lecture_wednesday_2

Introduction to Causal Inference and Causal Data Science

Day 3:
Addressing departures from randomisation

Bas B.L. Penning de Vries
Dept. Data Science & Biostatistics
Julius Center, UMC Utrecht

Any method may be used for confounding control:

Regression adjustment
(Propensity score) matching
G-computation
Inverse probability weighting (IPW)
Instrumental variable analysis
…

Choice should be influence in part by assumptions and estimand — propensity score matching and g-methods (IPW and g-computation) target quantities typically estimated in trials

Propensity score methods

A collection of methods based on the propensity score (PS):

PS stratification
Regression on the PS
PS matching
Inverse probability weighting (IPW)

Propensity score

The propensity score ps(L) is the conditional probability of (propensity for) exposure A given a specified set L of variables:

ps(L) = Pr(A = 1 | L)

It is common practice to impose a parametric model on the propensity score — popular choice is that the logit of ps(L) is a linear function of the variables in L.
Not a must! In modern causal inference, ps is considered a nuisance function — quantity used for inference but not of substantive interest + candidate for flexible machine learning methods (but see e.g., Alexander (2021) and Naimi et al (2021)).

Rosenbaum and Rubin (1983, Theorem 1) demonstrated balancing property:

conditional on ps(L), the distribution of L is the same among the treated (A = 1)
as it is among the untreated (A = 0).

More importantly, they showed (Theorem 3) conditional exchangeability given L implies conditional exchangeability given the propensity score ps(L).

Key balancing property of the propensity score

Y ^A=a ⫫ A | L ⇒ Y ^A=a ⫫ A | ps(L)

Powerful property: if a potentially high-dimensional L is sufficient to adjust for confounding, then so is the one-dimensional PS.

PS matching

Typical estimand is average treatment effect among treated (ATT), but exact estimand depends on implementation/variation.

Population

Crude comparison

ATT

E[Y ^A=1 | A = 1] = E[Y | A = 1] (consistency), but what about E[Y ^A=0 | A = 1]?
Y ^A=0 is unobserved for treated individuals — this is where (exact) matching on the propensity score comes in!

Exact propensity score matching

Exact matching on the propensity score means replacing the unobserved
outcomes Y ^A=0 of treated individuals with propensity score ps(L), with random draws Y_match from the observed outcomes Y of untreated individuals with the same ps(L):

Y_match | A = 1, ps(L) ~ Y | A = 0, ps(L)

If we could do exact matching, it turns out that, under conditional exchangeability, positivity and consistency, E[Y ^A=0 | A = 1] = E[Y_match | A = 1].
The causal contrast E[Y ^A=1 | A = 1] versus E[Y ^A=0 | A = 1] is thus identified by the observed contrast E[Y | A = 1] versus E[Y_match | A = 1] — this amounts to quantifying the crude association between the treated and the matched untreated individuals.

PS matching in practice

Problem with finite samples: exact matching often (nearly always) impossible for all treated individuals!
Solution: use approximate matching
- Many algorithms to choose from (e.g., greedy 1:1 nearest neightbour matching with distance defined as absolute difference between logit ps(L); with calliper)
But:
- Allowing matching of distant subjects may result in bias (residual confounding).
- Excluding subjects for whom no close-enough match can be found may result in poor precision (bias-variance trade-off) or selection bias.

After propensity score estimation and matching, standard confounding-naïve estimators can be used to estimate the treatment-outcome effect (e.g., OLS from a simple linear regression applied to the matched treated and untreated participants).
- Valid for point estimation assuming observations in the original sample are i.i.d.
- But naïve variance estimation is problematic, mainly due to matching-induced outcome dependence between treated and control units through the propensity score.
Alternative: bootstrap

IPW for binary treatments/exposures

Typical estimand is average treatment effect (ATE) among everyone in the target population.

Population

Crude comparison

ATE

Key idea: reweight treated and untreated subpopulations so that they look (in some respects) like the entire population.

Goal: create a pseudopopulation (by weighting the original population) whose treated and untreated subgroups each have the same distribution of covariates L and (more importantly) counterfactuals Y ^A=a as the original population.

Covariates L and counterfactual outcomes Y ^A=a are independent of treatment A in the pseudopopulation (marginal exchangeability).
Causal effect = crude association, under consistency

How: it turns out that the pseudopopulation with this property can be made under positivity and conditional exchangeability given L, by …

weighting treated individuals with 1 / ps(L)
and untreated individuals with 1 / (1 − ps(L)).

Treated

Sex	Treated	PS	Weight
Female	Yes	4/6	1.5
Female	No	4/6	3.0
Male	Yes	2/6	3.0
Male	No	2/6	1.5

Unreated

Modifications

For inference about average effect among the treated (ATT)
To accommodate more than two treatment levels
- Point interventions with more than two levels or interventions on time-varying variables
- But with many possible treatment regimes to consider: there may be a need to make modelling assumptions regarding the distribution of the outcome and the treatment(s) in the pseudopopulation (marginal structural modelling)
- ‘Stabilisation’ of weights recommended (to reduce their variability)
To address selective censoring or account for missing data

G-computation for binary treatments

Alternative to IPW that comes under same (non-parametric) identifiability conditions, for the same estimand (ATE)
Relies on outcome modelling rather than treatment modelling
Essentially “standardisation” of conditional quantities

Identification of average treatment effect (ATE) via g-computation

ATE

= E[Y ^A=1] − E[Y ^A=0]

(causal estimand)

E[Y ^A=a]

= E{E[Y ^A=a | L]}

(law of total expectation)

= E{E[Y ^A=a | L, A = a]}

(conditional exchangeability and positivity)

= E{E[Y | L, A = a]}

(consistency)

ATE

= E{E[Y | L, A = 1]} − E{E[Y | L, A = 0]}

This theoretical identification result naturally translates into the following (“g-computation”) estimator:

Model the conditional outcome mean: estimate the conditional mean E[Y | L, A = a] using an appropriate model (e.g., linear regression, logistic regression, or more flexible methods).
Obtain predicted potential outcomes: for each individual in the sample, use the model fit(s) to predict the outcome under treatment level a. This will give an estimate of Y ^A=a for everyone.
Average over the empirical distribution of L: take average as estimate of E[Y ^A=a].
Contrast treatment levels: repeat for all a and compute contrast (e.g., ATE).

The gist of g-estimation

Originally proposed as one of three classical methods, alongside g-computation and IPW, for causal inference about the effects of time-varying treatments.
We’ll discuss the special case of time-fixed treatments …

G-estimation for time-fixed treatments

Postulate model for contrast between treatments within levels of covariates — e.g.,

E[Y ^A=a − Y ^A=0 | L, A = a] = βa
Rewrite and apply consistency:

E[Y ^A=0 | L, A = a] = E[Y − βa | L, A = a]
Under conditional exchangeability E[Y ^A=0 | L, A = a] = E[Y ^A=0 | L], so right-hand side should be conditionally independent of a too! Search for model parameters that are compatible with this:

Search for β such that Cov(Y − βA, A | L) = 0

Extentions of g-methods to time-varying treatment settings

IPW, g-computation and g-estimation suitable to estimate causal effects of time-varying treatment effects — more on this later!

Statin-cancer example revisited (2)

Previous studies implicitly compared long-term statin users versus non-users — didn’t necessarily answer questions like …

What would be my 10-year cancer risk if I would start statin treatment now? And what if I wouldn’t?
What would be my 10-year cancer risk if I would start statin treatment now and adhered to it? And what if I wouldn’t start now or in the future? (Friday!)

Instrumental variable analysis

Previously discussed methods rely on the availability of a set of variables L conditional on which the treatment groups are exchangeable
The key motivation for instrumental variable analysis is a possible violation of this conditional exchangeability assumption

Motivating example: encouragement trial

For inference about effect of A on Y
(or per-protocol effect), consider:

Intention-to-treat analysis (❌ misclassification)
As-treated analysis (❌ confounding)
Per-protocol analysis (❌ confounding)