Overview
PSpower is a design-stage tool for planning
observational and randomized studies that estimate treatment effects
using propensity score (PS) weighting. It answers: how large does my
study need to be?
Two functions cover the most common outcome types:
power_ps() |
Continuous or binary |
Two-sided |
power_cox() |
Time-to-event (survival) |
One-sided |
This is a prospective planning tool. Computing power
after data are collected using the observed sample size (“post-hoc
power”) is uninformative and should not be done.
Inputs at a glance
Common inputs
Effect size — the standardized treatment effect \(\delta = \tau / \text{SD}(Y)\) for
continuous/binary outcomes, or the log hazard ratio \(\tau_0 = \log(\text{HR})\) for survival
outcomes. For continuous outcomes, \(|\delta|
= 0.2\) is considered small; 0.5 moderate; 0.8 large. For
survival, \(\text{HR} = 0.75\) gives
\(\tau_0 \approx -0.29\).
Treatment proportion r — the fraction of participants who
receive treatment, \(r = \Pr(Z = 1)\).
A balanced design uses \(r = 0.5\); in
observational data, \(r\) is estimated
from the study population.
Estimand — the scientific quantity of interest:
- ATE — average treatment effect across the entire
study population (inverse probability weights)
- ATT — average effect among the treated group
(weights for group 1)
- ATC — average effect among the control group
(weights for group 0); operationally equivalent to ATT with group labels
swapped (\(r \to 1 - r\), group 1 \(\leftrightarrow\) group 0). Supported
directly by
power_ps(); for power_cox(),
re-label groups and use estimand = "ATT".
- ATO — average effect in the overlap
population (overlap weights); the most efficient estimand when
covariate overlap is limited
In a randomized trial all four estimands coincide.
Continuous and binary outcomes
(power_ps)
Overlap coefficient φ — a number in \((0, 1)\) measuring how similar the PS
distributions of the two groups are. \(\phi =
1\) in a randomized trial; \(\phi <
1\) reflects confounding, and sample size increases as \(\phi\) decreases. Rule of thumb: \(\phi \geq 0.95\) good; \([0.90, 0.95)\) moderate; \([0.85, 0.90)\) poor; \(< 0.85\) very poor.
\(\phi\) can be estimated from a
pilot study or any dataset with fitted propensity scores using
overlap_coef(). When it is not yet known, supplying a
vector of plausible values (e.g., \(\phi \in
\{0.85, 0.90, 0.95\}\)) allows \(N\) to be computed across the plausible
range.
Confounder coefficient ρ² — the squared correlation between
the potential outcome \(Y(0)\) and the
PS linear predictor. When \(\rho^2 =
0\) (the default), the outcome is unrelated to the confounders
beyond the treatment indicator. A sensitivity analysis over \(\rho^2 \in [0, 0.05]\) is recommended;
values above 0.05 are uncommon in practice.
Survival outcomes (power_cox)
Event rates d₁ and d₀ — the probability of observing an
event during follow-up in the treated group (\(d_1\)) and the control group (\(d_0\)). These can be estimated from
registries, pilot data, or published survival curves. If \(d_0\) is omitted it is set equal to \(d_1\).
Study type — "rct" for a randomized trial or
"obs" for an observational study.
Overlap coefficient φ — required when
study_type = "obs"; the same quantity as in
power_ps(). See the rule of thumb and estimation guidance
under the continuous/binary section above.
Method — "robust" (default) uses the sandwich
variance, which accounts for the actual hazard ratio;
"schoenfeld" is the classical formula derived under a null
effect and may overestimate or underestimate the required \(N\) relative to the robust method.
Available only for randomized trials.
1. Continuous and binary outcomes: power_ps()
Single scenario
Suppose we plan a study with:
- Standardized effect \(\delta =
0.2\) (e.g., a 5-point difference in a symptom score with SD =
25)
- 50% of participants receive treatment (\(r
= 0.5\))
- Estimated overlap \(\phi = 0.90\)
(moderate)
- Target: 80% power, two-sided \(\alpha =
0.05\)
res <- power_ps(
effect_size = 0.2,
r = 0.5,
phi = 0.9,
estimand = "ATE",
power = 0.80
)
res
#> PSpower: power_ps
#> --------------------------------------------------
#> Estimand: ATE
#> Weights: inverse probability weights
#> Effect size: 0.200
#> Treatment proportion: 0.500
#> Overlap (phi): 0.900
#> Confounding (rho2): 0.000 [default]
#> --------------------------------------------------
#> Required sample size: 1058
#> Target power: 0.80 | sig. level: 0.050 | two-sided test
Multiple configurations
Vectors can be supplied to any input; all combinations are computed
automatically.
Effect of treatment allocation
The required sample size depends on how evenly participants are split
between treatment and control. The following evaluates five values of
\(r\) across three estimands (\(5 \times 3 = 15\) scenarios):
res_r <- power_ps(
effect_size = 0.2,
r = c(0.30, 0.40, 0.50, 0.60, 0.70),
phi = 0.90,
estimand = c("ATE", "ATT", "ATO"),
power = 0.80
)
plot(res_r)
\(N\) is minimized at \(r = 0.5\) for ATE; the ATT and ATC
estimands are not symmetric in \(r\) —
ATT favors a larger treated fraction, and vice versa for ATC.
Effect of covariate overlap
res_phi <- power_ps(
effect_size = 0.2,
r = 0.5,
phi = c(0.85, 0.90, 0.95),
estimand = c("ATE", "ATT", "ATO"),
power = 0.80
)
plot(res_phi)
summary(res_phi)
#> PSpower: power_ps
#> -------------------------------------------------------
#> Scenarios: 9
#> Varying: phi, estimand
#> Fixed: effect_size = 0.2, r = 0.5, rho2 = 0 [default]
#> Target power: 0.80 | sig. level: 0.050 | two-sided test
#> -------------------------------------------------------
#>
#> Sample size (N, ceiling integer) distribution:
#>
#> sample_size phi estimand
#> Min: 867 0.95 ATO
#> 25%: 958 0.9 ATO
#> 50%: 1058 0.9 ATE
#> 75%: 1330 0.9 ATT
#> Max: 1978 0.85 ATT
The ATO estimand consistently requires fewer participants than ATE,
especially under poor overlap.
Role of the confounder coefficient ρ²
The table below shows how \(N\)
changes across \(\rho^2 \in [0, 0.05]\)
for ATE with \(\phi = 0.90\). A
sensitivity analysis over this range is recommended; values above 0.05
are uncommon in practice.
power_ps(
effect_size = 0.2,
r = 0.5,
phi = 0.90,
rho2 = c(0, 0.01, 0.02, 0.05),
estimand = "ATE",
power = 0.80
)
#> PSpower: power_ps
#> --------------------------------------------------
#> Scenarios: 4
#> Varying: rho2
#> Fixed: effect_size = 0.2, r = 0.5, phi = 0.9, estimand = ATE
#> Target power: 0.80 | sig. level: 0.050 | two-sided test
#> --------------------------------------------------
#> rho2 sample_size
#> 0.00 1058
#> 0.01 1065
#> 0.02 1072
#> 0.05 1093
An upper bound on \(\rho^2\) can be
obtained from the \(R^2\) of a
regression of the outcome on the study covariates using historical
data.
3. Estimating propensity score overlap from data
If a pilot dataset or a similar published cohort is available, \(\phi\) can be estimated directly from
fitted propensity scores using overlap_coef().
set.seed(2026)
n <- 500
X <- rnorm(n)
ps <- plogis(0.5 * X) # fitted propensity scores from a PS model
Z <- rbinom(n, 1, ps)
overlap_coef(ps = ps, Z = Z)
#> $phi
#> [1] 0.9696848
#>
#> $r
#> [1] 0.49
When no data are available, use the rule of thumb above and compute
\(N\) across the plausible range (e.g.,
\(\phi \in \{0.85, 0.90, 0.95\}\)), as
shown in the examples above.
overlap_coef() also accepts Beta distribution parameters
directly, which is useful when the PS distribution has been summarized
in a published paper:
overlap_coef(a = 3, b = 2) # r = a/(a+b) = 0.60
#> $phi
#> [1] 0.9017928
#>
#> $r
#> [1] 0.6
