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When a study’s sample size is already fixed — the data are collected, the cohort is closed, the registry extract is final — asking “how big a sample do I need?” no longer makes sense. The question that remains is the reverse one: given the sample I actually have, what is the smallest true effect this study was capable of detecting? That reversed calculation is a sensitivity power analysis, and its output is the minimum detectable effect size (MDES).
Sensitivity power analysis is one of the four standard power-analysis modes alongside a priori, compromise, and post hoc power analysis (see Power Analysis and Sample Size Calculation for how all four relate). It solves the same four-parameter equation — significance level (α), statistical power (1 − β), sample size (N), and effect size — as an a priori calculation, just for a different unknown: N is fixed and known, and effect size is the value being solved for.
Why compute it instead of post hoc (observed) power
The instinct after a null or non-significant result is often to ask “was my study powered to find this effect?” using the effect size actually observed in the data. That calculation is post hoc (observed) power, and statisticians treat it as methodologically unsound: because power is a monotonic, one-to-one transformation of the p-value at a fixed N and α, observed power adds no information beyond the p-value itself — a non-significant result will always show “low” post hoc power by construction, regardless of the study’s actual sensitivity. The American Statistical Association’s 2016 statement on p-values and mainstream biostatistics guidance both discourage using observed effect size to compute power for this reason.
Sensitivity power analysis avoids the circularity because it never uses the observed effect size as an input. It answers a genuinely different, non-circular question: “holding α and target power fixed, what effect size would this N have let me detect?” — a property of the study’s design, not of what happened to come out of it. That is why sensitivity analysis, not post hoc power, is the accepted way to characterize what a completed, fixed-N study was actually equipped to find.
When to run a sensitivity power analysis
- Secondary analysis of existing data — a registry, biobank, EHR extract, or public dataset where the analyst has no control over N. Reporting the MDES tells readers what magnitude of effect the analysis could plausibly have found, framing a null result honestly.
- Underpowered or terminated studies — a trial stopped early for futility, slow accrual, or budget, where the pre-registered a priori calculation is now moot. The MDES for the achieved N is the honest replacement figure.
- Grant and manuscript reviewer requests — reviewers sometimes ask “was this study powered to detect a clinically meaningful effect?” after the fact. A sensitivity analysis answers that directly, and pairs naturally with a stated minimal clinically important difference or minimal important change benchmark: if the MDES is larger than the MCID, the study could only have detected effects beyond what clinicians would consider meaningful.
- Pre-registration transparency — some registered-report and protocol templates ask authors to report the smallest effect their planned N will detect, as a sensitivity check alongside the a priori calculation, particularly when N is constrained by cost or feasibility rather than chosen freely.
Worked example: two-group comparison
Take a two-independent-samples t-test with a fixed, already-collected sample of n = 64 participants per group (N = 128 total), a conventional two-tailed α = .05, and a target power of .80. Under Cohen’s (1988) normal-theory tables for this test family, n = 64 per group is the sample that gives 80% power to detect a medium effect, Cohen’s d ≈ 0.50. Run in reverse — N, α and power fixed, solving for d — the same relationship gives the sensitivity answer directly: this study’s MDES is d ≈ 0.50. Anything smaller than a medium effect, by Cohen’s own benchmarks (d = 0.2 small, 0.5 medium, 0.8 large), was not something this design could reliably distinguish from noise. See Effect Size for how these benchmarks are defined and when they do and don’t apply to a given field.
The exact MDES for any other test family — correlation, ANOVA, chi-square, logistic regression — follows the same logic (fix N, α, and power; solve the power function for effect size) but the closed-form relationship differs by test, which is why this is normally done in software rather than by hand.
Computing it: G*Power and R
G*Power
- Choose the correct test family and statistical test for your design (e.g., “t tests” → “Means: Difference between two independent means”).
- Under Type of power analysis, select Sensitivity: Compute required effect size — given α, power, and N. This is a distinct menu option from “A priori” (which solves for N) and “Post hoc” (which solves for power given an observed effect).
- Enter the fixed values you already have: sample size(s) per group, α err prob (typically .05), and Power (1 − β err prob), conventionally .80 or .90.
- G*Power returns the effect size (Cohen’s d, f, or the relevant metric for the chosen test) that satisfies those constraints — the MDES.
R (pwr package)
The same logic applies to any pwr.*.test() function in the pwr package: supply every argument except the effect size, and the function solves for whichever argument is left as NULL. For a two-sample t-test:
pwr.t.test(n = 64, sig.level = 0.05, power = 0.80,
type = "two.sample", alternative = "two.sided")
# returns d, the minimum detectable effect for n = 64/groupThe equivalent pattern holds for pwr.anova.test() (solve for f), pwr.r.test() (solve for r), and pwr.chisq.test() (solve for w) — leave the effect-size argument out and supply everything else.
Reporting a sensitivity power analysis
State, in the methods or limitations section: the fixed N actually analyzed, the α and power level used, the resulting MDES with its metric (Cohen’s d, r, f, odds ratio, etc.), and — where one exists — how that MDES compares to a pre-specified minimal important difference or the effect sizes reported in comparable published studies. A sensitivity analysis reported this way turns “we found no significant effect” into a falsifiable, quantified claim about what the study could and could not have detected, rather than an unqualified null.
Sensitivity power analysis vs. the other three modes
| Mode | Fixed inputs | Solves for | When used |
|---|---|---|---|
| A priori | α, power, expected effect size | Required N | Before data collection (grant/IRB planning) |
| Compromise | N, effect size, β/α error ratio | Optimal α and power trade-off | Fixed-resource planning |
| Sensitivity | α, power, fixed N | Minimum detectable effect size | After data collection, N already fixed |
| Post hoc (observed) | α, observed effect size, N | “Achieved” power | Discouraged — circular with the p-value |
For the full mechanics of a priori and compromise power analysis, including software walkthroughs, see Statistical Power Analysis & G*Power Sample Size Guide and Power Analysis and Sample Size Calculation: A Complete Guide. For related fixed-sample benchmarks used in outcome measurement rather than hypothesis testing, see Minimal Detectable Change (MDC) and Minimal Clinically Important Difference (MCID) — both answer “how big a change is meaningful,” a related but distinct question from “how big an effect could this design detect.”
Frequently asked questions
Is sensitivity power analysis the same as post hoc power?
No. Post hoc (observed) power plugs the study’s own observed effect size into the power formula, which makes it a direct, circular restatement of the p-value — a non-significant result always yields “low” post hoc power regardless of the study’s real sensitivity, so it tells you nothing new. Sensitivity power analysis never uses the observed effect; it fixes N, α, and a target power level chosen independently of the results, and solves for the smallest effect that combination could detect. That independence from the observed result is what makes it non-circular and defensible to report.
Can a sensitivity power analysis justify a null result?
It can contextualize one honestly, but it cannot prove the absence of an effect. Reporting “this study’s MDES was d = 0.50, and no effect that large was found” tells a reader precisely what magnitude of effect the design could and could not rule out — it does not establish that smaller true effects don’t exist, only that this particular study wasn’t equipped to detect them reliably.
Does the MDES change if I use a one-tailed test?
Yes. A one-tailed test at the same α concentrates the entire rejection region in one direction, which lowers the MDES (the design becomes more sensitive to effects in the specified direction) compared to a two-tailed test at the same α, N, and power. Most fields default to two-tailed tests unless there is a strong, pre-specified directional hypothesis, so specify the tail explicitly when reporting.
What software computes the minimum detectable effect size?
G*Power (free, cross-platform) has a dedicated “Sensitivity” analysis mode built into every test family it supports. R’s pwr package solves for effect size the same way by leaving that argument as NULL in any pwr.*.test() call. Both are standard, citable tools for this calculation.
