Causal Inference Opening Workshop - Testing Weak Nulls in Matched Observational Studies - Colin Fogarty, December 11, 2019

Testing Weak Nulls in Matched Observational
Studies
Colin Fogarty
Massachusetts Institute of Technology
December 11, 2019

Robustness of Randomization Tests
Randomization tests provide exact tests for sharp null hypotheses,
the most common of which is Fisher’s sharp null:
HF : yi (0) = yi (1) for all i
Colin Fogarty Sensitivity Analysis for Weak Nulls December 11, 2019

Concern: are randomization tests of sharp nulls prone to
misinterpretation?

misinterpretation?
Perhaps a researcher will use a randomization test, but then
think that she/he has evidence for the existence of a positive
average eﬀect if it rejects...

misinterpretation?
Perhaps a researcher will use a randomization test, but then
think that she/he has evidence for the existence of a positive
average eﬀect if it rejects...
Related issue: permutation tests are often viewed by
practitioners as nonparametric alternatives to t-tests when the
two samples don’t look normally distributed

Randomization Tests for Weak Nulls
Consider a completely randomized experiment under the ﬁnite
population model. Suppose I want to use a randomization test to
test the weak null hypothesis
HN : ¯y(1) − ¯y(0) = 0.

HN : ¯y(1) − ¯y(0) = 0.
Cannot use the randomization test based upon the observed
treated-minus-control diﬀerence in means, ˆτ

HN : ¯y(1) − ¯y(0) = 0.
Cannot use the randomization test based upon the observed
treated-minus-control diﬀerence in means, ˆτ
Can use the randomization test with the studentized diﬀerence
in means,
ˆτ
s2
T /nT + s2
C /nC
,
where s2
T and s2
C are the sample variances of the observed
outcomes in the treated and control groups

The studentized randomization test yields a single, uniﬁed mode of
inference under the ﬁnite population model that is
1 Exact under Fisher’s sharp null

2 Asymptotically conservative under Neyman’s weak null
Conservative inference is a fundamental property of inference
under the ﬁnite population model with heterogeneous eﬀects

2 Asymptotically conservative under Neyman’s weak null
Conservative inference is a fundamental property of inference
under the finite population model with heterogeneous effects
While the sharp and weak nulls are surely different, the studentized
randomization test obviates the distinction for practitioners
See Loh et al. (2017); Wu and Ding (2018). Also see Chung and
Romano (2013) for related developments for permutation tests.

What’s Diﬀerent for Observational Studies?
For randomized experiments, E(ˆτ) = 0 under both the sharp and
weak nulls. But regardless of assuming the sharp or weak null, we
would not expect E(ˆτ) = 0 in a matched observational study.
Why? Unmeasured confounding.

In observational studies, we assess the robustness of our study’s
ﬁndings to unmeasured confounding through a sensitivity analysis

We’ll review a model proposed by Rosenbaum for such an
analysis in matched observational studies
Existing methods under the model primarily focus on testing
sharp null hypotheses.

We’ll review a model proposed by Rosenbaum for such an
analysis in matched observational studies
Existing methods under the model primarily focus on testing
sharp null hypotheses.
What if eﬀect heterogeneity induces larger discrepancies between
¯y(1) − ¯y(0) and the worst-case expectation of ˆτ?

Notation for Matched Studies
The ith
of B matched sets has ni individuals. N = B
i=1 ni

The ith
i=1 ni
Individual j in set i has observed covariates, xij and an
unobserved covariate, 0 ≤ uij ≤ 1.

The ith
i=1 ni
Using an optimal, without-replacement matching algorithm,
treated and control individuals are placed into matched sets such
that xij ≈ xij .

The ith
i=1 ni
that xij ≈ xij .
Consider post-strata with one treated individual; ni − 1 controls
(what follows immediately extends to full matching).

The ith
i=1 ni
that xij ≈ xij .
yij (1) and yij (0) are the potential outcomes under treatment and
control for individual j in set i.

The ith
i=1 ni
that xij ≈ xij .
yij (1) and yij (0) are the potential outcomes under treatment and
control for individual j in set i.
τij = yij (1) − yij (0) is the treatment eﬀect for each individual

What do we observe?
Zij is the treatment indicator (1 treated, 0 control).

What do we observe?
Yij = Zij yij (1) + (1 − Zij )yij (0) is the observed response

What do we observe?
ˆτi = ni
j=1 Zij Yij − ni
j=1(1 − Zij )Yij /(ni − 1)

What do we observe?
ˆτi = ni
j=1 Zij Yij − ni
j=1(1 − Zij )Yij /(ni − 1)
¯τi = ni
j=1 τij /ni is the average of the treatment eﬀects in set i.

What do we observe?
ˆτi = ni
j=1 Zij Yij − ni
j=1(1 − Zij )Yij /(ni − 1)
¯τi = ni
¯τ = ni
j=1(ni /N)¯τi is the sample average treatment eﬀect.

What do we observe?
ˆτi = ni
j=1 Zij Yij − ni
j=1(1 − Zij )Yij /(ni − 1)
¯τi = ni
¯τ = ni
j=1(ni /N)¯τi is the sample average treatment eﬀect.
Let F = {yij (1), yij (0), xij , uij }, Ω = {z : ni
j=1 zij = 1},
Z = {Z ∈ Ω}
Inference moving forwards will condition upon F and Z.

A Simple Model for Hidden Bias
Let πij = pr(Zij = 1 | F). A simple model for hidden bias states that
πij = pr(Zij = 1 | xij , uij ), with 0 ≤ uij ≤ 1 and
log
πij
1 − πij
= κi + log(Γ)uij .

log
πij
1 − πij
Γ ≥ 1 controls the impact of hidden bias on treatment
assignment.

log
πij
1 − πij
Γ ≥ 1 controls the impact of hidden bias on treatment
assignment.
Equivalently, for individuals j, j in the same matched set i,
1
Γ
≤
πij (1 − πij )
πij (1 − πij )
≤ Γ

Let ij = pr(Zij = 1 | F, Z)

Conditions on the matched structure, removing dependence on
the nuisance parameters κi

At Γ = 1, ij = 1/ni .

At Γ = 1, ij = 1/ni .
Recovers a finely stratified experiment (one treated, ni − 1
controls, ni equiprobable assignments).
Entitles one to modes of inference justified under those designs

At Γ = 1, ij = 1/ni .
Recovers a finely stratified experiment (one treated, ni − 1
controls, ni equiprobable assignments).
Entitles one to modes of inference justified under those designs
Γ > 1 allows for departures from the idealized finely stratified
experiment that matching seeks to emulate,
ij =
exp{log(Γ)uij }
ni
k=1 exp{log(Γ)uik}

Sensitivity Analysis Under the Sharp Null
Let δij be the observed value of the treated-minus-control paired
diﬀerence if individual j receives the treatment.
δij = yij (1) −
j =j
yij (0)/(ni − 1),
Under the sharp null, yij (1) = yij (0) and the observed response Yij
impute the yij (0), and hence δij .

δij = yij (1) −
j =j
yij (0)/(ni − 1),
Consider the randomization distribution based upon ˆτ,
pr(ˆτ ≥ a | F, Z) =
z∈Ω
1{ˆτ ≥ a}pr(Z = z | F, Z)

δij = yij (1) −
j =j
yij (0)/(ni − 1),
Consider the randomization distribution based upon ˆτ,
pr(ˆτ ≥ a | F, Z) =
z∈Ω
1{ˆτ ≥ a}pr(Z = z | F, Z)
Even under the sharp null, we can’t directly use this randomization
distribution in observational studies because pr(Z = z | F, Z) is
unknown.

For a given value of Γ, a sensitivity analysis tries to construct a
random variable TΓ such that under the sharp null
pr(ˆτ ≥ a | F, Z) ≤ pr(TΓ ≥ a | F, Z)

pr(ˆτ ≥ a | F, Z) ≤ pr(TΓ ≥ a | F, Z)
This bounding random variable is used to upper bound p-values
Can be done exactly and tractably in a paired design
Not tractable for general matched designs, so one typically
resorts to asymptotic approximations (Gastwirth 2000)

pr(ˆτ ≥ a | F, Z) ≤ pr(TΓ ≥ a | F, Z)
This bounding random variable is used to upper bound p-values
Can be done exactly and tractably in a paired design
Not tractable for general matched designs, so one typically
resorts to asymptotic approximations (Gastwirth 2000)
Iteratively increase Γ until one can no longer reject the null. Attests
to the robustness of the study’s ﬁnding to hidden bias

A Uniﬁed Procedure?
Constructing the worst-case random variable makes heavy use of the
sharp null
Finding worst-case treatment assignment probabilities requires
knowledge of what the outcomes would have been for all
assignments

sharp null
assignments
Suppose we assume that the sensitivity model holds at Γ ≥ 1. Can
we ﬁnd a single, uniﬁed non-randomized hypothesis test ϕ(α, Γ) (1
if reject, 0 otherwise) such that, under suitable regularity conditions
and for any u

sharp null
assignments
and for any u
1 Under the sharp null, lim sup E{ϕ(α, Γ) | F, Z} ≤ α, with
equality possible

sharp null
assignments
and for any u
1 Under the sharp null, lim sup E{ϕ(α, Γ) | F, Z} ≤ α, with
equality possible
2 Under the weak null, lim sup E{ϕ(α, Γ) | F, Z} ≤ α, with
equality possible

Success for Paired Designs at Γ > 1
Deﬁne n new random variables
DΓi = ˆτi −
Γ − 1
1 + Γ
|ˆτi |,
For any Γ, ˆτi has worst-case expectation {(Γ − 1)/(1 + Γ)}|ˆτi |
under sharp null.
Consider the standard error estimate
se( ¯DΓ) =
1
n(n − 1)
n
i=1
(DΓi − ¯DΓ)2.

Deﬁne n new random variables
DΓi = ˆτi −
Γ − 1
1 + Γ
|ˆτi |,
For any Γ, ˆτi has worst-case expectation {(Γ − 1)/(1 + Γ)}|ˆτi |
under sharp null.
Consider the standard error estimate
se( ¯DΓ) =
1
n(n − 1)
n
i=1
(DΓi − ¯DΓ)2.
Consider the test
ϕ(α, Γ) = 1 ¯DΓ/se( ¯DΓ) ≥ Φ−1
(1 − α) ,
where Φ(·) is the standard normal CDF.

An Asymptotic Sensitivity Analysis for Neyman’s Null
Under mild regularity conditions, if the sensitivity model holds at Γ
and the weak null holds
lim sup
n→∞
E{ϕ(α, Γ) | F, Z} ≤ α.

An Asymptotic Sensitivity Analysis for Neyman’s Null
Under mild regularity conditions, if the sensitivity model holds at Γ
and the weak null holds
lim sup
n→∞
E{ϕ(α, Γ) | F, Z} ≤ α.
Furthermore, one can replace the standard normal reference
distribution with a studentized randomization distribution (with
biased assignment probabilities governed by Γ) such that
1 Under the sharp null, E{ϕ(α, Γ) | F, Z} ≤ α for any sample
size, with equality possible
2 Under the weak null, lim sup E{ϕ(α, Γ) | F, Z} ≤ α, with
equality possible
See Fogarty (2019+) for more details.

Γ > 1, General Matched Designs
For a given Γ, consider as a test statistic any monotone increasing
function of the stratumwise diﬀerences in means hΓ,ni
(ˆτi )
May depend upon Γ and ni

Γ > 1, General Matched Designs
For a given Γ, consider as a test statistic any monotone increasing
function of the stratumwise diﬀerences in means hΓ,ni
(ˆτi )
May depend upon Γ and ni
Let µΓi be the worst-case expectation for hΓ,ni
(ˆτi ) under the sharp
null hypothesis
Easy to compute through asymptotic separability (could also just
solve the LP).
µΓi is random under the weak null, varying with Z.

An Impossibility Result
Consider
E N−1
B
i=1
{hΓ,ni
(ˆτi ) − µΓi }
Bounded above by zero under sharp null if model holds at Γ,
with equality possible. What about the weak null?

An Impossibility Result
Consider
E N−1
B
i=1
{hΓ,ni
(ˆτi ) − µΓi }
Bounded above by zero under sharp null if model holds at Γ,
with equality possible. What about the weak null?
Failure to Control the Worst-Case Expectation
Suppose the sensitivity model holds at Γ. Then, for any choice of
functions {hΓni
}ni ≥2, there exist combinations of stratum sizes,
potential outcomes satisfying the weak null, and unmeasured
confounders u such that
lim inf
B→∞
N−1
E{hΓ,ni
(ˆτi ) − µΓi } > 0

Implications
The expectations cannot be simultaneously tightly controlled

Implications
If a sensitivity analysis could be asymptotically correct (rather
than conservative) under the sharp null, it must only be valid for
a subset of the weak null

Implications
If a sensitivity analysis is valid for the entirety of the weak null,
it must be unnecessarily conservative if only the sharp null holds.

Implications
If a sensitivity analysis is valid for the entirety of the weak null,
it must be unnecessarily conservative if only the sharp null holds.
Using a studentized test statistic does nothing to help! Has to
do with misalignment of worst-case expectations

Valid Inference for the Weak Null
For a given Γ, the conditional probabilities ij are bounded as
1
Γ(ni − 1) + 1
≤ ij ≤
Γ
ni − 1 + Γ
,
and are further constrained by ni
j=1 ij = 1.

Valid Inference for the Weak Null
For a given Γ, the conditional probabilities ij are bounded as
1
Γ(ni − 1) + 1
≤ ij ≤
Γ
ni − 1 + Γ
,
and are further constrained by ni
j=1 ij = 1.
If we knew ij , an unbiased estimator for ¯τi given F, Z would be
IPWi =
1
ni
ˆτi
ni
j=1 Zij ij
At Γ = 1, IPWi = ˆτi .
For Γ > 1, can’t use IPWi for any i as ij are unknown

Worst-Case Weighting
We can’t use IPWi as πi is unknown. Consider instead
WΓi =
1
ni

 ˆτi
Γ
ni −1+Γ
1(ˆτi ≥ 0) + 1
1+Γ(ni −1)
1(ˆτi < 0)

 .

Worst-Case Weighting
We can’t use IPWi as πi is unknown. Consider instead
WΓi =
1
ni

 ˆτi
Γ
ni −1+Γ
1(ˆτi ≥ 0) + 1
1+Γ(ni −1)
1(ˆτi < 0)

 .
Weights ˆτi by the worst-case assignment probability if the sensitivity
model holds at Γ
E(WΓi | F, Z) ≤ ¯τi
E
B
i=1
(ni /n)WΓi | F, Z ≤ ¯τ

Why Might This Be Conservative?
Suppose ni = 3, the potential values for ˆτi given F, Z are
δi1 = 5, δi2 = −2, δi3 = −3, and we conduct inference at Γ = 2.

What would the worst-case probabilities employed by W2i be?
W2i =
1
3
ˆτi
1
2
1(ˆτi ≥ 0) + 1
5
1(ˆτi < 0)
.

W2i =
1
3
ˆτi
1
2
1(ˆτi ≥ 0) + 1
5
1(ˆτi < 0)
.
δi1 = 5 ⇒ ∗
i1 = 1/2
δi2 = −2 ⇒ ∗
i2 = 1/5
δi3 = −3 ⇒ ∗
i3 = 1/5

W2i =
1
3
ˆτi
1
2
1(ˆτi ≥ 0) + 1
5
1(ˆτi < 0)
.
δi1 = 5 ⇒ ∗
i1 = 1/2
δi2 = −2 ⇒ ∗
i2 = 1/5
δi3 = −3 ⇒ ∗
i3 = 1/5
ni
j=1
∗
ij = 9/10. Not a probability distribution!
If model holds at Γ = 2, E(W2i | F, Z) < 0.

Incompatibility
In general, the worst-case IPW estimator WΓi does not generate
worst-case probabilities corresponding to a valid probability
distribution

Incompatibility
distribution
Under the sharp null, we can ﬁnd the worst-case probability
distribution because we know δij for j = 1, ..., ni

Incompatibility
distribution
WΓi is unduly conservative for sharp null, can be improved upon
by weighting with the worst-case distribution.

Incompatibility
distribution
Under the weak null, we can’t impose the constraint! We only
know δij for one of the ni individuals.

Incompatibility
distribution
Any attempt at adjusting ∗
ij runs the risk of yielding liberal
inference.

Incompatibility
distribution
Any attempt at adjusting ∗
ij runs the risk of yielding liberal
inference.
Incompatibility not a deﬁciency of matching - prevalent in many
modes of sensitivity analysis

An Alternative Weighting
Consider the alternative weighting
˜WΓi =
1
ni

 ˆτi
2Γ
ni (1+Γ)
1(ˆτi ≥ 0) + 2
ni (1+Γ)
1(ˆτi < 0)

 .

An Alternative Weighting
Consider the alternative weighting
˜WΓi =
1
ni

 ˆτi
2Γ
ni (1+Γ)
1(ˆτi ≥ 0) + 2
ni (1+Γ)
1(ˆτi < 0)

 .
Relative to before...
1
Γ(ni − 1) + 1
⇒
2
ni (1 + Γ)
(larger)
Γ
ni − 1 + Γ
⇒
2Γ
ni (1 + Γ)
(smaller)

Expectation under the Sharp Null
This weighting scheme has nice properties under the sharp null
The Worst-Case Expectation Under the Sharp Null
Suppose the sensitivity model holds at Γ. Then,
E( ˜WΓi | F, Z) ≤ 0,
and equality holds when uij = 1{δij ≥ 0}

E( ˜WΓi | F, Z) ≤ 0,
So, taking a weighted average, under the sharp null
E
B
i=1
(ni /N) ˜WΓi | F, Z ≤ 0,

E( ˜WΓi | F, Z) ≤ 0,
So, taking a weighted average, under the sharp null
E
B
i=1
(ni /N) ˜WΓi | F, Z ≤ 0,
Because it sharply bounds the expectation under the sharp null, it
must only do so for a subset of the weak null!

The Expectation Under the Weak Null
For other matched designs, if the weak null holds but the sharp null
does not,
E
B
i=1
(ni /N) ˜WΓi
≤ CΓ
B
i=1
(ni /N) 1 +
1 − Γ
Γ
pr(ˆτi ≥ ¯τi | F, Z) ¯τi

The Expectation Under the Weak Null
For other matched designs, if the weak null holds but the sharp null
does not,
E
B
i=1
(ni /N) ˜WΓi
≤ CΓ
B
i=1
(ni /N) 1 +
1 − Γ
Γ
pr(ˆτi ≥ ¯τi | F, Z) ¯τi
A suﬃcient condition for the worst-case expectation under the weak
null to be controlled under the weak null is that, for the worst-case
confounder,
cov {pr(ˆτi ≥ ¯τi | F, Z), ni ¯τi } ≥ 0,

Reasonable?
Consider uij = {δij ≥ ¯τi }. The suﬃcient condition can be rewritten as
cov
ni
ni
j=1{1(δij < ¯τi ) + Γ1(δij ≥ ¯τi )}
, ni (¯τi ) ≤ 0.

Reasonable?
cov
ni
ni
j=1{1(δij < ¯τi ) + Γ1(δij ≥ ¯τi )}
, ni (¯τi ) ≤ 0.
The ﬁrst term is a function of the stratum size ni and
ni
j=1 1(δij − ¯τi ≥ 0); Second term is ni ¯τi

Reasonable?
cov
ni
ni
j=1{1(δij < ¯τi ) + Γ1(δij ≥ ¯τi )}
, ni (¯τi ) ≤ 0.
ni
ni
ni
j=1(δij − ¯τi )¯τi = 0 for all i!

Reasonable?
cov
ni
ni
j=1{1(δij < ¯τi ) + Γ1(δij ≥ ¯τi )}
, ni (¯τi ) ≤ 0.
ni
ni
ni
Covariance may be nonzero only because residuals being uncorrelated
from ﬁtted values does not imply that functions of residuals are
uncorrelated from ﬁtted values

Reasonable?
cov
ni
ni
j=1{1(δij < ¯τi ) + Γ1(δij ≥ ¯τi )}
, ni (¯τi ) ≤ 0.
ni
ni
ni
Covariance may be nonzero only because residuals being uncorrelated
from ﬁtted values does not imply that functions of residuals are
uncorrelated from ﬁtted values
Do we care?

Reasonable?
Are we worried about hidden bias acting in this way?
To be anticonservative, skewness of δij − ¯τi must relate to values
of ¯τi

Reasonable?
Are we worried about hidden bias acting in this way?
To be anticonservative, skewness of δij − ¯τi must relate to values
of ¯τi
If we decide this doesn’t matter, sensitivity analysis using the test
statistic B
i=1(ni /N) ˜WΓi yields a single procedure that...
Controls the worst-case expectation under the sharp null, with
equality possible
Controls the worst-case expectation under a (potentially
benign?) subset of the weak null.
Variance estimation, asymptotic normality follows naturally from
Fogarty (2018); Pashley and Miratrix (2019+).

Conclusions
In randomized experiments, randomization tests using suitably
studentized diﬀerences-in-means typically yield a single mode of
inference for both the sharp (exact) and weak (asymptotic) nulls

Conclusions
Continues to be true in a sensitivity analysis in paired
observational studies with a careful choice of test statistic

Conclusions
Modes of inference cannot be uniﬁed in the same way when
conducting a sensitivity analysis for general forms of matching

Conclusions
If we truly require correctness over the entirety of the weak null,
we must be conservative if the sharp null is true

Conclusions
If we truly require correctness over the entirety of the weak null,
we must be conservative if the sharp null is true
Presented a test statistic that can be exact for the sharp null,
and valid over a potentially innocuous subset of the weak null

Thanks!
Fogarty, C.B. Testing weak nulls in matched observational
studies. arXiv.
Fogarty, C.B. (2019+) Studentized sensitivity analysis for the
sample average treatment effect in paired observational studies.
Journal of the American Statistical Association.
Fogarty, C.B. (2018) On mitigating the analytical limitations of
finely stratified experiments Journal of the Royal Statistical
Society, Series B.

Paired Designs
For paired designs, ˜WΓi equals WΓi , and is equivalent to the unifying
procedure described in Fogarty (2019+).
Paired Designs
Recall that DΓi = ˆτi − {(Γ − 1)/(1 + Γ)}|ˆτi |. ˜WΓi and DΓi are
proportional,
WΓi =
(1 + Γ)2
4Γ
DΓi ,
such that E B−1 B
i=1
˜WΓi | F, Z ≤ 0 under the weak null

The Propensity Score as a Nuisance Parameter
For inference on average treatment effects under strong ignorability,
the propensity score, e(x) = pr(Z = 1 | X = x), is a nuisance
parameter.
Not of primary interest, but important for inference
Inverse Propensity Weighting, AIPW,...
Use a plug-in estimator, ê(x)
Pay attention to the rate of convergence of ê(x) and its impact
on resulting inference
How does matching try to handle nuisance parameters?
A mix of conditioning (dealing with observed covariates) and
optimizing (sensitivity to unmeasured confounders)
Imperfections in the strategy will be be investigated here

A Simple Model with Hidden Bias
Suppose treatment is strongly ignorable given (X, U) where U is
unobserved, and consider the following model probability of
assignment to treatment
logit{pr(Z = 1 | X = x, U = u)} = xβ + log(Γ)u
x are observed covariates
u ∈ [0, 1] is an unobserved scalar covariate
Γ is a sensitivity parameter

Conditional Permutation Tests
Let Ω = {z : zT
x = a}, where a is observed value of ZT
x in the
study, and consider conditioning upon Z ∈ Ω
Rosenbaum (1984) with hidden bias
pr(Z = z | x, u, Z ∈ Ω) =
exp{log(Γ)zT
u}
b∈Ω exp{log(Γ)bT u}
ZT
x is suﬃcient for β, so conditioning removes dependence on β
Inference still depends on u
Example: Suppose x ∈ RN×B
, where the ith of B columns contains
an indicator for membership in the ith stratum
βi = stratum-speciﬁc slope
(ZT
x)i = number of treated in ith stratum

Similarities with Matching
Consider optimal matching without replacement.
ith matched set is of size ni
mi is the number of treated individuals in matched set i
Letting Ω = {z : ni
j=1 zij = mi }, Rosenbaum (2002) suggests using
the following model for biased treatment assignments
pr(Z = z | x, u, Z ∈ Ω) =
exp{log(Γ)zT
u}
b∈Ω exp{log(Γ)bT u}
,
aligning with the randomization distribution on the previous slide

Matching as a Search for Sufficiency
Modify the logit form to allow for nonlinearities in x
logit{pr(Z = 1 | X = x, U = u)} = ϕ(x) + log(Γ)u
In using the conditional inference described in Rosenbaum (2002), we
assume
1 ϕ(xij ) = ϕ(xik) for individuals j and k in the same matched set i
2 The matched structure would be invariant over z ∈ Ω
Of course, these won’t hold in practice
1 Discrepancies on ϕ(xij ) are inevitable (induce bias)
2 Had I observed a different z ∈ Ω, I may have attained a different
match (affects variance)
How much of a difference can this make? Working paper...

Causal Inference Opening Workshop - Testing Weak Nulls in Matched Observational Studies - Colin Fogarty, December 11, 2019

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Causal Inference Opening Workshop - Testing Weak Nulls in Matched Observational Studies - Colin Fogarty, December 11, 2019