Signature
ATE = sum_(x=1)^K (P_x * CATE_x)
| Inputs | Definition | Unit |
|---|---|---|
P_x | Share of the population of interest in subgroup x, written P(X = x) in the article | proportion of the population |
CATE_x | Conditional average treatment effect in subgroup x, the mean of Y(1) minus Y(0) among patients with characteristics x | the same as ATE |
ATE | Average treatment effect in the population of interest, on cost or on QALYs | the outcome's unit per patient, for example QALYs or £ |
|---|
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
ATE from subgroup shares and effects in one cell
Excel multiplies each subgroup's share by its effect and adds the products, with the shares and the effects in two named ranges of the same length.
=SUMPRODUCT(SubgroupShares,SubgroupEffects)
Assumptions
Subgroups cover the population once for the ATE
Every patient in the population of interest belongs to exactly one of the K subgroups, so the shares P_x sum to 1.
Subgroup shares from the decision population
The shares P_x describe the population covered by the decision, not the trial sample or the treated group. If trial participants differ from that population in characteristics that change the effect, such as baseline risk, the trial's mix gives a different average.
Subgroup effects on one scale for the ATE
Each CATE_x is measured on the same outcome, perspective, time horizon and discounting as the others, for example discounted QALYs per patient or cost per patient in one price year.
Worked examples
ATE on QALYs in the two-risk-group example
In the article's illustrative example 60% of patients are low risk with a QALY gain of 0.10 and 40% are high risk with a gain of 0.40. The population ATE is 0.22 QALYs per patient.
K = 2; P_x = [0.6,0.4]; CATE_x = [0.10,0.40]; ATE = 0.22
ATE on cost in the two-risk-group example
With cost increments of £3,000 in the low-risk group and £4,000 in the high-risk group, the ATE on cost is £3,400 per patient. Combined with the 0.22 QALYs above, the pairwise ICER (HE-FM-ICER-001) is about £15,455 per QALY and the incremental net monetary benefit at an illustrative £25,000 per QALY (HE-FM-NMB-002) is £2,100 per patient.
K = 2; P_x = [0.6,0.4]; CATE_x = [3000,4000]; ATE = 3400
Common errors
Averaging subgroup effects without population shares
A simple mean of the two subgroup effects gives 0.25 QALYs and £3,500, and an ICER of £14,000 per QALY instead of about £15,455, because it treats the high-risk group as half of the population when it is 40%.
Reading the ATE as every patient's gain
An ATE of 0.22 QALYs fits gains of 0.10 and 0.40 in two groups, or gains for some patients and losses for others. It describes the population and says nothing about which patients gain.
Sources
What If on standardisation over strata
Hernán MA, Robins JM. Causal Inference: What If. Boca Raton: Chapman & Hall/CRC; 2020. Section 2.3, which writes the marginal counterfactual risk as the average of the stratum-specific risks weighted by the proportion of the population in each stratum, and section 4.3, which notes that the average causal effect differs between populations with different prevalence of an effect modifier.
TSD 17 on the ATE as the effect of interest
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 ATE and states that it is the treatment effect typically of interest in NICE technology appraisals.
Canonical Identity
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