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206 Applications of the Roadmap
1 tmle_fit <- tmle (Y = Y , A = Z , W = W, Delta = 1 - Delta , Q.SL .
library = c(" SL.glm" , " tmle. SL.dbarts2 ", "SL. glmnet "), g1W =
g1S, g.Delta .SL. library = c ("SL. glm", "tmle .SL .dbarts .k.5" , "
SL.gam "), family = "gaus sian" , ...)
2
3 tmle_fit$ estimates$ ATE$ psi
4 tmle_fit$ estimates$ ATE$ CI
In the above R function, Y=Y is to assign outcome variable Y on the right
to argument Y on the left, A=Z is to assign treatment assignment Z to argu-
ment Z, W=W is to assign the expanded covariate vector, W (including covari-
ates in X and the corresponding missing-covariate indicators), to argument
W, Delta=1-Delta is to assign 1 minus missing indicator Δ in the data (0
for non-missing and 1 for missing) to argument Delta (1 for non-missing and
0 for missing), Q.SL.library is to specify a library of predictive models to
fit super learner for the outcome regression function, g1W=glS is the vector
of conditional treatment assignment probabilities, g(1|S)=P(Z =1|S)for
stratified RCT or g(1|S)=1/2 for 1:1 complete RCT, g.Delta.SL.library
is to specify a library of predictive models to fit super learner for the non-
missing probability function, and family specifies “gaussian” for continuous
outcome and “binomial” for binary outcome.
Step 8: Sensitivity analysis
Conduct sensitivity analysis to explore the robustness of the results from the
main estimator to deviations from its underlying assumptions.
Example 1 (continued). The identifiability assumptions are the consistency,
MAR, and positivity assumptions. For RCTs, the consistency assumption and
the positivity assumption are usually not worrisome. For the MAR assump-
tion, a series of reference-based imputation methods discussed in Chapter 10
can be carried out to explore the robustness of the results under some missing
not-at-random (MNAR) scenarios.
Step 9: Evidence
The last step is to interpret the results of the main analysis and the corre-
sponding sensitivity analysis. The findings from such interpretation are sum-
marized as evidence.
Example 1 (continued). If the results show that the treatment effect is
statistically significant under MAR and statistically significant under those
pre-specified MNAR scenarios, then the evidence is robust. However, if the
results show that the treatment effect is not statistically significant under
those pre-specified MNAR scenarios, then the evidence is not robust.
Applications to RCTs 207
12.2.2 Longitudinal RCTs
Step 1: Research question
Follow the PROTECT criteria to prepare the following items:
– Population: defined via inclusion/exclusion criteria;
– Response/Outcome variable: Y ;
– Treatment/Exposure variable: treatment 1 or 0;
– Counterfactual thinking: randomization and blinding;
– Time: outcome measured at t =1,...,T.
Research question/objective: to establish the existence, and estimate the
magnitude, of treatment effects of treatment 1 (throughout T decision points)
vs. treatment 0 (throughout T decision points) on the primary outcome mea-
sured at time T among subjects in the well-defined ITT population.
The pair of treatments under comparison can be replaced by any other two
feasible treatment regimes to be investigated by a SMART design discussed
in Chapter 5.
Step 2: Study design
Consider a complete RCT or a stratified RCT with stratification factor S,
with follow-up visits indicated by 1,...,T. In a typical double-blind RCT,
treatment 0 is a placebo.
Step 3: Causal estimand
Following Chapter 5, define a causal estimand in terms of potential outcomes,
taking into account the handling of the anticipated missing data and ICEs.
Example 2. In a stratified RCT, assume that there are two types of ICEs,
treatment dropout and analysis dropout, and assume that the hypotheti-
cal strategy—envisaging a hypothetical scenario in which no dropout would
occur—is applied to handle both treatment dropout and analysis dropout. Let
Z be the treatment assignment and let
Δ = (Δ(1),...,Δ(T )) be the vector of
the indicators of dropout occurrence at T measurement time-points. Assume
we are interested in the following causal estimand:
θ
∗
2
= E[Y
z=1,δ=0
] − E[Y
z=0,δ=0
].
Remark: As discussed in Chapter 4, if a different strategy (or a different
combination of strategies) is applied to handle ICEs, then the research ques-
tion in Step 1 needs to be revised accordingly. See the remark below Step 3
of Example 1 for more detail.
208 Applications of the Roadmap
Step 4: Data
Conduct sample size calculation that is aligned with the causal estimand,
using one of the methods discussed in Chapter 4. Then enroll patients. And
then collect data following the study protocol.
Example 2 (continued). Collect data consisting of X
i
(0),Z
i
, Δ
i
(t), (1 −
Δ
i
(t))
&
X
i
(t), (1 − Δ
i
(t))Y
i
(t),t =1,...,T − 1, Δ
i
(T ), (1 − Δ
i
(T ))Y
i
, i =
1,...,N,whereX (0) includes stratification factor S,
&
X(t) is a vector of
time-dependent covariates, Y (t) is outcome measured at t, t =1,...,T − 1,
and Y = Y (T ) is the outcome variable measured at T.LetX(t) include
both time-dependent covariates
&
X(t) and intermediate outcome Y (t), and let
X(0) = X(0) and X(t)=(X(0),...,X(t)), t =1, ···,T −1.
Step 5: Statistical estimand
Under the identifiability assumptions, translate the causal estimand into a
statistical estimand.
Example 2 (continued). Under the identifiability assumptions (consistency,
MAR, positivity), the causal estimand can be translated into the following
statistical estimand,
θ
2
= E
X∼F
1
E(Y |
X,Z =1, Δ=0)
− E
X∼F
0
E(Y |
X,Z =0, Δ=0)
,
where the outer expectation on the right-hand-side is over
X ∼F
j
and F
j
is
a distribution function of (X (0),...,X(T − 1)) defined as
P
F
j
(X(0) = x(0),...,X(T − 1) = x(T − 1))
=P{X(0) = x(0)}
T −1
+
t=1
P
X(t)=x(t)|X(t − 1) = x(t − 1),Z = j, Δ(t)=0
.
Step 6: Estimator
To estimate the statistical estimand, one of the estimation methods discussed
in Chapter 9 is selected.
Example 2 (continued). We will demonstrate the application of LTMLE in
Example 6. In this example and Example 3, we demonstrate the application of
an alternative method that combines the convenience of multiple imputation
(MI) and the strength of TMLE; referred to as MI+TMLE. An advantage
of using MI+TMLE is that it will be straightforward to conduct sensitivity
analysis for the MAR assumption under different imputation models that
reflect different MNAR scenarios.
Step 7: Estimate
Plugging the data into the proposed estimator, a point estimate is obtained,
along with standard error estimator, confidence estimate, and p-value.
Applications to RCTs 209
Example 2 (continued). According to the name, MI+TMLE consists of
two stages. In stage one, missing values are imputed using some multiple
imputation procedure under the MAR assumption (say, using the R package
“mice”). In stage two, TMLE is applied to the imputed dataset, in which there
is no need to specify arguments Delta or g.Delta.SL.library. Repeat the
process by M times and combine the results using Rubin’s rule.
Step 8: Sensitivity analysis
Conduct sensitivity analysis to explore the robustness of the results from the
main estimator to deviations from its underlying assumptions.
Example 2 (continued). The identifiability assumptions are the consistency,
MAR, and positivity assumptions. Reference-based imputation methods dis-
cussed in Chapter 10, including the copy-reference (CR) imputation and jump-
to-reference (J2R) imputation, are appropriate for conducting sensitivity anal-
ysis to explore the robustness of the results under MNAR scenarios.
To implement the CR imputation, consider the following steps. First, ob-
tain the subset consisting of all the subjects with Z
i
= 0 and all the subjects
with Z
i
=1andΔ
i
(T ) = 1, keeping the remaining set consisting of those
subjects with Z
i
=1andΔ
i
(T ) = 0 for later use. Second, apply some MI
procedure under the MAR assumption to the subset. Third, combine the im-
puted subset with the aforementioned set consisting of those subjects with
Z
i
=1andΔ
i
(T ) = 0. Fourth, apply TMLE to the final full dataset.
To implement the J2R imputation, consider the following steps. First, ob-
tain the subset consisting of all the subjects with Z
i
= 0 and all the subjects
with Z
i
=1andΔ
i
(T ) = 1, keeping the remaining set consisting of those
subjects with Z
i
=1andΔ
i
(T ) = 0 for later use. Second, delete the observed
outcome data in the subset for those subjects with Z
i
=1andΔ
i
(T )=1,
bearing in mind that these tentatively deleted data will be restored. Third,
apply some MI procedure under the MAR assumption to the modified subset.
Fourth, restore the tentatively-deleted outcome data back to the imputed sub-
set and combine the subset with the remained set consisting of those subjects
with Z
i
=1andΔ
i
(T ) = 0. Fifth, apply TMLE to the final full dataset.
Repeat the above process, either CR or J2R, by M timesandcombinethe
results using Rubin’s rule.
Step 9: Evidence
The last step is to interpret the results of the main analysis and the corre-
sponding sensitivity analysis.
Example 2 (continued). If the results show that the treatment effect is
statistically significant under MAR and statistically significant under those
pre-specified MNAR scenarios, then the evidence is robust. However, if the re-
sults show that the treatment effect is not significant under those pre-specified
MNAR scenarios, then the evidence is not robust.
210 Applications of the Roadmap
12.2.3 RCTs with time-to-event outcome
In the preceding subsection, the outcome variable is either continuous or bi-
nary. As discussed in Chapter 5, an RCT with time-to-event outcome can
be converted into a longitudinal RCT with repeatedly measured binary out-
come. Following this idea, we only need to slightly revise the roadmap in the
preceding subsection and apply it to RCTs with time-to-event outcome.
Step 1: Research question
Follow the PROTECT criteria to prepare the following items:
– Population: defined via inclusion/exclusion criteria;
– Response/Outcome variable: survival outcome Y ;
– Treatment/Exposure variable: treatment 1 or 0;
– Counterfactual thinking: randomization and blinding;
– Time: survival status measured at t =1,...,T.
Research question/objective: to establish the existence, and estimate the
magnitude, of treatment effects of treatment 1 vs. treatment 0 on the survival
rate at time T counting from the time of treatment initiation among all the
subjects in the well-defined ITT population.
Note that the population-level summary, which is the survival rate at time
T in the above, can be replaced by others, say restricted mean survival time
(RMST) limited by T .
Step 2: Study design
Consider a complete RCT or stratified RCT with stratification factor S.Ina
typical double-blind RCT, treatment 0 is placebo.
Step 3: Causal estimand
Following Chapter 5, define a causal estimand in terms of potential outcomes,
taking into account the handling of the anticipated censoring and ICEs.
Example 3. In a stratified RCT, assume that there are two types of ICEs,
treatment dropout and analysis dropout (also known as right censoring for
time-to-event outcome), and assume that the treatment policy strategy is ap-
plied to handle treatment dropout and the hypothetical strategy—envisaging
a hypothetical scenario in which censoring would not occur—is applied to
handle censoring. Since the treatment policy strategy is applied to handle
treatment dropout (say, treatment discontinuation, rescue medication), the
“T/E” component of the research question in Step 1 needs to be revised, in-
corporating treatment discontinuation and rescue medication as parts of the
treatments under comparison.
Let Z be the treatment assignment. Let Y
∗
be the time-to-death. Define
Y (t)=I(Y
∗
≤ t), with Y (t) = 1 indicating death status at t.LetΔ=
Applications to RCTs 211
(Δ(1),...,Δ(T −1)) be the vector of the indicators of censoring up to T −1. In
the definition of
Δ, Δ(T ) is excluded because Δ(T ) = 1 implies that Y (T )=0.
Let Y = Y (T ) be the primary outcome, which is the death/survival status at
T .Byconvention,ifY (t)=1,thenY (t
)=1andΔ(t
) = 0 for all t
≥ t.
Assume we are interested in the following causal estimand:
θ
∗
3
= E[Y
z=1,δ=0
] − E[Y
z=0,δ=0
].
An alternative version of Example 3. Same as Example 3, except that
the hypothetical strategy—envisaging a hypothetical scenario in which nei-
ther treatment dropout nor censoring would occur—is applied to handle both
treatment dropout and censoring. Let
&
Δ(t) = 1 if either treatment dropout or
censoring occurs at time t. The remaining steps will be the same except that
Δ(t) is replaced by
&
Δ(t).
Step 4: Data
Conduct sample size calculation that is aligned with the causal estimand.
Then enroll patients. And then collect data following the study protocol.
Example 3 (continued). Collect data consisting of X
i
(0),Z
i
, Δ
i
(t), (1 −
Δ
i
(t))X
i
(t), (1 − Δ
i
(t))Y
i
(t),t =1,...,T − 1, Δ
i
(T ), (1 − Δ
i
(T ))Y
i
, i =
1,...,N,whereX(0) includes stratification factor S, X(t) is vector of time-
dependent covariates, Y (t) is death/survival status at t, t =1,...,T −1, and
Y = Y (T ) is the death/survival status at T . By convention, if Y (t)=1,then
let Y (t
)=1,X(t
)=X(t), and Δ(t
) = 0 for all t
≥ t.
Step 5: Statistical estimand
Under the identifiability assumptions, translate the causal estimand into a
statistical estimand.
Example 3 (continued). Under the identifiability assumptions (consistency,
censoring at random (CAR), positivity), the causal estimand can be translated
into the following statistical estimand,
θ
3
= E
X∼F
1
E(Y |
X,Z =1, Δ=0)
− E
X∼F
0
E(Y |
X,Z =0, Δ=0)
,
where the outer expectation on the right-hand-side is over
X ∼F
j
and F
j
is
the same as the one defined in the preceding subsection.
Step 6: Estimator
To estimate the statistical estimand, one of the estimation methods discussed
in Chapter 9 is selected.
Example 3 (continued). We will demonstrate the application of LTMLE in
Example 6. Like in Example 2, in this example we consider MI+TMLE.
212 Applications of the Roadmap
Step 7: Estimate
Plugging the data into the proposed estimator, a point estimate is obtained,
along with standard error estimate, 95% confidence estimate, and p-value.
Example 3 (continued). Like in Example 2, MI+TMLE consists of two
stages. In stage one, censored values are imputed using some multiple impu-
tation procedure under the CAR assumption. After each imputation, we need
to revise the imputed values following the convention: if Y (t) = 1, then let
Y (t
)=1andX(t
)=X(t) for all t
≥ t. In stage two, TMLE is applied to
the imputed dataset. Repeat the process by M times and combine the results
using Rubin’s rule.
Step 8: Sensitivity analysis
Conduct sensitivity analysis to explore the robustness of the results from the
main estimator to deviations from its underlying assumptions.
Example 3 (continued). The identifiability assumptions are the consistency,
CAR, and positivity assumptions. Same as in Example 2, reference-based
imputation methods (say, CR and J2R imputation methods) are appropriate
methods for conducting sensitivity analysis to explore the robustness of the
results under censoring not at random (CNAR) scenarios.
Step 9: Evidence
The last step is to interpret the results of the main analysis and the corre-
sponding sensitivity analysis.
Example 3 (continued). If the main results show that the treatment effect
is statistically significant under CAR and statistically significant under those
pre-specified CNAR scenarios, then the evidence is robust. However, if the
treatment effect is not statistically significant under those pre-specified CNAR
scenarios, then the evidence is not robust.
12.3 Applications to Cohort Studies
Consider a sample of subjects to be enrolled in an observational study. Al-
though the sample is usually sampled by convenience, the subjects in the
sample can be assumed to be i.i.d. drawn from a population which is concep-
tually constructed—like the way by which a super-population is constructed.
Therefore, as in Chapters 6–9, we denote the sample size as n.
The main difference between an interventional study and an observational
cohort study is that the propensity score function is unknown in an observa-
tional study.
Applications to Cohort Studies 213
12.3.1 Cohort studies with a single follow-up
We have discussed this application in Chapter 11 in detail using a plasmode
case study. Here we briefly go through the steps.
Step 1: Research question
Follow the PROTECT criteria to prepare the following items:
– Population: defined via inclusion/exclusion criteria;
– Response/Outcome variable: Y ;
– Treatment/Exposure variable: treatment 1 or 0;
– Counterfactual thinking: confounding and confounders;
– Time: outcome measured at T .
Research question/objective: to establish the existence, and estimate the
magnitude, of treatment effects of treatment 1 vs. treatment 0 on the outcome
variable measured at time T from the time of treatment initiation among all
the subjects in the population.
Step 2: Study design
Consider a cohort study, consisting of two unmatched cohorts, one cohort
treated by treatment 1 and the other cohort treated by treatment 0. Baseline
is the time when the treatment is initiated and follow-up time T is the time
when the outcome variable is measured. The study can be prospective or
retrospective, which should be determined at this step.
Step 3: Causal estimand
Following Chapter 6, define a causal estimand in terms of potential outcomes.
Following Chapters 3–4, revise the causal estimand to take into account the
handling of missing data and ICEs.
Example 4. Assume that in a cohort study, there are two types of ICEs,
treatment discontinuation and analysis dropout, and assume that the hypo-
thetical strategy—envisaging a hypothetical scenario in which neither ICE
would occur—is applied to handle the ICEs. Let A be the treatment assign-
ment and let Δ be the indicator of any ICE. Assume we are interested in the
following causal estimand:
θ
∗
4
= E[Y
a=1,δ=0
] − E[Y
a=0,δ=0
].
Remark: As discussed in Chapter 4, if a different strategy (or a different
combination of strategies) is applied to handle ICEs, then the research ques-
tion in Step 1 needs to be revised accordingly. See the remark below Step 3
of Example 1 for more detail.
214 Applications of the Roadmap
Step 4: Data
If the cohort study is prospective, conduct sample size calculation that is
aligned with the causal estimand as discussed in Section 10.4. Then enroll
patients. And then collect data following the study protocol.
If the cohort study is retrospective, conduct feasibility analysis to deter-
mine the feasible sample size. Then obtain the data (say, from preexisting
databases) following the study protocol.
Example 4 (continued). Collect or obtain data
O = {(X
i
,A
i
, Δ
i
, (1 − Δ
i
)Y
i
,i=1,...,n},
where X consists of pre-treatment covariates that ensure the exchangeability
assumption and the MAR assumption be satisfied and covariates that are
believed to be associated with the outcome variable.
Step 5: Statistical estimand
Under the identifiability assumptions, translate the causal estimand into a
statistical estimand.
Example 4 (continued). Under the identifiability assumptions (consistency,
exchangeability, MAR, positivity), the causal estimand can be translated into
the following statistical estimand,
θ
4
= E[Y |X, A =1, Δ=0]−E[Y |X, A =0, Δ=0].
Step 6: Estimator
To estimate the statistical estimand, one of the estimation methods discussed
in Chapters 7–8 is selected.
Example 4 (continued). TMLE is doubly robust; it is consistent if either
the propensity score function and the non-missing probability function are
estimated consistently or the outcome regression function is estimated consis-
tently. TMLE is also an RAL estimator with the efficient influence function.
Step 7: Estimate
Plugging the data into the proposed estimator, a point estimate is obtained,
along with standard error estimate, 95% confidence estimate, and p-value.
Example 4 (continued). To implement TMLE, consider the following R
function “tmle.”
1 tmle_fit <- tmle (Y = Y , A = A , W = W, Delta = 1 - Delta , Q.SL .
library = c(" SL.glm" , " tmle. SL.dbarts2 ", "SL. glmnet "), g. SL.
library = c(" SL.glm" , " tmle.SL .dbarts .k.5" , "SL. gam"), g. Delta
.SL. library = c ("SL.glm ", " tmle.SL .dbarts .k.5", "SL. gam") ,
family = " gaussian ", ...)
2
3 tmle_fit$ estimates$ ATE$ psi
4 tmle_fit$ estimates$ ATE$ CI
Applications to Cohort Studies 215
The above R codes are the same as those in Example 1, except that
g.SL.library is used to replace g1W to specify a library of predictive models
to fit super learner for the propensity score function, which is unknown in the
observational cohort study.
Step 8: Sensitivity analysis
Conduct sensitivity analysis to explore the robustness of the results from the
main estimator to deviations from its underlying assumptions.
Example 4 (continued). The identifiability assumptions are the consis-
tency, exchangeability, MAR, and positivity assumptions. Among them, the
exchangeability assumption (a.k.a., the no-unmeasured confounding assump-
tion) is the most crucial one for non-randomized studies. For example, E-value
is an appropriate method to explore the robustness of the results if there are
unmeasured confounders.
Step 9: Evidence
The last step is to interpret the results of the main analysis and the corre-
sponding sensitivity analysis. The findings from such interpretation are sum-
marized as evidence.
Example 4 (continued). In order to interpret the E-value, some threshold
(say, 2) should be pre-specified in the protocol that is agreed upon among
the stakeholders. If the E-value is larger than or equal to the pre-specified
threshold, the main result can be claimed to be robust. Otherwise, the main
result is not robust.
12.3.2 Externally controlled trials
An externally controlled trials (ECTs) can be considered as a special point-
exposure cohort study, which consists of two cohorts—the treated cohort is
the single-arm clinical trial and the control cohort is formed externally. The
control cohort may be formed based on the placebo arms of some historical
RCTs or may be formed based on external real-world data (RWD).
But there are some differences between the ECT to be discussed soon in
the following and the typical cohort study that is discussed in the preceding
example. In the preceding cohort study, the population consists of all the
subjects to be treated by either the investigative treatment or the control
treatment, and therefore the estimand is the average treatment effect (ATE).
In the ECT to be discussed soon, the population consists of all the subjects
that are treated by the investigative treatment in the real world, and therefore
the estimand is the average treatment effect among the treated (ATT).
Step 1: Research question
Follow the PROTECT criteria to prepare the following items: