Concept Architecture
Concept
Theoretically, a Critical Value is a threshold value from a probability distribution that defines the boundary between the acceptance and rejection regions of a statistical hypothesis test. It is derived from statistical decision theory and frequentist inference, providing the criterion for determining whether an observed test statistic is sufficiently extreme to reject the null hypothesis. In health economics, critical values are used extensively in hypothesis testing, confidence interval estimation, regression analysis and clinical trial evaluation.
Mathematically, a critical value is determined from the sampling distribution of a test statistic for a specified significance level (�) and, where applicable, the appropriate degrees of freedom. Depending on the statistical procedure, critical values are obtained from distributions such as the standard normal, Student's t, chi-square or F distribution. They define the probability threshold beyond which observed results are considered statistically inconsistent with the null hypothesis.
In practice, critical values are calculated before or during statistical analysis using statistical software or probability tables. They are compared with calculated test statistics to determine statistical significance and are also used to construct confidence intervals and establish decision thresholds in health economic evaluations, epidemiological studies and health technology assessments.
Purpose
Used to determine statistical significance, define rejection regions for hypothesis tests, construct confidence intervals and support statistical decision-making in health economic analyses.
Mathematical Formulae
Primary Formula
Reject H? if
|T| > Critical Value
where T is the calculated test statistic.
Supporting Formulae
For the standard normal distribution:
Critical Value = z??�??
For Student's t-distribution:
Critical Value = t??�??,df
For the chi-square distribution:
Critical Value = ?�??�,df
For the F-distribution:
Critical Value = F??�,df?,df?
Related Mathematical Methods
Hypothesis Testing
Confidence Interval Estimation
Student's t-Test
Chi-Square Test
Analysis of Variance
Standard Normal Distribution
Statistical Decision Theory
Example
A health economist compares the mean annual healthcare costs of two treatment groups using a two-sided t-test with a significance level of 0.05.
Degrees of freedom = 98
Critical value:
t?.???,?? = 1.984
If the calculated test statistic equals 2.37, then:
2.37 > 1.984
The null hypothesis is rejected, indicating a statistically significant difference between the treatment groups at the 5% significance level.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| NORM.S.INV | =NORM.S.INV(0.975) | Calculate standard normal critical value |
| T.INV.2T | =T.INV.2T(0.05,98) | Calculate two-sided t critical value |
| CHISQ.INV.RT | =CHISQ.INV.RT(0.05,10) | Calculate chi-square critical value |
| F.INV.RT | =F.INV.RT(0.05,5,120) | Calculate F-distribution critical value |
VBA (Optional)
Automate calculation of critical values for multiple statistical tests and generate hypothesis testing summaries for health economic analyses.
Sources
Casella G, Berger RL. Statistical Inference.
Rice JA. Mathematical Statistics and Data Analysis.
Altman DG. Practical Statistics for Medical Research.
Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes.
Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
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 critical value?
The specific threshold of a test statistic separating the region where a null hypothesis is rejected from where it is not.
Source: Neyman & Pearson 1933
What does a critical value mark in hypothesis testing?
A critical value is the threshold a test statistic must cross for the null hypothesis to be rejected, dividing the values that count as too surprising from those that do not. It is set by the chosen significance level, so that the chance of a test statistic falling beyond it, when the null hypothesis is true, equals that level. Comparing the observed statistic against this fixed boundary gives the test its decision. Marking the boundary of rejection is what it does. Kirkwood and Sterne (2003) describe this.
Source: Kirkwood & Sterne 2003
How is a critical value determined?
A critical value is determined by the chosen significance level, the alpha, and the distribution of the test statistic under the null hypothesis, together with whether the test is one-sided or two-sided. It is the value of the statistic that leaves a tail probability equal to the significance level, so that the chance of the statistic exceeding it under the null equals alpha. So a critical value is determined from the null distribution and the significance level, being the threshold beyond which results are considered too improbable under the null to be attributed to chance, and it depends on the test, the alpha, and the number of tails considered.
Source: Neyman & Pearson 1933
How is a critical value used in hypothesis testing?
A critical value is used in hypothesis testing by comparing it with the test statistic calculated from the data: if the test statistic is more extreme than the critical value, falling in the rejection region, the null hypothesis is rejected; otherwise it is not. This provides a decision rule for the test. So a critical value is used as the threshold in the decision rule of a hypothesis test, determining whether the observed result is extreme enough to reject the null, and comparing the test statistic with the critical value is an equivalent alternative to comparing the p-value with the significance level.
Source: Neyman & Pearson 1933
How does a critical value relate to the significance level?
A critical value relates to the significance level in that it is derived from it: the significance level, alpha, sets the probability of rejecting the null when it is true, and the critical value is the threshold that leaves exactly that tail probability under the null distribution. A smaller alpha gives a more extreme critical value. So the critical value and the significance level are directly linked, with the critical value being the point on the test statistic's scale corresponding to the chosen alpha, which means selecting a significance level fixes the critical value, and the two express the same decision boundary in different terms.
Source: Neyman & Pearson 1933
How does a critical value relate to the p-value?
A critical value and the p-value are two equivalent ways of reaching the same test decision: the critical value approach compares the test statistic with a threshold set by the significance level, while the p-value approach computes the probability of a result as extreme as observed under the null and compares it with the significance level. Rejecting because the statistic exceeds the critical value is equivalent to rejecting because the p-value is below alpha. So a critical value relates to the p-value as an alternative formulation of the same test, with both leading to the identical conclusion, one working on the scale of the test statistic and the other on the probability scale.
Source: Neyman & Pearson 1933
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 12 Dec 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/critical-value
- Term code
- HE-ES-SA-041
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