Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Authors: Ahmed BOUGHDIRI
Julie Josse
Erwan Scornet
Venue: ICML 2025 poster (2025)
Keywords: Causal inference
Observational data
Risk Ratio
Primary area: general_machine_learning->causality

Abstract: The Risk Difference (RD), an absolute measure of effect, is widely used and well-studied in both randomized controlled trials (RCTs) and observational studies. Complementary to the RD, the Risk Ratio (RR), as a relative measure, is critical for a comprehensive understanding of intervention effects: RD can downplay small absolute changes, while RR can highlight them. Despite its significance, the theoretical study of RR has received less attention, particularly in observational settings. This paper addresses this gap by tackling the estimation of RR in observational data. We propose several RR estimators and establish their theoretical properties, including asymptotic normality and confidence intervals. Through analyses on simulated and real-world datasets, we evaluate the performance of these estimators in terms of bias, efficiency, and robustness to generative data models. We also examine the coverage and length of the associated confidence intervals. Due to the non-linear nature of RR, influence function theory yields two distinct efficient estimators with different convergence assumptions. Based on theoretical and empirical insights, we recommend, among all estimators, one of the two doubly-robust estimators, which, intriguingly, challenges conventional expectations.

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Ahmed Boughdiri! Julie Josse'
Erwan Scornet
Abstract
from extraneous or undesirable factors. The medical guide- line CONSORT (Moher et al., 2010) recommends reporting the treatment effect with relative measures like the Risk Ratio (RR) along with absolute measures like the Risk Dif- ference (RD) to provide a more comprehensive understand- ing of the effect and its implications, as neither measure alone offers a complete picture. Indeed, selecting one mea- over another carries several implication. For instance, with 3% baseline mortality reduced to 1% by treatment, RD shows 2% drop, while RR shows controls have three times the risk: RD suggests a small effect; RR highlights a larger one. In addition, (Naylor et al., 1992) and (Forrow et al., 1992) demonstrated that physicians inclination to treat patients, based on their perception of therapeutic im- pact, is influenced by the scale utilized present clinical effects. Finally, the treatment effect may be heterogeneous in one scale, i.e. the treatment effect varies according to patient characteristics, but homogeneous in another scale (Rothman, 201 1), which significantly disrupts interpretation. (Colnet et al., 2024) discusses causal measure properties with a focus on generalization of the treatment effect from a trial to a target population.
The Risk Difference (RD), an absolute measure of effect, is widely used and well-studied in both randomized controlled trials (RCTs) and obser- vational studies. Complementary to the RD, the Risk Ratio (RR), as relative measure, is critical for comprehensive understanding of interven- tion effects: RD can downplay small absolute changes, while RR can highlight them. Despite its significance, the theoretical study of RR has re- ceived less attention, partieularly in observational settings. This paper addresses this gap by tack- ling the estimation of RR in observational data. We propose several RR estimators and establish their theoretical properties, including asymptotic normality and confidence intervals. Through anal- yses on simulated and real-world datasets, we evaluate the performance of these estimators in terms of bias, efficiency, and robustness to gener- ative data models. We also examine the coverage and length of the associated confidence intervals. Due to the non-linear nature of RR, influence function theory yields two distinct efficient es- timators with different convergence assumptions. Based theoretical and empirical insights, we recommend, among all estimators, one of the two doubly-robust estimators, which, intriguingly, challenges conventional expectations.
Consequently, both RD and RR measures are widely used in the analysis of clinical trial data as explained in Malenka et al. (1993). Sinclair & Bracken (1994) and Nakayama et al. (1998). The Risk Ratio is particularly relevant in scenar- ios where outcomes are always either positive or negative (Malenka al., 1993) and in cases where both expected potential outcomes being compared are small, as it is more stable and interpretable than the RD. Furthermore, in cases of rare treatment occurrences, the Risk Ratio closely approx- imates the Odds Ratio (OR), further enhancing its relevance and utility in clinical analyses (Schechtman, 2002). Bar- ratt et al. (2004) recommend using the RR from clinical trials with an estimation of the individual patient baseline to provide the right treatment.
1. Introduction
Treatment effect estimation in trials. Modern evidence- based medicine prioritizes Randomized Controlled Trials (RCTs) the cornerstone of clinical evidence. Random- ization in RCTs allows for the quantification of the average treatment effect (ATE) by removing confounding influences
'Equal contribution 'INRIA Sophia-Antipolis 'Sorbonne Uni- versité and Université Paris Cité. Correspondence to: Ahmed Boughdiri <ahmed. boughdiri inria.fr>.
Treatment effect estimation in observational data. De- spite being the gold standard to assess treatment effects, RCTs may face limitations due to stringent eligibility crite- ria, unrealistic real-world compliance, short study durations, and limited sample sizes. Medical journals such as JAMA Dombins-Domingo, 2024) and others (Hernan & Robins,
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Quantifying Treatment Effeets: Estimating Risk Ratios via Observational Studies
2016) have advocated the use of real-world data, often re- ferred to observational data, to provide additional sources of evidence. These data sets are typically less expensive to collect, more representative of the target population, and usually encompass large sample sizes.
analyses allow us build asymptotic confidence intervals. In Section 4, we evaluate all estimators on observational data, and study the empirical properties of confidence in- tervals in terms of coverage and lengths. In Section 5, we extend our analysis to a semi-synthetic and a real-world dataset. We recommend the RR-AIPW due to (t) less strin- gent assumptions needed to ensure asymptotic normality, (ii) empirical performances (consistent and with small asymptotical variance), (iit) the ease of implementation, as ratio of two simple AIPW estimators.
In the context of observational studies, various estimators exist to measure the treatment effect, mainly on an abso- lute scale. Different methods such as re-weighting using Inverse Probability Weighting (IPW) (Hirano et al., 2003), outcome modeling with the G-formula, and doubly-robust approaches like Augmented Inverse Probability Weighting (AIPW) (Robins, 1986) aim to estimate the RD while han- dling confounding effects.
Related work- Estimation of RR. To the best of our knowledge, Cornfield (1956) was the first to propose an estimate of the RR, together with exact and asymptotic con- fidence intervals, for binary responses, in RCT scenario, followed by Kupper et al. (1975); Katz et al. (1978); Bailey (1987); Morris & Gardner (1988); Sato (1992). Considering a logistic model, Schouten et al. (1993) propose a RR esti- mator. Later on, exact confidence intervals were derived by Wang & Shan (2015). Recently, Inverse Propensity Weight- ing schemes have been used in different study designs to estimate the Odds Ratios (Staus et al., 2022) or directly the RR in some simple settings (Hernán & Robins, 2006). Be- sides, Secudo-Poisson and pseudo normal distribution have been proposed with IPW strategies to estimate RD and RR in clinical trials (Noma et al., 2023). Other methods include G-formula approaches (Dukes & Vansteelandt, 2018) and semiparametrie density estimation (Kennedy et al., 2023).
However, to the best of our knowledge, there exist no work proposing or using estimators of the ATE measured with the Risk Ratio in observational studies based on (non- Opaametrric estimation (G-formula or AIPW), nor deriva- tions of their theoretical properties. A clear gap exists, need- ing robust estimators and analyses to improve treatment effect assessments and support medical recommendations in observational studies. Noteworthily, compared to the Risk Difference, studying the Risk Ratio induces additional technical difficulties, due to its non-linen oinner nature.
Contributions. In this paper, we propose and analyze dif- ferent estimators of the Risk Ratio in observational studies. Considering first the well studied RCT setting in Section 2, we analyze the first RR estimator introduced by Cornfield (1956), establishing a Central Limit Theorem and asymp- totic confidence intervals. We prove that in RCT adjusting for covariates or estimating the probability of being treated reduces the estimator variance. As the probability of being treated varies across individuals in observational studies, the above estimator is no longer valid. In Section 3, we detail different estimators that can be used for estimating the RR in observational studies: Inverse Propensity Weighting (RR- IPW), G-formula (RR-G) and two doubly-robust estimators. For the first two, we prove their asymptotic normality when the nuisance functions (surface responses and propensity score) are known or estimated via parametric models. Our analyses show that estimating nuisance function increases the variance of RR-G and decreases that of RR-IPW. Be- sides, using influence function theory (see, e.g., Kennedy, 2022), we derive two doubly-robust estimators, RR-OS, for one-step correction and RR- AIPW based on estimating equations. Contrary to the RD, due to the non-linearity of the RR, both estimators differ. We prove that they are both asymptotically unbiased and have the minimal variance among all asymptotically unbiased estimators. Surprisingly, the RR-AIPW estimator turns out to be a plug-in version of AIPW estimators for both the numerators and denomina- tors, which requires weaker assumptions than RR-OS to be asymptotically normal. For all estimators, our asymptotic
In observational studies, one can mention the work of Richardson et al. Yadllowyyy et al. (2021) and Shir- vaikar & Holmes (2023) who focus on estimators of the conditional average treatment effect (CATE) for the RR. Curth et al. (2020) introduces an "IF-learning" approach with pseudo outcome regression and derive the influence function for the CATE of the RR. However, since the expec- tation of a ratio is not the ratio of the expectations, CATE es- timations do not directly yield ATE for the RR. This departs from the RD, for which the ATE is simply the expectation of the CATE.
2. A Well-Known Risk Ratio Estimator in RCT
Problem setting Following the potential outcome frame- work (see Rubin, 1974; Splawa-Neyman al., 1990), we consider the random variables (X, (0), Y y("), where X E Rp denotes covariates describing patient, T is the treatment assignment (T = 1 when the treatment is given to an individual, T = 0 otherwise) and (0) (resp. (()) is the outcome of interest, describing the status of a patient with- out treatment (and with treatment respectively). In practice, we do not have access simultaneously to ((1) and ( and
2

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
we only observe
Assumption 2.2 (Bernoulli Trial). We assume:
Y TY + (1-T)Y(o)
(i) IgnorabbittyIExchangeallltttt T TT (2
Causal effect measures are functions of the joint distribution of potential outcomes (see Pearl, 2009). In particular, the Risk Difference (RD) and the Risk Ratio (RR) contrast the two states as followed
(ii) SUTVA: Y = =TY() + (1-T)r
(iii) iid.: (X,, Ti,,T.,Y, V. y,
NEE Lid. P.
(iv) Trial Positivity: Each participant has a fixed proba-
E TRR E
bility e E (0, 1) of assignment: IP[T; = 1] = e.
TRD = - [ and
(1)
To ensure our estimates are valid, we need to guarantee the existence of the ratio we aim to estimate.
The aim of this paper is to propose and study estimators of ThE for binary and continuous outcomes. Indeed, all our the- oretical results (except Proposition 3.9) are valid for binary and continuous outcomes. To estimate TER, we assume to be given an i.idd dataset (X1, Y.)..... (X., T,,Y,,,,))) Yn).
Assumption 2.3 (Outcome positivity). We suppose that both ( and y()) are squared integrable and that E [ 0.
Proposition 2.4 (Asymptotic normality of fRRN). Grant Assumption 2.2 and Assumption 2.3, the Risk Ratio Neyman estimator is asymptotically unbiased and satisfies
The most simple estimator of the risk ratio consists in re- placing expectations by empirical means in Thx (Equation I). Such an estimator has already been proposed outside the causal inference framework, with confidence intervals for the binary case (where E {0, 1). see Katz et al., 1978; Bailey, 1987). the potential outcome framework, inspired by the Neyman estimator of the Risk Difference (Splawa- Neyman et al., 1990), we call this estimator the Risk Ratio Neyman estimator.
VT (FRRN - Taa) NN (0, VARN)
(5)
where
Var (Y) VEEN = TRR e (1 e) 1013
Var (Y(1)
+
Definition 2.1 (Risk Ratio Neyman estimator). Let N = Ex T; and No- = n - n, The Risk Ratio Neyman esti- mator, denoted ARRN. is defined as
Proposition 2.4 establishes the asymptotic normality of the RR-N estimator, simple ratio of mean estimates, which leads to asymptotic confidence intervals (CI). Indeed, ac- cording to Proposition 2.4, for all a E (0, 1), a (1 - a) asymptotic confidence interval for The is given by
1 E - T.Y
RRRN
R2 2i=1 (1 T:JY.
No
FRKN + 21-a/2 VERN//1
(6)
the denominator is nonzero and otherwise.
With our notation, the 95% confidence interval for Tha for binary outcomes (Katz et al., 1978) takes the form
where VERN is any estimation of VERN. Throughout this paper, based on the Central Limit Theorems we establish, we will consider such CI. The properties of the different CI are studied in Section 4.2.
F exp(- 221 -20 FURN exp(#1
(3)
where 2-a/22 is the 1 -x/2 quantile of a standard Gaussian (0,01)) 1) and
Contrary to Katz et al. (1978), Proposition 2.4 valid for both continuous and binary outcomes. However, consider- ing binary outcomes in Proposition 2.4 leads to an asymp- totic confidence interval equivalent to that presented in Katz et al. (1978) (see Appendix 7.2.3). Deriving a Central Limit Theorem for log( (fRRR) instead of ARRn would lead to the exact same CI (see Appendix 7.2.4).
1
1
I
on V
Th DELTTY N1 E (1 T.YY,
No
(4)
In the sequel, we establish under which theoretical assump- tions the RR-N an accurate estimator of the Risk Ratio in Randomized Clinical Trials.
Probability of receiving treatment As the probability of treatment e is known in an RCT, one could be tempted consider what we call the Risk Ratio Horvitz-Thomast mator (in reference of the Risk Difference Horvitz-Thomson estimator, see Horvitz & Thompson, 1952) defined as
RCT randomly assign treatment patients in order eval- uate treatment effects. We focus on the Bernoulli design, one of the most widely used RCT designs (Rubin, 1974; Im- bens & Rubin, 2015), where each participant has the same probability e E (0, 1) of being treated, independently of the treatments of others.
esti-
1 DE T.Y. Di=l
TRRAT
n D I--
(1-T:)
3

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
if E < n and 0 otherwise. Indeed, the frequency of treatments assignments in the sample may be different from the actual probability of receiving treatment e. Similarly to what Hirano et al. (2003); Hahn (1998); Robins et al. (1992) noticed for the RD, we prove in Appendix 7.2.2 that opting for è over e in the Risk Ratio estimator (thereby employing the RR-N instead of the RR-HT) results in a reduced asymptotic variance, with a larger reduction when is close asymptotic variance of FRR.HT, we have
Definition 3.2 (RR-IPW). Grant Assumption 2.3 and As-
sumption 3.1. Given an estimator 0 é(-) < of the propensity score e(z) IP [T' 1|X = z). the Risk Ratio IPW, denoted by fRr.pWa is defined as
T.Y: n 2i 8(J)
TRRIPW
1 15 (1-7.1
one. More precisely, letting VEKHII the
Proposition 3.3 demonstrates the asymptotic normality of the Oracle Ratio IPW estimator, defined as the RR-IPW but where é(-) is replaced by the oracle propensity score e(-). Proposition 3.3 (RR-IPW asymptotic normality). Grant Assumptions 2.3 and 3.1. Then the Oracle Risk Ratio IPW is asymptoticallyy unbiased and satisfies
VERRN = VERHTT - - -e)
3. Risk Ratio Estimators in Observational Studies
A key difference between RCTs and observational studies is the handling of confounding variables. If not properly addressed, these can distort the true causal association be- tween exposure and outcome due to their correlation with both. Therefore, estimating the Risk Ratio in observational studies is more complex than in RCTs, as randomization assumptions do not apply (i.e. the propensity score now depends on the covariates X).
vn (TTR.Iew TRR)
- NNO0, (0, VRR.IE))
(9)
VERJPW (r
(O-10)
where
+
"AR
Note that when the propensity score is constant, one can retrieve the variance of the RR-HT as expected. Note also that the asymptotic variance may be large, due to strata on which the propensity score is close to zero or one. In other words, correct estimation difficult when some subpopulations are unlikely be treated (or untreated).
Assumption 3.1 (Observational Study Identifiability As- sumptions). We assume:
(i) Unconfoundednnes/Coniiiii Exchangeability: (L))), LL T I.
If assume a logistic model for the true propensity score and estimate it using maximum likelihood estimation (MLE), the variance of the RR-IPW can be derived.
iii) Overlap/Posittivtty: 3n E (0, 1/2) such that 7 S P[T = 1[x] < 1- - 77 almost surely.
Assumption 3.4. We assume TE [ is positive definite, X is Sub-Gaussian, and for all X E RP,
(iii) SUTVA: Y (1) + ())
P(T = 1|X) = (1 exp( B2 - 300 )
(iv) i.idd.: The data is (Xi, T; (( y(1)
Jejn iid P.
where 3x0 == (6% 32, E RR+11
Unconfoundedness means that after accounting for known confounding variables, no hidden factors affect both treat- ment assignment and outcomes. It is a relaxed form of exchangeability.
For any positive semi-definite matrix A and any vector X, let || vXT TXX.
Proposition 3.5 (Asymptotics of fRR under a logistic model). Under Assumptions 3.1 and 3.4, the RR-IPW esti- mator, with the propensity score estimated via MLE, satisfies
RR-N and RR-HT estimators cannot be used in the context of observational studies, since they are built on the assump- tion of constant propensity score. But the RR-N estimator can be extended to observational studies follows.
Vul(FRRIPW TRR) S NO VRR
2
3.1. Risk Ratio Inverse Propensity Weighting (RR-IPW)
with ViR MLE. VRR.IPW
TRR
G10 E[Y (0)
E[Y
cp1
Q-11"
Treatment effect in observational studies can be estimated via reweighting individuals by the inverse of their propensity score, thus giving more weights to people who are very likely/unlikely to be treated. Such method, called Inverse Propensity Weighting (IPW, see Hirano et al., 2003) for estimating the Risk Difference, can be straightforwardly extended to build Risk Ratio estimators.
where C10 = B[ e(X) and Q =
co1 = E[X(1 e(
with X = (1,
The variance of fRR,IPW is notably smaller than that of the oracle estimator TRR.PP-- While this might initially seem counterintuitive, similar observations have been made in RCT, as highlighted in studies by Hirano et al. (2003); Hahn

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
(1998); Robins al. (1992) and in observational studies by Lunceford & Davidian (2004b). Choosing é over e in the Risk Ratio estimator (thus using IRW instead of TRR.Pw) leads to a reduction in asymptotic variance.
where we assume that Y 2 for some c. Proposition 3.9 (Asymptotic normality of fRR,OLS). Grant Assumptions 3.1 and 3.8. Then, the RR G-formula estimator FRR.OLS that uses linear regression to estimate p(c) satisfies
VulfuR.OLs - TRR) 4 N (0, VRR TeR,oLs), where, letting = -[[X|T = t and Ex- = Var(XXT tt),
3.2. Risk Ratio G-formula estimator (RR-G)
For all (z, Rº x {0, let (z) E [r be the surface response of the potential outcome. Assume that we have at our disposal two estimators (:))(-) and (+))()) which respectively estimate ((o) (-) and (-))(-). We then employ the ratio of these two potential outcome esti- mations to compute the Risk Ratio. This method, termed the plug-in G-formula or outcome- based modeling, was first introduced by Robins (1986) for the Risk Difference.
VRR-OLS TRN
B(13 E |
8(00
+ +
E
|Y
Σ
(1 -1--cp c)2 -
1 e² el11
1o 1 (1 - e) Y
eB y(a)
2
The variance of RR-OLS can be decomposed in two terms: the first term corresponds to the oracle variance of RR- G, that is V the second term appears due to the estimation of response surfaces via OLS. Contrary to RR- IPW, the variance of the oracle estimator for the G-formula is smaller than that of the OLS estimator. Note that if we use RR-OLS in a Bernoulli Trial, then one can show that even in an RCT setting, adjusting for covariates is beneficial as the variance of the RR-OLS is smaller than the variance of RR-N. These results are provided in Appendix 7.3.4.
Definition 3.6 (RR G-formula). Given two estimators (xo)( (-) and (.)) the Risk Ratio G-formula estimator, denoted GRRG: is defined as
1 A n1
(10)
TRR.G
E. (X,,'
n
1 E Mr (X,) # 0 and zero otherwise.
The properties of RR-G depend on the estimators (r) and P()) We analyze in the following the behavior of Or- acle Risk Ratio G-formula estimator defined as TRK. E
3.3. Risk Ratio One-step estimator (RR-OS)
= ((n)(( (X.))((H M(,))))X.))...
Proposition 3.7 (Asymptotic Normality of TİRG)- Grant Assumptions 2.2 and 2.3. Then, the Oracle Risk Ratio G- formula estimator, Tin.o asymptotically unbiased and satisfies
A popular estimator for the RD is the augmented in- verse probability weighted estimator (AIPW, see Robins et al., 1992). AIPW combines the properties of G-formula and IPW estimators and is doubly-robust in the sense that it is consistent as soon either the propensity or outcome models are correctly specified. By calculating the influence function of the statistical estimand rixo = E [E [Y[T = X] - E I 0, X]] we obtain an effi- cient estimator: it has no asymptotic bias and the minimal asymptotic variance (Kennedy, 2022). Therefore, to esti- mate the Risk Ratio (RR), a natural approach is to derive an efficient estimator using semi-parametric theory (Tsiatis, 2006), presented below.
V (TReG - TxR) $ N (0, VERA),
(11)
(X) B|Ya]
(X)
where VIRG= = T Var
E 2
Proposition 3.7 establishes that the Oracle Risk Ratio G- formula estimator is asymptotically normal. Surprisingly, in the case where there is no effect (i.e. TRR = 1), the asymptotic variance driven by the variance of the Risk Difference on each strata determined by X, namely Var(pay (X) ((o) (X)). By considering the Oracle RR- remove the additional randomness
Definition 3.10 (Crossfitted RR-OS). For all t E {0. 1] and all x, let (a (x) = E | = x] and (() = P [ = f|X = z). We denote I
G instead of RR-G, related to the estimation of the surface responses. It is thus likely that the true variance of RR-G is larger than that of Oracle RR-G. as shown below.
nI....... let
T2, Ix be a partition of I. Let ( (X) and
I k (X) be estimators of ((() and e()) built on the sam- (())
ple I- * = I For all t {0, 1}. let
Assuming a linear model for y and estimating both re- sponse surfaces (o) and fi() using ordinary least squares, the variance of the RR-G can be derived.
1 K AIPAA
Y - 1 (X,)
I MO
(X,) +
1,
n
k=
(()
(Xi)
Assumption 3.8 (Linear model). For all / E {0, 1}.
(12)
Y c + X Bto + [[X] = μ B = [arls a2.
K Σ Σ
and
fas
1T (Xi),
(13)
n
R= 1EIi
5

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
The crossfitted Risk Ratio One-Step (RR-OS) estimator TRR.OS is defined as
The RR-AIPW is simply the ratio of two one-step estima- tors, one for E [yu] and one for EE [Y[O]. This method may seem simplistic at first glance, since approximating both the numerator and denominator usually results in non-zero asymptotic bias. However, RR-AIPW is derived via the estimating equation method using influence function theory, which results in an efficient (asymptotically unbi- ased) estimator. Note that in the case of the Risk Difference (RD), both approaches (One-step bias correction and esti- mating equation) yield the same AIPW estimator. However, because our statistical estimand for the Risk Ratio is nonlin- ear, the resulting estimators differ. It remains that they are both efficient, as shown below.
FaiprLo
TRR.OS
1
FG.0
FG.
f0.0
Considering the statistical estimand v'ER - V we obtained the estimator RR-OS, which is efficient when nuisance components are estimated via cross-fitting (Cher- nozhukov et al., 2017) and with non- oarramamtriic methods.
Proposition 3.11 (Asymptotic normality of fRR-OS). Grant Assumption 2.3 and Assumption 3.1. Assume that for all 1 S k S K, and for all t {0, 1}.
Proposition 3.13 (RR AIPW asymptotic normality). Grant Assumptions 2.3 and 3.1. Assume that (14) holds and that, for all 1 < k S K and all I E {0, 10,
B (Ki (x) (x)) (((
(14)
k 2l (x) -B (x]o 1/4)
(15)
E (Ais (x)-m((x)) =o(1), =001),
=((===xxx- =o())
(17)
= a(( (x) * (x)) (x)x)) (16)
Then, the crossfitted Risk Ratio AIPW estimator is asymptotically unbiased and satisfies
with n S
with v S ()) S n) (see Assumption 3.1). Then the
One-Step estimator is asymptotically unbiased and satisfies
I (FRRAAIP TRR) - N (0, VaROs),
Vn (fRnoos - TRR) S N (0, Vnn.os,
where Van.OS defined in Proposition 3.11.
Assumptions in Proposition 3.13 are the same as those used in the Risk Difference AIPW estimator (Wager, 2020) to achieve double robustness. Specifically, RR-AIPW ben- efits from weak double robustness, meaning consistency is maintained as long as either the outcome model or the propensity score model is estimated consistently. This con- trasts with RR-OS, which lacks this flexibility: it requires consistency of both outcome models (Jo, j1) or the joint consistency of (e, (o) to achieve reliable inference. Ad- ditionally, condition 14 often referred to as a risk decay condition ensures strong double robustness for RR-AIPW. This property ensures asymptotic normality when both the propensity score and outcome regression converge at suffi- ciently fast rates. We recommend RR-AIPW over RR-OS, not only because both share identical efficiency in their asymptotic distributions, but also because RR-AIPW op- erates under weaker assumptions. RR-AIPW's strong and weak double robustness properties provide greater resilience to model misspecification compared to RR-OS.
where
VEROS
(x)
Mco) (X)
Var
2
E [
E [y[0]]
Var (x())x)
Var (Y
+ EF
E
[(()) [r
(1 e(X)) |
This estimator is efficient: its asymptotic variance is min- imal. The semi parametric theory develops efficient esti- mators by compensating for the first-order bias (Kennedy, 2022), this can be achieved either by estimating and subtract- ing the first-order bias, leading to the RR-OS estimator or by finding values for the target parameter and nuisance pa- rameters that solve the estimating equation (see A. A.Shhul,r, 2024, for the RD case), resulting in RR-AIPW presented below (calculations are detailed in Appendix 7.3.6).
3.4. Risk Ratio Augmented Inverse Propensity Weighting (RR-AIPW)
4. Simulation
Definition 3.12 (Crossfitted RR-AIPW). The Risk Ratio AIPW crossfitted is defined as
Simulations for RCT are provided in Appendix 8. For ob- servational studies, we generate datasets (X, (X,T, (()), (1)) according to the general model
i
RRRAIPP
Auwoo
(()) m (X) + b(X) + E(1) (w) 6(X) =
P = 1|x] e(X), with E(o) N (0,
where 7APwO, and fxPPA are defined in 3.10.
6

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Lunceford Linear Logistic
Esfmmtions
100
300
10000
RI-N
Foned
Forwel RINAAFWW Estimators
Logadic
Linwar
RRN-APWW
Figure 1. Risk Ratio estimators computed for Linear/Logistic DGP, with 3000 repetitions.
Wager C
30
100
1000
Eximmtionn
10000
RR
Forest RR IPW
Forest RR
Forest RR 0S
Forest RR AIP!!
Logistic RR IPW
Lintar RR
Liher RR Os
Lincar RR NPW
Estimatoss
Figure Risk Ratio estimators computed for a con-Linerr-Lgistee DGP, with 3000 repetitions.
where m(.), b(.), and e(.) respectively correspond to the treatment effect, the baseline and the propensity score. The Risk Ratio can be expressed TRR [Y( /E [y[0] E. [m(X)) /E [6X)] + 1. We com- pare the performances of all estimators defined in Sec- tion 3 where nuisance components (regression surfaces and propensity score) are estimated via parametric (lin- car/logistic regression) or non-parnnn non-aaametric methods (random forests). More details are provided in Appendix
0.25(1 -X))). Conditional on X3, (X1, V1, X2, 122T N(xx, E), where:
1
0.5
-0.5
00.5
1
0.5
1
-05
-0.5
A = -
Σ
-0.5
0.5
0.5
0.5
0.5
0.5
Results are depicted in Figure 1. Only confounding vari- ables are used inputs in the different estimators. As expected, since the generative process is linear, methods that use parametric estimators (logistic/linear regression) outperform those using non- parametric approaches (random forests) in finite-sample settings. While all methods (ex- cept RR-N) converge
4.1. Linear and Logistic DGP
The first observational tuta-generating process (DGP), in- troduced in Lunceford & Davidian (2004a), uses linear out- come models (treatment effect and baseline) and a logistic propensity score, defined m (X, V) 2 with:
correct RR, methods based on parametric estimators exhibit a faster rate of convergence and are unbiased (except for Logistic RR-IPW) even for small sample sizes. Indeed, random forests not suited for linear generative process and require here more than 10000 samples to estimate correctly the RR. All in all, when the outcome modelling and the propensity scores are linear, the two doubly-robust estimators (RR-AIPW and RR-OS) and the RR-G, all based on linear estimators, achieve the
e(X) = (1 exp(-8. X) 3n - (-1,,,,,,,,,111111 B. = (-0.6, 0.6,
b(X.V)= [x, VJ,
Covariates X = (X1,X,, X3) influence treatment and response, while -= (V, V2, V²) only affect the response. [X, V] are jointly distributed, with X3 ~ (ernoull(00.2) and V/ (ernoull((P(V, = X3) 0.75Xx +

---

To better illustrate the practical application and behavior of our estimators, we include a real-world study from Mayer et al. (2020) involving 8,270 patients with traumatic brain injury (TBD, using data extracted from the Traumabase. Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
best performances: they are unbiased, even for small sample sizes, and converge quickly to the true RR.
Turning the Non-Linear/LLoiistie DGP, Figure 4 shows the CI length and coverage for the same estimators. In this setting, only the Linear RR-AIPW estimator maintains ac- ceptable coverage-liked because propensity scores are well-estimated, whereas modeling complex, non-linear re- sponse surfaces is difficult with either linear models or forest-based methods when n = 1000. As before, For- est RR-G and Forest RR-AIPW struggle with coverage. and the RR-N estimator again shows almost no coverage, reflecting its limitations in observational settings. More sim- ulations on a Non-LinearrNon-LLogisse DGP are available in Appendix 8.2,
Furthermore, both Linear RR-OS and RR-AIPW estimators give very similar results. More simulations on a Non-Linear and (Non-)Logistie DGPs are available in 8.2. As antici- pated, estimators with non-parametric nuisance functions outperform their parametric counterparts when the DGP deviates from linear or logistic structures.
4.1.1. I. NON-LINEAR AND LOGISTIC DGP
We use semi- parametric setup (see Nie & Wager, 2020) with non-linear baseline models, a constant treatment effect and a logistic propensity score:
5. Real- World Experiment
b(x) = 2 log (1 + X))
((x) = 1/ (111111+/ XxtXxs and m(X)= = 1,
where X ~ N (0, Idxa). Results are presented in Figure 2. The Forest IPW and Linear G-formula estimators yield poor RR estimates for the largest sample size. The forest IPW uses random forests to estimate logistic model, which may still converge, but at a slower rate than other methods. The Linear G-formula employs linear regressions to estimate the response surfaces, potentially leading to an irreducible asymptotic bias. The Forest RR-G, Forest RR-AIPW, and Forest RR-OS estimators converge slowly to the true RR. This simulation highlights the doubly robust properties of the Linear RR-AIPW and Linear RR-OS estimators: they target the true RR even at small sample sizes, they have at least one well-specified model.
The Traumabase is a continuously updated database that collects comprehensive clinical data from the scene of the accident through to hospital discharge. The causal effect of tranexamic acid (TXA) on 28-day mortality is estimated by adjusting for 17 confounding variables. These variables include key metries for severe trauma cases, such as systolic and diastolic blood pressure, heart rate, oxygen saturation, and details of interventions. We subsample the real data to obtain different sample sizes, with results averaged over 5,000 simulations for each sample size. Since this a real dataset. the true value of TRR is unknown. semiisynthettic simulations, in which the treatment and potential outcomes are generated (allowing us to know the true RR value), are described in detail in 8.3. Results are displayed in Figure 5,
4.2. Confidence intervals (CI)
As observed, most estimators yield values larger than one, suggesting a potential deleterious effect of TXA (increased mortality). This trend aligns with findings from previous studies (c.g.. (Mayer et al., 2020)). Notably, Forest RR-IPW is the only estimator indicating beneficial effect of the treatment. RR-N, which is not suited for use with obser- vational data, produces a higher value compared to other estimators. Interestingly, estimators based on linear models (Linear RR-AIPW, Linear RR-G, Linear RR-OS) exhibit the largest variances, particularly for small sample sizes.
In both the Linear/Logistic and Non- Non-Linear/Logistiie data- generating processes (DGPs), we build asymptotic 95% CI for the RR-AIPW, RR-G and RR-N estimators based on their asymptotic normality. Variances were estimated via equa- tion 136, 33, 47, 52 and 30. In Figure 3, we present the dis- tribution of the length and coverage (probability that the CI contains the risk ratio) for each estimator (300 repetitions). IPW was excluded for its poor performance (excessively large Cls), caused by propensity scores close to 0 or 1 (see Figure 8). As expected, RR-N CI has nearly zero coverage due to Ron-RCT setting. The Forest RR-G and RR-AIPW confidence intervals also exhibit poor coverage, which is in agreement with the Linear/Logistic DGP. In contrast, Linear RR-G and RR-AIPW demonstrate good coverage. Note that the CI for OLS RR-G, built on Proposition 3.9, has better coverage than the Linear RR-G method, as it includes the uncertainty due to linear estimations. Although only the RR AIPW has coverage above 95%, the OLS RR-G has a shorter average predicted length compared to the Linear RR AIPW.
6. Conclusion
Quantifying treatment effects presents challenges, since dif- ferent measures may lead to different understanding of the same phenomenon. In our study, we focus on one of these measures, the Risk Ratio and introduced several estimators, valid in RCT or observational studies. Using dedicated math- ematical tools (influence function theory, M-estimation), we establish their asymptotic normality, limiting variance and
8

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Logistic
60
0.993
0.293
01775
Figure Coverage (left) and Length (right) of asymptotic CI derived from Scction and Scction with n 1000 and 300 repetitions.
Wager
0.
0.1
66
02-
16-M
Forelt MSHAPM
RM-4PW
10-
Fonudd BdvelaW
Figure Average coverage (left) average length (right) asymptotic confidence interval derived from Section 2 Section for different estimators with 1000 and 300 repetitions for Non-Linear and Logistie DGP.
Estimates ovolution across sample sibe
Sample Stze
2000
2003
POTESC
Figure 5. RR estimations with weighting, outcome based and augmented estimators as function of the sample size for the Traumabase. Parametric (Linear) and non parametric (Forest) estimations of nuisance displayed.
derive asymptotic confidence intervals. Empirical evalua- tions show that RR-N and RR-IPW have poor performances. Either Linear Forest RR-AIPW (or RR-OS) show similar (good) behaviors to estimate the Risk Ratio, with the best theoretical guarantees among all studied estimators. Since RR-AIPW requires fewer assumption and is simpler to com- pute, we would recommend to use RR-AIPW. As for the doubly-robust approaches, G-formula competitive, with performances that depend the setting and the estima- tion method used for the nuisance components. Identifying guidelines establishing when linear nuisance components should be used instead of non parammerriic ones still remains an open problem. In practice, observational studies may
be used to generalize the treatment effect from RCT pop- ulation the general population of interest. Our work is first step toward proposing procedures to generalize the Risk Ratio to general populations.
A Aeknowledgments
This work has been done the frame the PEPR SMATCH project and has benefited from a governmental grant managed by the Agence Nationale de la Recherche der the France 2030 programme, reference ANR-22-PESN. 0003.

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Impact Statement
Dukes, O. and Vansteclandt, S. A note on g-estimation of causal risk ratios. American journal epidemiology, 187 (5):1079-1884, 2018.
This paper proposes and studies estimators for the Risk Ra- tio in observational studies, addressing a methodological gap in causal inference. The Risk Ratio is widely used in clinical and epidemiological research assess relative treat- ment effects. However, it is important to consider the Risk Ratio alongside other effect measures, as each provides different perspective on the data and may have different implications for practice. As these methods rely on stan- can be
Forrow, L., Taylor, W. C., and Arnol,, R. M. Absolutely relative: How research results summarized can affect treatment decisions. The American Journal of Medicine, 92(2):21-124, 1992. ISSN 0002-9343. doi: https//b6ioo//000000000099999119 URL ht:p:::/www.sccenneddiieeee scderse//ricde/////0033339220200
dard causal assumptions such as ignorability difficult to satisfy in practice-caution is needed when in- terpreting the results. Violations of these assumptions may lead to biased estimates and incorrect conclusions about treatment effects.
Hahn, On the role of the propensity score in effi- cient semiparametric estimation of average treatment effects. Econometrica, 66(2):315-332, 1998. URL www.ppp....g/eeeeeeeeeeee
https://BconPapgee ecm: 11
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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
7. Appendix
7.1. Preliminary results
Since we are studying the asymptotic properties of the risk ratio, we cannot directly apply a central limit theorem as in (Wager, 2020). We will therefore rely on Theorem 7.1 to prove most of our asymptotic results.
Theorem 7.1 (Asymptotic normality of the ratio of two estimators). Let (Zi, Zo) be i..dd random variables, go
and 91 two functions square integrable such that E 190( Zi) = To and [g1 (Z,)] = TL where To # 0. Then, we have that
91 (Zi)
4 N (0, Vine
1=
(2)
70
where
V
T1
2
Var
91 (Z)
go (Z)
TO
71
T0
Proof. We rely on M-estimation theory to prove Theorem 7.1. Let
90 (2i)
90 (0 (2) 0o
0, =
91 (Z,)
and ((2,0)= =
(z) - 01
(18)
O1 (2.)
01 0280
1 (2,)
where 0- -0o. 01, 02). We have that
1 12
go go(
Σ
90 Zzz)
Σ
90 (Z:) Σ
90 (Z²) =0,
71
j=1
and similarly
Σ
gl (1,),)
Σ
1
1 n
91 (Zi)
= - (Z.)
Σ 9و (2,) 0.
71
1=1
Besides,
n Σ 91 (Z,)
Σ
91 (Z₁)
Σ 90(2;) 0.
Σド= 90 (0,)))
2=1
j=1
Gathering the three previous equalities, obtain
(Z,, = 0,
(19)
which proves that θ, is an M-estimator of type vi (see Stefanski & Boos, 2002). Furthermore, letting 0x (To, Tı, TT)
simple calculations show that
E (Z, 0.
(20)
Since the first two components of db are linear with respect to fy and 0₁ and since the third component is linear with respect Oz. On defined above the only value satisfying (20). Define
T8g 00 10=00
A (0xz) E
and =E = -E -
(21)
13

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
We now check the conditions of Theorem 7.2 in Stefanski & Boos (2002). First, let compute A (0xc) and B 30 Since
Obb -1 d0
0
0
(Z,
-1
(22)
80
we obtain
(000)
-1
(23)
70
which leads to
A- (0,0) =
(24)
Regarding B (0x)), elementary calculations show that
-
(go(2) )0)²
(go(Z) To) (9
-1)
0
(2,00,))(22000))) (90(2) - TO) (2) - T1)
(2) 71)
0
which leads to
B(0x)) ( Cov (Z) (Z)
Var [Z)]
91 Cov (go (Z), (Z))
0
Var (gn (Z) 0
0
0
Based on the previous calculations, we have
(2, 0) and its first two partial derivatives with respect exist for all and for all @ in the neighborhood of 0
For each in the neighborhood 0oc. we have for all i, k E {0,
0² (2 0) 5
00,00,
and is integrable.
A(0so) exists and nonsingular.
(0xx)) exists and is finite.
Since we have:
(T, and 0, 2 0.c.
Then the conditions of Theorem 7.2 in Stefanski & Boos (2002) are satisfied, we have
v (O,- 0x) $ N
where
- (AO0)))
Var (go [2)]
Cov (go(Z), (Z))
Cov(gx(2).g112
Var(gn(2)
7a
Cov (go (Z). 91 (Z))
Var 191 (Z)]
Cov( g) (2))
Var (2))
(25)
7
To
Cov(ga( (2))
(Z))
Cov(gel (Z).g)
Var la (2)
V
RR
14

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
with
11
91 [1)
90(2)
Var
(26)
RR
To
T1
TO
In particular,
Di=1 1=
91 (2,)
Vn
4 N (0, V
(27)
(2,2,)
50
Assumption 7.2. We suppose that both (0) and ( are squared integrable and that IE [ X E [ X > 0.
Theorem 7.3 (Finite sample bias and variance of the ratio of two estimators). Let (Z) and To Z) be two unbiased
estimators Tı and 0 where Z = (Z,...., Zn) be i..dd random variables. We assume that Mo 2 To Z) 2 mo
(Ti(Z)] < Mi We also assume that Var(Ti (Z)) Op (H) and Var(To Z)) = Op (+) Then, have that
Ti 71 To(Z)
Ti To
Mi Mo
Mo
+ 1)
Bias
E
To
70
nmg
m0
and
(Ti(Z)
T1 (Z)
2
To(Z)
MM
[Modd
+1)
Var
Var
S
(_m mg
To(Z)
nmg
To
Proof. We rely the multivariate version of Taylor's theorem prove Theorem 7.3. We first introduce the multi-index notation:
|a| = a1 + ... + an. a! = ajl .e.....oe and -ppl
x = 2 2
and
glals f Daf dr Iz
|a << k
Let be the ratio function
f: R x R. R (I1, F22 ب 21/22.
Since f is two times continuously differrentablee then one can derive an exact formula for the remainder in terms of 2nd order partial derivatives of f. Namely, we define x = (II, x2) and for a E R x R
DO (a) a)) + Rx+1(x)),
f(x) Σ
(28)
S al
with
Σ (x- a) |
31 / (1 - 1) Da f(a + t(x aa) dt.
Rx+1(x))
18|=++1
Bias:
15

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Computing 28 for the ratio function with (Ti (T), -T1, To) and taking the expectation gives
E (1)
T0( (Ti(Z) - - T0) (Tu(Z) -a) + (2( (Ti(Z)
- ((1),T)) +
(T)
0f(1i. To)
-B[((n,
E (TT -1)
(Zi(Z)
0f(TI, TO)
E (To((2) - To)1 + E 2 (T)),
(Z)
11
In order to produce Theorem 7.3, we just need to show that E ((())))))) Op (1). To do so, we first compute R2 (T)
() (Z) -1)) dt
(Z), = (ZT - -0)²
(To + t(To(Z) =o))3
(2,
2(To(Z) - -o) (o) - 72) /
1-t
(To + t(To(Z) To))2 dt
(3(T1(2),6(22)
Since assume that Tz(Z) mo > 0 and that (Z) S M1 have:
(1 t)(T1 + t(T((Z) - r1))
(TT 2(To(Z) r0)2
dt
(To ((To(Z) To))3
(1 ( dt
(Too(Z) - T0)2
min To)3
Mi
(Tb(Z) )
m
C
Similarly, have:
-0) m1) /
1-t
= 2(76(Z) (2),
(To ((To(Z) To))2 dt
=2](T((2) (T) m)
(To + ((To(T) - -0)) dl
1-1
< <((T(2)) CcL (T) 71)
Finally get that:
E ||00://|6666000|3333332322221111
C1 + (,1
Ci V)r((b (Cy//Vr(TT(2))) Var(Ti(Z))
+
C.M M2MoMi
16

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Quantifying Treatment Effects:: Estimating Risk Ratios via Observational Studies
Since have that Var(Ti (Z)) =Op (+) and Var(To(Z))
O2 (H), can conclude by:
(Z) TI
(Z)
71
M M
Mo
Bias
[o( To To(Z)
TO
mmg
Variance:
Let us begin by expanding the variance of the function f:
Var (2), To ()) = IE [11T]222){{()))E((((()0)2))1111
Next, apply Taylor's expansion around the means Ti and To:
E ( To) 0S(TI, ro) (Ti - -
(1, (To(Z) To)
(Z) (2)(T((2),To(2)) (2)
(T)
Simplify by focusing on the first-order derivatives and residual terms:
(ff(). (Z) - -1) (0) (To(Z) -D)
= E (
(Z)
(2)
+ (T) To (2)) (2)
Decompose the variance into linear, cross-term, and residual contributions:
@f(n) (Ti(Z) - af(m), To) (To(Z) - -o))2
(Z)
(T)
Tf((T..T) To) (Zi -1)
+ 2E
(Z)
0s ( TO)
(Z) (To(Z) - mm) E (
+ Var ( 2), To()))
Finally, re-express the result using simplified ratio of variances:
T1
2
Var
(Z
To(Z)
+ Var(Rg(T((Z),TT((2)
TO
We now focus on Var (Z)) (T))
Var (ar(Ra(T To(2)) x ( ((8(((2)))
< 2E [1] (2)
We first focus the first term:
< (C)) 3
E (1
[ s CRMEE [(To - [o]
To(Z) s Mo
$ (ar(To()))
For the second term we have:
E
< (T) (T))) 71
Ti (Z), (Z) bounded
s / Var (T))
17

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Hence get that:
2M nun
(MoMM mg
(2)( -
+1
7.2. Proofs of Section 2
7.2.1. RISK RATIO NEYMAN ESTIMATOR
Proof of Proposition 2.4.
Asymptotic Bias and Variance: we proceed with M-estimations to prove asymptotic bias and variance of the Ratio Neyman estimator, we first define the following:
Σ T Yi 1
40(())
(1 (Y 00)
)
On
En==
and (TT,Y,0)= = 4/1 (0)
T 11)
(29)
TR-N.r
(0)
01 0200
where 0 01, 02)
Next, we verify that for 0 = ( Er: Y,, 1 Σn,=1 Yo, FR-N.n), we have:
mi
(Ti, Yú, 0..) - O.
We begin by demonstrating this for tt:
Y: 1 Σ Y
de (T,YY, =
Σ T.
111 T==1
1=1
1
T
Y: 11 1=1
1=1
T.Y
1
Σ T TJY,
m 1=1
1=1
Σ T.Y >>T T.Y
1=1
0.
Similarly, we can show:
n
Σ
2o Yr, 0.) O.
Moreover, by construction:
(T Y, O.. = 0.
Thus, we have established that θ, is an M-estimator of type 1 (see Stefanski Boos, 2002). Given that we are a Bernoulli
18

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Trial, we now demonstrate that E (T, Y, 0x)) where 0x (B[Y B TRR). Therefore, have:
(0 = T (Y - -EI
T (y(1)) - [ -
(by SUTVA)
(7) yra)) - kyaaaa
(by ignorability)
Similarly, we can show:
[Vo() 0.
Furthermore, we have:
E [00((> = - TRR 0.
At this point, we note that 0n the only value of 0 such that IE [[, 0. We proceed by defining:
A (0xz)
E
~=
and B(0x) = (,,
08
Next, we check the conditions of Theorem 7.2 in Stefanski & Boos (2002). First, compute A (0xc) and B (0 Since:
-(1 T)
0
(Z, 0)
-T
08
0g
Oo
we obtain:
-(1 e)
0
0
A (0xo)=
0
-
0
TRn
-
E[
which leads to:
0
0
A-'(00))
0
0
1
Regarding B (0x)), elementary calculations show that:
(Zz, (1Z 0z)
((1 - (Y (0)))) - E((
(1 (1- () - E|Y ( - -XXXX--XXX)XX---XXXXXX- E(
(T(Y - -(r()))
0
0
which leads to:
(1 e) Var [Y
0 )
B(0x²)=
e Var [au)
Based on the previous calculations, we have:
(b, 0) and first two partial derivatives with respect exist for all and for all 0 in the neighborhood of A.o.
19

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
For each 0 in the neighborhood of oo have for all i, (0,
8
1h (x(2,, (z, 0) S
08,00
and is integrable.
30) exists and is nonsingular.
30) exists and is finite.
Since we have:
(T, = and 0, 2 0xc
Then, the conditions of Theorem 7.2 Stefanski Boos (2002) are satisfied, we have:
vn O 000) S N (0, B(0w))(((w)))))))))
where:
Var (03)
Var 103
(1-e)
voll-e)
A(0,z) ((00)
Var (1)
Var (2)
0
e5o
Var(y
Var y01
VR-N
To(1-e)
with:
Var (Y
Var (Y
VR-N THR eEEl(1)j
+
(1 e){(Y())p
In particular, obtain:
Vn (FRRNa TaR) 4 N (0, VERN)
Finally, note that:
Var (Y
Var (Y)
VR-N ThR
eBlY(uj
(1-e) e) [ 10]
([((((0)22 (
cE
(1 - (-
TRR
VR-HT e(l e)
As consequence an estimator VR-N can be derived
Er= -
1 E - Er
VR-N = +2
+ n
(30)
TRKNa
Y.)=
(1 (2 Y,)
Optimal choice of e: the optimal value of copt is the one that minimizes the variance of the Ratio Neyman estimator. Therefore, we need solve:
Var (Y(I)
+ Var (Y(0))
inf T eE(0.1)
eE [1]]
(1 - e) (
20

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Noting that the variance want minimize is convex in e, we derive the variance and it to find Copt We have:
C1
Co
Copt
2
(1
Copt )²
where C Var ())(
(ar(y
rf(ca]= and Co
If Var(y()) Byr
Var(y
Cppt 0.5
otherwise:
C1 VCiCo E (0,
Copt
C1 Co.
7.2.2. RISK RATIO HORVITZ-THOMSON ESTIMATOR Definition 7.4 (Risk Ratío Horvitz-Thomson estimator). Grant Assumption 2.2 and Assumption 2.3. The Risk Ratio Horvitz-Thomson estimator denoted FRRMTT is defined as,
T.Y. ΣM
*RR.IT.
(31)
(1-T,Y 21=1 I-
if EE Ti < n and 0 otherwise.
Within the context of Bernoulli trial, Proposition 7.5 proves that the Risk Ratio Horvitz-Thompson estimator is asymptoti- cally unbiased and normally distributed.
Proposition 7.5 (Asymptotic normality of FRKHTT)). Under Assumption 2.2 and Assumption 2.3, the Risk Ratio Horvitz- Thompson estimator asymptotically unbiased and satisfies
Vn (FRAHT. -RR) 4 N (0, VRRNT)
(32)
(e( [r(u)] eB(r(s)
E (
where VIRHT
TBR
(1
we assume that for all i, M 2 Y 2
0 and 0 < Σn T3 n, we also have:
2M e)s
Bias( frnn, NT. a
3m3
4M e)
Var(Faa
nmm511
Proof of Proposition 7.5.
Asymptotic Bias and Variance. Let Z; = (T,, Y2) and define go (Z,) (1-T.)Y, the expectation of gı (Zz):
and 91 (Z,) T,Y. First, we evaluate
1-t
T.Y.
19
(Z.)
E
(by i.i.d)
T.Y (1)
E
(by SUTVA)
T IE
E
(by ignorability)
[1)
(by Trial positivity)
21

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Quantifying Treatment Effects:: Estimating Risk Ratios via Observational Studies
Similarly, we find the expectation of (Z,):
E [Z,)l E [ O.
Thus, according to Theorem 7.1, we have v (FRR.TT,.
Tan) S N (0, VER.HT), with
T1
91(Z)
90(Z)
VER-HT
Var
T0
TI
70
TY
(1 T)Y
2 RRR
Var
eE
[YA]
(1 e) ( YYO]
Next, we evaluate the variance terms separately:
TY cR [r(u]]
Var
=
E [ 2 e²
Var (TY)
E (TY))] - E(Tr))
E ]
E (x) [((x)?
(I' is binary)
[ra]]
E T (riw)) (
(by SUTVA)
[raj² 22
eE (xri)) 27 - = (x(1)))))
(by ignorability)
[ra]]
( 27
eB ra] 2 1.
Similarly, we find the variance of the second term:
E (Y(00) 27
(1 TYY
Var
1.
eI- Y
e)B [Y00]]
Finally, we compute the covariance between the two terms:
TY
(1 T)Y
Cov (1- T)Y)
Cov
eE
[1)
(1 E) [ro]]]
eE [Y (1 e) [
( m - -E[TY]
eE [L) e1 y0)
E [YY (1- T]
eB [ e1 [
-1
Using Bienayme's identity, we finally obtain:
22

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
(ra)² 2
E (ri0))²] 2
E
VER.HT 22
eB [y(u]²
(1 - ee- [r(o]]²
As consequence estimator VRR-HT can be derived:
11 -1= Y?
1 ET; Y?
+2 V VRR-HI HT
(33)
RRIHTT
2T: Yi)
(1- 8) ( ΣT,
Finite sample Bias and Variance. Let (Z) First, consider the variance of Ti (Z):
1 E== T.Y and (Z) = 1/2 E (1TA)
where Z = (Z),... Zn).
Var(Ti(Z)
Var (T)Y;)
(by i..d)
NE²
[E (T:Y.))
- E [T
ne²
(Y eB [ya]]
Thus Var(Ti(Z)) = O2 (1/n) and similarly Var(To(Z)) = -Op (1/r). Next, we show that To(Z) is bounded:
Ti)Y
Tz(Z)
1-6
1=1
Y.
n
T:)
(since Y, 2 m 0)
m
(as
T < <)).
e)
Similarly, we also have the upper bound
1 12
Y
TZ
11
e
[1 T.
Y
(since T' is binary)
M
(since Y; < M).
Similarly, have (Z( M Therefore, we have shown that Ti (Z) and TZ(Z) unbiased estimators of E [y()] and [y > O. respectively. We also established that M/e 2 To ( 2 m/( - e) > 0 and [T](Z] S M/e. Furthermore,
23

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
pointed that (ar(Ti(Z)) =0p (+) and Var(To(Z)) =0 ( Applying Theorem 7.3, obtain:
-1 (M(1 - c)
2M - -e)²
[RR ITT. TRR| -
+1
m
me
mm
and
2M -e)*
M [ e)
4M (1-11 e)
Var(fra. HT..) - VER 2
+1 S mmm
nmm222
m2e22
Optimal choice of The optimal value of Copt is the one that minimizes the variance of the Ratio Horvitz-Thomson estimator. Therefore, we need to solve:
E (r
E (r())) P]]
inf (E(1)))
TRK
el [v(uj]2
(1 - e) [y[0]]²
Noting that the variance we want minimize is convex in e, we can derive the variance and it find Copt. We have:
C1
Co
Cpt
(1
Copt) )2
where Ci
and Co
If Var(y(1) Var()(()), B(YCU) x[rio??
Capt = 0.5
otherwise:
E (r(u))} 2
E [0(0) v E (r())) E (r) 2 E [ E [Y
Copt
(0. 1)
E (x(1)) 2
(0))] (r))))) 3 E [ry)
7.2.3. LINK WITH EXISTING ASYMPTOTIC CONFIDENCE INTERVALS According to Proposition 2.4, a (1 a) asymptotic confidence interval for TRR is given by
AARN + V ViRNe1-00 -a/2
(34)
with VARx an estimator of
Var (y
Var (x(
VIRN 720
(1)2
- (-=))E(y(33)
24

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Now, assume that y y E {0,1] with associated probabilities Poy = Po and P = Pr. In this setting. the variance VERN takes the form
VERN
Var (Y
Var (Y
72
72
e
(1 e) (
(1 PI)
+ Po po)
TRR
Nipl
Nopo
P
1- Pa
TRR
Nip1
+
NoPo
1
1
TR
N1P1
N1
Nopo
No
1
1
TRR
N1p1
N1
NoPo
No
An estimation of such quantity can be constructed by replacing (resp. po) by (1/N1) E T.Y, (resp. (1/No) Ei= (1 T)),), which leads to
VIKNN
1
11
n
"RR
T.Y;
N1
Σ (1
T:)Y,
No
(35)
Thus, a (1 a) asymptotic confidence interval for TAn is given by
1
1
1
TRUNa + 21 21-0//2uRNNa
(36)
E T,Y,
Ni
Ei= (1-Ti)Y,
No
PRKNa
1 21
En DiLITYY,
N1
+
E.
(1-T))))
No
(37)
Finally, since e is equivalent to 1 + x near x = 0, the above interval is equivalent to that given by (3), which concludes the proof.
7.2.4. DELTA METHOD WITH log FUNCTION According to Proposition 2.4, we know that
Vn (fRRNa TKR) $ N (0, VERN),
(38)
where
VIRN = 2 Var (Y()) Var (Y(0)
(
(1 - e)
Using the Delta method, with the function 0 + log(f), we obtain
/n (log(FuR.Na) (og((Tu))) $ N (0, (1/TER) / ?
(39)
Thus, a (1 - a) asymptotic confidence interval for log(TRR) is given by
VERN
log(fruxn.) 21-0/2
(40)
Letting VIRRRN- = V (1 a)) asymptotic confidence interval for TaK is
Yog RR.
ARRNN exp
(41)
72
25

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Now, note that, if y yau E {0,1] with Ply = = Pc. we have
Var (2)
Var (Y
Vo KRN
(42)
(1 e E (o))
PI PI)
Po(1 Po)
(43)
epi
(1 e)Pp
1
1
+
(44)
cp1
c epo
Hence,
Vug RR.
1
1
(45)
enp1
enpo
n(1 e)
which can be estimated replacing ne (resp. n(1 e)) by Ni = En= T, (resp. No = n N1 and emp1 (resp. enpo) by
Ex=1 Y,T, (resp. Σi Y₁(1 T.)). Replacing Von RR. n by such an estimate the asymptotic confidence interval (41)
leads to the
formula presented in Equation (3).
7.3. Proofs of Section 3
7.3.1. RISK RATIO INVERSE PROPENSITY WEIGHTING
Proof of Proposition 3.3.
Asymptotic bias and variance of the oracle Risk Ratio IPW estimator Recall that the oracle Risk Ratio IPW is defined as
T.Y 111-T: Σ
TJY, -.yT)
'RRIPW
e(x,)
(1--((x)) c(X.)
where the propensity score is assumed to be known. Let us define gi(Z) TY /e()(() and with Z (X,T.Y). Since
=0)==(1) 900 -11-T)\((1-e(X)
M 7
M
1 S
91
and go (2)
the function go and øi are bounded from above and below and thus square integrable. Besides, E [go (2.)] E [Y and 19 (2.)) E [y]] We can therefore apply Thcorem 7.3 and conclude that
Vn((TRRIP
TRK) NO, VIRRWW). );
where
T.Y: c(X,)
(1 U-T--Y I-e(X.) Y()
V VR..PP
TRR Var
(46)
E
Y
E
Moreover,
TY
TY
27
TY 2
Var
E
- E
e(X)
e(X)
{(X)
T
E [ 2
E
e(X)²
e(X)² R ((((())))
E [yaee
R
[rraa} E
E
E
x
e(X)
E (ra))
E [y())]
E
e(X)
2
E3
EE
e(X)
26

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Similarly
(1 (1TY) 6
(Y(10)
EE [y10)]
Var
= E
Additionally, the covariance satisfies
TY (1 (1-7)8)
e(X) E [
TY
(
(1 (1-
E [000
Cov
e(X) 1 - e(X)
- e(X)
TY (1 T
E y
(1-T)Y
T)Y
c(X) 1 e(x)
1 - e(x)
(0)
TY
+ [xa]]
[
[e(X)
- E y
[y(0)]
Therefore, we get that
2 c(X)
E [
E
1-c(X)
VIRIIP
TRx
E
Yuj
E
[0)
As consequence an estimator VRR- rPW can be derived:
2
1 ΣT,1
n ET= (I(((a)
VRR-IPW- 1PW
22
11
E(2,)
(47)
RRRIPWA
T1E Yi)
( Er; vi)
Since we have E
TY [(X]
E
(r<t)ya] e(X)
Finite sample bias and variance of the oracle Risk Ratio IPW estimator Let (2(Z) Ex= T.Y. and To(Z)
e(Xi)
(1-T)), where (Zi...., Zn). We first show that Var(Ti(Z)) Op (2) and Var(Tu(Z)) = =0 (t)::
1-)(X.)
1
n
T.Y; Xi)
Var(Ti(Z))
Var
n
el
T.Y;
Var
(by i.i.d))
2
e(X,)
T.Y;
T.Y
E
E
(by law of total expectation)
n
e(x,)
(((x,)
E (2) 2
E
e(x,)
n
Op
Similarly, Var(To(Z)) Op (t). And we also have:
T.Y;
[y()]
(Z)) IE
(((x))
27

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
E [Z) = (1- (
1-e(X,)) = E (
Therefore, we showed that (2) and (2 are respectively unbiased estimators of [Ya]] and IE [Y(0)] > such that Var(Ti(Z)) = Op (1) and Var(To ()) = Op (H.) By assumption,
m
(z
M and Ti S
M
7
"
thus [Z) and Ti (Z) are bounded. Applying Theorem 7.3, we obtain
21 - 7))2
E RR IPw..] TRRl S
mm3173
and
(1 - 17)
(Far(FRR.H..)) VRR
7.3.2. RISK RATIO INVERSE PROPENSITY WEIGHTING IN LOGISTIC MODELS Proof of Proposition 3.5. The likelihood function L (B) is
(B) = II 1=1
P = t x== x=x))= = II e (I: (1- e (z1;3))²
where we define e 3x {1 + exp(-x -ppp-xTa0 -0]) Now, taking minus the logarithm of this expression, the log likelihood function, denoted In L (B), we obtain,
In (g)
Σ 4 log(e(X;;3))) + (1 4:) log(1 e(X;;))).
The minimization of this quantity be obtained when looking for the root of the derivative, so that obtain the following expression,
T, e
a
In (B) = Σ
e (X;
0
(X,; (1 (X,; B]) 03.
0
T - e
a
In L (B3 = Σ
e (X.; B).
03,
eX,; (1 (X: 03,
i=1
Therefore,
a In (g) = Σ eT:-e(X;3)) and
n
à In (g) = - x, (T. -e (Xx 031
"
(48)
033
1=1
In particular, if we apply 48 for the maximum likelihood estimator Br, and if define X == (1, X), we have:
28

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
0
Xi Ti - (xx:Ba)) 0.
In (33
=0
0g
13-1,
Let Z = SX, Y) and (B, 00. b1, 02), we define 1o and 0, as:
- -
(1-THY 0o
and 0n fiww , Rpw.1 fiw 1/ww.// .0
-
1-1(x-19)
xx,,)) - 01
01 - 0200
where fPww1 = 1 E
TY im1 .(,,
and fiww. = 1 Ei= (1-TY 1-c(XX
One can note that:
1 (Zi, 0m)
Ti X2 Σ 1=1
-i- -e
(x): O.
n
(1 Y
1 n
(1 (1-3
Σ 2 (z. 0m) = Σ
Σ
0.
1 c Xi, (nn
n1 j=l 1 c(X;, Bm)
T,Y,
1 Σ T,Y,
S/s (n) 0n)= Σ
=0.
((Xi,
n j=l e( Bnn)
2m1
Σ (z 0m) = Σ
fIw 1
7Pw.1
ÂPw.O .0 =
1=1
fuww,1
Gathering the three previous equalities, we obtain
(Z 0m.) = 0,
(49)
which proves that On is an M-estimator of type vi (see Stefanski & Boos, 2002). Furthermore, letting 0x0 (Bx,, E [y[[]] E [yū]] E [y[c]]] /E [)],,,, we can compute the following quantities:
E [1b (2, 0x) = E X ((TT)(()))
X E [T e((X | xx]
= E
(Law of Total Probability)
E
x (E x - el ()))
(e(X) is function of X)
0
(Definition of e(X))
Furthermore, note that:
29

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
)
1 Σ
T.Y. e(X.))
E
11 (2, 000
71
and the following holds:
T.Y:
T.Y,
E
(Linearity of expectation)
<(X;)
T1
e((;)
(Y
E
(Independence and consistency)
e(X)
(1)
E
E
XX
(Law of Total Probability)
e(X)
E
1
E
(T)
x
(e(X) depends on X)
e(x)
E
E
y(u
|x]
E xx]
(No confounding assumption)
e(X)
E
E
(1)) xx]
(Definition of e(X))
E[y
This shows that E 0. Similarly, can show that E (v, 0x)] 0. Finally, we also have (2, 000) Therefore,
0.
E [Z,
(50)
We now show that Ox defined above is the unique value that satisfies 50. Let
L(B) = - E T In(e(X,3)) + (1 (1-1) T) In(1 - In(1-
e(x,,))
A direct calculation shows that
X e(x, (1 cxx, VoL(3) = E x (e(X, r)] and V3l(3) - E x X
Since ETT X] e(X, Boo). at 300.
vBL(Bx) = E x
making 830 a stationary point. Furthermore, using overlap we have e(X, 3) - e(X, 2 n²² therefore Vo E IR++1;
v 1 || // (| (1- e(X, (1-e(x,)))
2 2 u 2 2
Since we assumed that E x x T
is positive definite, the Hessian (3L(3) is positive definite, so L(8) is strictly convex. Hence there is unique global minimizer of L(9); since 3ox is critical point, it must be that unique minimizer. Consequently, any solution to
E
X
must equal 8xc. Since the second and third two components s are linear with respect to 6g and 0₂ and since the forth component linear with respect to 02 Ox the only value satisfying (50).
30

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Quantifying Treatment Effects::
Estimating Risk Ratios via Observational Studies
We want show that for every 0 in neighborhood of 6o all the components of the second derivatives
0² t (z, 0) 0²0
are integrable for all E (1,4). Since 0 = (3,000 01, 02), we need to show that for k E (1,4) and 8, E {0,2) the following quantities are integrable
0²
0²
8² vn (z, 0)
v2 (2. 0)
(z 0)
Un (z, 0)
00,00,
00,0
0030,
0
One can note that the first three quantities are bounded by and therefore integrable. Hence, suffices to consider
0²
8 2h (2 0)
where & E (1,2,3), since 2/1 does not depend on 01. For k 1, direct calculation yields
02
03 1 (z, 0)
Xm e(x, (3. 2ex, Xx X1 Xmm
By Cauchy-Schwer or Hölder's inequality,
E X X
s [Xx)) 1/2 E E{(X)) ( 99
Since X is sub-Gaussian, it has finite moments of all orders, implying integrability of Xx X, Xm/ For
3, we similarly get
82 2/2 (z, 0)
S Y exp(2 XTp(2XT)))
() 1
02
and
02
2 Wz(2, 0)
s Y exp(2 3) 3) XT
020
Hence, it remains to show that
Y exp(228 B) Xx X
is finite. By Cauchy-Schwarr
E y exp(2 X B) N Xi]
s [E [B[oxp(4X X (3) (X, x7)3
Since X is sub-Gaussian, its exponential moments are finite. Specifically, for some o >
Texp(Av [X)] s exp
xa 3 AY R. VER,
which guarantees (S X [)] is finite. Moreover, Gaussian random variables have finite polynomial moments, so E[(X, X] is also finite. Therefore,
Y exp(2 B) Xx X
is integrable.
Collecting these results, we conclude that every second derivative neighborhood 0
aog Ub (z, 0) is integrable for all k E {1,4] in
g2
31

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Define
Oし 08
(
E
and B( E (2
x=
Next, verify the conditions of Theorem 7.2 in Stefanski & Boos (2002). To do so, compute A (0xz) and B (0x0) Since
- XX O
( X
-1
(2.0)
[=)(X
(51)
00
TY(1-((X.)) X
-1
c(X,3)
02
00
We obtain
Q
C10
-1
A0oo)
Gp1
0
-1
0
E [Y(o)]
E E
where:
Q E -
C10 E|| ((x)) and coi E[(((1e(())) ((
which using Schur complement leads to:
Q
0
0
0
-1
0
0
- (0xc=
C0 Q-1
0
-1
ETY [
BD
ciu E
y
Col
Q-1
E|y [
(
Y
E
Regarding B (0xz), elementary calculations show that
C10 Var
Co1 [ (1)
B(0x))
-c10
[1-THY
T-e(X)
(01)
0
G
[y [0]
Var
TY
a(x)
0
Based on the previous calculations, have
(2, 0) and its first two partial derivatives with respect to exist for all and for all 0 in the neighborhood of 9.c.
00g For each 0 in the neighborhood of loo. we have for all & E [1,4]
0² (2, 0) is integrable.
A(0x0) exists and is nonsingular.
B(0x0) exists and finite.
32

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Since have:
(T,, =0 and On S 0x.
Then the conditions of Theorem 7.2 in Stefanski & Boos (2002) are satisfied, we have
vn (0n, 000) S
Since we are only interested in the bottom right term of the sandwich term we only need to compute u A40x)(0))) B(0a))((()) ua+3 where u443 is the last vector canonical basis vector of Id+3, Hence,
[(000) d43,043
=12::40001111000014110001111 ud44
= -= ((0x))
Noting that T A(0x) = ((aYON
(FYCN) 1
TT 9i0
Ba Sol
T
Q
BYUL
where we used that
(Q-1)T= = since Q is symmetric. We defining
u T 1043 = = W2, W3, Wa)
where:
E (1)]
(
W1
[0) c1 + [[](O)]
Q 2
c
Co EY( 0,
Wº
(1) T +
CYY())
110
(()
Col
Q- + Var
(1 (1-T)Y
1 (1-e(X) e(x)
E B E[Y(0)
+ E [raa]
W3
(1) T (O)= Ci0 EI (0)
C01 Q-'c0 co1
(1)
Var
c(X)
TY
(0)
[()
and W4- 0. Finally, gathering all the terms we have:
W [[Y W [{Y(O))
-B0m)
d43,d+3
[0)
(1) 1
E[
[0]2 C10 (O) CO1
C10 E[)(
(I) TT
E| (0))²
(O)
Co1
Cu1
E|Y (0)
(1-TIY)
E|((1)
Var
(
+ Var
+
T--(X))
(L (0)))²
[[)(1
E[Y (0))2
33

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
One can note that:
Var ( (1 (1-1)
(
Var
e(x)
TY
(1) (Y)
(D) BY
(1-TY Var
I-e(X)
Var
x(X) +
TY
1 (1-e(X)
OY
(0)))
+
E (
E[((1)}
TRR
02 i-c(X)
E
I
E(X)
FRR
E
[r
E
[
where for the last equality used that:
TY
(Y
[yr]]
(1 - T)Y (1-TT))
(Y(0))2
E [x00)]
Var
= E
- E
and Var
= E
e(X)
e(X)
- e(X)
e(X)
Finally using calculation we did previously we get that:
(2 E
e(x)
I-e(X)
C10 EY)
Cu1
MLR MLFE = -RR
E
E
Y(U)
-RR
E Y
VINc-Pw
7.3.3. RISK RATIO G FORMULA ESTIMATOR
Proof of Proposition 3.7.
Asymptotic bias and variance of the oracle risk ratio G formula estimator Recall that the oracle risk ratio G formula is defined as
((n Li (io) (X;)²
RR.G.In
where the response surfaces и(0) and ((1) are assumed be known. Let define (Z) = (())(X,) and go(Z) = (((o (X,) with 7 = X. Since 91 (Z) and (Z) are bounded, they are square integrable. We also have that E. [go (Z:) = E. [y and E [gi (Z:)] = E { We can therefore apply Theorem 7.3 and conclude that
Vn(TTRGa TRR) - NO, VERRG),
where VERG
TRR Var
xi(
Ha( E E(y
As consequence estimator VRR. can be derived:
(x(()) 2
22 TRRGa
(X,)
(x(X,)
(X,,
VRR.G =
Σ
(52)
IE Y,
ΣT.
Y,
ΣT, Y,
75
Finite sample bias and variance of the oracle ratio G formula estimator Let Ti(Z) Σn M()) and To(Z)
34

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
1 Ei=1 (Xi) where Z (X1,.... Nn). We first show that (ar(T](Z)) =0p (+) and Var(To(Z)) Op (H)
1
n Σ Emoo(xs)
Var(Ti(Z))
Var
n²
1
(ar(H(n)(XX)))
(by i...d))
n
E [(mn))()))) - )
(by law of total expectation)
M E 2
(ma)((,)) S M)
1
Op
Similarly, Var(To(Z)) = Op (1/n). Since we also have that
E T [2) = [yo] E (T)( = E [00)
Therefore, we showed that (Z)) and To Z) are unbiased estimators of [y and E [r
0 such that Var(Ti
Op (1/n) and Var(To ()) = Op (1/n). We also have that To (Z) and Ti (Z) are bounded:
mo S (o [ < Mo and T2 S M1
Applying Theorem 7.3, under Assumption 7.2 obtain:
2M, M²
2M M,8M,,MM (M, + Mo)
E uT TRR| 1
and
|Var (fr.ffR, VIR
nmn 3
m
7.3.4. RISK RATIO G-FORMULA IN LINEAR MODELS
Lemma 7.6 (see, e.g. (Seber & Lee, 2012)). Grant Assumption 3.8 linear model. Let Y(t) = (C(t), P1o)) E Rd+1 and (1, X). We reurrange the Y; and Zi so that the first n, observations correspond to = 1. We then define Y1 VI....;Y,,)) and Yo (Yns+1..... Y,)" as well as Z₁ (Zi...., I,,))" and Zo (Zni+1...., Zn) Then
fort E 10, the linear model can be formulated
( = zT T())
700) + E(())
BlE [2 = 0, Varle Z2 02,
and the least square estimator is given
7(0)
Z, Z₁
Z, Y,
74
Proposition 7.7. Grunt Assumption 3.8. Let EEin T:)/n and for all E (0, 1}.
((1) Em 11= 11
(53)
Defining [ and E, Var(X = t), we have
Vn 0xc) N (O,E))
35

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
where
Var T
%,00 Z(1)
F 0)
(1-c)
0
0
0
Var[2|T=1]
E|ZTT - 1]
0
0
0
0
0,
110)
000
7(0)
E
0
0
qQQ
0
0
I-e
1(1)
7(1)
Qq
0
0
0
0
ê
0
0
0
e(1
c)
1 + v E 'n
E
with Q
Ex
U
Proof. Using M-estimation theory to prove asymptotic normality of the 0n, we first define the following:
Wo 2, (TT
(1 (2- -
T(Z - -1))
(T,2,0)
(2
=
(1 (T -Z (02 - -o())
1/ (T,
(Ze((1) ZZ" (0s ((1)))
(T,
T 04
where 0 = (06. 01, 20 03. 01). We still have that ,n = (Z Z(1), Ro), f(1) é) is an M-estimator of type vù (see Stefanski Boos, 2002) since
UT, 2,, 2,, =0
We now demonstrate that E [U x0) = 0. We directly have that E (T, Y, 0 x] = 0. For the other terms we have:
E [bb (T, (T, x0 = ET (2 - E|2 1])]
[T - 1)) [T]]
E (E
=1]]
[T E
- 1]]]
IP 1B
E| 1J)
0
We also have that:
[bb (T, 0x) = E (TZE())
= E LE (Te 2
= E [ZE [ca] 7,, 1]]
= 0.
Similarly, we can show:
E [T,, Z, 0].)] [00]] =0 and [T, (, =0
At this point, we note that since (T, Z, 0) a linear function 0, A00 is the only value of such that [ Z,0)] =
We proceed by defining:
Ori 20
(0xz) E
and B(0.o)
00=0.0
Next, we check the conditions of Theorem 7.2 in Stefanski & Boos (2002). First, we compute A (foc) and B (f%.). Since:
(1
T)
0
0
T
0
Og
00 (T, 2,0)
-(1
T)ZZT
-T77
0
-1
36

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
obtain:
(1
0
0
e
0
0
A 20,z)=
-(1
e
Qo
0
where
Qt =
[
7
0
0
eQ1
0
0
0
which leads to:
0
0
0
1-c
0
0
0
0
A- (0x0)
0
0
Qu
0
0
0
0
0
0
0
0
-1
Regarding B(0,,), since we have 7(1 - T) = 0, elementary calculations show that:
B(0xz 1,,2 = B30xx)2 )2.1 =0 B00~) = (80x0))
B(0 = o( )4,1 =
and
B(0 3,3 = B(0 3,2 = 0.
Besides
B(0 E [T B| =
E [T E 1))(2 E 1
E [TE [Z -
=[z]T=1)) -)
[T' E [2 -E [])(z 112--27--122--37-1111---1111
Var [2| = 1],
and similarly,
B(0x))1, = (1 e) Var [ = 0]
We can also note that:
B(0xo)4,4 E 22
TZ [1] E TZZ 201
E TE ZZ
(1)
PT 1] E ZZ & TT =1
= = 77 ( T = 1,7
T
= =002E E
[= = ===
and similarly,
B(Ox)3333 = (1 - e)o2Qoo
Finally,
((0x0)2,4 B(0
E. [T² =[ZT= [w)
E. [T - F - =1])2T<p
=I [T 1 - -E = Eny|T =
eE [ | 1))2 E[qulT 1,2] T [T
0,
37

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
and similarly,
B(0 B(0xx)3,) O.
We also have that:
b,, (0 )5,
[ =[
E [ -
- - -1 -
=e |Z|T = - cF - - = 1| -1 =
and similarly,
B(0cc)1,5 B(0 55.1 O.
We also have that
O0 B(0 5,,4 F. [(T e) [()
E. [())<(0)) eF (TZ
(1 e) [TZE(o)]
(1 e) LE [T) 12]]
(1 e)E ZB [c()) Z 1]]
and similarly,
B(0 3,5 B(0 5,, 0.
Gathering all calculations, and since B(0x0)5.5 5.5 = e(1 -00)),, e), we have
(1 e) Var [ = 0]
0 Var [
0
1]
0
B(0oo)=
0
0
(1
Qu
0
0
0
eo Q
0
0
0
e(1 e)
Based on the previous calculations, have:
(2, 0) and its first two partial derivatives with respect exist for all and for all 0 in the neighborhood of 6...
For each 0 in the neighborhood of Ooc. we have for all i, k {0,2):
82
wha (z, 0) 5
00,00,
and is integrable.
A0000) exists and nonsingular.
B0xx) exists and is finite.
Since we have:
2 (T,, Z,, 0,) 0 and 0,, Ps 0...
Then, the conditions of Theorem 7.2 Stefanski & Boos (2002) are satisfied, we have:
Vn (... 0m0) $ N (0. =(00))00)
38

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
where:
Var(2|T==0)
0
0
(1-e)
Var[2[T=1]
0
0
0
A(0,0)-' ((0))(00))
Q
0
0
0
o
Q
0
0
0
e(1
e)
Proposition 7.8 (asymptotical normality of FRR,OLS). Assume have linear model then we have:
Vu(iRRROI TRR) S. (0, VRR.OIS)
with
1+ (1 (1-e) -oll
+ e² ||1 nol (1- e) [ (0)
VRR-OLS TRR
((0)
o²
[
E
E
Y
Proof. Let Ba) and G(j) be the parameters obtained via fitting an ordinary least square method on the treated individuals only, that is
(B(1), Gay) E arg min
E - -(() - Xi) 1T;
(54)
(a))P() 1=1
Similarly, let (0) and êgo) be the parameters obtained via fitting an ordinary least square method on the control individuals only, that
(Bio) E arg min ((o)
Evr -c(o)
c(o) - -B0 11,;
(55)
An estimator of the RR using the G-formula approach is thus given by
Em cc)) + X,T (w)
RR.OLS
(56)
Σn (1o) + X,T 3000)
ê(1) + +XTB11)
(57)
2 + (T
Besides, note that assuming a linear model implies that
9(1) + E[X] B()
tRR.OLS
(58)
C(u) E[ B())
39

---

Quantifying Treatment Effects:: Estimating Risk Ratios via Observational Studies
Let, for all i, 2, X1) and Tir) (c) for all j E {(,1)) Expanding the following difference, have:
VTi(TRROOSS - -RR) Vni ( 4 + (()
(a) + B[X] 8())
(59)
C(O) XT (B)(
((0) + [[[] B10)
VT
(0)) (0()))
+x1 +X1B())
1
1
(60)
((()) + ETT
((0) + [[ B(0)
Vn
3100 (Qn) + +X (())1) -(1) E[X]] 9(1))
(61)
((0) + [[X]
vn (TT 9(1))
(62)
(T B[Z] B[Z]Tr()) T())
Vn.
(0
((1) (ZT(())
B[Z] B(Z)T()))
(63)
B[Z] 7())
Z
(1) ((1)
nn ZT 110 (0) E[Z] T 7(0)
(B(Z) ((0) 2T T())
(64)
/n
(Z (()) E[2] 7())
(65)
B|Z] 7(0
Besides, we have
2 - E|2] = eZ(1) + -1-
e) - e)] 0]
(1 e) (Z(1) E 0]) +e (Z EEZIT =1]) (Z(1) Z(o)) (e e)
= =c(On - -,x))
where ( <= [11 -e) elat1, Oa+1, Oat1, (2(1) % E (+++))x4(++1) and
(1)
F 0)
Z(1)
E
O,
5(0)
0x
7(0) 7(1)
711)
e
Note that for all E {0,
Z ZTAO 310
[ -E[Z] 7(0)
910 (7 B(Z]) + B[2] (f(e) 7()
5(0, - 0x)) + B[Z]T T (ft) 7(0)
=TT0 - 0xx),
with
(1 e)ito
e910
dqt)
1 1
t= [
(Z Z T500
40

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Therefore
2 T (1))
Va((TRROOLS - -RRR = vn
(E|Z]* 7() 2T 9(o1)
2T (0) [||] 7(0)
Vn
(Z))())
7(1) E|Z] T 7()))
E3 2 7(0)
vn
Ü (1)) (0n - 0zo)
E Z Y(0)
T (1)
Vnl 7 7(0)
d(u) (0n 0xo)
3100 B|2] 7(0)
7(0)
Therefore, we get that
2T (T))))
Vn(TRROOSS TRR) vn
a(())
610)
(0n 0xc)
7(0)
Z (1) E[Z] 70)
According to the Law Large Numbers,
2T 7()) (T)(
EiZ] 7(1)
E (() 6())
(<(0)
P
a(1)
[(0)
2T (T) B[2] 7(0)
E E M()
E[Z] 7()
E[Z] 70)
with, for all t (0, 1}.
(1 (1-e) e)
e7(()
1 1=
o(()
2=1 [
(E
[| 1] B[Z[T = 01) 7
and
(1-e)7(1)
(1-e)
(1
E %()
E, To
E[2] 70)
E,2] T(o)
[[[2] 7(1) E[2]
E[2]
0
3|2 TrO0)
(66)
(())
B(2]
[[7] Tre))
2:0n [B)
Yra
E 7003
E[ 700
According to Proposition 7.7, letting Q E.. [2 1 =1 1 for all (0, 1), have
Var[Z|T==0
0
(1-e)
22rl2
0
Vn(On. -Ox)) 4 (0,E) where Σ
0
0
1-e
0
0
0
0
0
c(1
c)
By Slutsky's theorem,
Z 7())
6(()) )
Vn
â(1)
(0n, 0x) $ SN [I
(67)
E|
7()
TT
E[Z] T(o)

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
We now compute the covariance matrix
E
7(1)
7(0)
7(1)
7(0)
(1
e)
e
(68)
2
B|Z]
7(1)
E
7(0)
Var[2[T==]]
B[Z] 7(1)
E|Z] ?(0)
Var[2|/==1]
T[2]
o²
Z
0²
E
+ +e(1 ee1-e)
7(1)
7(0)
(69)
e
7(0)
e
7(2)
9-
[E|Z]
(1))
7(1)
E E|Z]
7(0)
AAT
where A E|Z T
B|Z 7 0). This variance be rewritten as follows. Summing the first two terms and the
last term in (69) leads
701
7())
E|Z] 7(1)
(1)
(())
where = (1 - e) Var / = 0) +e Var T 1) +) e)A Let us prove that J Var( (Z). Letting Z, the components of 7 for all 1 si S d + 1, by the law of total covariance, we have
CoviZ,, 2,] = E[Cov
+ 24 (7,111
(70)
with, since TE (0,1)
T (1 (1- c 2,, 2,
(71)
Besides, since B[2] (1 e) TT e T 1]. we can compute the deviations from the unconditional mean:
Z T= 0] E = E|Z T 0| ((1 e)E|Z T 0] cE|z T- T= 1))
(1 (1- - - T
= -e ( 1 E| T = 0J)
= -GA.
and
E T= 1] B[Z = -1 T= +E TT=11 1))
(1 e) (E[Z 1] T 0))
=11 e)A.
Now, we can compute the second term in (70)
Cov[B/2//7,,|1,,177) = -[18111111-
(72)
1 -
(73)
+(1 -00-01/77000000222222000------0-
(74)
c(1 - +
(75)
e(1 - -)
(76)
Consequently, according (70),
Cov[Z,, Zy] e Cov Cov 2z,
1] +
-(1-)))) e)
(77)
which leads to
= (1-e) Var (2 T =- +E Var =
(78)
Similarly the last two remaining terms we have for t E {0, 1):
E E
1
|B |
Z
700
(E Y(r) )2
1
E[2] Q. Q
E
7())²
42

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Note that we have = = 0] and that for & [0,1),
", 1| ( (
T
U Ei
-u Ex
) (1)
m1.) =1,
- Vz
Ex
and
)
1+ vi 1i
- (m)-)
11
-21
=1+ + |1/1 - vells."
and
(1)
(1 U E
-h x-
B = 1) Q, | - =
vi
- vt
v1-1)
1.
Therefore, we have
T eo E[2] = - 100
1))
=<|||B|Z[7 = ')1lo, + (1 61 + 20(1 - e)[B = = 11)0,
= (1 e)² e² (1+ 11 - No ( + 2e(1 c)
= =1+ + e²1111 - vol voll? xo
and similarly B[2] Q- = 1 + (1 e)211 Poll Finally, noting that for all t E {0,
00
Var [X] )
E[Z]J70 7(0) [y(0]]
and Var [2] =
we have, letting E = Var [X]
B()) E[
B(o) E [
0 1 + (1 -
e)
volls-
1 1 e² | 1o 20122,1
VIRGGUOS = -2
TRR
eE [y[u]]
+
(1 - e) [y
E
Lemma 7.9 (Comparison of the asymptotic variances of TRRN and ÁRR. under a linear model). Grant Assumption 2.2, Assumption 2.3 and Assumption 3.8. Recalling that VaRsOIS (resp. VERGous) is the asymptotic variance of the G-formula when oracle surface responses used (resp. when they estimated via OLS), we have
/1|5tn|1 1 + +
||| 1 + 0 - (1-e)E \(0)2
VERN
TAR
(79)
eE
[(a)]
(1
B(1)
3(0)
VERcOS The
0²
CE (eElr
(1 e)B (moope))
(80)
E [y
E
[
ViRo + T
1
1
(81)
cE 2
(1 - e)E |(1o)
and
3(1)
3 (1- (1-e)H]
VRRN VeRG,OLS Txx
e(1 e)
N 0.
(82)
cE [YU)
e)E
Y
43

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Proof of Lemma 7.9.
First equality The variance of (*) satisfies
Var[Y(o)] = VVrleeu) + x (1) + E()
Var[x B/4) =()
c(t) is constant Bienaymé's s identity
Var[X* ((4) Varle(o)) + Cov(X E())
= [18()]|æ +
(by linear model)
since
Cov(X" E()) = B[xT (())(()) - E[X B(o) (t))
=E[X" (o X]] E[X (to) [[
(by total expectation)
O,
E [ = 0,
and, using Eve's law, Var[s()] B[Varle()) Var[Ele 02 Thus, VERN satisfies
(Vr(Y
Var(Y
VRRN TRk cE[y(Up
(83)
(1-e) E
|| + + cE (1- E (
|1 11 + o²
TRR
(84)
Second and third equality According Proposition 3.9,
1 (1 (1-e)?llus -voll
+ e² 1 vol 2
VER,GOLS
8(1) E
B
+ 0
TRR
[
E
Y
eF [y(]]²
(1 - e)E [y
Since we are in a RCT setting, we have that v1 = =o and Ei = Vo = E. Therefore
V VRR.GOL
B(1)
B(o)
1 + 0²
(ARlrry)² (1 - e)TE [V(0]]
2
E
E
[Y
E
The first term corresponds to the Oracle variance the G-formula. Indeed, for all t E {0, 1}.
Varluro)XX)) = Var (0)
Var[B[c(o + B[X B[o| + [X]
Var(c E[X [[]]X]]]
-Var[E(X (MrTE(TT(001X1 [1]
VVr[x TBeerl
= -13
44

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Besides, the covariance between 141 (xx) and Mo (x) satisfies
(x)(m -Blun)
=B((eay + X (C)) + x B(o))] - JE
=Bleay + _((n)_ B(oo] + B|c Ba))
+ +E|x B(o) X B(n) - E[x (E|
-E[X B B(n)
=E
X
Σ X
Σ
= -(1)))BB11)>>.
Therefore,
(X)
Ho (X)
(ar(Hn(())
War(Ho(( (X))
Cov(Ho(X (X), 111 (X))
2 VIR.G TRR Var
E y
E Y
TRR
E [1]
E
E (()) E [YD]
B(1)
(O)
(3r0)= B(1)s
TRR
2
IE [Y
+
E
[[co]
2 E [Y [ra]]
9(1)
B1o0
TRR
[r)]]
EE
Y (0)
Σ
Last inequality A simple computation leads to
VRRN
VERG
||s +
|| oo
TRR
eE
[r(]]]
e)E [r
2
3(1)
B(0)
+ t
+
(1 - e) [(oo]
E (
E
[0)
elR [rcu]2
|1: 1
||
11
2(B()) Bo,)
e
e
+
E (r)
1 e
2
E [[rW] E. [Y
3(1)
B(00)
c(1 e)
eF [u]]
(1 E
(
7.3.5. RISK RATIO ONE-STEP ESTIMATOR
Proof of Definition 3.10. We will use (Kennedy, 2022: 2015) notation in this proof. If you are familiar how to compute an influence function, note that it is very similar to compute the derivative of function. We define our estimand quantity
E
E
T
= 1,
11
E
1/0
45

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Quantifying Treatment Effects:: Estimating Risk Ratios via Observational Studies
We now compute the influence function yp of
ub.
1/11
IIR (4/1) 1/o
TIF (160)
x/1
= IIF
(+b)
IF
250
%
IF (41)
(4/)
40
According to Example 2 in (Kennedy, 2022), we have
(15)) = (1( + Y (x())
1/2
e(x)
II (160) = mo(x) + +1 Y no (X)
and
1/0-
e(X)
Therefore,
IF (11)
IF (Vo)
8
10
v0
40
(X)
c(x)
(X)
4/1
po(X) + (1 - T) - V-
1-e(X)
400
VD
1/10
(I(X) T (x)
mo(X) + (1 T) x-
c(x)
1/
I-e(X)
4/0
400
[1 T (X)
(o(X) + (1 7) I-((X)
(X)
e(X)
v0
200
As referenced in (Kennedy, 2022) regarding the semiparametric von Mises expansion, consider the functional th P -> TR, where P' represents the true data distribution and P its estimation. The expansion is formulated as:
*(P) (PP) / (zz: P)d(P- P)) + R2(P,
(85)
for all distributions P and P. The influence function (p(:; P), associated with vh, is function with mean and finite variance as defined by (Tsiatis, 2006)
/ 1l
/ / oo,
and and (2, P) denotes a second-order remainder term. According to the expansion in (85). most plug-in estimators (PP) biased the first order, evidenced by:
(86)
= =PP) + / ((2; + R2(P, P):
since I ((; (z) = 0. Therefore, a first-order approximation of (PP) is given by ((P) + I p(z; P))P(=) which can
be estimated
PRR-OS = -/+ ((4))
i=1
(X,) + Tí (X.)(X.))
Ho(X,) + (1 - T₂) (X,)(X,)
n
1 +
e(X;)
1-e(X,)
3/0
Σn (Xx) + (1 - TT) Yi-)o(X,)
1 (1))) + T, Yi-)
1
1-e(X.)
e(Xi)
400
30
En. iin (Xi:
En (X,) +
m (X,) + (x(x)
1 1-6(X.)
e(X.)
1
1= (Xi)
Ei=rAo(X,,
46

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Proof of Proposition 3.11.
Asymptotic bias and variance of the cross-fitted One-step estimator Recall that
(P) Ppp P YY X, T = 1]] Ep [E Y X,T = O]]
4/1
(87)
4'
((P) DE in (X:) 1/1
(88)
E. (X)) D0
(Z; (X,, + Ti Yí- (X.))
(o() + (1 T.) Y-- (X,)
((X,)
(-2(X,)
(89)
4/o
vo
where P represents the true underlying data distribution and P the distribution where oracle quantities have been replaced by plug-in estimates. We express (6()) as follows:
((P) ((x:P))P(() RR(P, P).
where R2 encapsulates higher order remainder terms.
To elucidate, we rearrange find 4 ((p) (P)):
- - (PP) R2(P, - / p(2; p(z;P)dP(2)
(Zz P) 1
n1 Σ P)
(0<;; p(( PI)
- aP(:)
R2 (2 P).
Recalling that ARR-OS = (p)( 1/ Σ (0;;; P) and TRR (P), we have
TRR = (p) + Σ (Z, P) (P))
ARR.OS
(90)
(lZ P)
(91)
7
(o< P) - - - ((((::)(()))))))))))))))))) - Pi) aP()))
(92)
n
+ (P..
(93)
The first term is sample average of centered i.i.d. terms since, by definition (86), J y(=; P)dP(z) = According to the central limit theorem, it converges normally distributed random variable with variance Var((p(Z))/n.
Following the work of (Vaart, 1998), consider the second term in (93), that
a Σ
- (p(:: -
pP)(s)) aP(=).
Since estimator is built on cross-fitting strategy with K folds Iy... .Ik, containing respectively n....., ng observa- tions, the above quantity may be written as
K
-12;;---))<<(;pP)))) / ( p02: p(2: dP(z).
A IET
47

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
where p-k corresponds data distribution where oracle quantity replaced by plug-in estimates built all observations except those in Ig- We denote this set of observations I-k. We let ( = sp( Zz P-k) and
U = (p(kk) P) -
(94)
where pP(x) is the empirical measure over In. The quantity of interest can thus be written as
1 K
1
-Z))) - / (p(x;)) dP(z)
Σ
nkUk.
(95)
T1
nu k=1
A= TE
The expectation and variance of Uk satisfy
E
UU I = E
(p()) - P) (0-h - 4) I-
(96)
E
( + - 4) I b -E [P( - ()) [-+]
(97)
E [B(x<(()) p()) -R (Z) (x(2))
(98)
0,
(99)
and
Var [Uk II [] = Var
(P(()) - (8 - (p) I
P)((
(100)
Var [P(k (p- () - -P( * - -+) I-
(101)
1 Σ (p~ (2i) - ((2i)) I
VVr
(102)
TIk
i=1
1 Var [ (Z) (( | I+]
(103)
71k
v (()) (()))² I]
(104)
7&
Let > O. Applying Chebyshev's inequality leads to
Un ETUR I=
P
2 a
I_n
(105)
UUk
km]
Unl
1
IP
2a a
I_
(106)
Var
Im
a
Thus,
llAl v
IP
2 a I_k
<P
2 a
I_k
5
(107)
EE (()) x(Z))² I_m]
Var
I-
which leads to Finally, taking the expectation on both sides leads to
E (()=*(2) - (Z))2 I
P ((UR| Vng 2 a I-
(108)
a²
E [ +() - ( (2
P ((UR) Vnk 2 a) 5
(109)
a²
48

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
According to (95), the quantity of interest takes the form
K
K
7
Uk
(110)
k=
Hence,
E ( * (Z) -Z) (Z))²
Tkk
n2 vn.
vn
U
I a
2 1
(111)
n2
a²
Therefore,
THk vni
E
( (0=*((2) (22))
P
[UA
2
(112)
a²
k=1
k=1
K
7k
K
E
[
k (7) (Z)) 2
P
vn
Unl S ak
2
(113)
a²
which proves that k=1 U2 or (z) as K is fixed and -- is L consistent.
Regarding the last term, note that
R2(P, P) = (b(((()))))))) -PP))((P) (PP) -)())
+ / P)dP(s)
(114)
=v(P) - x(P)+ + Blp(Z; P)]
(115)
(X) + T (x) 8(X)
(o(X) + (1 T) b Y-
-V(P) - (PP) +E
T-E(X)
(116)
100
(X) + T (X)
Ao(X) + (1 - T) Y An
E
E
((P) (P))
+
x(X)
1- 6())
(117)
S
120
[x( - (X) + T Y
(P) - x(P)+ +
x(X)
+ 1/1
(118)
30
4/0
E [xo(X) 1o (X) + (1 7) (x)
I-e(X)
(119)
100
400
Note that
E (x) - mi() + - (x)
E
((()) (M - - (x)))
(120)
((X)
Positivity
E (X) -xx()))((X)- -
(121)
x E [a(xx) - -((x)) 1/2
(a1(X) - ()
Cauchy-Schward
E
(122)
7
= 0p
Vn
(123)
Similarly,
Pn (X) -i(() + Y - (X)
EE
=0p
(124)
8(X)
49

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
For the last term in (119), since vi /11/100 and vi
/ /vo.
1/1
4'0
1/0
4/1
+ 2/
1
4/0
4/0
4
Vu
150
1/1 20
t0
+ 2b
200
who
1ho v0
5o tho
4
vi 1/0
)
((
1
1
1
1/1 - 4/1)
+
Vo
440
v0
40
((vo
v/0(1/1
1/2
- 20800 clo) vb)))
vodio
By assumption, we have
(vo -0) - =bi) = E (X)
- x1(x]]
(125)
s (LE (x)( NN (X)) (x) - (I(X))
(126)
Op
(127)
Ti
and
(6o - --00 ('o) (E [xo( E [xo(X)))²
Op
vn
By assumption, IE [(oo(X) [x())) tends to zero. Thus, v0
BB tends to tho [x)o(X)] Thus,
$ tb +
W1
1 10
=
Vn
40
which implies that R) =op (n 112). Finally,
1
1
Vn (fRROS TKR)
(Z; P) + +p
Vn
and thus
Vn (Fress -TRR) AN (0, Var(p)))
where
[I(X) + (X)
Mx(X) + (1 - T) (X)
Var(p) = Var
e(X)
T-e(X)
(128)
40
(1(X) + T (X) c(X)
(X( + (1 T
Y s4n x)
2 Var
1 x(x)
(129)
(i)
To (Z)
- Var
(130)
EE [1)]
E (0)
50

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Using Bienaymé's identity, we get
(i(2) (0(2) =
(X)
(X)
T(Y (x))
(1- (0
Var
Var
+ Var
B [
E [y
E [Y
[
E [[ e(x)
E [Y(0)]] (1 e(X))
(131)
(1(X)
T(Y -1
(1 T) - -o((X))
Cov
(132)
E Y
E Y
E y e(X)
E | (1 - (eeee(()
The second term can be rewritten as
T(Y (x)
Var
(133)
E ] e(x)
E [Y(0] (1 e(x))
T(Y (2(()) B[yWu]e[X)
(1 (1-T)) T) - mo(x))
T(Y -1()) E E((()]((X) E [ (1 coxee(())
(1 -1- (1-T)
Var
Var
Cov
E Y(0) (1 e(X))
(134)
with
T(Y ((Y-)1(X)))
| ( (1()))
[
T(Y - (x)))) E [Y(U] c(X)
Var
E [ e(x)
=E
[r e(X)
-E
(135)
For the first term in (135),
T Y (x)
E
e c(X)B [ra]]]
27
- (x))
Consistency
c(X)E [r
y()) (X)) e(X)E [Y(]]]
E
E
X
Total expectation
2
Y - (x)) c(X)L [
E
E
T
XX
T is binary
y(1) - (X
E
E
1 1[T-1)
X
T written an indicator
e(X)E [y()]]
1
L(r=1) (r()) (x))) xx]
E
E
e(X) is function of X
e(X)E (())
Var (X
E
IE [|T==> x]
Uncounf. & ur (.) is func. of X
el e(X)= [r]]]
Var (x)
E
e [yaan x(())
Definition of e(X)
Var (x
E
e(X)E 2
51

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
For the second term in (135),
T(Y- (x)) rra] e(x)
E
T(Y (x()
E
Consistency
E [Y() e(x)
T y() - (X)
E
E
x
Total expectation
e(X)E [Y(]]
e(X IE YQJ] E ((Y(1)) 1u1 (X)) x]
E
e(X) is function of X
e(X)IE [Yaa] (x)(X) -1 -
e(X)
E
Uncounf. & (1(() is func. of X
0.
Therefore
T(Y - -xx))(
Var (Y(I))
Var
=E
E ([r(j]e(x)
c(X)E [(1)
and similarly
(1 - TY ( mo(())
Var (0 (1 e(X)) [Y [loo)]
Var
E [ (1 e(X))
Besides,
T(Y - 11 (X)) (1 -Y- -
T(Y mi (X)) (1 - - mo()))
Cov
E
E Y()) e(x) E ( (1 e(x))
E (araa e(X) IE ( (1-)(X))
T(Y- - (z(())
(1 - -)
E
E
E [V()]
E [ (1 e(X))
0.
Gathering all these results into (134), we obtain
T(Y (x)))
(1 (1-77) (- mo()))
Var (Y
Var (r
Var
E
+ +E
E B[ e(X)
E [ (1-e
e(X)E (1)]
(1 - e(x)) [Y (00))
In order to rewrite the last term in (132), note that
Cov (M((X), T(Y - -1 (X))
(X) T(Y - (i(())
T(Y-
- u1 (X))
E
[X]]
e(X)
e(X)
e(X)
T(Y - (x)(
E
(X)
e(X)
E
EE
(TY(1) us (X
e(X)
(I
T(Y(1) mI())) X
E
E
e(X)
e(x)
(1(()e(X)
E
(mr(X) - vx
c(X)
0.
52

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Similar calculations leads
(X) [
(x( T(Y - (I())) (1
(1 (1- (Y-T)(Y- - -1-x)) [ (1 - e(x))
Cov
-0
[y E [ e(x)
Finally
Mi (x)
(x)
Var (Y
Var
V RR.OS
Var
+ E
+ E
THR
E [Y
E [
e(X)E {
(1-=e(X)) (1 - e(X)) IB [Y [0)]
An estimator Vnn,os can be derived as follows:
2
22
V Vnnos
RRR.O..n
A
Σ A
(136)
π1
n
where
A f1(1) (,(0)
5(1) S(0)
with the intermediate for 10 quantities defined as:
Y, (X,,) (t(t) = (X,,)) 1T= (X,,
and Š(t) = M (t)
j=l
7.3.6. RISK RATIO AUGMENTED INVERSE PROPENSITY WEIGHTING
Proof of Definition 3.1 We use the derivations established the proof of Proposition 3.11. Indeed, we showed in Section 7.3.5 that the influence function of db
- can be written:
3B 7 =
p) 111 (X) + (X) e(X)
Mo(X) + (1 T) (x)
I-c(X)
do
4'o
Using Equation (85), and knowing that p(2; P)dP(() = 0, we have:
-(P))((P) - / Pzxx ((lP))(P-PPCC P)(2) + R2(P, P)
P) P))((()
p(Z;; P) - 1 72 Σ P(
n
(p(Z;; () - - (p(x; p(z;))) aP(z)
(2(P.P).
As outlined (A. Schuler,,, 2024), in the estimating equation approach, we assume that the efficient influence function for any given distribution depends solely on the target parameter 1b and a set of nuisance parameters 7). Therefore, instead
53

---

Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
of denoting the efficient influence function as s(;; P), we can express it as (b; n)). If the influence function can be represented in this form, we proceed by first estimating ý = (é, (o) with crossfitting. For any fixed value 7) find 0, that is
value & such that Pa v.n
P1 (X;) + Ti Y. (x.)
(x,) + (1 T,) Y3 Do (X,)
((X))
((x))
0,
71 1=1
1/0
1/0
which implies
E (X,) T: Y. ((-
e(X,)
D No (X;) + (1 T Y (x.)
(-8(X.)
Hereafter, we propose proof of Proposition 3.13 which does not use the influence function theory
Proof of Proposition 3.13.
Asymptotic bias and variance of the crossfitted Ratio AIPW estimator In this alternative proof, we further assume that Var [X] S o² for some 0. Recall that we want to analyze / (FRRAIPW
TRR). Letting
(z,,) 7:(%
e(X:)
RRRAIPW.
(137)
TRRAIPPW
E (X,) (1 - (Y)
1-e(X.)
AIPW
be the oracle version of FRKAIPW where the propensity score and both response surfaces assumed to be known, rewrite
RRRAIPP
TRK
RR.AIPW
ARR. +
'RRAIPPW
TBR-
(138)
Regarding the first term in (138), we have
ARR.
TRRAIPW
(139)
TRR
TRR.IPP,
(140)
FRR.IIW O
AIPM D
- (FRRAIPW.O)
(TRR.AIP..) .) -1
FRR.IPP..
(141)
+ (TÉR.AIPW.)) u)
(FKR.AIPW,
THR
TRRAIPW.
(142)
( (FRR.IIWW.) o)
(TRR.AIPW.)) .) T *RR.AIO..)
(143)
(T RRRAIPW. o)
(FRR.AIPW. TRR.AIP.... ))/
(144)
We now show that
1
FRRAIPRK
TRR.ATP..1
Op
and
FRR.AIPW.O TIR. ol -p
Vn
The following decomposition holds
TT (Xi).
- -M (X.) - T; Y; 41 (X,)
n TRRAIPP. TRR.AIP..1 .1
Σ
[1 (X,) + T:
Vn
((X,)
(Xi)
SEI.
Further denoted A&
Σ
(17 I. [1 (Xi) - -mi (X.) (1-((x))
Ti
Vn
SEI.
e(x)))
Further denoted B,
+
Vn
Σ Ti
(Y, (Y: 11 (xx,))
(((x))
TEI
(Xx) - (x,) ( (e(x)) )
Further denoted O
Σ Ti
[1T
Vn
TET
54

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
In the following, we prove that the first two terms tend to zero in L.2.
Regarding A One can show that the expectation of A / V is null:
E
A k
I
1
Σ
E
(11 (X,) - 14 (x.) (1
T
II=
k
72
C (X,,)
VEI
LT
E
( - (X) - -MI (x))
(1 (1-
T
1I- k
i..d.
(X)
[R
(17 * (x) - mn (x))
(1-
T
E
E
|X,T
II_K k
(X)
IZx
E
(nf-* (x) in (x)) E
{11-
e e(x)
T
|X,I
I7- k
[Z
(NI (x) mi (x))
(
e(x) (x)
)
EE
1I= k.
O.
We will make use of this results in several calculations. Now,
E
A
I_ k
=
Var
1
Σ
(1f=+(x,) - 1/2 (x.) (1-
c (x,)) )) |-+
Tj
TET
1
Var
Σ
I- (MI-* (X.)) - (xx))(xxxx)x (x.) (1
e (xi)) ))
Ti
I
LiEII
1
Σ Var
[11
I. *(x:) mm (X.) (1 (1- (Xxi)
Ti
I
idd
n
SEI
[
(1i [x (1i+t(3) (X) - -M1 (x3) (1- xx/))
T
))
2
E
I
n²
Ik
[11 - (X)-M1 (x) (1- e(x))
T
))
2
E
E
|X,I_A + I_N
n2
[T&|
I. (X) - m (x)) 2 E.
[[11- e e(x)
T
2
E
|X |I
n?
[Zk]
E
(Oi (X) in (x))
e (xx) E (e (X) - -T2² X I m)
1
I
n2
[IR| n²
E (x( e (X))
E
(AT (X) in (X))
I-
e e(x) 2
[ZR
E
(af (X) - an (x))
(((x) -1 1) |I-
n²
[
(AT (X) - 11 (x))² ?
E
Overlap.
nyn
Taking the expectation, we obtain
1 vn
Inl Ia mm
(nf (X) (x) /1 (x))
E
E
(145)
that is
IE
[AA)*
27
<
1 E
(ot (X) (x)-11(1 (x))
(146)
55

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Thus A converges zero in L and thus in probability.
Regarding D The second term B can also be controlled using similar arguments. By assumption,
7 - ((x) 1 2
2
Thus,
1 e(X)- e(()
2 (e(X) -(())
((X)
e(X)
((X)
112
Derivations are very close to the ones for the first term, noting that,
E
Ti
(X 6z (X,) e 1 exx1))
11 (X:))
I
II=
= =0,
I
so that.
Bk
2
1
E
I-K
k
Var
Ti -
- M1 (x:))
2I= (Xi) e(Xi)
II
vTi
71
iEIN
1
Var
Ti Ti(Y; -M (Xi))
[
iid
(I*(X,)
e <(Xi)
1 2
E
T M1 (X))²
II=
n²
00 xX) e(X)
4|ZA|
E
T (Y 11 (X)) (6- (X) e(x))² II-A
n
4|Ta|
[Y 1n (X))² (8 (X) (x))) |_m]
Sicne T S
1/'n2
E
(Y 1n (X))² (6 (X) - (X)) 2 |x, |
n/22
4|IN||
IE
(r in (x))² 11, 1 (0 (X) - -((x))² 2 |
n/'n
4|Tx|
E Var [V]X] (e= (X) -xxx)* [I
n22
4 Var [V|X] E (c- (X) - e (x)) I
[
n22
Taking the expectation on both sides, since Var [X] S 0², we get
B
40 I E
E
(éT v xx)==xxx+-++-0 - (x))² [Im]
(147)
17 An2
which leads to
E
(Bh)* 27
<
402
(
- (x) |
(148)
Since, by assumption, the right-hand side term converges to zero, B converges to zero in 1,2
56

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Regarding C Regarding the last term, the approach is different and will involve another assumption the product of residuals. More precisely,
Ti [i] - (x.) - (x ( (g (X:) e(x)) 1
=
=EZu
Vn 1
T (pT+ (X,) - mm (x:) (c(X)) E +xxi)))
E
172 n
TE
V
||T
[1q (X) - (x)) (e(x) [x)]]]
E.
172
VT
[| [11 (x) 11 (x)) (e(x) - ( -x)11
E
112
vn -
11 (x)) 21 B -e (1f- (x) (2))-M((3)
E
-
EI
1
which tends to zero by assumption. Each term A Bh, and C has been shown to be bounded by term in or()). Thus,
K
vn PRRAIPP PRRAIP
=
Σ
A
B
C
(149)
k=1
tends to zero in probability. Similarly, one can show that
vn FRRAIPW. ARPP.
P O.
(150)
According to (144), since for all E {0, 1), ARPwL tends to AIRM which is lower and upper bounded, we have
FRRAIPP.
nn ARRAIP
TRR.AIPW
S
FRR.IPPW.O - TRRAIPW,
PKR.AIPW. O RRLAIPP.
+ Vn
F.RRAIPW.
TRR.AIPW.
TRRAIPW.D
which tends to zero.
Regarding the second term in (138), use Theorem 7.1 with 91 (Z) μ1 (X) + T(Y-4:00 (X))
and go =M(X)) + e(X)
(1-7) ()=())))o((X)) where Z (T,X, YYY. Hence, we have that (Z) is square integrable:
(x))(X)
( (X))
E [[nn (2) < 2E [(m) (X) + 2E
e(X)
where E (X))] = ()r( is finite. Using Consistency, Unconfoundedness, and definition or μu (X)
B| X, T = 1], simple calculations show that
E ( Y (x))
27
Y()
(X) 27
E
Consistency
c(X)
e(X)
T (L) u1 (X)
I
1 X
Total expectation
e(X)
(y) (p())p(()) (x))) 2
=
e(X)
(ar(
1
57

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Similarly, we can show that 9o(2) (Z) is square integrable. Since E [Z) E | and E [ox( E [Ya], we can
apply Theorem 7.1 and conclude that
VTi(TRRAIP TRRR + NO VRR.OS).
(151)
Finally,
Vi(FAIPW TRR)
VII(FRRAIPW
TRRAIPW
VII(TRR
TRR
4 N (0, VRROOS)
where
[41 (X)
(X)
Var (r
+E Var (I (X)
V KROS
*RR
Var
E [Y(U]
[
c(X)E [y(U]]
(1 e(X)) [ [r(1]]
58

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
8. Simulation
For the simulations we have implemented all estimators Python using Scikit-Learn for our regression and classification models. All our experiments were run on a 8GB MI Mac. The propensity scores is estimated based on the provided training data and covariate names. Depending on the chosen method, it either uses logistic regression with a high regularization parameter (parametric) a random forest classifier with parameters determined by the training data size (non-param parametrie) The response surface is estimated based on the training data, covariate names, the method (parametric or non-parametrie), and whether the response is binary or continuous. For parametric methods. it uses stochastic gradient descent classifier for binary responses and linear regression model for continuous responses. For non-parametric methods, it employs a random forest classifier for binary responses and a random forest regressor for continuous responses. Both methods fit the model using the training data to estimate the respective scores and surfaces, enabling flexible handling of various datasets and assumptions for causal inference analysis.
8.1. Randomized Controlled Trials
In this part we will simulate Randomized Controlled Trials (RCT) and test the following Ratio estimators: Ratio Neyman, Ratio Horvitz Thomson and the Ratio G-formula. Since we in a Randomized Controlled Trials, the propensity score e(.) constant.
8.1.1. 1. LINEAR RCT
The first DGP has linear outcome models (linear treatment effect and the baseline). The data is generated using:
m(X) = (cI cb) + (B1 - -B0]
Co = 6, = 12
6(X) = Co + 00 X
B1 = (2. -5, -2
e(X) = 0.5
Bo (3, -,,1,,4444 22,
Linear RCT
30
4.0
100
300
3.5
1000
3000
10000
Estimations
2.0
1.5
1.0
0.5
RR-HT
RR-N
Forest RR-G
Regression RR-G
Estimators
Figure 6. Comparison of RCT estimators in Linear RCT
Given that X has mean, it follows that TRk = c1 /c/ = 2. This scenario aligns with the linear setting outlined in Assumption 3.8. Referring to Figure 6, as proved in the previous sections all estimators converge to the true Risk Ratio as increases. Additionally, within this linear framework as per Lemma 7.9, the variance of the Neyman estimator exceeds the one of the G-formula. In such a linear environment, the parametric G-formula performs better than its non- paramerrtt counterpart. Additionally, the Ratio Neyman estimator demonstrates lower variance compared to the Horvitz- Horvitz-Thomson Thomson estimator as indicated in Equation (5).
59

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
8.1.2. NON-LINEAR RCT
This DGP is also Randomized Controlled Trials however, the outcomes are not linear this time:
X3
m(X) sin(Xi) X2 +
X4 + log + 1) + X +
b(X) max(X, X X3, 0) min(X4 X6. and c(X) 0.5
Non-Linear RCT
30
100
300
1000
3000
10000
Estimations
RHT
RN
Forest RG
Regression RG
Estimators
Figure 7. Comparison of RCT estimators in Non-Linear RCT
The presence trigonometric, exponential, logarithmic, and polynomial terms makes this setting non-linear. It's important to note that since we are in a Randomized Controlled Trial (RCT), the propensity function remains constant. As the sample size (n) increases, all proposed estimators converge. bias can be seen in 7 but decreases to 0 as (n) increases as predicted in previous sections. Linear regression struggles with small n values, failing capture the intricate relationships between features and non-linearities. On the other hand, Random Forest, non-maramtttic method, excels in capturing these complexities by segmenting the feature space and predicting based on response averages within those segments. However, predicting the complex function can be challenging, the Neyman estimator might outperform the G-formula, particularly when both parametric and non-parametric responses may lack consistency. Although we do not fall in assumptions of Equation (5) the Ratio Neyman estimator demonstrates lower variance compared to the Thoomz estimator.
8.2. Observational Studies
8.2. 1. NON-LINEAR AND NON-LOGISTIC DGP
We use the same simulations as in (Nie & Wager, 2020) using nonlinear models for every quantity, as detailed below, with X Unif(0, 1)6
m(X) = sin (xX +2( - 0.5)² 0.5)3 + - (X1 + X2)
b(X) (X, xx2) /2
e(X) max 10.1, min(sin (mX1) 0.9))
Results are presented Figure At first glance, all methods seem to have similar performances. However, estimators based on parametric estimators (last four) fail to converge to the correct quantity. They present an intrinsic bias, which does
60

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
Fequency
Values of
Figure 8. Histogram of the propensity score of Logistic DGPs
Wager
2.5
100
3.0
2000
10000
Edhreoor
Estimstors
Figure Estimations of Risk Ratio weighting, outcome based and augmented estimators as a function of sample size the mon-Linnea--n Lineea-non--oogiseeee DGP. Parametric (Regression) and non parametric (Forest) estimations of nuisance are displayed.
Wager A
332
330
325-
66.
320
Figure 10. Average coverage (left) and average length (right) of asymptotic confidence interval derived from Section 2 and Section 3 for different estimators with 1000 and 300 repetitions for Non-Linear and Non-Logistic DGP.
vanish as the sample size increases. This was expected as linear methods are unable model the complex non-linear generative process of this simulation. On the other hand, methods that employ random forests estimators achieve good performances: they consistent and unbias even for small sample sizes. Note that RIPW has larger variance than the other methods, with a small bias for very small sample sizes. Therefore, the G-formula and the two doubly-robust estimators that use random forests are competitive in this setting. Here again, both double robust estimators give similar performances. No estimator achieves 95% coverage, which is expected given the non-linear, non-logistic DGP. Linear estimators, such as Linear RR-G and RR-AIPW struggle to accurately estimate the nuisance functions in this context. Additionally, the limited number of observations prevents the Forest estimators from converging effectively.
61

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Quantifying Treatment Effects: Estimating Risk Ratios via Observational Studies
8.3. Semi-synthetic simulations
In our semi-synthetic experiment, we retain the original co-variates, X, and first regenerate the treatment assignment, T, using logistic regression (see Section 8.3 for random forests classifier) trained on the original 17 co-variates to predict the observed treatment. Based the estimated probabilities from this model, we sample binary treatment from Bernoulli distribution. Next, we simulate potential outcomes using two separate logistic regressions: one for the treated group and another for the control group, incorporating all co-variates. This process yields the potential outcomes, y and ( for each individual, enabling us to compute the true relative risk, TRR = 1.2. Finally, we estimate the relative risk and its variance using bootstrap resampling across different sample sizes (300, 1000, and 3000). Results are displayed in Figure 11. All estimators except Neyman appear to converge the true value of the Risk Ratio. As anticipated, doubly-robust estimators consistent and efficient (smallest asymptotic variance).
Estimates evolution
Semple Sioe
The
2000
25
3002
20
10-
05-
00
Estimato
Figure 11. RR estimations with weighting, outcome based and augmented estimators as function of the sample size for the logistic Semi-synthetic data. Parametric (Lincar) and non parametric (Forest) estimations of nuisance are displayed.
In this approach, we replace logistic regression with random forest classifiers model the treatment T and the potential outcomes (2) and (0),, As observed, all estimators, except for RR-N, converge toward the true value of the Risk Ratio. Moreover, since both the treatment assignment and potential outcomes generated using Random Forest, estimators that also leverage Random Forest demonstrate reduced bias and lower variance. Conversely, linear estimators exhibit higher bias, likely due to model misspecification.
Estimates evolution scress sample 1i2ss
3000
4.3
0.4
Purest RKow
Purest Me
Perest Wh- 33
Firest Rh APW antivater
LegRik KMMP
Linear
Unter MH-0
Linear NAAPP
Figure 12. RR estimations with weighting, outcome based and augmented estimators as function of the sample size for the Semi- Semi-synthetic data, Parametric (Linear) non parametric (Forest) estimations nuisance are displayed.
62


Reviews:
[{"review_id": "scbytKKfnQ", "reviewer": "Reviewer_KPHb", "summary": "Authors focus on risk ratio (RR), a measure of treatment effectiveness complementary to the more common \"risk difference\". They first analyze the standard estimators for RR in an RCT and derive some new asymptotic normality/variance results, such as for continuous outcomes in addition to binary outcomes. They then develop IPW, regression function, and AIPW estimators for the RR in the context of an observational study where one needs to adjust for confounders and to estimate nuisance functions. They derive asymptotic unbiasedness and normality results of their estimators under the assumption that the true underlying outcome and propensity score models are (generalized) linear models and derive an expression for the asymptotic variance. They also derive, using the influence function theory, and efficient AIPW estimator.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your positive review. We are glad that you found the paper well-written, well-structured with interesting results for practitioners. We also appreciate your suggestion regarding the use of histograms to illustrate the asymptotic normality of our estimators. We agree that this would provide a clearer visual understanding of the results and plan to incorporate it in the next revision.", "claims_and_evidence": "Theoretical results are well presented and supported.", "methods_and_evaluation_criteria": "Yes. See experiments and benchmarks section below.", "theoretical_claims": "I did not check in detail. The parts I quickly skimmed looked fine and the proofs in the appendix seem to be well-written.", "experimental_designs_or_analyses": "Synthetic setup makes sense and supports the asymptotic unbiasedness results of proposed algorithms, except for the Logistic RR-IPW. Although that is not surprising to me as IPW estimators almost always have very high variance with finite samples in practice. Authors could also plot histograms instead of boxplots to demonstrate asymptotic normality of their estimators.", "relation_to_broader_scientific_literature": "Formal results on RR estimators could be very useful to practitioners in practice. Methodologically, most of the techniques used are standard and does not reveal new insights, except may be for the results that require the derivation of influence functions for the RR estimand.", "essential_references_not_discussed": "I am not an expert on the literature about RR, but was satisfied with how the related work was covered.", "other_strengths_and_weaknesses": "The paper is very well-written and structured. It is self-contained in terms of the statistical methodology it uses, which could be very helpful to those who are not very familiar this type of (e.g., CLT) results/derivations.", "other_comments_or_suggestions": "NA", "questions_for_authors": "NA", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "wcG2RuybKr", "reviewer": "Reviewer_V8v7", "summary": "The authors discussed theory of risk ratio estimation in observational data. Several RR estimators are proposed with theoretical investigation, including asymptotic normality and confidence intervals. Two doubly robust estimators are proposed and the authors recommended the use of one of them.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful and constructive review. We appreciate your positive comments on the clarity of the writing and the contribution of our work to the causal inference literature. Please find our responses to your comments below:\n\n1. **Regarding the assumption $E[Y(0) \\mid X] > 0$:**\n\n   Our initial motivation for introducing it was to establish finite-sample bias and variance results, which are currently included only in the appendix. However, for the main theoretical results, we only need to assume $E[Y(0)] > 0$ to ensure that the risk ratio is well-defined.\n\n   We agree that this assumption may not always hold, but in practice, many biological quantities are positive (e.g., blood concentrations). Besides, if the variable $Y(0)$ is binary, $E[Y(0)] > 0$ holds as soon as one observation satisfies $Y_i(0) = 1$, which seems to be a reasonable assumption.\n\n   In the revised manuscript, we will move the stronger assumption $E[Y(0) \\mid X] > 0$ to the appendix (where it is needed) and replace it in the main text with the weaker and more general condition $E[Y(0)] > 0$. We will also add the above discussion.\n\n2. **On model misspecification in simulations:**\n\n   We completely agree that comparing the estimators under various forms of model misspecification is important. Due to space constraints, we were unable to include this analysis in the main body of the paper, but we incorporated two relevant scenarios in Appendix 8.2:\n\n   - **Wager C model (Appendix 8.2.1):**  \n     Both response functions are nonlinear, while the propensity score is logistic. In this setting, Logistic RR-IPW and Linear RR-AIPW and RR-OS (using linear and logistic regressions) remain asymptotically unbiased, while Forest RR-OS and Forest RR-AIPW are biased for small sample sizes due to the difficulty of forests in estimating logistic functions with few observations.\n\n   - **Wager A model (Appendix 8.2.2):**  \n     Both response functions and the propensity score are nonlinear/non-logistic. All linear estimators are biased in this case, but estimators that use Random Forests to estimate the nuisance functions exhibit decreasing bias as the sample size increases.\n\n   If the paper is accepted, we will use the additional page to move this analysis into the main paper.\n\n3. **On the implementation of variance estimators:**\n\n   Thank you for pointing this out. In fact, we designed variance estimation procedures to construct the confidence intervals reported in the experiments, but we recognize that the implementation details were not highlighted enough.\n\n   In the current version, the relevant formulas are in the appendix (in Section 4.2). We will elaborate on this and add a dedicated section in the Appendix. In addition, we will better highlight in the main text that we have estimated all variances, as they can all be expressed using estimable quantities, and that we used these estimations in all our simulations.\n\n4. **Redundancy in Assumption 2.2:**\n\n   You are totally right. We will revise this assumption to remove the redundancy. Thank you for pointing this out!", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "Yes, I have checked all the proof.", "experimental_designs_or_analyses": "Yes - the experimental design makes sense to me.", "relation_to_broader_scientific_literature": "Adding contribution to estimation of risk ratio in observation study.", "essential_references_not_discussed": "Not aware of any.", "other_strengths_and_weaknesses": "Strength: the work is very clearly written, and is a good addition to the cauasl inference literature for estimation of RR. I am very familiar with the observational study literature and the results for the different estimators are pretty natural. Overall I am positive about the paper and I will skip technical questions here. \n\nWeakness: \n1. Though in assumption it is assumes that conditional expectation of $Y(0)$ given $X$ is greater than zero, this is never guaranteed in finite sample using say AIPW. How should one control for this? \n2. in term of simulation, I am interested in seeing the performance of RR-OS estimator and RR-AIPW under some wrong model misspecifications. \n3. the whole paper did not discuss the implementation of several variance estimators. For example, how do we implement $V_{RR,OS}$.", "other_comments_or_suggestions": "1. Assumption 2.2, part (iii): independence between treatment assignment is already guaranteed by iid part.", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "XJD8xxPhOD", "reviewer": "Reviewer_7moV", "summary": "The authors propose several estimators for the average risk ratio (RR). They begin by analyzing the RR version of the Neyman estimator under standard causal inference assumptions, proving its asymptotic normality and deriving an expression for its variance. They then extend this analysis to an RR variant of the Horvitz-Thompson estimator.\n\nNext, the authors refine their results by assuming the true propensity scores follow a logistic model and fitting estimators using maximum likelihood estimation (MLE). They further establish the asymptotic normality and derive an explicit variance expression for an RR version of the G-formula—first using the true feature probabilities and then with an estimator where the conditional expectation is obtained via least squares.\n\nThe paper concludes with simulations and a real-world application using a dataset on traumatic brain injury.\n\nStrengths\n-The paper is well-executed, providing a thorough set of theoretical results and strong empirical validation, including a real-world case study.\n\nWeaknesses\n-The properties of RR estimators may already be well known, albeit I am not a causal inference expert.", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "- **Clarifying the Novelty and Contribution:**  \n  We thank you for acknowledging the thoroughness of both the theoretical development and empirical validation. We would like to take this opportunity to clarify the main contribution of the paper.\n\n  One of the main contributions of this paper is to provide a comprehensive theoretical and empirical analysis of RR estimators, including new ones like RR-OS and RR-AIPW, in the context of observational studies. While RR is a well-established estimand, its estimation introduces distinct challenges due to its non-linearity.\n\n  To the best of our knowledge, there has not been a unified and rigorous asymptotic analysis of IPW, G-formula, or AIPW estimators for the marginal RR in the literature. Our work aims to fill this gap by deriving explicit variance expressions, exploring efficiency bounds, and validating the estimators both theoretically and empirically in both continuous and binary outcome settings.\n\n- **On the Work by Rose and van der Laan (2014):**  \n  The cited work by Rose and van der Laan focuses on Targeted Maximum Likelihood Estimation (TMLE) for the risk difference in case-control studies. While this is an important contribution, the setting considered there differs from the randomized and observational study framework analyzed in our work and focuses on binary outcomes. Furthermore, while TMLE is a flexible approach, our paper focuses on IPW, AIPW, and G-formula estimators, whose asymptotic properties have not, to the best of our knowledge, been rigorously established in the context of marginal RR estimation using observational data — also for continuous outcomes. We thank the reviewer for pointing out this reference; we will mention it in the conclusion as a potential research direction for finite sample studies.\n\n- **On the Mention of Kennedy’s Work:**  \n  We are not entirely sure what specific paper by Kennedy you refer to. We have read several papers by Kennedy, three of which are cited in our work, but we did not encounter specific studies by Kennedy (or anyone else) analyzing different estimators of the marginal risk ratio. That said, it is worth noting that the conditional risk ratio is not directly collapsible. This means that estimating the conditional RR generally does not provide a valid estimator for the marginal RR. In contrast, our work specifically targets estimators for the marginal RR and provides a detailed asymptotic analysis under standard causal inference assumptions.\n\nWe hope that the above points help to dispel your doubts about the novelty of our work.", "claims_and_evidence": "Refer to summary", "methods_and_evaluation_criteria": "Refer to summary", "theoretical_claims": "Refer to summary", "experimental_designs_or_analyses": "Refer to summary", "relation_to_broader_scientific_literature": "In the introduction of this paper by Rose and Van Der Laan (2014) https://sci-hub.ru/https://doi.org/10.1093/aje/kwt318 , they talk about how there are methods to estimate the risk-ratio. Also I think Kennedy has a paper where he proves properties of the risk ratio condoned on the features.", "essential_references_not_discussed": "refer to Broader Scientific Literature", "other_strengths_and_weaknesses": "Refer to summary", "other_comments_or_suggestions": "No extra comments", "questions_for_authors": "No questions for the authors", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "MJDsG0xMA3", "reviewer": "Reviewer_2hoQ", "summary": "This paper develops novel estimators for the Risk Ratio (RR) in observational studies to accommodate confounding in non-randomized settings. It introduces estimators based on inverse propensity weighting, the G-formula, and doubly robust techniques, and establishes their asymptotic normality. Simulation studies and a real-world application demonstrate that their estimators yield valid and efficient treatment effect estimates.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "- **Figure Caption Clarity:**  \n  Thank you for pointing this out. We agree that the figure captions could be clearer. In particular, we will revise the caption to explicitly mention that the colors represent the \"Sample Size\".\n\n- **Doubly Robust Conditions:**  \n  You are absolutely right that in classical doubly robust settings, it is sufficient for only one of the nuisance estimators (either the outcome regression or the propensity score) to be consistent for the overall estimator to remain consistent. This is commonly referred to as *weak double robustness*.\n\n  As discussed in Wager's *Causal Inference Book* ([Section 3](https://web.stanford.edu/~swager/causal_inf_book.pdf)), it is helpful to distinguish between **weak** and **strong** double robustness:\n\n  - **Weak double robustness** refers to the property that the estimator is consistent if either the outcome model or the propensity score model is estimated consistently. However, weak double robustness does not, in general, guarantee asymptotic normality or valid confidence intervals under model misspecification.\n  - **Strong double robustness**, on the other hand, refers to the setting where asymptotic normality can be achieved if both nuisance functions are estimated consistently at sufficiently fast rates. Specifically for the classical RD setting, this holds when both  $\\hat \\mu_{(t)}$ and $\\hat{e}$ are estimated with root-mean squared error rates decaying faster than $n^{-\\alpha_{\\mu_t}}$ and $n^{-\\alpha_e}$ respectively, and when $\\alpha_{\\mu_t} + \\alpha_e \\geq \\frac{1}{2}$ for each $t \\in \\{0, 1\\}$. Then AIPW is asymptotically normal.\n\n  In our paper, we establish *strong double robustness* for RR-AIPW. This result relies on both Equation (17) and Equation (14), which are strictly equivalent to the classical conditions for strong double robustness. It is worth noting that weak double robustness is also achieved for RR-AIPW but not for RR-OS since we always need $\\hat{\\mu}_{(0)}$ to be consistent.\n\n- **Low Baseline Risk:**  \n  Thank you for raising this important point. When the baseline risk is extremely low, the risk ratio becomes inherently unstable, often resulting in wide confidence intervals. We have explored this empirically, and the results are shown in the table below.\n\n  As the baseline risk approaches zero, the estimated RR can become extremely large, while the absolute risk difference remains small: different causal measures lead to different interpretations, thus suggesting that several causal measures should be used simultaneously to properly understand the impact of a treatment.\n\n  This case also illustrates why it is important to report both an absolute measure (e.g., risk difference) and a relative one (e.g., risk ratio), as they provide complementary perspectives on treatment effects—particularly when baseline risks are extreme.\n\n  | **Baseline** | **RR**   | **Estimator**       | **Coverage (mean ± std)** | **Length (mean ± std)**   |\n  |--------------|----------|---------------------|----------------------------|----------------------------|\n  | 2.5          | 1.77     | Linear RR-G         | 0.878 ± 0.113              | 0.099 ± 0.002              |\n  | 2.5          | 1.77     | OLS RR-G            | 0.948 ± 0.050              | 0.120 ± 0.003              |\n  | 2.5          | 1.77     | Linear RR-AIPW      | 0.968 ± 0.047              | 0.185 ± 0.007              |\n  | 0.045        | 15.3     | Linear RR-G         | 0.992 ± 0.027              | 3.728 ± 0.513              |\n  | 0.045        | 15.3     | OLS RR-G            | 0.984 ± 0.037              | 3.037 ± 0.435              |\n  | 0.045        | 15.3     | Linear RR-AIPW      | 1.000 ± 0.000              | 4.734 ± 0.735              |\n  | 0.0018       | 112.1    | Linear RR-G         | 0.987 ± 0.022              | 884.427 ± 206.672          |\n  | 0.0018       | 112.1    | OLS RR-G            | 0.978 ± 0.032              | 1026.384 ± 214.754         |\n  | 0.0018       | 112.1    | Linear RR-AIPW      | 0.972 ± 0.036              | 2073.229 ± 509.709         |\n\n  **Table:** Coverage and length of confidence intervals for different RR estimators across varying baseline risks.", "claims_and_evidence": "The claims made in the paper are well supported by both theoretical derivations and simulation studies.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria are standard and make sense for the problem at hand.", "theoretical_claims": "I did not check the correctness of any theoretical claims.", "experimental_designs_or_analyses": "N/A", "relation_to_broader_scientific_literature": "The paper’s key contributions are well presented and well connected to the broader literature on treatment effect estimation.", "essential_references_not_discussed": "All the essential references were discussed.", "other_strengths_and_weaknesses": "The writing of figure captions could be improved for greater clarity and context. \n\nFor example, the caption for Figure 1 does not explain what the different colors represent.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "- Equation (17) requires consistency on both nuisance function estimators, yet in classic doubly robust settings it is often sufficient for only one to be consistent. Could you explain why both conditions are imposed here—does the non-linear nature of the risk ratio functional necessitate that both the propensity score and outcome model be estimated consistenly, or could a relaxation where only one of them is consistently estimated still yield valid inference?\n- When the baseline risk is very low, the risk ratio may be unstable. Could you provide additional simulations to show the coverage of your estimator as the baseline risk approaches zero?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper studies the estimation of the risk ratio in observational studies under the assumption of confoundedness. This propose and compare several estimators for the risk ratio. However, the proposed estimators and the corresponding conclusions are fairly standard and closely parallel existing work on estimating the risk difference. 

Four reviewers evaluated this work, but there was some discrepancy in their scores (1, 4, 4, 4). After discussing with the reviewers, the key concern centers on whether prior work has addressed a similar problem.  We found a paper that addresses an estimand (Lemma 1(a) of [1]) similar to the risk ratio. The work in [1] proposes a semiparametric efficient (doubly robust) estimator for this estimand. Since this is an ArXiv paper unpublished, we decided to accept this paper. However, the authors are encouraged to incorporate the reviewers' suggestions and include a brief discussion with [1] in the final version. 


[1] Tian and Wu. Semiparametric Efficient Inference for the Probability of Necessary and Sufficient Causation. arXiv, 2024.
Decision: Accept (poster)