# Planning a MAIC from the estimator’s influence function, not from n or
Kish ESS
Ahmad Sofi-Mahmudi
2026-09-23

# Abstract

**Background.** A MAIC’s precision depends on overlap, the balancing
constraints, the outcome and the allocation, not on nominal sample size
alone. Catalog problem OVL-06 asks whether a precision calculation made
before any data exist can predict what the analysis will achieve.

**Methods.** For each of 72 scenarios (total size 150 to 1000, target
shift, 2 or 5 covariates, 1:1 or 2:1 allocation, event prevalence 0.1 or
0.3) we computed three planned standard errors from the posited source
and target laws: nominal $n$; $n$ replaced by each arm’s Kish ESS; and
the estimator’s influence-function variance, evaluated on a large draw
from the posited laws. Each was compared with the standard deviation
MAIC achieved over 1000 replicates.

**Results.** The influence-function plan was within 10% of the achieved
SD in 85% of scenarios (median ratio 0.963), Kish ESS in 58% (median
0.912), nominal $n$ in 21% (median 0.797). The Kish plan understated the
achieved SD in 99% of scenarios. Every miss of the influence-function
plan was at 150 patients with 10% prevalence, where the asymptotic
variance does not yet hold.

**Conclusion.** Planning in Kish ESS is optimistic, not pessimistic, and
misses by more than 10% in four scenarios of ten. The estimator’s own
influence function, which a planner can evaluate at posited
distributions, predicts achieved precision within 10% except where
events are too few for any asymptotic calculation.

# The problem

Kish’s ESS is a functional of the weights alone; it ignores allocation,
outcome variance and the estimation of the weights, and it is recorded
as likely an underestimate of the true effective sample size
([1](#ref-phillippo2018)). The variance of a weighted estimator is
$\mathbb{E}[\psi^2]/n$ for its influence function $\psi$, which carries
all of those things, and it can be computed from posited source and
target laws before any data exist. The catalog’s refuting sentence is
that nominal $n$ with a crude overlap adjustment predicts achieved
precision well enough for planning.

# Design

Registered protocol: `protocol.md`. Individual-data trial A versus C
with $d$ independent normal covariates; binary outcome
$\operatorname{logit} p = \operatorname{logit}(\pi) + 0.4\sum_j x_j/\sqrt d + A(-0.5 + 0.4x_1)$;
MAIC on means to a target at Mahalanobis distance $s$. Estimand: the SD
of the MAIC estimate of the marginal log odds ratio, achieved over 1000
replicates. Plans: nominal, $1/(n_1p_1(1-p_1)) + 1/(n_0p_0(1-p_0))$;
Kish, the same with $n_a$ multiplied by the arm’s population ESS
fraction; influence function, the stacked sandwich $A^{-1}BA^{-\top}/n$
over weights and arm risks evaluated on 100,000 draws from the posited
laws.

# Results

<div id="fig-plans">

![](figures/fig1-plans.png)

Figure 1: Planned SE over achieved SD for each plan, by target shift,
total size and prevalence. Dotted lines mark 10%.

</div>

| plan               | within 10% | median ratio | range          |
|--------------------|-----------:|-------------:|----------------|
| nominal $n$        |        21% |        0.797 | 0.541 to 0.985 |
| Kish ESS           |        58% |        0.912 | 0.745 to 1.004 |
| influence function |        85% |        0.963 | 0.827 to 1.023 |

Both simpler plans are optimistic, increasingly so with the shift
(<a href="#fig-plans" class="quarto-xref">Figure 1</a>): Kish ESS
corrects nominal $n$ for weight concentration but not for how the
weights interact with the outcome and the allocation. The
influence-function plan’s 11 misses were all at 150 patients and 10%
prevalence, about eight events per arm, and its median ratio rose from
0.908 at 150 patients to 0.981 at 1000.

# What this does not answer

The planner is assumed to know the source and target laws; misspecifying
them is a separate error not studied. Binary outcome and anchored-side
precision only: the unanchored bias floor, which no sample size removes,
and survival outcomes, where the unit is weighted risk-set information,
are not run. MAIC on means only. Peer review has not been done.

# References

<div id="refs" class="references csl-bib-body">

<div id="ref-phillippo2018" class="csl-entry">

<span class="csl-left-margin">1.
</span><span class="csl-right-inline">David M. Phillippo, A. E. Ades,
Sofia Dias, Stephen Palmer, Keith R. Abrams, Nicky J. Welton. Methods
for population-adjusted indirect comparisons in health technology
appraisal. Medical Decision Making. 2018;38(2):200–11.
doi:[10.1177/0272989X17725740](https://doi.org/10.1177/0272989X17725740)</span>

</div>

</div>