}:{{\mathcal{K}}}\not\in{{\mathcal{F}}_{\scaleto{+}{4pt}}}\\}$,
respectively, denoted by
$\hat{\tau}_{\delta{\scaleto{\times}{4pt}},{\scaleto{+}{4pt}}}$ and
$\hat{\tau}_{{\delta{\scaleto{\times}{4pt}}},{\scaleto{-}{4pt}}}$. Theorem 3
gives two sufficient conditions for $\tilde{\gamma}_{{\scaleto{+}{4pt}}}$ to
recover exactly
$\hat{\tau}_{\delta{\scaleto{\times}{4pt}},{\scaleto{+}{4pt}}}$, requiring
orthogonality of $F_{\scaleto{+}{4pt}}$ and $F_{\scaleto{-}{4pt}}$ either in
original form or after centered by the column averages. These conditions do
not hold in general unless the design is balanced and the factorial effects
are the standard ones under the equal weighting scheme. This generalizes the
intuition from the $2^{2}$ and $2^{3}$ cases to general $K$.

######  Corollary 1.

Under the $2^{K}$ experiment,
$\tilde{\gamma}_{{\scaleto{+}{4pt}}}=\hat{\gamma}_{{\scaleto{+}{4pt}}}$ if (i)
$\delta_{k}=1/2$ for all $k=1,\ldots,K$ and (ii) $N_{z}=N/Q$ for all
$z\in\mathcal{T}$.

We can drop the balance condition (ii) in Corollary 1 if we use the weighted
least squares fit with weights $1/N_{Z_{i}}$ for $i=1,\ldots,N$. We relegate
the details to §A.5 in the Supplementary Material and focus on the ordinary
least squares fit in the main text.

Despite the loss of exact recovery of the moment estimator
$\hat{\tau}_{\delta{\scaleto{\times}{4pt}},{\scaleto{+}{4pt}}}$ when
$D\hat{\gamma}_{\scaleto{-}{4pt}}\neq 0$, the intuition from the $2^{2}$ and
$2^{3}$ experiments extends here and ensures the unbiasedness of
$\tilde{\gamma}_{{\scaleto{+}{4pt}}}$ in the absence of the nuisance effects.

######  Condition 4.

The nuisance effects are zero, that is,
$\tau_{{\scaleto{\mathcal{K}}{4.5pt}},\pi}=0$ for all
${{\mathcal{K}}}\not\in{{\mathcal{F}}_{\scaleto{+}{4pt}}}$.

######  Theorem 4.

Under the completely randomized $2^{K}$ experiment, the coefficients from (8)
satisfy
$E(\tilde{\gamma}_{{\scaleto{+}{4pt}}})={\tau_{\delta{\scaleto{\times}{4pt}},{\scaleto{+}{4pt}}}}+D{\tau_{\delta{\scaleto{\times}{4pt}},{\scaleto{-}{4pt}}}}$
and
$\textup{cov}(\tilde{\gamma}_{{\scaleto{+}{4pt}}})=(I,D)G_{\delta{\scaleto{\times}{4pt}}}\textup{cov}(\hat{Y})G_{\delta{\scaleto{\times}{4pt}}}^{\mathrm{\scriptscriptstyle
T}}(I,D)^{\mathrm{\scriptscriptstyle T}}$. Further assume Condition 1 and
Condition 4 with $\pi=\delta_{\scaleto{\times}{4pt}}$. Then
$E(\tilde{\gamma}_{{\scaleto{+}{4pt}}})={\tau_{\delta{\scaleto{\times}{4pt}},{\scaleto{+}{4pt}}}}$,
and $\tilde{\gamma}_{{\scaleto{+}{4pt}}}$ is asymptotically Normal with
$N\\{\tilde{\Psi}_{{\scaleto{+}{4pt}}}-\textup{cov}(\tilde{\gamma}_{{\scaleto{+}{4pt}}})\\}=\Delta+o_{\text{p}}(1)$,
where
$\Delta=(I,D)G_{\delta{\scaleto{\times}{4pt}}}SG_{\delta{\scaleto{\times}{4pt}}}^{\mathrm{\scriptscriptstyle
T}}(I,D)^{\mathrm{\scriptscriptstyle T}}\geq 0$.

Theorem 4 justifies the Wald-type inference of
$\tau_{\delta{\scaleto{\times}{4pt}},{\scaleto{+}{4pt}}}$ from the unsaturated
specification (8) when the nuisance effects omitted indeed do not exist. The
resulting $\tilde{\gamma}_{{\scaleto{+}{4pt}}}$ is both unbiased and
consistent for estimating
$\tau_{\delta{\scaleto{\times}{4pt}},{\scaleto{+}{4pt}}}$, with the robust
covariance $\tilde{\Psi}_{{\scaleto{+}{4pt}}}$ affording an asymptotically
conservative estimator for the true sampling covariance. The proof of Theorem
1 further shows that the intercept from (8) is an unbiased estimator of a
weighted average of $\bar{Y}(z)$ instead of a contrast and is thus non-zero in
general. This suggests the necessity to include the intercept in the
unsaturated specification for the satisfaction of Condition 4. One limitation
of (8), again, lies in its requirement on the product weighting scheme.
Juxtaposing Condition 3, Condition 4, and Theorem 2 together ensures that the
result of Theorem 4 extends to $\tau_{\pi,{\scaleto{+}{4pt}}}$ for all
coherent $\pi$ in the absence of three-way interactions.

######  Remark 2.

The constant treatment effects condition further ensures
$\textup{cov}(\tilde{\gamma}_{{\scaleto{+}{4pt}}})\leq\textup{cov}(\hat{\gamma}_{{\scaleto{+}{4pt}}})$
such that the estimator from (8) has smaller sampling covariance compared with
that from (7). This, together with Theorem 4, illustrates the bias-variance
trade-off between the saturated and unsaturated regressions from the design-
based perspective. This result, however, does not hold without the constant
treatment effects assumption. We give a counterexample in §A.4 in the
Supplementary Material.

The assumption of no nuisance effects can never be verified exactly in
practice. Extra caution is thus needed when applying unsaturated
specifications to unbalanced designs or estimands other than the standard
effects. The saturated specification is, in this sense, a safer choice when
the sample size permits. When the number of treatment combinations $Q=2^{K}$
is large relative to the sample size $N$, however, the saturated regression is
subject to substantial finite-sample variability, and leaves the unsaturated
regressions possibly more attractive alternatives for finite-sample inference.
Even if the nuisance effects are not exactly zero, depending on our belief of
the data generating process, the gain in finite-sample precision by the
unsaturated regressions can still outweigh the bias as long as the omitted
nuisance effects, most likely some higher-order interactions, are reasonably
small, ensuring a smaller mean squared error overall.

Alternatively, lasso and ridge regression afford attractive alternatives when
the saturated regression is not possible. Indeed, discussion so far holds with
a given unsaturated specification (8). It is desirable to have a data-driven
specification with both model selection and post-selection inference (Chipman
et al. 1997; Espinosa et al. 2016; Egami and Imai 2019). Although these topics
have been discussed extensively under the classic linear model, analogous
results are largely unexplored under the design-based framework. We leave this
to future work.

##  Discussion and recommendations

We wrap up this article with three practical implications of our findings in
terms of the $2^{K}$ experiment. The intuition extends to the general
$Q_{1}\times\cdots\times Q_{K}$ experiment with minimal modification as shown
in §A of the Supplementary Material. First, the definition of the general
factorial effects greatly broadens the range of estimands that could be
considered under factorial experiments, enabling flexible weighting schemes to
accommodate context-specific concerns. Second, location-shifted factor-based
regression affords a convenient way to recover the moment estimators of the
general factorial effects from least squares, with the corresponding robust
covariance being an asymptotically conservative estimator of the true sampling
covariance. This enables large-sample Wald-type inference from least-squares
outputs. With more than two factors, factor-based regression is capable of
estimating general factorial effects under product weighting schemes, yet
regains generality in the absence of three-way interactions. Third,
unsaturated regressions reduce sampling variances under the constant treatment
effects assumption, but are subject to non-diminishing biases when the no
nuisance effects condition is violated. Importantly, our theory is design-
based without requiring any stochastic models for the potential outcomes.

We focused on complete randomization due to its own wide applications.
Clarifying the above important issues in this basic experiment affords a proof
of concept for other more complex experiments. The definitions of the general
factorial effects remain unchanged, and the correspondence between the least-
squares outputs and moment estimators is purely numeric and thus holds under
any randomization mechanism. The appropriateness of the Wald-type inference,
on the other hand, is assignment specific and requires modifications under
different randomization mechanisms. We conjecture that the theory extends to
experiments with non-constant treatment probabilities (Mukerjee et al. 2018)
if we weight the least-squares procedure by the inverse of the treatment
probability. We leave this to future work.

## Supplementary material

The Supplementary Material contains the results for the general
$Q_{1}\times\cdots\times Q_{K}$ factorial experiment, details omitted in the
main text, and numerical examples.

## Acknowledgment

We thank the three reviewers, Avi Feller, Cheng Gao, and Nicole Pashley for
constructive comments. Peng Ding was partially supported by the U.S. National
Science Foundation.

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