Signature
P_2 = exp(-x / 2)
| Inputs | Definition | Unit |
|---|---|---|
x | Observed test statistic referred to chi-square on two degrees of freedom, zero or above | none |
P_2 | Probability that a chi-square variable with two degrees of freedom exceeds x, the p-value for an observed statistic x | probability |
|---|
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 p-value in one cell
With the statistic in a cell named Stat, the formula returns the right-tail probability.
=EXP(-Stat/2)
Assumptions
Two degrees of freedom for the closed-form chi-square tail
The closed form holds only for k = 2, for example two extra parameters in a likelihood ratio test or Cochran's Q from three studies. Other degrees of freedom need CHISQ.DIST.RT or the computational function on this page.
Worked examples
Generalised gamma against exponential with a statistic of 9.8
Twice the log-likelihood gain of 4.9 gives 9.8 on two degrees of freedom and a p-value of about 0.0074, the article's step 3.
x = 9.8; P_2 = 0.0074
Cochran's Q of 6.867 from three trials
The three-trial Q from the Aggregate Data Meta-Analysis page gives a p-value of about 0.032, below both 0.05 and the more lenient 0.10, the article's step 4.
x = 6.867; P_2 = 0.0323
Statistic at the two-degree-of-freedom critical value
A statistic equal to the 5% critical value of about 5.9915 returns a p-value of 0.05.
x = 5.9915; P_2 = 0.05
Common errors
Two-degree-of-freedom closed form used for another chi-square reference
Applied to the Weibull against exponential statistic of 9.0, which has one degree of freedom, the closed form gives about 0.0111 instead of 0.0027, overstating the p-value about fourfold.
Sources
Chi-square density with two degrees of freedom in the R documentation
R Core Team. The (non-central) chi-squared distribution: Chisquare, package stats. R Documentation; accessed 2 October 2026. Details: the density formula, which for n = 2 is one half of e^(minus x/2), the exponential density with mean 2 whose right tail is e^(minus x/2); the documentation also gives P[X > x] through pchisq with lower.tail = FALSE.
Excel CHISQ.DIST.RT right-tailed chi-square probability
Microsoft. CHISQ.DIST.RT function. Microsoft Support; accessed 2 October 2026. Returns the right-tailed probability of the chi-squared distribution; a degrees of freedom argument that is not a whole number is truncated.
Canonical Identity
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