Personalized Decision Making - A Conceptual Introduction (Scott Mueller and Judea Pearl, 2022)

Ashley / 2023-01-12

Abstract #

This paper

clarifies the distinction between
- personalized decision making, which targets the behavurir of a specific individual,
- population-based decision making, which concerns a sub-population resembling that individual,
explains why personalized decision making leads to more informed dicisions.

The authors argued that combining experimental and observational studies can

obtain valuable information about individual behavior,
improve decisions over those obtained from experimental studies alone.

1. Introducation #

Purpose:
- conceptual understanding of the distinction betwen the personalized and population-based decision making
- the advantages of the personalized decision making
- how the personalized decision making could be achieved
Distinction
- formally captured in two causal effects
  - Individual Treatment Effect (ITE) $$
    ITE(u) = Y(1, u) - Y(0, u) $$ where $Y(x, u)$ stands for the outcome that individual $u$ would attain had decision $x \in {1, 0}$
  - Conditional Average Treatment Effect (CATE) $$
    CATE(u) = E[Y(1, u^{'}) - Y(0, u^{'}) | C(u^{'}) = C(u)] $$ where $C(u)$ stands for a vector of characteristics observedon individual $u$, and the average is taken over all units $u^{'}$ that share these characteristics.
- $ITE(u)$ and $CATE(u)$ lead to different decision strategies.
- Although $ITE(u)$ is in general not identifiable, informative bounds can be obtained by combiningg experimental and observational studies. Informative bounds can improve decisions (that would otherwise be taken using $CATE(u)$).
Organization of the paper
- Section 2: an extreme example with two surprising findings
  - popultion data are capable of providing decisive information about individual responses
  - non-experimental data (usually discarded as bias-prone) can add information regarding individual responses beyond that provided by a Randomized Controlled Trials (RCT) alone
- Section 3: generalize the two surprising findings from section 2, using a more realistic example
  - how critical decisions can be made using the information obtained and their ramification (to the targeted individual and to a population-minded policy maker)
- Section 4: casts the findings in section 3 in a numerical setting, allowing for quantitative application of the magnitude involved; this analysis leads to actionable policies that guarantee risk-free benifits in certain populations
  - explains the mathematics that give rise to the results featured in section 3, as well as the assumptions upon which they depend
  - confounding in observational studies, is shown to be helpful in narrowing down the probabilities of benefit and harm
- Section 5: discuss related topics such as monotonicity, number needed to treat and probability of harm
- Section 6: provides annotated bibliography of the source papers, including realted works

2. Preliminary Semi-qualitative Example #

Target: how an individual would react if given treatment and if denied treatment?
- no individual can be subjected to both a treatment and its denial –> its responses function must be inferred from population data (originating from one or several studies) –> to what degree can population data inform us about an individual response?
Two conceptual hurdles:
- why should population data provide any information about whatsoever on the individual response?
- why should non-experimental data add any information (regarding individual response) to what we can learn with an RCT alone?
A simple example that demonstrates the hurdles: an RCT found no difference between treatment (drug) and control (placebo), say 10% in both treatment and control groups die, while the rest (90%) survive. This led to the conclusion that the drug is ineffective, but also leaves uncertain between (at least) two competing models:
- Model-1: the drug has no effect whatsoever on any individual.
- Model-2: the drug saves 10% of the population and kills the another 10%.
  –>
- From a polcy maker viewpoint, the two models may be deemed equivalent, the drug has zero average effect on the target population.
- From an individual viewpoint, the two models differ substaintially in the sets of risks and opportunities they offer:
  - Model-1, the drug is useless but safe.
  - Model-2, the drug may be deemed dangerous by some and life-saver by others.
    –> To see how such attitudes may emerge, assume that the drug also provides temporary pain relief
  - Model-1 would be deemed desirable and safe for all
  - Model-2 will scare away those who do noe urgently need the pain relief, while offering a glimpse of hope to those whose suffering has become unbearable, and who would be ready to risk death for the chance (10%) of recovery. (Hoping, of course, they are among the lucky beneficiaries.)
The second theme of the paper – the crucial role of obervational studies.
- The above simple example show that supplementing the RCT with an observational study on the same population (conducted, for example, by an independent survey of patients who have the option of taking or avoiding the drug) would allow us to decide between the two models, totally changing our undrestanding of what risks await an individual taking the drug.
- Consider an extreme case where the observational study shows 100% survival in both drug-chosing and drug-avoiding patients, as if patient knew in advance where danger lies and manaed to avoid it.
  –> Such a finding immediately rules out Model-1 which claims no treatment effect on any individual.
  The same argument applies when the probability of survival among option-having individuals is not precisely 100% but simply higher (or lower) tahn the probability of survival in the RCT. Using RCT alone, in the contrast, we were unable to rule out Model-1 or even to distinguish Model-1 from Model-2.
- Consider another example/edge case wehre Model-2, rather than Model-1, is rule out as impossible. Assume the observational study informs us that all those who chose the drug died and all who avoided the drug survived. It seems that drug-choosers were truly dumb while drug-avoiders knew precisely what’s good for them.
  –> No one can be cured by the drug, contrary to Model-2.
Summary: up to now, it is demonstrated how certain combinations of observational and experimental data can provide information on indiviudal behavior that each study alone cannot.
Next: we are ready to go to a more realistic motiviating example which
- establishes individual behavior for any combination of observational and experimental data, and, moreover,
- demonstrate critical decision making ramifications of the information obtained.

3. Motivating Numerical Example #

Example: the effect of a drug on two subpopulations, males and females. The drug is found to be somewhat effective for both males and females and, in additional, deaths are found to occur in the observational study as well.
- An RCT is conducted to evaluate the efficacy of the drug and is found to be 28% effective in both males and females.
- However, adding observational data proves men to react remarkedly different than women, calling for two different treatment policies in the two groups.
  - A women has a 28% chance of benefiting from the drug and no danger at all of being harmed by it.
  - A men has a 49% chance of benefiting from it and as much as a 21% chance of dying because of it.

4. How the Results Were Obtained #

Notations
- $y_t$ stands for recovery among the RCT treatment group.
- $y_c$ stands for recovery among the RCT control group.
- $y'$ stands for non-recovery.
- $P (y_t | Gender)$ stands for the causal effect for the treatment group.
- $P (y_c | Gender)$ stands for the causal effect for the control group.
- $P (y | t, Gender)$ stands for recovery among teh drug-choosers in the observational study.
- $P (y | c, Gender)$ stands for recovery among the drug-avoiders in teh observational study.
Target
$$
P(benefit) = P(y_t, y_c')
$$
The probability that an individual would both recover if assigned to the RCT treatment arm and die if assigned to control.
Consistency assumption
$$
P(y_t | t) = P(y | t), P(y_c | c) = P(y | c)
$$
The probability that a drug-chooser would recover in the treatment arm of the RCT, $P(y_t | t), is the same as the probability of the recovert in the observational study, P(y | t).
An individual response to treatment depends entirely on biological factors, unaffected by the settings in which treatment is taken.
In other words, the outcome of a person choosing the drug would be the same had this person been assigned to the treatment grou in an RCT study.

Bounds on the probability of benefit
$$
max{0, P(y_t) - P(y_c), P(y) - P(y_c), P(y_t) - P(y)} ≤ P(benefit) ≤ min{P(y_t), P(y_c'), P(t, y) + P(c, y'), P(y_t) - P(y_c) + P(t, y') + P(c, y)}
$$

Applying these expressions to the feamale data from Table 2
Table 2: Female survival and recovery data

		Survivals	Deaths	Total
Experimental	do(drug)	489 (49%)	511 (51%)	1,000 (50%)
Experimental	do(no drug)	210 (21%)	790 (79%)	1,000 (50%)
Observational	drug	378 (27%)	1,022 (73%)	1,400 (70%)
Observational	no drug	420 (70%)	180 (30%)	600 (30%)

P(y_t) - P(y_c) = 0.489 - 0.210 = 0.279
P(y) - P(y_c) = (378 + 420)/(1400 + 600) - 0.21 = 0.399 - 0.21 = 0.189
P(y_t) - P(y) = 0.489 - (378 + 420)/(1400 + 600) = 0.489 - 0.399 = 0.09
P(y_t) = 0.489
P(y_c') = 0.79
P(t, y) + P(c, y') = 0.7 * 0.27 + 0.3 * 0.30 = 0.279
P(y_t) - P(y_c) + P(t, y') + P(c, y) = 0.489 - 0.21 + 0.7 * 0.73 + 0.3 * 0.7 = 1

max{0, 0.279, 0.189, 0.09} ≤ P(benefit|female) ≤ min{0.489, 0.79, 0.279, 1}
0.279 ≤ P(benefit|female) ≤ 0.279
P(benefit|female) = 0.279

Applying these expression to the male data from Table 3
Table 3 Male survival and recovery data

		Survivals	Deaths	Total
Experimental	do(drug)	490 (49%)	510 (51%)	1,000 (50%)
Experimental	do(no drug)	210 (21%)	790 (79%)	1,000 (50%)
Observational	drug	980 (70%)	420 (30%)	1,400 (70%)
Observational	no drug	420 (70%)	180 (30%)	600 (30%)

P(y_t) - P(y_c) = 0.49 - 0.21 = 0.28
P(y) - P(y_c) = (980 + 420)/(1400 + 600) - 0.21 = 0.7 - 0.21 = 0.49
P(y_t) - P(y) = 0.49 - (980 + 420)/(1400 + 600) = 0.49 - 0.7 = -0.21
P(y_t) = 0.49
P(y_c') = 0.79
P(t, y) + P(c, y') = 0.7 * 0.7 + 0.3 * 0.3 = 0.58
P(y_t) - P(y_c) + P(t, y') + P(c, y) = 0.49 - 0.21 + 0.7 * 0.3 + 0.3 * 0.7 = 0.7

max{0, 0.28, 0.49, -0.21} ≤ P(benefit|female) ≤ min{0.49, 0.79, 0.58, 0.7}
0.49 ≤ P(benefit|male) ≤ 0.49
P(benefit|male) = 0.49

If only observational data is available, the bounds would be
0 ≤ P(benefit|female) ≤ 0.279
0 ≤ P(benefit|male) ≤ 0.58
If only experimental data is available, the bounds would be
0.279 ≤ P(benefit|female) ≤ 0.489
0.28 ≤ P(benefit|male) ≤ 0.49
The overlap between the female bounds using observational data and the female bounds using experimental data, is the point estimate. The full bounds formula is not necessary.
However, the intersection of the male bounds using observational data and the male bounds using experimental data, does not provide us with point estimate. The full bounds formula is necessary.
–> The mechanism of combining ovservational data and experimental data work so well. What’s behind this?
- Observational data incorporates individuals' whims, and whims are proxies for hidden factors that may affect that individuals' response to the treatments.

5. Monotonicity, Probability of Harm, Number needed to Treat, and Other Results #

Under waht condition will RCT results constitute a point estimate for our target quantity, P(benefit)?
- this occurs under a condition called monotonicity (Pearl, 1999)
  namely, when the treatment cannot harm any individual,
  formally, P(y_t', y_c) = P(harm) = 0.
  A general relationshiop between P(harm), P(benefit) and ATE:
  P(harm) = P(benefit) - ATE
  - under monotonicity (i.e., P(harm) = 0), P(benefit) conincides with ATE
  - when monotonicity does not hold, P(harm) can be computed from P(benefit) and ATE
  * P(harm | female) = P(benefit | female) - CATE(female) = 0.279 - 0.279 = 0
  * P(harm | male) = P(benefit | male) - CATE(male) = 0.49 - 0.28 = 0.21
Number Needed to Treat (NNT): “The number of persons needed to be treated, on average, to prevent one more event”.
- How NNT should be estimated?
  $$
  max{1/P(y_t), 1/P(y_c')} ≤ NNT ≤ 1/(P(y_t) - P(y_c))
  $$
The bounds on the probability of harm.
$$
max{0, P(y_c) - P(y_t), P(y) - P(y_t), P(y_c) - P(y)} ≤ P(harm) ≤ min{P(y_c), P(y_t'), P(t, y') + P(c, y), P(y_c) - P(y_t) + P(t, y) + P(c, y')}
$$
The necessary test for monotonicity.
$$
P(y_t) ≥ P(y) ≥ P(y_c)
$$

Skipped

7. Conclusion #

The theoretical results in this paper show that, by combining observational data and experimental data, individual causal effects can be bounded, which sometimes can be quite narrow and allow us to make accurate personalized decisions.
The authors projected that these methods provide the key for the next-generation personalized decision making.

Last modified on 2023-01-12