From IPTW to IPSW: Who Gets the Treatment, Who Gets Into the Study

อ่านฉบับภาษาไทย (Thai version)

Abstract

IPTW and IPSW use the same inverse-probability logic for two different problems. Inverse probability of treatment weighting (IPTW) asks who received which treatment. It models the probability of treatment given baseline covariates and builds a pseudo-population in which treated and control patients are comparable, which protects internal validity. Inverse probability of sampling weighting (IPSW) asks who entered the study. It models the sampling score, the probability of study participation given covariates, and reweights participants so that the sample resembles a target population, which serves generalisability. If young patients enter a trial with probability 0.80 and elderly multimorbid patients with probability 0.10, each elderly participant gets a weight of 10 and each young one 1.25. A randomised trial whose patients average 55 years, applied to patients who average 72, has a sampling problem, not a treatment problem. This article follows propensity-score weighting as a family, the two doors into study and treatment, and the two parallel workflows.

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One drug question, two different problems

A physician wants to know whether Drug A lowers mortality compared with Drug B. Two kinds of evidence are at hand. In the hospital database, the drug each patient got depended on how old and how sick that patient looked. A randomised trial compared the two drugs fairly, but it recruited patients unlike the ones on the ward.

Both problems can be handled by weighting each patient by one over a probability. They are not the same probability. Choosing the wrong one answers the wrong question.

Propensity-score weighting is a broader idea than IPTW

A useful distinction in terms comes first.

$$\boxed{\text{Propensity-score weighting} = \text{a family of weighting approaches}}$$

IPTW is one member of that family. Another member is overlap weighting, which gives each patient the probability of the treatment they did not receive. Another is matching weights, which mimic one-to-one propensity-score matching. These approaches target somewhat different populations or weighting structures [2, 3].

So:

$$\boxed{\text{IPTW} \subset \text{propensity-score weighting}}$$

Weighting by one over a probability is a general move. The same move can be pointed at a different probability, and that is where IPSW comes in.

Then what is IPSW?

IPSW usually means:

$$\boxed{\text{Inverse Probability of Sampling Weighting}}$$

It uses the same broad inverse-probability principle as IPTW, but it addresses a different problem. The difference lies in the question each weight asks.

The problem IPSW tries to solve

Suppose a randomised controlled trial (RCT) has excellent internal validity: its comparison is free of bias for the people inside it. Within the trial, treatment A versus treatment B may be well randomised.

$$\text{Treatment A} \leftrightarrow \text{Treatment B}$$

But trial participants might differ substantially from the population where we ultimately want to apply the results, the target population. For example:

Trial participants

• younger
• fewer comorbidities
• treated at academic hospitals

Target population

• older
• more multimorbidity
• treated at community hospitals

Then:

$$\text{Trial sample} \neq \text{Target population}$$

The treatment effect estimated in the trial may therefore not automatically represent the treatment effect in the target population. IPSW was developed to address this sampling or generalisability problem, where generalisability means whether a result holds in the target population [4, 5]. It weights trial participants according to their probability of being represented in the study.

The sampling score

For IPSW, we model something like:

$$P(S = 1 \mid X)$$

Here $S = 1$ means the person was included in the study or trial, and $X$ holds the characteristics related to study participation. This probability is the sampling score. It is estimated from data that hold both the study participants and people from the target population [4, 5].

This is fundamentally different from the IPTW propensity score:

$$P(A = 1 \mid X)$$

Compare the two side by side.

IPTW

$$\boxed{P(\text{Treatment} \mid X)}$$

IPSW

$$\boxed{P(\text{Study participation} \mid X)}$$

The first models treatment assignment. The second models study or sample selection.

Intuition behind IPSW: who is underrepresented

Imagine that young patients are very likely to enter a clinical trial, but elderly multimorbid patients are rarely recruited. To see what the weights do, picture a target population of 1,000 patients: 500 young and 500 elderly multimorbid.

1. The sampling scores

   \[ P(S = 1 \mid \text{Young}) = 0.80, \quad P(S = 1 \mid \text{Elderly}) = 0.10 \]

   Of the 1,000, the trial enrols 400 young and 50 elderly patients. The trial is 88.9% young, while the target is half young.

2. One over the sampling score

   \[ w = \frac{1}{0.80} = 1.25, \quad w = \frac{1}{0.10} = 10 \]

   The first weight is for a young participant and the second for an elderly participant. The arithmetic is the same as in IPTW. Only the probability has changed, from the chance of the treatment received to the chance of being in the study.

3. The weighted trial

   \[ 400 \times 1.25 = 500, \quad 50 \times 10 = 500 \]

   Each elderly participant now stands for 10 elderly patients in the target. The weighted trial has the target's half-and-half age mix again.

Result: Underrepresented in the study, more sampling weight. The elderly participant is a type of patient that is uncommon in the trial but more relevant in the target population, so IPSW gives that participant greater statistical representation. The weighted trial then resembles the target population only on the measured characteristics used in the sampling model, here age group.

IPTW versus IPSW: the core difference

A clinical example: Drug A versus Drug B

Suppose we want to study whether Drug A reduces mortality compared with Drug B. The same question can run into two different problems, and each calls for a different weight.

Problem 1: treatment confounding

In our hospital database:

• Drug A patients are older.
• Drug A patients are more severely ill.
• Drug B patients are younger.

We need IPTW, because the problem is:

$$P(\text{Treatment} \mid X)$$

We want treated and control patients to become more comparable.

Problem 2: generalising an RCT

Now suppose we already have an RCT comparing Drug A with Drug B. Randomisation means the treatment groups inside the trial are comparable. However:

• RCT patients have a mean age of 55 years.
• Real-world patients have a mean age of 72 years.
• Frail patients were rarely enrolled.

Now our concern is not primarily treatment assignment. The concern is:

$$P(\text{Trial participation} \mid X)$$

We may therefore use IPSW to reweight the trial population toward the target clinical population.

Two different selection mechanisms

The example gives a useful causal hierarchy. There are potentially two different questions.

The first question: who gets the treatment?

$$X \rightarrow A$$

This motivates IPTW.

The second question: who gets into the study?

$$X \rightarrow S$$

This motivates IPSW.

Therefore the two methods should not be treated as competitors. They solve different problems.

A useful mental model: two doors

Imagine two doors.

Door 1: entering the study. Some people enter; others do not.

$$\text{Population} \xrightarrow{\text{Sampling}} \text{Study}$$

This is the domain of IPSW.

Door 2: receiving treatment. Among study participants, some receive A and others receive B.

$$\text{Study} \xrightarrow{\text{Treatment assignment}} \text{A/B}$$

This is the domain of IPTW. The diagram below puts the two doors in order, from the target population to the outcome.

⤢ click to enlarge

What each weight is trying to reconstruct

IPTW creates a pseudo-population with comparable treatment groups. Its question is: what would the treatment comparison look like if measured baseline characteristics did not systematically determine treatment assignment?

IPSW creates a weighted study sample resembling a target population. Its question is: what would the study results look like if the study participants represented the target population more appropriately?

The final integrated workflow

The two weights run as two parallel pipelines. Each step feeds the next.

A typical IPTW analysis

1. Measured baseline covariates
2. $P(\text{Treatment} \mid X)$
3. IPTW
4. Weighted pseudo-population
5. Balance, overlap and weight diagnostics
6. Weighted outcome analysis

An IPSW generalisability analysis

1. Study and target population characteristics
2. $P(\text{Study participation} \mid X)$
3. IPSW
4. Study sample reweighted toward the target population
5. Generalised or transported treatment effect [6, 7]

Final memory box

Propensity score for treatment

$$\boxed{PS = P(\text{Treatment} \mid X)}$$

IPTW reweights people according to treatment assignment. Its purpose is to improve treated-control comparability.

IPSW reweights people according to study participation. Its purpose is to make the study sample better represent the target population.

Common misreadings and their fixes

• “A randomised trial never needs weighting.”

  Randomisation makes the arms comparable inside the trial, which is internal validity. It says nothing about whether the participants resemble the patients the result will be applied to.

  Fix: Ask who entered the trial. If participants differ from the target population, as with a mean age of 55 against 72 or frail patients rarely enrolled, consider IPSW toward that target.

• “IPTW and IPSW are rival methods, so pick the better one.”

  They answer different questions. IPTW deals with the path from covariates to treatment, and IPSW with the path from covariates to study entry.

  Fix: Name the mechanism first. For who gets the treatment ($X \rightarrow A$), use IPTW; for who gets into the study ($X \rightarrow S$), use IPSW.

• “The sampling score is just the propensity score.”

  The propensity score models treatment assignment, $P(A = 1 \mid X)$. The sampling score models study selection, $P(S = 1 \mid X)$.

  Fix: Fit the sampling model on study participants and target-population members together, with participation as the outcome and the characteristics related to participation as predictors.

• “Weighting by the sampling score makes the trial represent everyone.”

  The weighted trial resembles the target population only on the measured characteristics used in the sampling model.

  Fix: List the characteristics on which the trial and the target differ, put them in the sampling model, and name the target population before reading the weighted result.

• “IPSW is checked the way IPTW is, by balance between the arms.”

  IPSW changes whom the study sample represents, not how the arms compare with each other.

  Fix: Check covariate balance between the weighted study sample and the target population.

Glossary

propensity score (propensity score (คะแนนแนวโน้มการได้รับการรักษา))

e(X) = P(A = 1 given X), the probability of receiving the treatment given baseline covariates.

propensity-score weighting

A family of weighting approaches built on the propensity score; IPTW, overlap weighting and matching weights are members.

IPTW (การถ่วงน้ำหนักด้วยส่วนกลับของความน่าจะเป็นในการได้รับการรักษา (IPTW))

Inverse probability of treatment weighting: each patient weighted by one over the probability of the treatment received, to make treated and control patients comparable.

pseudo-population (ประชากรเสมือน (pseudo-population))

The weighted sample IPTW creates, in which the measured covariates are balanced between treatment groups.

overlap weighting (น้ำหนักแบบทับซ้อน (overlap weight))

A propensity-score weighting approach that gives each patient the probability of the treatment they did not receive.

matching weights

A propensity-score weighting approach that mimics one-to-one propensity-score matching.

IPSW (การถ่วงน้ำหนักด้วยส่วนกลับของความน่าจะเป็นในการถูกคัดเลือก (IPSW))

Inverse probability of sampling weighting: study participants weighted by one over the sampling score, so the study sample better represents a target population.

sampling score (คะแนนการถูกคัดเลือกเข้าการศึกษา (sampling score))

P(S = 1 given X), the probability of being included in the study given the characteristics related to participation.

internal validity

Whether a study's comparison is free of bias for the people inside the study.

generalisability

Whether a study result holds in the target population; also called external validity.

target population

The population where we ultimately want to apply the study results.

References

1. Austin PC, Stuart EA. Moving towards best practice when using inverse probability of treatment weighting (IPTW) using the propensity score to estimate causal treatment effects in observational studies. Stat Med. 2015;34(28):3661-3679. doi:10.1002/sim.6607 https://doi.org/10.1002/sim.6607
2. Li F, Morgan KL, Zaslavsky AM. Balancing covariates via propensity score weighting. J Am Stat Assoc. 2018;113(521):390-400. doi:10.1080/01621459.2016.1260466 https://doi.org/10.1080/01621459.2016.1260466
3. Desai RJ, Franklin JM. Alternative approaches for confounding adjustment in observational studies using weighting based on the propensity score: a primer for practitioners. BMJ. 2019;367:l5657. doi:10.1136/bmj.l5657 https://doi.org/10.1136/bmj.l5657
4. Cole SR, Stuart EA. Generalizing evidence from randomized clinical trials to target populations: the ACTG 320 trial. Am J Epidemiol. 2010;172:107-115. doi:10.1093/aje/kwq084 https://doi.org/10.1093/aje/kwq084
5. Stuart EA, Cole SR, Bradshaw CP, Leaf PJ. The use of propensity scores to assess the generalizability of results from randomized trials. J R Stat Soc Ser A Stat Soc. 2011;174(2):369-386. doi:10.1111/j.1467-985x.2010.00673.x https://doi.org/10.1111/j.1467-985x.2010.00673.x
6. Lesko CR, Buchanan AL, Westreich D, Edwards JK, Hudgens MG, Cole SR. Generalizing study results: a potential outcomes perspective. Epidemiology. 2017;28(4):553-561. doi:10.1097/ede.0000000000000664 https://doi.org/10.1097/ede.0000000000000664
7. Dahabreh IJ, Robertson SE, Tchetgen EJ, Stuart EA, Hernán MA. Generalizing causal inferences from individuals in randomized trials to all trial-eligible individuals. Biometrics. 2019;75(2):685-694. doi:10.1111/biom.13009 https://doi.org/10.1111/biom.13009

Key takeaways

• Propensity-score weighting is a family of weighting approaches; IPTW is one member, beside overlap weighting and matching weights.
• IPTW models who received which treatment, the propensity score; IPSW models who entered the study, the sampling score.
• A participant from a group that rarely enters the study gets more sampling weight: sampling scores of 0.80 and 0.10 give weights of 1.25 and 10.
• IPTW builds comparability between treatment groups; IPSW builds generalisability to a target population, but only on the characteristics in the sampling model.
• The two are not competitors: IPTW asks why this patient got this treatment, IPSW asks why this patient got into this study.

Related in the wiki: [[iptw-guide]]