Concept Architecture
Concept
Theoretically, Test Statistic is a mathematical quantity calculated from sample data that summarises the evidence against a null hypothesis. It provides the basis for statistical hypothesis testing by measuring the degree of agreement between observed data and the assumptions of the null hypothesis. Different statistical tests employ different test statistics, each with a known sampling distribution under the null hypothesis.
Mathematically, a test statistic is a function of the observed sample and follows a specified probability distribution when the null hypothesis is true. Common examples include the t-statistic, z-statistic, chi-square statistic, and F-statistic. The observed value is compared with a critical value or converted into a p-value to determine whether the observed evidence is sufficiently inconsistent with the null hypothesis.
In practice, test statistics are fundamental to statistical inference in health economics. They are used to compare treatments, evaluate regression coefficients, assess model parameters, analyse cost and effectiveness outcomes, and support evidence generation for health technology assessment. Nearly every inferential statistical procedure relies on the calculation and interpretation of an appropriate test statistic.
Purpose
Used to quantify the evidence against a null hypothesis by comparing observed sample data with its expected sampling distribution, supporting hypothesis testing, confidence interval construction, and statistical inference.
Mathematical Formulae
Primary Formula
General form:
Test Statistic = (Observed Estimate ? Hypothesised Value) � Standard Error
Supporting Formulae
t-statistic:
t = (x? ? ??) � (s/�n)
z-statistic:
z = (x? ? ??) � (�/�n)
Chi-square statistic:
?� = ?[(O? ? E?)� � E?]
F-statistic:
F = MSBetween � MSWithin
Related Mathematical Methods
- Hypothesis Testing
- Student's t-Test
- z-Test
- Chi-Square Test
- F-Test
- Analysis of Variance
- Confidence Interval
- p-Value
Example
A health economist compares mean annual healthcare costs between two treatment groups.
Observed sample mean:
�5,450
Hypothesised mean:
�5,000
Standard error:
�180
The test statistic is:
t = (5,450 ? 5,000) � 180
= 2.50
This value is compared with the appropriate t-distribution to determine the corresponding p-value and assess whether the observed cost difference is statistically significant.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| ABS | =ABS((A2-B2)/C2) | Calculate the magnitude of a test statistic. |
| T.DIST.2T | =T.DIST.2T(ABS(D2),15) | Calculate the two-sided p-value from a t-statistic. |
| NORM.S.DIST | =1-NORM.S.DIST(ABS(D2),TRUE) | Calculate p-values for z-tests. |
| CHISQ.DIST.RT | =CHISQ.DIST.RT(A2,DF) | Calculate right-tail probabilities for chi-square statistics. |
| F.DIST.RT | =F.DIST.RT(A2,DF1,DF2) | Calculate p-values for F-statistics. |
VBA (Optional)
Automate calculation of common test statistics, associated p-values, critical values, and hypothesis test summaries for health economic analyses.
Sources
- Fisher RA. Statistical Methods for Research Workers.
- Neyman J, Pearson ES. On the Problem of the Most Efficient Tests of Statistical Hypotheses. Philosophical Transactions of the Royal Society A. 1933.
- Casella G, Berger RL. Statistical Inference.
- Rice JA. Mathematical Statistics and Data Analysis.
- Agresti A. Foundations of Linear and Generalized Linear Models.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Drummond MF, et al. Methods for the Economic Evaluation of Health Care Programmes.
Related Concepts (2)
Library
Publications
1
Bayesian Methods in Health Economics — Gianluca Baio, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
An overview of Bayesian statistical methods for the analysis of health economic data, covering economic evaluation concepts, statistical cost-effectiveness analysis, Bayesian computation and MCMC, and applied health economic evaluation.
BookView source →
Frequently Asked Questions (6)
What is a test statistic?
A single numerical value calculated from sample data during a hypothesis test, used to judge whether to reject the null hypothesis.
Source: Neyman & Pearson 1933
What does a test statistic condense the data into?
A test statistic condenses the data of a hypothesis test into a single number that measures how far the observed result departs from what the null hypothesis predicts. Its value is compared with the distribution it would follow if the null were true, and that comparison yields the p-value, the probability of a result at least as extreme. Familiar examples include the t statistic and the chi-square statistic. Summarising the evidence against the null in one figure is its role. Kirkwood and Sterne (2003) describe this.
Source: Kirkwood & Sterne 2003
How is a test statistic used?
A test statistic is used by computing it from the data and assessing how extreme it is under the null hypothesis, either by comparing it with a critical value marking the rejection region or by calculating the p-value, the probability of a statistic as extreme as observed under the null. If it is sufficiently extreme, the null is rejected. So a test statistic is used as the basis for the test decision, with its value determining whether the data are improbable enough under the null to reject it, and its known distribution under the null is what allows critical values and p-values to be derived for judging significance.
Source: Neyman & Pearson 1933
What are examples of test statistics?
Examples of test statistics include the t-statistic, comparing a mean or difference with its standard error, referred to the t-distribution; the z-statistic, referred to the normal distribution; the chi-square statistic for categorical data; and the F-statistic for comparing variances, as in analysis of variance. So test statistics take various forms suited to different tests and data, each summarising the data relevant to the hypothesis and having a known distribution under the null, which is why the appropriate test statistic is chosen for the analysis, with its reference distribution providing the critical values and p-values needed to reach a conclusion.
Source: Neyman & Pearson 1933
How does a test statistic relate to the p-value?
A test statistic relates to the p-value in that the p-value is computed from the test statistic as the probability, under the null hypothesis, of obtaining a value as extreme as or more extreme than the one observed. So the test statistic and the p-value are directly connected, with the test statistic summarising the data and the p-value expressing how extreme it is under the null on a probability scale, which is why comparing the test statistic with a critical value is equivalent to comparing the p-value with the significance level, both determining whether the result is statistically significant.
Source: Neyman & Pearson 1933
How is the distribution of a test statistic determined?
The distribution of a test statistic under the null hypothesis is determined by the statistical theory of the test and the assumptions made, so that, if the null is true, the statistic follows a known distribution, such as the t, normal, chi-square, or F distribution, from which critical values and p-values are obtained. So the distribution of a test statistic under the null is what makes the test possible, since knowing how the statistic would behave if the null held allows the observed value to be judged, which is why the assumptions underlying that distribution matter, as violations can invalidate the reference distribution and hence the test.
Source: Neyman & Pearson 1933
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 25 Dec 2025
Content version: 1.0.0
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- Persistent URI
- https://healtheconomics.wiki/concept/test-statistic
- Term code
- HE-ES-SA-216
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