Estimate; do not invent a null.
Prevalence, distribution and service-description questions usually need a precise population, definition, timeframe and confidence interval rather than formal H₀/H₁.
Statistical foundations · questions and hypotheses
Write a hypothesis that fits the question.
A hypothesis is a proposed, testable answer, not the starting label for every study and not a statement that a statistical test can prove.
Use this when: you are unsure whether your question needs H₀ and H₁.
Bring: who, what comparison, which outcome and when.
You will decide: write a clear pre-specified hypothesis, or explain why one is not needed.
Before you begin
Define the question before testing an answer.
A broad topic such as “home blood-pressure monitoring” is not yet a hypothesis. First decide what population, comparison, outcome, timeframe and effect the study can estimate.
Helpful first: complete the PECO/PICO part of the research question and analysis brief. If you do not have one yet, the worked examples here show why it matters.
By the end you can
- distinguish a topic, aim, question and hypothesis;
- decide when a formal hypothesis is useful;
- write H₀ and H₁ on the correct effect scale;
- choose direction and sidedness before outcome inspection;
- explain alpha, Type I/II error and power without claiming proof.
1 · The sequence
Which comes first: research question or hypothesis?
Recommended route
Substantive problem or topic → focused research question → hypothesis when appropriate → analysis plan.
Theory or prior evidence may generate an initial hypothesis. That proposed answer must still be translated into a precise question and operationalised before testing. Confirmatory hypotheses and their analysis plan are pre-specified before outcome results are inspected.
Pre-specify the proposed answer.
Use a justified hypothesis tied to one primary question, target effect and model before outcome inspection.
Label discovery honestly.
Explore signals or assess predictive performance without retrofitting a confirmatory hypothesis after seeing results. Validate what is found.
Home blood-pressure monitoring.
To estimate the effect of supported home monitoring on systolic blood pressure at 12 weeks.
Among adults with hypertension, does supported home monitoring, compared with usual care, change mean systolic blood pressure after 12 weeks?
The intervention-minus-usual-care mean difference at 12 weeks differs from zero.
2 · Write the pair
State H₀ and H₁ on the scale you will analyse.
Supported home monitoring and blood pressure.
- Operational question
- Among adults with hypertension, does supported home monitoring, compared with usual care, change mean systolic blood pressure at 12 weeks?
- Target effect
- Intervention-minus-usual-care mean difference in 12-week systolic blood pressure, adjusted for baseline blood pressure as pre-specified.
- H₀
- The adjusted mean difference equals 0 mmHg.
- H₁
- The adjusted mean difference does not equal 0 mmHg. This is non-directional.
- Analysis brief
- Record eligibility, random allocation, intervention, comparator, continuous outcome, 12-week timeframe, model, missing-data plan, assumptions and reporting before outcome inspection.
Intervention-minus-usual-care mean blood pressure.
- Question
- What is the adjusted mean difference in 12-week systolic blood pressure between randomised groups?
- Non-directional H₁
- H₀: mean difference = 0. H₁: mean difference ≠ 0. This is the defensible default when either direction would matter.
- Directional H₁
- H₁: mean difference < 0 is defensible only if prior evidence makes the opposite direction irrelevant to the confirmatory decision and it is fixed before outcomes are inspected.
- Analysis
- Group means, adjusted mean difference and 95% CI from the pre-specified model. Effect size and uncertainty remain primary.
A coefficient is a conditional comparison.
- Point null
- H₀ can restrict the target parameter to one value: a named coefficient equals 0, or its exponentiated ratio equals 1. Other nuisance parameters in the full model remain unknown.
- Composite null
- A null can allow a range of target values. For example, it may include all values at or below a non-inferiority margin. State the parameter space rather than relying on shorthand.
- Interpretation
- The coefficient hypothesis is conditional on the covariate set, functional forms, reference groups, analysis population and model assumptions.
- Confounding
- Covariates belong because of the question, design and causal structure, not because a univariable p-value crossed a threshold.
Neither H₀ nor H₁ is proved. A frequentist test may reject H₀ or fail to reject H₀, conditional on the design, model and assumptions. Failure to reject is not proof of no effect; rejection is not proof of H₁, clinical importance or causation.
3 · Decisions before testing
Sidedness, error rates and power belong in the plan.
Use a two-sided alternative when departures in either direction matter. A one-sided test must be justified and pre-specified; an effect in the opposite direction is still clinically real even if the chosen test does not count it.
Alpha is a pre-specified long-run probability of rejecting H₀ when H₀ is true under repeated use of the procedure. It is not the probability that this rejection is wrong.
Type II error is failing to reject H₀ for a specified alternative effect. Power is 1 minus that error probability and depends on the effect considered important, variability, sample size, design and analysis.
Many outcomes, subgroups, time points or models create many chances for apparently unusual results. Identify the primary question and plan any adjustment or hierarchy before looking.
Non-significant is not equivalent.
A conventional superiority test that fails to reject H₀ does not demonstrate equivalence or non-inferiority. Those designs specify a clinically justified margin, appropriate hypotheses and analysis before the study; the confidence interval is then compared with that margin.
4 · Beginner self-check
Choose the defensible statement.
“What proportion of clinic patients met their blood-pressure target last month?” Must it have H₀ and H₁?
No. It is a descriptive estimation question. Report the pre-defined numerator, denominator, percentage and confidence interval.
“The adjusted OR was 0.64, p = 0.055, so H₀ is true.” Correct?
No. Fail to reject H₀ at a pre-specified 0.05 rule if that rule applies; do not accept or prove H₀. The 95% CI of 0.41 to 1.01 remains compatible with appreciably lower odds through to almost no difference.
After seeing p = 0.03, a student writes “we predicted an association.” What is wrong?
The hypothesis is post hoc. Label the result exploratory and seek validation. Do not present a result-driven statement as pre-specified confirmation.
“H₁: home monitoring improves health.” Is this ready to test?
No. It lacks a population, comparator, outcome definition, timeframe and target effect. “Health” must be replaced by a defined measured outcome.
5 · Thesis and project narrative
Let the same hypothesis travel through the project.
Build the substantive rationale and focused question from prior evidence; state the hypothesis only when appropriate.
Define PECO/PICO, target effect, H₀/H₁, sidedness, alpha where relevant, model, assumptions and pre-specification.
Report the estimate and confidence interval before the p-value. Say reject or fail to reject only if that decision rule matters.
Answer the question with magnitude and uncertainty; consider bias, confounding and clinical meaning without claiming proof.
Go deeper · authoritative medical-research resources
Optional direct sources for question formulation, trial estimands, testing principles and non-inferiority designs.
Determining the scope and questions
Cochrane guidance on structured review questions, outcomes and comparisons.
Open Cochrane Handbook chapter 2ICH E9 statistical principles
International regulatory principles for hypotheses, estimands, Type I/II error, multiplicity and analysis sets.
Open the ICH E9 guidelineStatistics at Square One: study design
A clinical introduction to choosing statistical questions and procedures in relation to study design.
Open the BMJ chapterCONSORT extension for non-inferiority
Reporting guidance that makes the margin, hypotheses and analysis distinct from an ordinary non-significant superiority test.
Open the CONSORT extension