A C ROSS -VALIDATED TARGETED MAXIMUM LIKELIHOOD ESTIMATOR FOR DATA-ADAPTIVE EXPERIMENT SELECTION APPLIED TO THE AUGMENTATION OF RCT C ONTROL ARMS

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A CROSS-VALIDATED TARGETED MAXIMUM LIKELIHOOD

ESTIMATOR FOR DATA-ADAPTIVE EXPERIMENT SELECTION

APPLIED TO THE AUGMENTATION OF RCT CONTROL ARMS

WITH EXTERNAL DATA

A PREPRINT

Lauren Eyler Dang, Jens Magelund Tarp, Trine Julie Abrahamsen, Kajsa Kvist, John B Buse,

Maya Petersen,Mark van der Laan

February 21, 2023

ABSTRACT

Augmenting the control arm of a randomized controlled trial (RCT) with external data may increase

power at the risk of introducing bias. Existing data fusion estimators generally rely on stringent

assumptions or may have decreased coverage or power in the presence of bias. Framing the problem

as one of data-adaptive experiment selection, potential experiments include the RCT only or the RCT

combined with different candidate real-world datasets. To select and analyze the experiment with

the optimal bias-variance tradeoff, we develop a novel experiment-selector cross-validated targeted

maximum likelihood estimator (ES-CVTMLE). The ES-CVTMLE uses two bias estimates: 1) a

function of the difference in conditional mean outcome under control between the RCT and combined

experiments and 2) an estimate of the average treatment effect on a negative control outcome (NCO).

We deﬁne the asymptotic distribution of the ES-CVTMLE under varying magnitudes of bias and

construct conﬁdence intervals by Monte Carlo simulation. In simulations involving violations of

identiﬁcation assumptions, the ES-CVTMLE had better coverage than test-then-pool approaches and

an NCO-based bias adjustment approach and higher power than one implementation of a Bayesian

dynamic borrowing approach. We further demonstrate the ability of the ES-CVTMLE to distinguish

biased from unbiased external controls through a re-analysis of the effect of liraglutide on glycemic

control from the LEADER trial. The ES-CVTMLE has the potential to improve power while providing

relatively robust inference for future hybrid RCT-RWD studies.

1 Introduction

In 2016, the United States Congress passed the 21st Century Cures Act (114th Congress, 2016) with the aim of

improving the efﬁciency of medical product development. In response, the U.S. Food and Drug Administration

established its “Framework for FDA’s Real-World Evidence Program" (FDA, 2018), which considers how data collected

outside the scope of the randomized controlled trial (RCT) may be used in the regulatory approval process. One such

application is the use of real world healthcare data (RWD) to augment or replace the control arm of an RCT. The FDA

has considered the use of such external controls in situations “when randomized trials are infeasible or impractical",

“unethical", or there is a “lack of equipoise" (Rivera, 2021). Conducting an adequately-powered RCT may be infeasible

when target populations are small, as in the case of rare diseases (Rivera, 2021; Franklin et al., 2020; Jahanshahi et al.,

2021). In other circumstances, research ethics may dictate that a trial control arm consist of the minimum number of

patients necessary, as in the case of severe diseases without effective treatments (Rivera, 2021; Ghadessi et al., 2020;

Dejardin et al., 2018) or in pediatric approvals of drugs that have previously been shown to be safe and efﬁcacious in

adults (Rivera, 2021; Dejardin et al., 2018; Viele et al., 2014). With the growing availability of observational data from

sources such as registries, claims databases, electronic health records, or the control arms of previously-conducted trials,

the power of such studies could potentially be improved while randomizing fewer people to control status if we were

able to incorporate real-world data in the analysis (Schmidli et al., 2014; Colnet et al., 2021).

arXiv:2210.05802v3 [stat.ME] 20 Feb 2023

Experiment-Selector CV-TMLE A PREPRINT

Yet combining these data types comes with the risk of introducing bias from multiple sources, including measurement

error, selection bias, and confounding (Bareinboim and Pearl, 2016). Bareinboim and Pearl (2016) previously deﬁned a

structural causal model-based framework for causal inference when multiple data sources are utilized, a setting known

as data fusion. Using directed acyclic graphs (DAGs), this framework helps researchers understand what assumptions,

testable or untestable, must be made in order to identify a causal effect from the combined data. By introducing

observational data, we can no longer rely on randomization to satisfy the assumption that there are no unmeasured

common causes of the intervention and the outcome in the pooled data. Furthermore, causal identiﬁcation of the

average treatment effect is generally precluded if the conditional expectations of the counterfactual outcomes given the

measured covariates are different for those in the trial compared to those in the RWD (Rudolph and van der Laan, 2017).

Such a difference can occur for a number of reasons, including changes in medical care over time or health beneﬁts

simply from being enrolled in a clinical trial (Ghadessi et al., 2020; Viele et al., 2014; Pocock, 1976; Chow et al.,

2013). In 1976, Stuart Pocock developed a set of criteria for evaluating whether historical control groups are sufﬁciently

comparable to trial controls such that the suspected bias of combining data sources would be small (Pocock, 1976). We

are not limited to historical information, however, but could also incorporate data from prospectively followed cohorts

in established health care systems. These and other considerations proposed by subsequent authors are vital when

designing a hybrid randomized-RWD study to make included populations similar, minimize measurement error, and

measure relevant confounding factors (FDA, 2018; Franklin et al., 2020; Ghadessi et al., 2020).

Despite careful consideration of an appropriate real-world control group, the possibility of bias remains, casting doubt

on whether effect estimates from combined RCT-RWD analyses are truly causal. A growing number of data fusion

estimators — discussed in Related Literature below — attempt to estimate the bias from including RWD in order

to decide whether to incorporate RWD or how to weight RWD in a combined analysis. A key insight from this

literature is that there is an inherent tradeoff between maximizing power when unbiased external data are available

and maintaining close to nominal coverage across the spectrum of potential magnitudes of RWD bias (Chen et al.,

2021; Oberst et al., 2022). The strengths and limitations of existing methods led us to consider an alternate approach to

augmenting the control arm of an RCT with external data that incorporates multiple estimates of bias to boost potential

power gains while providing robust inference despite violations of necessary identiﬁcation assumptions. Framing the

decision of whether to integrate RWD (and by extension, which RWD to integrate) as a problem of data-adaptive

experiment selection, we develop a novel cross-validated targeted maximum likelihood estimator for this context that 1)

incorporates an estimate of the average treatment effect on a negative control outcome (NCO) into the bias estimate,

2) uses cross-validation to separate bias estimation from effect estimation, and 3) constructs conﬁdence intervals by

sampling from the estimated limit distribution of this estimator, where the sampling process includes an estimate of the

bias, further promoting accurate inference.

The remainder of this paper is organized as follows. In Section 2, we discuss related data fusion estimators. In Section

3, we introduce the problem of data-adaptive experiment selection and discuss issues of causal identiﬁcation, including

estimation of bias due to inclusion of RWD. In Section 4, we introduce potential criteria for including RWD based on

optimizing the bias-variance tradeoff and utilizing the estimated effect of treatment on an NCO. In Section 5, we develop

an extension of the cross-validated targeted maximum likelihood estimator (CV-TMLE) (Zheng and van der Laan, 2010;

Hubbard et al., 2016) for this new context of data-adaptive experiment selection and deﬁne the limit distribution of this

estimator under varying amounts of bias. In Section 6, we set up a simulation to assess the performance of our estimator

and describe four potential comparator methods: two test-then-pool approaches (Viele et al., 2014), one method of

Bayesian dynamic borrowing (Schmidli et al., 2014), and a difference-in-differences (DID) approach to adjusting for

bias based on a negative control outcome (Sofer et al., 2016; Shi et al., 2020b). We also introduce a CV-TMLE based

version of this DID method. In Section 7, we compare the causal coverage, power, bias, variance, and mean squared

error of the experiment-selector CV-TMLE to these four methods as well as to a CV-TMLE and t-test for the RCT only.

In Section 8, we demonstrate the use of the experiment-selector CV-TMLE to distinguish biased from unbiased external

controls in a real data analysis of the effect of liraglutide versus placebo on improvement in glycemic control in the

Central/South America subgroup of the LEADER trial.

2 Related Literature

A growing literature highlights different strategies for combined RCT-RWD analyses. One set of approaches, known as

Bayesian dynamic borrowing, generates a prior distribution of the RCT control parameter based on external control

data, with different approaches to down-weighting the observational information (Pocock, 1976; Ibrahim and Chen,

2000; Hobbs et al., 2012; Schmidli et al., 2014). These methods generally require assumptions on the distributions of

the involved parameters, which may signiﬁcantly impact the effect estimates (Galwey, 2017; Dejardin et al., 2018).

While these methods can decrease bias compared to pooling alone, multiple studies have noted either increased type 1

error or decreased power when there is heterogeneity between the historical and RCT control groups (Dejardin et al.,

2018; Viele et al., 2014; Galwey, 2017; Cuffe, 2011; Harun et al., 2020).

Experiment-Selector CV-TMLE A PREPRINT

This tradeoff between the ability to increase power with unbiased external data and the ability to control type 1 error

across all potential magnitudes of bias has also been noted in the frequentist literature (Chen et al., 2021; Oberst et al.,

2022). A simple “test-then-pool" strategy for combining RCT and RWD, described by Viele et al. (2014), involves a

hypothesis test that the mean outcomes are equal in the RCT and RWD control arms; datasets are only combined if

the null hypothesis of the test is not rejected. However, when the RCT is small, tests for inclusion of RWD are also

underpowered, and so observational controls may be inappropriately included even when the test’s null hypothesis is

not rejected (Li et al., 2020). Thus, such approaches are subject to inﬂated type 1 error in exactly the settings in which

inclusion of external controls is of greatest interest.

Subsequently, several estimators that are more conservative in their ability to maintain nominal type 1 error control

have been proposed. For example, Rosenman et al. (2020) have built on the work of Green and Strawderman (1991)

in adapting the James-Stein shrinkage estimator (Stein, 1956) to weight RCT and RWD effect estimates in order

to estimate stratum-speciﬁc average treatment effects. Another set of methods aims to minimize the mean squared

error of a combined RCT-RWD estimator, with various criteria for including RWD or for deﬁning optimal weighted

combinations of RCT and RWD (Yang et al., 2020; Chen et al., 2021; Cheng and Cai, 2021; Oberst et al., 2022). These

studies reveal the challenge of optimizing the bias-variance tradeoff when bias must be estimated. Oberst et al. (2022)

note that estimators that decrease variance most with unbiased RWD also tend to have the largest increase in relative

mean squared error compared to the RCT only when biased RWD is considered. Similarly, Chen et al. (2021) show that

if the magnitude of bias introduced by incorporating RWD is unknown, the optimal minimax conﬁdence interval length

for their anchored thresholding estimator is achieved by an RCT-only estimator, again demonstrating that both power

gains and guaranteed type I error control should not be expected. Yang et al. (2020), Chen et al. (2021), and Cheng

and Cai (2021) introduce tuning parameters for their estimators to modify this balance. Because no estimator is likely

to outperform all others both by maximizing power and maintaining appropriate type 1 error in all settings, different

estimators may be beneﬁcial in different contexts where one or the other of these factors is a greater priority. While

these methods focus on estimating either the conditional average treatment effect (Yang et al., 2020; Cheng and Cai,

2021) or the average treatment effect (Chen et al., 2021; Oberst et al., 2022) in contexts when treatment is available in

the external data, in this paper we focus on the setting where a medication has yet to be approved in the real world.

An alternate approach to estimating bias, used mostly for observational data analyses, involves the use of an NCO.

Because the treatment does not affect an NCO, evidence of an association between the treatment and this outcome is

indicative of bias (Lipsitch et al., 2010). Authors including Sofer et al. (2016), Shi et al. (2020a), and Miao et al. (2020)

have developed methods of bias adjustment using an NCO. Yet because there may be unmeasured factors that confound

the relationship between the treatment and the true outcome that do not confound the relationship between the treatment

and the NCO, an NCO-based bias estimate near zero does not rule out residual bias (Lipsitch et al., 2010).

In summary, methods that estimate bias to evaluate whether to include RWD or how to weight RWD in a combined

analysis most commonly rely either on a comparison of mean outcomes or effect estimates between RCT and RWD

(e.g. Viele et al. (2014); Schmidli et al. (2014); Yang et al. (2020); Oberst et al. (2022)) or on the estimated average

treatment effect on an NCO (e.g. Shi et al. (2020b)). The latter approach requires additional assumptions regarding

the quality of the NCO (Shi et al., 2020b; Lipsitch et al., 2010). Bias estimation is a challenge for both approaches,

leading to a tradeoff between the probability that information from unbiased RWD is included and the probability that

information from biased RWD is excluded (Chen et al., 2021; Oberst et al., 2022). We discuss both options for bias

estimation and our proposal to combine information from both sources below.

3 Causal Roadmap for Hybrid RCT-RWD Trials

In this section, we follow the causal inference roadmap described by Petersen and van der Laan (2014) to explain this

data fusion challenge. Please refer to Supplementary Table 1 in Appendix 1 for a list of symbols used in this manuscript.

For a hybrid RCT-RWD study, let

indicate the experiment being analyzed, where

si= 0

indicates that individual

participated in an RCT,

si∈ {1, ..., K}

indicates that individual

participated in one of

potential observational

cohorts, and

S∈ {0, s}

indicates an experiment combining an RCT with dataset

. We have a binary intervention,

, a

set of baseline covariates,

, and an outcome

may affect inclusion in the RCT versus RWD. Assignment to

active treatment,

, is randomized with probability

for those in the RCT and set to

(standard of care) for those in

the RWD, because the treatment has yet to be approved. Thus,

is only affected by

and

, not directly by

any exogenous error.

may be affected by

, and potentially also directly by

. The unmeasured exogenous

errors

U= (UW, US, UY)

for each of these variables could potentially be dependent. The full data then consist of both

endogenous and exogenous variables. Our observed data are

independent and identically distributed observations

Oi= (Wi, Si, Ai, Yi)with true distribution P0.

A common causal target parameter for RCTs is the average treatment effect (ATE). With multiple available datasets,

there are multiple possible experiments we could run to evaluate the ATE for the population represented by that

Experiment-Selector CV-TMLE A PREPRINT

experiment, where each experiment includes S=0 with or without external control dataset

. With counterfactual

outcomes (Neyman, 1923) deﬁned as the outcome an individual would have had if they had received treatment (

) or

standard of care (

), there are thus multiple potential causal parameters that we could target, one for each potential

experiment:

ΨF

s(PU,O) = EW|S∈{0,s}[E(Y1−Y0|W, S ∈ {0, s})] for s∈ {0, ..., K}.

3.1 Identiﬁcation

Next, we discuss whether each of the potential causal parameters, ΨF

s(PU,O), is identiﬁable from the observed data.

Lemma 1:

For each experiment with

S∈ {0, s}

, under

Assumptions 1 and 2a-b

below, the causal ATE,

ΨF

s(PU,O)

is identiﬁable from the observed data by the g-computation formula (Robins, 1986), with statistical estimand

Ψs(P0) = EW|S∈{0,s}[E0[Y|A= 1, S ∈ {0, s}, W ]−E0[Y|A= 0, S ∈ {0, s}, W ]].(1)

Assumption 1

(Positivity (e.g. Hernan (2006); Petersen et al. (2012))):

P(A=a|W=w, S ∈ {0, s})>0

for all

a∈A

and all

for which

P(W=w, S ∈ {0, s})>0

.This assumption is true in the RCT by design and may

be satisﬁed for other experiments by removing RWD controls whose

covariates do not have support in the trial

population.

Assumption 2 (Mean Exchangeability (e.g. Rudolph and van der Laan (2017); Dahabreh et al. (2019b))):

As described by Rudolph and van der Laan (2017) and subsequently named by Dahabreh et al. (2019b),

Assumption 2a

(“Mean exchangeability in the trial" (Dahabreh et al., 2019b)):

E[Ya|W, S = 0, A =a] =

E[Ya|W, S = 0]. This assumption is also true by the design of the RCT.

Assumption 2b

(“Mean exchangeability over

" (Dahabreh et al., 2019b)):

E[Ya|W, S = 0] = E[Ya|W, S ∈ {0, s}]

for every

a∈A

Assumption 2b

may be violated if unmeasured factors affect trial inclusion or if being in the RCT

directly affects adherence or outcomes (Rudolph and van der Laan, 2017; Dahabreh et al., 2019a). Dahabreh et al.

(2019a) note that

Assumption 2b

is more likely to be true for pragmatic RCTs integrated with RWD from the same

healthcare system. Nonetheless, we may not be certain whether Assumption 2b is violated in practice.

3.2 Bias Estimation

One approach to concerns about violations of

Assumption 2b

would be to target a causal parameter that we know is

identiﬁable from the observed data. As noted by Hartman et al. (2015), Balzer (2017), and Dahabreh et al. (2019c),

we may consider interventions not only on treatment assignment but also on trial participation. The difference in the

outcomes an individual would have had if they had received active treatment and been in the RCT (

Ya=1,s=0

) compared

to if they had received standard of care and been in the RCT (

Ya=0,s=0

), averaged over a distribution of covariates

that are represented in the trial, gives a causal ATE of A on Y in the population deﬁned by that experiment. Under

Assumptions 1 and 2a

, this “ATE-RCT" parameter for any experiment,

ΨF

s(PU,O) = EW|S∈{0,s}[E(Ya=1,s=0 −

Ya=0,s=0|W, S ∈ {0, s})], is equal to the following statistical estimand:

Ψs(P0) = EW|S∈{0,s}[E0[Y|A= 1, S = 0, W ]−E0[Y|A= 0, S = 0, W ]].(2)

Nonetheless, by estimating this parameter, we would not gain efﬁciency compared to estimating the sample average

treatment effect for the RCT only (Balzer et al., 2015).

Another general approach to addressing concerns regarding violations of

Assumption 2b

would be to estimate the

causal gap or bias due to inclusion of external controls. In order to further explore this option, we consider two causal

gaps as the difference between one of our two potential causal parameters and the statistical estimand

Ψs(P0)

for a

given experiment with S∈ {0, s}:

1. Causal Gap 1: ΨF

s(PU,O)−Ψs(P0)

2. Causal Gap 2: ˜

ΨF

s(PU,O)−Ψs(P0)

While these causal gaps are functions of the full and observed data, we can estimate a statistical gap that is only a

function of the observed data as

Ψ#

s(P0)=Ψs(P0)−˜

Ψs(P0)

=EW|S∈{0,s}[E0[Y|A= 0, S = 0, W ]] −EW|S∈{0,s}[E0[Y|A= 0, S ∈ {0, s}, W ]] (3)

Lemma 2: Causal and Statistical Gaps for an experiment with S∈ {0, s}

If Assumption 2b is true, then ΨF

s(PU,O) = ˜

ΨF

s(PU,O),Ψ#

s(P0)=0

Causal Gap 1: ΨF

s(PU,O)−Ψs(P0)=0

Causal Gap 2: ˜

ΨF

s(PU,O)−Ψs(P0)=0

Experiment-Selector CV-TMLE A PREPRINT

Ψ#

s(P0)

may thus be used as evidence of whether

Assumption 2b

is violated. If we were to bias correct our estimate

Ψs(P0)

by subtracting

Ψ#

s(P0)

, we would again be estimating

Ψs(P0)

, with no gain in efﬁciency compared to

estimating the sample ATE from the RCT only (Balzer et al., 2015). Nonetheless, the information from estimating

Ψ#

s(P0)may still be incorporated into an experiment selector, s?

n, discussed below.

4 Potential Experiment Selection Criteria

A natural goal for experiment selection would be to optimize the bias-variance tradeoff for estimating a causal ATE.

Such an approach of determining combinations of RCT and RWD that minimize the estimated mean squared error is

taken by Yang et al. (2020), Cheng and Cai (2021), Chen et al. (2021), and Oberst et al. (2022). Next, we discuss the

challenge of selecting a truly optimal experiment when bias must be estimated from the data. We then introduce a novel

experiment selector that incorporates bias estimates based on both the primary outcome and a negative control outcome.

Ideally, we would like to construct a selector that is equivalent to the oracle selector of the experiment that optimizes

the bias-variance tradeoff for our target parameter:

s0=argmin

σ2

D∗

Ψs

n+ (Ψ#

s(P0))2

where

D∗

Ψs(O) =

I(S∈{0,s})

P(S∈{0,s})(( I(A=1)

0(A=1|W,S∈{0,s})−I(A=0)

0(A=0|W,S∈{0,s}))(Y−Q{0,s}

0(S∈ {0, s}, A, W ))

+Q{0,s}

0(S∈ {0, s},1, W )−Q{0,s}

0(S∈ {0, s},0, W )−Ψs(P0))

is the efﬁcient inﬂuence curve of

Ψs(P0)

Q{0,s}

0(S∈ {0, s}, A, W ) = E0[Y|S∈ {0, s}, A, W ]

, and

0(A=

a|W, S ∈ {0, s}) = P0(A=a|W, S ∈ {0, s}). Our statistical estimand of interest is then Ψs0(P0).

The primary challenge is that

must be estimated. We thus deﬁne an empirical bias squared plus variance (

“b2v")

selector,

n=argmin

ˆσ2

D∗

Ψs

n+ ( ˆ

Ψ#

s(Pn))2(4)

If, for a given experiment with

S∈ {0, s}

Ψ#

s(P0)

were given and small relative to the standard error of the ATE

estimator for that experiment, nominal coverage would be expected for the causal target parameter. If bias were large

relative to the standard error of the ATE estimator for the RCT, then the RWD would be rejected, and only the RCT

would be analyzed. One threat to valid inference using this experiment selection criterion is the case where bias is of the

same order as the standard error

σD∗

Ψs/√n

, risking decreased coverage. We could require a smaller magnitude of bias

by putting a penalty term in the denominator of the variance as

n=argmin

ˆσ2

D∗

Ψs

/(n∗c(n)) + ( ˆ

Ψ#

s(Pn))2

where

c(n)

is either a constant or some function of

. A similar approach is taken by Cheng and Cai (2021) who multiply

the bias term by a penalty and determine optimal weights for RCT and RWD estimators via L1-penalized regression.

However, ﬁnite sample variability may lead to overestimation of bias for unbiased RWD and underestimation of bias

similar in magnitude to

σD∗

Ψs/√n

. In order to make

c(n)

large enough to prevent selecting RWD that would introduce

bias of a magnitude that could decrease coverage for the causal parameter, we would also prevent unbiased RWD from

being included in a large proportion of samples.

This challenge exists for any method that bases inclusion of RWD on differences in the mean or conditional mean

outcome under control for a small RCT control arm versus a RWD population. It also suggests that having additional

knowledge beyond this information may help the selector distinguish between RWD that would introduce varying

degrees of bias. Intuitively, if we are not willing to assume mean exchangeability, information available in the RCT

alone is insufﬁcient to estimate bias from including real world data in the analysis precisely enough to guarantee

inclusion of extra unbiased controls and exclusion of additional controls that could bias the effect estimate; if the RCT

contained this precise information about bias, we would be able to estimate the ATE of A on Y from the RCT precisely

enough to not require the real world data at all. Conversely, if we were willing to assume mean exchangeability, then

simply pooling RCT and RWD would provide optimal power gains but also fully relinquish the protection to inference

afforded by randomization.

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ACROSS-VALIDATEDTARGETEDMAXIMUMLIKELIHOODESTIMATORFORDATA-ADAPTIVEEXPERIMENTSELECTIONAPPLIEDTOTHEAUGMENTATIONOFRCTCONTROLARMSWITHEXTERNALDATAAPREPRINTLaurenEylerDang,JensMagelundTarp,TrineJulieAbrahamsen,KajsaKvist,JohnBBuse,MayaPetersen,MarkvanderLaanFebruary21,2023ABSTRACTAugmentingthecontrolarmof...

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A C ROSS -VALIDATED TARGETED MAXIMUM LIKELIHOOD ESTIMATOR FOR DATA-ADAPTIVE EXPERIMENT SELECTION APPLIED TO THE AUGMENTATION OF RCT C ONTROL ARMS

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