Signature
ATT = sum_(x=1)^K (P_x * e_x * CATE_x) / sum_(x=1)^K (P_x * e_x)
| Inputs | Definition | Unit |
|---|---|---|
P_x | Share of the whole population in subgroup x, treated or not | proportion of the population |
e_x | Probability that a patient in subgroup x receives the treatment in current practice, the subgroup's propensity score | probability |
CATE_x | Conditional average treatment effect in subgroup x, as in HE-FM-ATE-001 | the same as ATT |
ATT | Average treatment effect among the patients who received the treatment, on cost or on QALYs | the outcome's unit per treated patient |
|---|
KNumber of non-overlapping subgroups that together make up the population (count)
Function
Average treatment effect function
Maps the two potential outcomes of each patient in a stated population, Y(1) under the treatment and Y(0) under the comparator, to the mean of their difference. Only one potential outcome is observed for any patient, so the mean is computed from subgroup averages or by reweighting observed outcomes under stated conditions. Conditioning on the treated gives the average effect on the treated (ATT), and conditioning on baseline characteristics gives the conditional average effect (CATE). Applied to costs and to QALYs, the two averages are the incremental cost and incremental QALYs used by the pairwise ICER (HE-FM-ICER-001) and the incremental net monetary benefit (HE-FM-NMB-002).
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Implementations
Excel
ATT from subgroup shares, propensities and effects
Excel weights each subgroup effect by its population share and its probability of treatment, then divides by the treated share of the population. The three named ranges hold one value per subgroup.
=SUMPRODUCT(SubgroupShares,Propensity,SubgroupEffects)/SUMPRODUCT(SubgroupShares,Propensity)
Assumptions
Subgroup effect applies to its treated members
Within each subgroup the treated patients have the same average effect as the untreated, so CATE_x describes the treated members of subgroup x. This holds when the subgroups capture the characteristics that drive both treatment and effect, the ignorability condition in TSD 17.
Treatment probabilities from the practice described
Each e_x is the share of subgroup x treated in the practice whose treated group is being described, for example 25% of low-risk and 87.5% of high-risk patients in the article's example. At least one e_x is above zero.
Worked examples
ATT on QALYs under confounding by indication
Clinicians treat 25% of the low-risk and 87.5% of the high-risk patients, so the treated group is 30% low risk and 70% high risk. The ATT is 0.31 QALYs, above the ATE of 0.22, because the treated group leans towards the patients who gain most.
K = 2; P_x = [0.6,0.4]; e_x = [0.25,0.875]; CATE_x = [0.10,0.40]; ATT = 0.31
ATT on cost under confounding by indication
With the same treatment probabilities the ATT on cost is £3,700 per treated patient. With the 0.31 QALYs above it gives about £11,935 per QALY for the treated group, against about £15,455 for the whole population.
K = 2; P_x = [0.6,0.4]; e_x = [0.25,0.875]; CATE_x = [3000,4000]; ATT = 3700
Common errors
Using the ATT for a whole-population decision
The ATT of 0.31 QALYs and £3,700 gives about £11,935 per QALY and an incremental net monetary benefit of £4,050 at £25,000 per QALY. A decision to treat the whole population needs the ATE instead: about £15,455 per QALY and £2,100. Matched analyses often report the ATT, so the estimand is checked before the figures enter a model.
Sources
TSD 17 on the ATT and selection on heterogeneity
Faria R, Hernández Alava M, Manca A, Wailoo AJ. NICE DSU Technical Support Document 17: the use of observational data to inform estimates of treatment effectiveness in technology appraisal: methods for comparative individual patient data. Sheffield: Decision Support Unit, ScHARR; 2015. Section 2.1.2, which defines the ATT as the effect for those who are treated and states that selection into treatment driven by the determinants of heterogeneity makes the ATE and ATT differ.
What If on the average effect in the treated
Hernán MA, Robins JM. Causal Inference: What If. Boca Raton: Chapman & Hall/CRC; 2020. Fine Point 4.1, which defines the average causal effect in the treated and notes that it differs from the population effect when individual causal effects are distributed differently in the treated and the untreated.
Canonical Identity
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