Signature
x_crit = -2 * log(alpha)
| Inputs | Definition | Unit |
|---|---|---|
alpha | Right-tail significance level, for example 0.05 or 0.10 | probability |
x_crit | Value that a chi-square variable with two degrees of freedom exceeds with probability alpha | none |
|---|
Function
Chi-square distribution of a sum of squared standard normal deviates
Maps the degrees of freedom k to the distribution of the sum of k independent squared standard normal variables. Health economic models meet it as the reference for test statistics, each read from the right-hand tail: a likelihood ratio between nested survival models, Cochran's Q for heterogeneity in a meta-analysis and Pearson's statistic for calibration targets. It is the gamma distribution with shape k/2 and scale 2. The formulae for Cochran's Q, I-squared and the between-study variance are on the Aggregate Data Meta-Analysis page (HE-FM-ADMA-002 to HE-FM-ADMA-004). AIC and BIC, which TSD 14 prefers for choosing between survival curves, are on the Akaike Information Criterion page (HE-FM-AIC-001 and HE-FM-AIC-004) and the Bayesian Information Criterion page (HE-FM-BIC-001).
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Implementations
Excel
Two-degree-of-freedom chi-square critical value in one cell
With the level in a cell named Alpha, the formula returns the critical value. LN is the natural logarithm.
=-2*LN(Alpha)
Assumptions
Right-tail level on two degrees of freedom for the closed-form critical value
The level applies to the right-hand tail, as in every chi-square test on this page, and the reference has exactly two degrees of freedom.
Worked examples
Two-degree-of-freedom critical value at the 5% level
Minus 2 times the natural logarithm of 0.05 is about 5.991, the critical value in the article.
alpha = 0.05; x_crit = 5.9915
Two-degree-of-freedom critical value at the 10% level
At the more lenient level sometimes used for heterogeneity tests the critical value is about 4.605.
alpha = 0.10; x_crit = 4.6052
Two-degree-of-freedom critical value at the 1% level
At the 1% level the critical value is about 9.210, so the statistic of 9.8 for the generalised gamma against the exponential exceeds it.
alpha = 0.01; x_crit = 9.2103
Common errors
Base-10 logarithm in the two-degree-of-freedom critical value
Excel LOG uses base 10 by default and gives about 2.602 at the 5% level instead of 5.991, so a statistic of 4 would be declared significant when its p-value is about 0.135.
Sources
Excel CHISQ.INV.RT inverse right-tailed chi-square probability
Microsoft. CHISQ.INV.RT function. Microsoft Support; accessed 2 October 2026. Returns the inverse of the right-tailed probability of the chi-squared distribution, the critical value for a chosen probability.
Canonical Identity
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