Concept Architecture
Concept
Theoretically, the Null Hypothesis is a formal statistical proposition stating that a specified effect, association, difference or model parameter is absent, equal to a reference value or otherwise constrained according to the research question. It provides the benchmark against which observed data are evaluated in frequentist hypothesis testing. The null hypothesis exists to establish a precise probabilistic statement that can be subjected to statistical evidence and either rejected or not rejected.
Mathematically, the null hypothesis is represented as a restriction on one or more population parameters, commonly expressed as equality to a specified value. A test statistic is calculated from the sample data and its distribution under the null hypothesis is used to determine the probability of observing a result at least as extreme as the one obtained. Decisions are based on a predefined significance level, although failure to reject the null hypothesis does not establish that it is true.
In practice, the null hypothesis is specified before statistical testing and evaluated using methods such as t-tests, Wald tests, likelihood ratio tests, chi-squared tests and permutation tests. In health economics, it is used to test hypotheses concerning differences in mean costs, health outcomes, treatment effects, regression coefficients and resource utilisation. Its interpretation should be accompanied by effect estimates, confidence intervals and consideration of clinical or economic relevance.
Purpose
Used to define the reference proposition in statistical hypothesis testing, evaluate evidence for treatment effects or associations and support formal statistical inference in health economic research.
Mathematical Formulae
Primary Formula
For a single parameter:
H?: ? = ??
Against a two-sided alternative:
H?: ? ? ??
where:
- H? = null hypothesis
- H? = alternative hypothesis
- ? = population parameter
- ?? = specified reference value
Supporting Formulae
Generic standardised test statistic:
Z = (?? ? ??) / SE(??)
Decision rule:
Reject H? if p � �
Type I error probability:
P(Reject H? | H? true) = �
Type II error probability:
P(Do not reject H? | H? true) = ?
Statistical power:
Power = 1 ? ?
Related Mathematical Methods
- Alternative Hypothesis
- Significance Testing
- p-Value
- Confidence Interval
- Type I Error
- Type II Error
- Statistical Power
- Wald Test
- Likelihood Ratio Test
Example
A health economist compares mean annual healthcare costs between a new intervention and standard care.
The hypotheses are:
H?: ?? ? ?? = 0
H?: ?? ? ?? ? 0
The estimated mean cost difference is:
�420
The standard error is:
�180
The test statistic is:
Z = 420 / 180
Z = 2.33
The corresponding two-sided p-value is approximately 0.020.
At a significance level of � = 0.05, the null hypothesis is rejected. The analysis provides evidence that mean annual healthcare costs differ between the intervention and standard-care groups, although the economic importance of the �420 difference requires separate interpretation.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| T.TEST | =T.TEST(B2:B101,C2:C101,2,3) | Test the null hypothesis of equal mean costs between two independent groups. |
| Z.TEST | =Z.TEST(B2:B101,0) | Test whether a sample mean differs from a specified reference value when the relevant assumptions are met. |
| CHISQ.TEST | =CHISQ.TEST(B2:C3,E2:F3) | Test independence between treatment group and a categorical health outcome. |
| NORM.S.DIST | =2*(1-NORM.S.DIST(ABS(D2),TRUE)) | Calculate a two-sided p-value from a standard normal test statistic. |
| IF | =IF(E2<=0.05,"Reject H0","Do not reject H0") | Apply a predefined significance threshold to a hypothesis test. |
VBA (Optional)
A VBA routine can automate hypothesis testing across multiple health economic outcomes and report test statistics, p-values, confidence intervals and decision rules.
Sources
- Neyman J, Pearson ES. On the Problem of the Most Efficient Tests of Statistical Hypotheses. Philosophical Transactions of the Royal Society of London. Series A. 1933;231:289?337.
- Fisher RA. Statistical Methods for Research Workers. Oliver and Boyd.
- Casella G, Berger RL. Statistical Inference. Cengage Learning.
- Lehmann EL, Romano JP. Testing Statistical Hypotheses. Springer.
- Altman DG. Practical Statistics for Medical Research. Chapman & Hall.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is the null hypothesis?
In hypothesis testing, the default assumption of no true effect or difference between the groups or conditions being compared.
Source: Neyman & Pearson 1933
What does the null hypothesis assume before a test?
The null hypothesis is the starting assumption in a statistical test that there is no real effect or difference between the groups being compared. The test then asks how compatible the observed data are with this assumption, and if the data would be very unlikely under it, the null is rejected in favour of an alternative. It can only ever be rejected, never proven, because failing to find evidence against it does not establish that it is true. Presuming no effect until the data say otherwise is its role. Kirkwood and Sterne (2003) describe this.
Source: Kirkwood & Sterne 2003
What is the role of the null hypothesis?
The role of the null hypothesis is to provide a specific, testable reference of no effect against which the data are evaluated: the test computes how compatible the data are with the null, and if they are sufficiently incompatible, the null is rejected in favour of the alternative. So the null hypothesis serves as the baseline that a statistical test seeks to refute, framing the question so that evidence can be quantified, since tests assess the probability of the data under the null and reject it when that probability is low, which structures the inference around disproving the assumption of no effect rather than directly proving an effect.
Source: Neyman & Pearson 1933
How is the null hypothesis tested?
The null hypothesis is tested by computing a test statistic from the data and determining how extreme it would be if the null were true, either by comparing it with a critical value or by calculating a p-value, and rejecting the null if the result is sufficiently unlikely under it, at a chosen significance level. So the null hypothesis is tested by assessing the compatibility of the data with it and rejecting it when the data are sufficiently improbable under the null, with the significance level controlling the risk of wrongly rejecting a true null, which yields a decision about whether the evidence supports the alternative over the null.
Source: Neyman & Pearson 1933
Why can the null hypothesis only be rejected, not proven?
The null hypothesis can be rejected but not proven because a hypothesis test assesses whether the data are incompatible with the null, and failing to reject it means only that there is insufficient evidence against it, not that it is true, since the absence of evidence is not evidence of absence. So a non-significant result does not prove the null hypothesis, as the effect may be real but undetected, for example due to low power, which is why one speaks of failing to reject rather than accepting the null, reflecting that a test can provide evidence against the null but cannot establish that no effect exists.
Source: Neyman & Pearson 1933
How does the null hypothesis relate to the alternative hypothesis?
The null hypothesis relates to the alternative hypothesis as the two competing propositions in a test: the null states no effect, and the alternative states that an effect exists, with the test weighing the evidence between them and rejecting the null taken as support for the alternative. So the null and alternative hypotheses are complementary, framing the question a test addresses, with the null the default of no effect and the alternative the claim of an effect, and because the test can only reject or fail to reject the null, support for the alternative comes indirectly through evidence against the null rather than by proving the alternative directly.
Source: Neyman & Pearson 1933
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 19 Dec 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/null-hypothesis
- Term code
- HE-ES-SA-140
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