Familywise Error Rate and the Testing Family

Why the Testing Family Matters

A genomic analysis may test thousands of claims: whether each gene is associated with a trait, whether each variant is associated with an outcome, or whether particular treatment groups differ. A statement about error control is incomplete until it specifies which hypotheses count together. The collection matters because familywise error rate describes the chance of making at least one false rejection within that collection.

For example, testing each of 20 genes against a single trait defines a different collection of claims from testing those same 20 genes against five traits. The data might be identical, but the scientific scope of the conclusions is not. A testing family is therefore defined by the inferential claims the analysis is meant to support, not just by a software output or a convenient subset of results.

Definition: A testing family, denoted $\mathcal{F}$, is a specified collection of null hypotheses considered together for a set of inferential claims. A false rejection occurs when a procedure rejects a null hypothesis that is true. The familywise error rate (FWER) is the probability of at least one false rejection in $\mathcal{F}$: $$ \mathrm{FWER}=P(V\geq 1) =P\left(\bigcup_{i\in I_0}\{\text{$H_i$ is rejected}\}\right), $$ where $I_0$ is the set of indices of the true null hypotheses in the family and $V$ is the number of true null hypotheses rejected.

The probability is defined over repeated samples under a specified data-generating situation and testing procedure. In a particular data set, $V$ is a realized count: it is the number of rejected hypotheses that happen to be true. FWER is not that observed count and is not the probability that a particular rejected hypothesis is false.

If every null hypothesis in the family is false, then $I_0$ is empty and no rejection can be a false rejection, so the FWER is zero. If some null hypotheses are true, the FWER depends on the chance of rejecting at least one of those true nulls. The identities of the false null hypotheses do not make their correct rejections into errors.

Families in Gene and Trait Analyses

A hypothesis should correspond to a specific claim. For a gene-by-trait analysis, one possible claim is that a specified gene is associated with a specified trait in a specified population under a specified model. Testing each of $G$ genes against each of $T$ traits gives $G\times T$ gene-trait hypotheses if all pairs are included. The family is those hypotheses only if the planned inferential goal concerns all those gene-trait claims.

A different scientific goal can define a different family. If the analysis is intended to make claims only about one pre-specified trait, tests for the other traits may not belong to that particular family. But if results across traits will be examined and reported as evidence for associations, those claims cannot be made irrelevant merely by focusing afterward on whichever trait produced the smallest p-value. The scope needs to reflect the intended conclusions, including claims that could be selected after seeing the results.

Worked Example: Counting Gene-by-Trait Hypotheses

Suppose an investigator plans to test 12 genes against each of 3 traits. The intended conclusions may name any gene-trait association found. Define the family and count its hypotheses.

For each gene there are three trait-specific null hypotheses. Across the 12 genes, the family size is $$ 12\times 3=36. $$ Thus $\mathcal{F}$ consists of the 36 gene-trait null hypotheses, not just 12 gene-level hypotheses. For instance, “gene 4 is associated with trait 2” and “gene 4 is associated with trait 3” are distinct claims and correspond to distinct null hypotheses.

Suppose, hypothetically, that the true state is that 34 of these null hypotheses are true and 2 are false. If a particular analysis rejects 5 hypotheses, and 1 of those 5 is a true null, then the realized false-rejection count is $V=1$. This describes the outcome in that data set. It does not by itself give the FWER: to calculate FWER, we need the repeated-sampling probabilities of rejecting the true nulls under the specified procedure.

The family would be different if the intended claim concerned only the 12 gene-specific hypotheses for one designated trait. In that case, the other two traits would not be part of that family. The important point is to specify the claims before interpreting the error probability.

Computing the Probability of Any False Rejection

Let $R_i$ be the event that the test of $H_i$ rejects. The event of at least one false rejection is the union of the rejection events for true null hypotheses. For two true nulls, the addition rule gives $$ P(R_1\cup R_2)=P(R_1)+P(R_2)-P(R_1\cap R_2). $$ The subtraction is necessary because outcomes in which both nulls are rejected are counted in each marginal probability. With more hypotheses, the same principle applies: the FWER is the probability of the union, not the sum of marginal rejection probabilities in general.

A useful simplification is available when the tests of the true null hypotheses are independent. If there are $m_0$ true nulls and each has probability $\alpha$ of rejection, the probability of no false rejection is $(1-\alpha)^{m_0}$. Taking the complement gives the FWER. This calculation requires both independence of the relevant rejection events and the stated marginal rejection probability; it is not a universal formula for dependent genomic tests.

Worked Example: Three Independent True Nulls

Consider three hypotheses that are all true. Suppose each test rejects its null with probability $0.05$, and assume the three rejection events are independent. Find the FWER.

A false rejection occurs if at least one of the three tests rejects. It is easier to first calculate the probability that none rejects. For each test, the probability of not rejecting its true null is $1-0.05=0.95$. Independence gives $$ P(\text{no false rejections})=0.95^3=0.857375. $$ Therefore, $$ \mathrm{FWER}=1-0.857375=0.142625\approx 0.1426. $$ Under these specific assumptions, the chance of at least one false rejection is about 14.26%. This is not the probability that any particular rejection is false; it is the probability of one or more false rejections in the three-test family over repeated samples.

The independence assumption is important. Genomic tests can be dependent, for example because genes or traits are correlated. Without independence, multiplying the three non-rejection probabilities is not justified. The event-union definition still applies.

Worked Example: Dependent Tests and the Union Event

Suppose a family contains three true null hypotheses. In a hypothetical repeated-sampling model, let $R_i$ denote rejection of the $i$th true null. Assume the event probabilities are $P(R_1)=0.040$, $P(R_2)=0.030$, $P(R_3)=0.020$; the pairwise intersection probabilities are $P(R_1\cap R_2)=0.010$, $P(R_1\cap R_3)=0.005$, and $P(R_2\cap R_3)=0.004$; and $P(R_1\cap R_2\cap R_3)=0.002$. Calculate the FWER.

For three events, inclusion-exclusion gives the probability of their union: add the three single-event probabilities, subtract the three pairwise intersections, then add the triple intersection: $$ \begin{aligned} \mathrm{FWER} &=P(R_1\cup R_2\cup R_3)\\ &=0.040+0.030+0.020-0.010-0.005-0.004+0.002\\ &=0.073. \end{aligned} $$ The FWER is 7.3% under this specified joint distribution. The calculation uses the overlap probabilities, so it does not assume the rejection events are independent. The marginal probabilities alone would not determine the answer.

Planned Contrasts Define a Family of Claims

A family need not consist of gene-specific hypotheses. Suppose a study compares four treatment groups, labelled control, A, B, and C, on a continuous outcome. A planned contrast is a specified linear combination of group means, such as $\mu_A-\mu_{\text{control}}$. Each contrast has its own null hypothesis. If the analysis is intended to make separate claims about A versus control, B versus control, and C versus control, those three hypotheses form a natural family.

The choice of family depends on the planned claims. If the only question is whether any treatment group differs from control, a single overall test of that question is a different inferential target from three separate claims naming which treatments differ. An overall test and individual contrasts are not interchangeable: the former may establish evidence of some difference without identifying which particular contrast accounts for it.

Worked Example: Defining a Family of Planned Contrasts

Suppose a hypothetical study has four groups: control, low dose, medium dose, and high dose. The protocol specifies three contrasts against control: $$ \mu_{\text{low}}-\mu_{\text{control}},\qquad \mu_{\text{medium}}-\mu_{\text{control}},\qquad \mu_{\text{high}}-\mu_{\text{control}}. $$ If all three estimates may support separate treatment claims, the family contains three null hypotheses, one for each contrast. It does not contain the six possible pairwise comparisons among four groups, because those comparisons were not the specified claims in this analysis.

For a hypothetical result, suppose the three p-values are 0.012, 0.041, and 0.200, and the procedure under consideration rejects p-values below 0.05. It rejects the first two hypotheses and not the third. If the first contrast null is true and the second is false, then exactly one of the two rejections is false, so $V=1$ in this data set. The p-values and decisions alone do not tell us the probability of at least one false rejection across repeated samples; that probability depends on the true hypotheses and the joint behavior of the tests.

Common Pitfalls and Practical Checks

A common mistake is to say, “Each test uses a 5% error rate, so the familywise error rate is 5%.” That reasoning confuses the marginal false-rejection probability for one true null with the probability of at least one false rejection across a family. As the three-independent-tests example shows, the latter can exceed 5%. The exact value depends on the number of true nulls, their marginal rejection probabilities, and the dependence among rejection events.

Another mistake is to define the family as whichever hypotheses were reported as significant. That uses the results to decide which tests count, making the stated scope too narrow when the analysis considered more claims. A practical check is to write down the scientific claim attached to every hypothesis and ask which of those claims the analysis may support. Include the hypotheses whose outcomes could lead to reported conclusions, rather than only those that happened to be rejected.

A single omnibus test can be an appropriate alternative when the scientific question is one overall claim, while a collection of individual tests addresses several specific claims. The choice is about the inferential question, not merely convenience. Likewise, testing one trait at a time can address a narrower goal than testing many traits, but it does not justify ignoring other traits if conclusions will be drawn from whichever results look most compelling.

Before describing FWER, verify that the family is explicit, the null hypotheses in it are identifiable, and the error probability is tied to a stated testing procedure and repeated-sampling situation. Check whether the tests are dependent before using any calculation that assumes independence. Finally, keep the probability statement in context: “the chance of at least one false rejection among these specified gene-trait hypotheses” is informative; “the chance the results are wrong” is not.

Key takeaway: FWER is the probability of at least one false rejection among a defined family of hypotheses. Defining that family requires identifying the scientific claims being considered; the observed number of rejections or the list of significant results does not define it.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

An analysis tests 8 genes against 2 traits, and any gene-trait pair may be reported as an association. How many hypotheses are in the family?
In a family of five hypotheses, two are true nulls. A data set rejects both true nulls and one of the three false nulls. What is the realized value of $V$, and does it equal the FWER?
Two true null hypotheses each have rejection probability 0.04, and their rejection events are independent. Calculate the FWER.
Why can the marginal rejection probabilities alone be insufficient to calculate FWER for dependent tests?
A study specifies three treatment-versus-control contrasts but reports only the two with the smallest p-values. What should determine whether the family has two or three hypotheses?

Solutions. (1) The family has $8\times 2=16$ gene-trait hypotheses. (2) Both rejected true nulls are false rejections, so $V=2$. This realized count is not the FWER, which is a repeated-sampling probability. (3) The probability of no false rejection is $0.96^2=0.9216$, so $\mathrm{FWER}=1-0.9216=0.0784$, or 7.84%. (4) The probability of the union depends on how often the rejection events overlap; marginal probabilities do not specify those intersections. (5) The intended inferential claims determine the family. If all three planned contrasts could support treatment claims, the family contains three hypotheses even if only two results are reported.