Elementary and Intersection Null Hypotheses

Why the Scope of a Null Hypothesis Matters

A multi-arm trial can ask several treatment-versus-control questions at once. Before interpreting any result, it is necessary to state exactly which claim is being tested: does a null hypothesis concern one treatment comparison, or does it assert that a whole set of comparisons has no effect? Those are different hypotheses, even when they are built from the same treatment effects.

The earlier material on defining hypothesis families established that a family consists of the related claims for which conclusions are to be considered together. Here the focus is the logical structure of those claims. The distinction matters because a statement about every treatment in a set is stronger than a statement about an average, and different subsets of comparisons lead to different intersection hypotheses.

Definition: Suppose a trial has \(K\) experimental treatments and one control. Let \(\theta_j\) be the prespecified treatment effect for treatment \(j\) compared with control, for \(j=1,\ldots,K\). An elementary null hypothesis is a null hypothesis about one comparison, such as \(H_j:\theta_j=0\). For any nonempty subset \(I\subseteq\{1,\ldots,K\}\), the corresponding intersection null hypothesis is \(H_I=\bigcap_{j\in I}H_j\), meaning that every elementary null in the subset is true.

The parameter \(\theta_j\) must be defined for the scientific question at hand. For example, it could be a difference in mean outcome at a specified follow-up time, with positive values indicating better outcomes for treatment \(j\). The endpoint, population, follow-up time, and contrast direction should be specified consistently across the comparisons. In the examples below, the effects are differences in mean outcome, treatment minus control, measured in points; zero represents no difference in population means.

The elementary hypothesis \(H_j:\theta_j=0\) is a claim about one treatment. Its alternative might be two-sided, \(H_{a,j}:\theta_j\ne 0\), or directional, such as \(H_{a,j}:\theta_j>0\), depending on the prespecified scientific question. The corresponding intersection null concerns all the component nulls at once. For a subset \(I\), it says \(\theta_j=0\) for every \(j\in I\), not merely that the effects in \(I\) balance out.

Writing and Reading an Intersection

Let \(I\) contain some of the treatment labels. The intersection \(H_I\) is true only when each \(H_j\) with \(j\in I\) is true. If even one component null is false, then the intersection null is false. This follows directly from the meaning of an intersection: an outcome belongs to the intersection only if it belongs to every set being intersected.

H_I=\bigcap_{j\in I}H_j \quad\Longleftrightarrow\quad \theta_j=0\text{ for every }j\in I.

For example, if \(I=\{1,3\}\), then \(H_I=H_1\cap H_3\) asserts both \(\theta_1=0\) and \(\theta_3=0\). It makes no assertion about treatment 2 if treatment 2 is not in \(I\). The intersection over all \(K\) comparisons is often called the global or complete null:

H_{\{1,\ldots,K\}}=\bigcap_{j=1}^{K}H_j \quad\Longleftrightarrow\quad \theta_1=\theta_2=\cdots=\theta_K=0.

The alternative to an intersection null is a union of component alternatives. For two-sided elementary alternatives, the complement of the claim that all effects equal zero is the claim that at least one effect differs from zero. More generally, the alternative depends on the component alternatives that were specified; it should not be silently changed from two-sided to directional.

\left(\bigcap_{j\in I}H_j\right)^c = \bigcup_{j\in I}H_j^c.

This identity is a logical statement, not a testing rule. It says what parameter configurations make the intersection null false. It does not say how to calculate a test statistic, adjust a p-value, or control an error rate.

Worked Example: Naming Individual and Joint Claims

Consider a hypothetical trial with three experimental treatments, A, B, and C, compared with one control. The prespecified effect is the difference in mean symptom-score improvement, in points, with a positive difference favoring the experimental treatment. Write the elementary nulls and the intersection null for A and C.

Step 1: Define the parameters. Let \(\theta_A\), \(\theta_B\), and \(\theta_C\) denote the population mean improvement for each treatment minus the population mean improvement for control. The same outcome definition and follow-up time are used for all three comparisons.

Step 2: State the elementary nulls. The three elementary null hypotheses are \(H_A:\theta_A=0\), \(H_B:\theta_B=0\), and \(H_C:\theta_C=0\). Each one concerns only its named treatment-versus-control comparison.

Step 3: Form the requested intersection. For the subset \(I=\{A,C\}\), \(H_I=H_A\cap H_C\), so \(H_I\) asserts \(\theta_A=0\) and \(\theta_C=0\). It does not assert anything about \(\theta_B\). The global null would instead be \(H_A\cap H_B\cap H_C\).

The definitions do not require the treatment effect estimates to be independent. In a shared-control design, two estimated differences may be statistically dependent because both use data from the same control group. That affects the joint sampling distribution used in a test, but it does not change which parameter values satisfy this intersection null.

What an Intersection Null Is Not

An intersection null is sometimes confused with a null hypothesis about an average, a sum, or a pooled treatment effect. These claims generally differ. For instance, \(\theta_1+\theta_2=0\) allows \(\theta_1\) to be positive and \(\theta_2\) negative. By contrast, \(H_1\cap H_2\) requires both effects to equal zero separately.

The distinction also applies when the treatment estimates are weighted. A weighted average may be useful for a different scientific question, but an average of zero does not establish that every component effect is zero. The exact parameter restriction must match the claim being made.

Worked Example: A Zero Average Does Not Give an Intersection Null

Suppose a hypothetical three-treatment trial has population treatment effects, measured as mean improvement differences in points, \(\theta_A=2\), \(\theta_B=-2\), and \(\theta_C=0\). Consider the equally weighted average of the effects for A and B.

Calculate the average.

\frac{\theta_A+\theta_B}{2} = \frac{2+(-2)}{2} = 0\text{ points}.

The average for A and B is zero, but \(H_A\cap H_B\) is false: \(\theta_A=2\ne 0\) and \(\theta_B=-2\ne 0\). The full intersection \(H_A\cap H_B\cap H_C\) is also false, even though \(\theta_C=0\), because both A and B have nonzero effects. A test or estimate aimed at the average answers a different question from one about the intersection.

The values are hypothetical population effects used to illustrate the logic, not estimates from a real study. The example needs no assumption about how the effects were estimated: it is the parameter values themselves that determine whether the hypotheses hold.

From Trial Questions to Hypotheses

A reliable way to formulate these hypotheses is to begin with the scientific claim, define the comparison parameter, and only then write the null. In a trial with a common control, each treatment effect uses that control as its reference, but the individual hypotheses remain separate claims. Combining several of them with an intersection symbol adds the requirement that all those component claims hold simultaneously.

Specify each comparison.
Identify the treatment, common control, outcome, population, follow-up time, and direction of the effect parameter.

Write the elementary nulls.
For each treatment \(j\), state the parameter restriction for that single comparison, such as \(H_j:\theta_j=0\).

Select the subset.
Name the set \(I\) of treatment comparisons relevant to the joint claim. Do not include a comparison merely because its data share the same control.

Translate the intersection.
Write \(H_I=\bigcap_{j\in I}H_j\), then state in words that every selected treatment effect equals zero. Check that no average or unselected arm has entered the claim.

Worked Example: Interpreting a Selected Subset

A hypothetical four-arm study compares treatments A, B, C, and D with a common control on a continuous outcome measured in millimetres. Let \(\theta_j\) be the treatment-minus-control difference in population mean change, with the same endpoint and follow-up for all arms. The research team wants to describe the joint null for A, C, and D, but not B.

The elementary nulls for the selected comparisons are \(H_A:\theta_A=0\), \(H_C:\theta_C=0\), and \(H_D:\theta_D=0\). Their intersection is \(H_{\{A,C,D\}}=H_A\cap H_C\cap H_D\). Written as parameter restrictions, this is \(\theta_A=\theta_C=\theta_D=0\) millimetres.

Treatment B is absent from this intersection, so its effect can be zero or nonzero without changing whether \(H_{\{A,C,D\}}\) holds. For example, the parameter configuration \((\theta_A,\theta_B,\theta_C,\theta_D)=(0,5,0,0)\) mm satisfies this selected intersection, although it does not satisfy the global null. Conversely, \((0,5,1,0)\) mm does not satisfy the selected intersection because \(\theta_C=1\) mm is nonzero. These conclusions are direct substitutions into the restrictions; no distributional assumption is needed to determine them.

Assumptions, Diagnostics, and Limits

The definitions above are logical descriptions of parameter sets, so they do not depend on normality, equal variances, independent treatment-effect estimates, or a particular test statistic. To give the parameters clinical meaning, however, the comparisons must be well defined: the outcome, target population, treatment conditions, control condition, and measurement time need to be specified. If the scientific estimand is a difference in means, that estimand must also be appropriate for the outcome and question.

The analysis used to test a hypothesis will have its own assumptions, such as assumptions about the outcome distribution or the trial design. Those assumptions must be checked for the chosen method; they are not implied by writing \(H_j\) or \(H_I\). In particular, shared control data can make estimated treatment contrasts dependent. A test of a joint hypothesis must account for the relevant joint distribution, but dependence does not make the intersection notation invalid.

Several practical checks can reveal a formulation problem before any test is performed:

Confirm that every \(\theta_j\) uses the same prespecified outcome scale and follow-up when the joint claim is intended to compare like quantities.
Check that the index set \(I\) contains exactly the comparisons named in the scientific claim.
Translate the symbolic hypothesis back into words. If the words say “all selected effects are zero” but the equation only sets their average to zero, the hypotheses do not match.
Distinguish “all effects equal zero” from “at least one effect is nonzero.” These are opposite sides of the same intersection claim, not interchangeable descriptions.
Keep the logical hypothesis separate from the method used to test it. A valid definition alone does not establish that a chosen test has appropriate error properties.

An elementary hypothesis is useful for stating a claim about one specific treatment comparison. An intersection hypothesis is useful when the claim requires several specified elementary nulls to hold together. A union of hypotheses, by contrast, describes configurations satisfying at least one component null; it is not the same as the intersection requiring every component null. These forms of claims should not be substituted for one another.

The main limitation of this notation is that it specifies what is being claimed, not how strongly the data support the claim. A hypothesis statement does not provide an estimate, uncertainty interval, p-value, or decision rule. Those require a statistical analysis suited to the design and the selected hypothesis. For now, the essential task is to formulate the parameter restrictions without changing the scientific question.

Key takeaway: An elementary null concerns one treatment comparison. An intersection null is true only when every elementary null in its specified subset is true. A zero average, a shared control, or a nonzero effect in an unselected arm does not change that definition.

Check Your Understanding

Answer each question by identifying the exact parameter restrictions. Solutions are included after the questions.

For three treatments with effects \(\theta_1,\theta_2,\theta_3\), write the intersection null for treatments 1 and 3. Does it restrict \(\theta_2\)?
If \(\theta_1=3\) and \(\theta_2=-3\), does \(H_1\cap H_2\) hold because their average is zero? Explain.
For four comparisons, what parameter restrictions define the global null?
Can dependence between two effect estimates from a shared control change which parameter values satisfy \(H_1\cap H_2\)? Explain.

Solutions. (1) \(H_1\cap H_3\) requires \(\theta_1=0\) and \(\theta_3=0\); it places no restriction on \(\theta_2\). (2) No. The average is \((3+(-3))/2=0\), but neither component is zero, so \(H_1\cap H_2\) is false. (3) The global null requires \(\theta_1=\theta_2=\theta_3=\theta_4=0\). (4) No. Dependence may affect the sampling distribution and the analysis of a test, but the intersection is defined by the parameter restrictions \(\theta_1=0\) and \(\theta_2=0\).