A uniﬁed analysis of regression adjustment in randomized experiments Katarzyna RelugaTing Yeand Qingyuan Zhao

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A uniﬁed analysis of regression adjustment in

randomized experiments

Katarzyna Reluga,∗Ting Ye†and Qingyuan Zhao‡

Abstract

Regression adjustment is broadly applied in randomized trials under the premise

that it usually improves the precision of a treatment eﬀect estimator. However, pre-

vious work has shown that this is not always true. To further understand this phe-

nomenon, we develop a uniﬁed comparison of the asymptotic variance of a class of

linear regression-adjusted estimators. Our analysis is based on the classical theory

for linear regression with heteroscedastic errors and thus does not assume that the

postulated linear model is correct. For a completely randomized binary treatment,

we provide suﬃcient conditions under which some regression-adjusted estimators are

guaranteed to be more asymptotically eﬃcient than others. We explore other settings

such as general treatment assignment mechanisms and generalized linear models, and

ﬁnd that the variance dominance phenomenon no longer occurs.

Keywords: Average treatment eﬀect; Randomized controlled trials; Covariate adjustment;

Heteroscedasticity.

∗Division of Biostatistics, School of Public Health, University of California, Berkeley, U.S.A. E-mail:

katarzyna.reluga@berkeley.edu.

†Department of Biostatistics, University of Washington, 3980 15th Avenue NE, Box 351617, Seattle,

WA 98195, U.S.A. E-mail: tingye1@uw.edu.

‡Department of Pure Mathematics and Mathematical Statistics, University of Cambridge,

Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, U.K. E-mail:

qyzhao@statslab.cam.ac.uk.

The authors gratefully acknowledge support from the Swiss National Science Foundation for the project

P2GEP2-195898.

arXiv:2210.04360v1 [stat.ME] 9 Oct 2022

1 Introduction

Randomized experiments are the gold standard to answer questions about causality. Many

researchers use multiple linear regression with a treatment indicator and some baseline

covariates to analyze randomized experiments, in which the treatment coeﬃcient is often

interpreted as a causal eﬀect. In some ﬁelds, this is known as the “analysis of covariance”

(ANCOVA), which was ﬁrst proposed by Fisher (1932) to unify “two very widely appli-

cable procedures known as regression and analysis of variance”. This common practice is

motivated by the belief that regression adjustments can increase precision if covariates in

the regression are predictive of the outcome.

However, as pointed out by many authors, this is not always true especially when there

is a lot of treatment eﬀect heterogeneity. Regression adjustment in randomized experi-

ments has been studied in two diﬀerent frameworks, namely the ﬁnite-population potential

outcome model (Neyman, 1923; Rubin, 1974) and the super-population model that as-

sumes the experimental units are drawn independently from an inﬁnite population (see e.g.

Imbens and Rubin, 2015, Chapter 7). Three estimators have been extensively studied in

the literature: the simple diﬀerence-in-means or analysis of variance (ANOVA) estimator;

the ANCOVA estimator that includes covariate main eﬀects; and the regression-adjusted

estimator that includes covariate main eﬀects and all treatment-covariate interactions. The

last one is termed as the analysis of heterogeneous covariance (ANHECOVA) estimator by

Ye et al. (2022). The main conclusions about the asymptotic eﬃciency of these estimators

are the same, regardless of whether the potential outcome model (Freedman, 2008a,b; Scho-

chet, 2010; Lin, 2013; Guo and Basse, 2021) or super-population model (Koch et al., 1998;

Yang and Tsiatis, 2001; Tsiatis et al., 2008; Schochet, 2010; Rubin and van der Laan, 2011;

Ye et al., 2022) is used. Consider two estimators ˆ

β1and ˆ

β2that converge to the same limit.

We say that ˆ

β1(asymptotically) uniformly dominates ˆ

β2if the (asymptotic) variance of ˆ

β1

is always smaller or equal than that of ˆ

β2, no matter what the underlying distribution is. In

both the potential outcome model and the super-population model, it has been found that

ANHECOVA uniformly dominates the other two, but, somewhat surprisingly, ANCOVA

does not uniformly dominate ANOVA.

A major limitation of the existing analysis of regression adjustment is that the investiga-

tions are restricted to speciﬁc estimators and provides limited insights into the phenomenon

of uniform dominance. The variance calculations are often quite technical, which further

make the theoretical results less accessible to practitioners. Furthermore, the existing lit-

erature does not tell us whether including all treatment-covariate interactions is preferred

in other cases such as stratiﬁed experiments and generalized linear models.

In this article, we provide a uniﬁed analysis for a large class of linear-regression adjusted

estimators. Besides the estimators mentioned above, our theory also applies to regression

estimators with some coeﬃcients ﬁxed (such as the diﬀerence-in-diﬀerences estimator) or

with treatment-covariate interactions only. By a simple application of the textbook theory

for linear regression with heteroscedastic errors, this analysis not only recovers the known

relationships between ANOVA, ANCOVA, and ANHECOVA, but also immediately pro-

vides a suﬃcient condition for uniform dominance when the expectation of the covariates

is known (see Theorem 1 below). In the more practical situation when the covariate ex-

pectation is unknown, a slightly diﬀerent suﬃcient condition is obtained (see Theorem 2

below). This condition shows that, for example, the so-called lagged-dependent-variable

regression estimator is more eﬃcient than the diﬀerence-in-diﬀerences estimator in random-

ized experiments, despite them having a bracketing relationship in observational studies

(Ding and Li, 2019). This uniﬁed analysis allows us to explore whether the uniform domi-

nance extends to more complicated settings and provide numerical counterexamples. Some

further remarks are provided at the end of this article, whereas proofs of the technical

Lemmas can be found in Appendix A.

2 Linear regression adjustment in randomized trials

Consider a random sample {(Ai, Xi, Yi)}n

i=1 of nunits, where Ai∈ {0,1}is a binary

treatment indicator, Xi= (Xi1, Xi2, . . . , Xip)T∈Rpis a vector of unit covariates observed

before treatment assignment, and Yi∈Ris a real-valued outcome of the unit. We assume

that (Ai, XT

i, Yi), i = 1, . . . , n is independent and identically distributed, which is often a

good approximation when the units are randomly sampled from a large population. To

simplify the notation, we drop the subscript iwhen referring to a generic unit from the

population.

Unless mentioned otherwise, we assume that each unit receives the treatment indepen-

dently with equal probability pr(A= 1 |X) = π, where 0< π < 1is a known constant. In

other words, treatment is assigned by a simple Bernoulli trial, which approximates random

sampling without replacement that is often studied in the ﬁnite-population model (Freed-

man, 2008a,b; Lin, 2013). Under this assignment mechanism and standard assumptions in

causal inference, the average treatment eﬀect βATE =E[Y(1) −Y(0)], where Y(a)is the

potential outcome of unit iunder treatment level a, can be identiﬁed as (see e.g. Imbens

and Rubin, 2015, Chapter 7):

βATE =E(Y|A= 1) −E(Y|A= 0).(1)

In this article, we consider the following class of regression adjusted estimators of βATE.

Let Γ=Γ(1) × ··· × Γ(p)⊆Rpand ∆=∆(1) × ··· × ∆(p)⊆Rpbe two user-speciﬁed sets,

where the individual components Γ(j)and ∆(j)are either the real line Ror a singleton.

Deﬁne the constrained ordinary least squares estimator as

θ= (ˆα, ˆ

β, ˆγ, ˆ

δ) = arg min

γ∈Γ,δ∈∆

i=1 {Yi−α−βAi−γTXi−Ai(δTXi)}2.(2)

We sometimes use the notation ˆ

θ(Γ,∆) (and similarly for the components of ˆ

θ) to emphasize

the dependence of the estimator on the sets Γand ∆. Lemma 1 in Section 3.1 shows that ˆ

is a reasonable estimator of βATE when the covariates are centered, i.e. E(X) = 0; otherwise

βATE can be estimated by ˜

β=ˆ

β+ˆ

δT¯

X, where ¯

X=Pn

i=1 Xi/n. Before examining the

asymptotic properties of ˆ

βand ˜

β, we give several examples in the class of estimators (2).

Example 1. The ANOVA, ANCOVA, ANHECOVA estimators correspond to setting Γ =

∆ = {0};Γ = Rpand ∆ = {0};Γ = Rpand ∆ = Rp.

Example 2. In some applications, the covariate vector Xinclude the baseline value of

the response before the treatment is assigned (let us call it Y0). For simplicity, suppose

the ﬁrst entry of Xis Y0, so X= (X1=Y0, X2, . . . , Xp)T. The diﬀerence-in-diﬀerences

estimator corresponds to setting Γ = {1} × Rp−1and ∆ = {0} × Rp−1, while the lagged-

dependent-variable regression estimator corresponds to setting Γ⊆Rpand ∆ = {0}×Rp−1.

In observational studies, these two estimators rely on diﬀerent identiﬁcation assumptions

(Ding and Li, 2019) and may converge to diﬀerent limits. In the randomized experiment

described above, both estimators should converge to the average treatment eﬀect, but we are

unaware of any comparison of their statistical eﬃciency in presence of covariates besides Y0.

3 A uniﬁed analysis of linear regression-adjusted esti-

mators

3.1 Covariates with known expectation

We ﬁrst consider estimation of βATE when the covariates Xhave known expectation. As

will be seen in a moment, the proof of uniform dominance is fairly straightforward in this

case.

Consider the population counterpart to (2):

θ= (α, β, γ, δ) = arg min

γ∈Γ,δ∈∆

E{Y−α−βA −γTX−A(δTX)}2.(3)

Clearly, θ=θ(Γ,∆), and we often suppress the dependence of θon (Γ,∆) if it is clear from

the context.

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AuniedanalysisofregressionadjustmentinrandomizedexperimentsKatarzynaReluga,*TingYeandQingyuanZhaoAbstractRegressionadjustmentisbroadlyappliedinrandomizedtrialsunderthepremisethatitusuallyimprovestheprecisionofatreatmenteectestimator.However,pre-viousworkhasshownthatthisisnotalwaystrue.Tofurtheru...

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A uniﬁed analysis of regression adjustment in randomized experiments Katarzyna RelugaTing Yeand Qingyuan Zhao

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