Testing many hypotheses creates more opportunities for chance findings. To control the risk, first define which tests belong to the same analysis family, then choose a target: familywise error rate (FWER) when any false positive is unacceptable, or false discovery rate (FDR) when controlling the expected share of false findings among discoveries is the right goal. No adjustment can replace a sound study plan or transparent reporting.
Why testing more hypotheses raises the risk of false positives
Each statistical test has a chance of rejecting a true null hypothesis. Run more tests, and there are more opportunities for a low p-value to appear by chance. A nominal significance level applies to an individual test; by itself, it does not protect the entire set of results.
The overall chance of one or more false positives depends on both the number of tests and how they are related. Tests on correlated outcomes, for example, do not behave like independent tests, so a simple calculation that assumes independence should not be treated as a universal estimate.
Multiplicity can arise from testing several outcomes, producing multiple p-values, repeatedly checking data as it accumulates, or conducting unplanned analyses after seeing results. These are among the sources of multiplicity discussed in a 2015 review by Streiner: “Best (but oft-forgotten) practices: the multiple problems of multiplicity—whether and how to correct for many statistical tests”.
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Define the analysis family before choosing a correction
An analysis family is the group of hypotheses that together support the claims a reader could select from. It might include several outcomes, treatment comparisons, subgroups, or planned analyses. If a researcher reports only the most favorable results from a broader set, correcting only that selected subset can understate the multiplicity that informed the selection.
Decide which tests belong together in light of the scientific claims, not merely because they appear in separate tables or were run by different software commands. If some tests are separated into distinct families, explain why their claims or decisions are genuinely distinct. Prespecifying the family and primary outcomes helps make that rationale assessable.
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Choose between FWER and FDR
FWER and FDR answer different questions. FWER concerns the probability of making one or more false rejections in a defined family. FDR concerns the expected proportion of false discoveries among the hypotheses rejected. Neither target is universally preferable; select one based on what a false positive would mean for the decision and how the results will be used.
| Target | Question it controls | When it may fit | Trade-off or condition |
|---|---|---|---|
| FWER | What is the chance of at least one false rejection in this family? | When even one false-positive claim could have serious consequences. | FWER-oriented procedures can be conservative and reduce power. |
| FDR | What is the expected share of false findings among the rejected hypotheses? | When screening a broad set of hypotheses for follow-up and a controlled share of false discoveries is acceptable. | It permits some false discoveries; guarantees depend on the procedure’s assumptions about the tests. |
Bonferroni, Holm, and Benjamini–Hochberg
Bonferroni and Holm: control the familywise error rate
Bonferroni is a straightforward FWER-oriented correction. Holm’s procedure is a sequential step-down alternative that also targets FWER. Both are options when the objective is to limit the probability of any false rejection in the specified family. FWER methods can cost power, so a nonsignificant adjusted result should not be interpreted as proof that there is no effect.
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Benjamini–Hochberg: control the false discovery rate
Benjamini and Hochberg introduced FDR as an alternative to FWER. Their 1995 paper describes the aim as “controlling the expected proportion of falsely rejected hypotheses — the false discovery rate” (“Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing”). The original result establishes FDR control for independent test statistics. Do not assume that guarantee automatically applies to every dependence structure.
FDR can be a useful target when many findings are candidates for further investigation and the objective is to control the expected proportion of false discoveries, rather than to ensure that a family contains no false rejection. The choice is about the acceptable error, not simply which procedure produces more significant results.
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Dependence-aware and resampling approaches
When tests are dependent, the method must match the dependence structure and the guarantee needed. Later discussion of FDR methods addresses dependence and the conditions under which control is obtained (Benjamini, “Discovering the false discovery rate,” 2010). Resampling methods can also be used in procedures aimed at either FWER or FDR. For a field-specific example, a comparative review of functional neuroimaging discusses Bonferroni, random-field, and permutation approaches to FWER control (“Controlling the familywise error rate in functional neuroimaging: a comparative review”).
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.A practical workflow for controlling multiplicity
- Define the claims and family. Before examining results, identify the outcomes and hypotheses that could support the same scientific claim. Document why any tests are treated as separate families.
- Separate primary from exploratory work. Specify primary hypotheses in advance where possible, and state how exploratory analyses will be identified and reported.
- Choose the error target. Decide whether the priority is limiting the chance of any false positive in the family (FWER) or controlling the expected proportion of false discoveries among rejections (FDR).
- Match the procedure to the design. Consider the number and dependence of tests, the consequences of errors, and whether hypotheses were prespecified or selected after viewing data. State the target level and the procedure used.
- Report the full analysis. Provide effect estimates and uncertainty alongside adjusted results. Disclose outcomes, analyses, interim looks, and post hoc work, rather than presenting only favorable tests.
What a correction cannot fix
A multiplicity adjustment controls a specified error target only under the method’s assumptions and for the family to which it is applied. It cannot correct biased measurement, poor study design, selective reporting, or p-hacking, and it does not make a post hoc result confirmatory. Nor does statistical significance explain the practical size or importance of an effect: report estimates and uncertainty so readers can assess those separately.
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Whether and how to correct has been debated, as Streiner’s 2015 review notes. The defensible response is to state the decision rule, explain the family and target, and report the analyses transparently—not to treat one correction as a universal fix.
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