Chi-Square Distribution

Chi-Square Distribution as the Reference for Likelihood Ratio, Heterogeneity and Fit Tests

Health economists rarely fit a chi-square distribution to data. They meet it as the yardstick against which test statistics are judged, for example the gain in log-likelihood when a survival curve gains a parameter, the spread of trial results in a meta-analysis, or the gap between model predictions and calibration targets. The distribution is built from squared normal deviations and is a special case of the gamma distribution, and its degrees of freedom are counted differently in each of those uses.

Building the distribution from squared normal deviations

The chi-square distribution describes what happens when independent standard normal deviations are squared and added up. Each squared deviation is non-negative, so the total can never fall below zero, and adding more terms moves the distribution to the right and spreads it out. The number of terms is the single parameter, the degrees of freedom, as follows.

$$X = \sum_{i=1}^{k} Z_i^2 \sim \chi^2_k$$

where $X$ is the chi-square variable, $Z_1, \dots, Z_k$ are independent standard normal variables and $k$ is the degrees of freedom. The density for $x > 0$ is given below.

$$f(x; k) = \frac{1}{2^{k/2}\,\Gamma(k/2)}\, x^{k/2 - 1}\, e^{-x/2}$$

where $f(x; k)$ is the probability density at $x$, $\Gamma$ is the gamma function and $k$ is the degrees of freedom. The mean is $E(X) = k$ and the variance is $\mathrm{Var}(X) = 2k$, so a statistic that follows $\chi^2_k$ when the null hypothesis holds has expected value $k$. A value far above $k$ is what produces a small p-value, and every chi-square test on this page reads its p-value from the right-hand tail.

Links to the normal and gamma distributions

With one degree of freedom the chi-square distribution is the distribution of a single squared normal deviate. That is why a two-sided test at the 5% level based on a normal z statistic and a chi-square test on one degree of freedom give the same answer, because the squared normal critical value, $z^2 = 1.96^2 = 3.8416$, matches the 95th percentile of $\chi^2_1$, which is 3.841. A Wald test of one parameter can therefore be reported either as a z statistic or as its square.

The chi-square distribution is also a special case of the gamma distribution, with shape $k/2$ and scale 2. When $k = 2$ the shape is 1 and the gamma reduces to an exponential distribution with mean 2, so the right-tail probability has a closed form.

$$P(\chi^2_2 > x) = e^{-x/2}$$

where $P(\chi^2_2 > x)$ is the probability that a chi-square variable with two degrees of freedom exceeds $x$. The 5% critical value on two degrees of freedom is therefore $-2\ln(0.05) = 5.991$. As $k$ grows, the skewness, $\sqrt{8/k}$, falls towards zero, and because the variable is a sum of $k$ independent terms the central limit theorem makes it close to a normal distribution with the same mean and variance.

Likelihood ratio tests between nested survival models

In health economic modelling, the chi-square distribution is often met through the likelihood ratio test. Two models are nested when one is the other with some parameters fixed, and the test asks whether freeing those parameters improves the fit by more than chance. The statistic is twice the gain in maximised log-likelihood.

$$LR = 2\left(\ell_1 - \ell_0\right)$$

where $LR$ is the likelihood ratio statistic, $\ell_1$ is the maximised log-likelihood of the larger model and $\ell_0$ is that of the restricted model. Under Wilks' theorem, in large samples, under the null model and provided certain regularity conditions hold, $LR$ is compared with a chi-square distribution whose degrees of freedom equal the difference in the number of free parameters.

Among parametric survival models, standard statistical theory treats the exponential as the Weibull with its shape fixed at 1, and the generalised gamma as a three-parameter model that contains the Weibull, exponential and log normal as special cases. The NICE Decision Support Unit's Technical Support Document 14 (TSD 14) describes the same special cases in its sections on each distribution, but it does not recommend likelihood ratio tests for choosing a survival curve. Its section on fit statistics states that the negative 2 log likelihood is only suitable for comparing nested models, that different parametric models using different probability distributions cannot be nested within one another, and that the test is therefore not suitable for assessing the fit of alternative parametric models and has been used erroneously in past NICE technology appraisals. For the statistical comparison of survival curves, TSD 14 points instead to the Akaike information criterion and the Bayesian information criterion, which allow models that are not nested and penalise extra parameters. The nested tests in the worked example below are therefore shown as standard statistics, not as the curve-selection method TSD 14 sets out.

Cochran's Q and heterogeneity in meta-analysis

When trial results are pooled to give a model's treatment effect, the chi-square distribution supplies the test for heterogeneity. Cochran's Q sums each study's weighted squared deviation from the pooled estimate, and its p-value comes from a chi-square distribution with $k - 1$ degrees of freedom, where $k$ here is the number of studies. The formulas for Q, I-squared and the between-study variance are set out on the aggregate data meta-analysis page and are not repeated here.

The chi-square reading has a known weakness. Higgins and colleagues note that the test is poor at detecting true heterogeneity when studies are few, and the Cochrane Handbook warns that a non-significant result must not be taken as evidence of no heterogeneity, which is why a significance level of 0.10 rather than 0.05 is sometimes used. With many studies the test has the opposite problem and detects small amounts of between-study heterogeneity that may not matter clinically.

Goodness of fit when calibrating decision models

Calibration adjusts uncertain model parameters until model outputs agree with observed targets such as incidence or prevalence by age. Each candidate parameter set needs a score for how far its outputs sit from the targets. For counts, Pearson's statistic is one such score.

$$X^2 = \sum_{j=1}^{m} \frac{(O_j - E_j)^2}{E_j}$$

where $X^2$ is the Pearson statistic, $O_j$ is the observed count for target $j$, $E_j$ is the count predicted by the model for that target and $m$ is the number of targets. Each squared gap is scaled by the predicted count, so a gap of 7 cases matters more for a target of 45 than for a target of 450.

In a summary of options for calibration, Enns and colleagues list chi-squared alongside likelihood and log-likelihood as probabilistic measures of fit to individual targets, with absolute or squared difference and mean squared error as distance measures. Karnon and Vanni compared measures in probabilistic calibration and found that the chi-squared measure separated parameter sets by accuracy to a far greater degree than a likelihood measure. When the statistic is read against a chi-square distribution, as in a formal goodness of fit test, the NIST/SEMATECH handbook reduces the degrees of freedom for each parameter estimated from the same counts and asks for expected counts of at least 5 in each bin.

Worked example with three survival curves, one heterogeneity test and one calibration score

The figures below are illustrative. A modelling team fits exponential, Weibull and generalised gamma curves to progression-free survival in one trial arm, obtaining maximised log-likelihoods of -612.8, -608.3 and -607.9 with one, two and three parameters.

1. Weibull against exponential. The models differ by one parameter, so the reference is $\chi^2_1$, as follows.

$$LR_{W,E} = 2 \times (-608.3 - (-612.8)) = 2 \times 4.5 = 9.0$$

where $LR_{W,E}$ is the likelihood ratio statistic for the Weibull against the exponential. The value exceeds the critical value of 3.841, and because $\sqrt{9.0} = 3.0$ the p-value equals the two-sided normal tail beyond 3, about 0.0027. The constant hazard of the exponential is rejected.

2. Generalised gamma against Weibull. One further parameter gives the following.

$$LR_{G,W} = 2 \times (-607.9 - (-608.3)) = 2 \times 0.4 = 0.8$$

where $LR_{G,W}$ is the statistic for the generalised gamma against the Weibull. On one degree of freedom the p-value is about 0.37, so the third parameter adds nothing detectable within the trial data.

3. Generalised gamma against exponential. Two parameters separate these models, so the closed-form tail applies.

$$P(\chi^2_2 > 9.8) = e^{-9.8/2} = e^{-4.9} = 0.0074$$

where 9.8 is twice the log-likelihood gain of 4.9. Taken together, the three tests favour the Weibull among the nested candidates. They say nothing about a log-logistic or log normal alternative compared with the Weibull, and nothing about the plausibility of the extrapolated tail; in an appraisal, TSD 14 would direct the statistical comparison of all candidate curves to AIC and BIC.

4. Heterogeneity of the pooled effect. The illustrative three-trial example on the aggregate data meta-analysis page gives $Q = 6.867$ on 2 degrees of freedom. Using the same closed form, the p-value is $e^{-3.4335} = 0.032$, below both 0.05 and the more lenient 0.10, so the chi-square test agrees with the high I-squared reported there.

5. Scoring a calibrated model. A cancer model with two calibrated parameters predicts 45, 88, 112 and 105 new cases in four age bands where 52, 81, 120 and 97 were observed, as follows.

$$X^2 = \frac{(52-45)^2}{45} + \frac{(81-88)^2}{88} + \frac{(120-112)^2}{112} + \frac{(97-105)^2}{105} = 1.089 + 0.557 + 0.571 + 0.610 = 2.827$$

where each term is one age band's scaled squared gap. A second parameter set predicting 50, 85, 125 and 90 scores 0.080 + 0.188 + 0.200 + 0.544 = 1.012 with the terms rounded (1.013 unrounded) and would be ranked ahead. Read against $\chi^2_4$, which treats the four counts as independent with expectations fixed in advance, the first set's p-value would be about 0.59. Both sets total 350, the same as the observed counts, and both parameters were tuned to these targets, so applying the NIST/SEMATECH rule of subtracting the estimated parameters plus one leaves a single degree of freedom and a p-value of about 0.09. The same score gives very different p-values under two defensible counts of degrees of freedom, which is why it is better used to rank candidate sets than as a formal test.

Computing tail probabilities in Excel and R

Model work needs two operations, a right-tail probability for an observed statistic and a critical value for a chosen level. In Excel, CHISQ.DIST.RT(9, 1) returns about 0.0027, the p-value in step 1, and CHISQ.INV.RT(0.05, 1) returns 3.841. Microsoft's documentation states that a non-integer degrees of freedom argument is truncated, whereas R's pchisq(9, df = 1, lower.tail = FALSE) accepts non-integer degrees of freedom.

Where the chi-square reference breaks down

The chi-square reference for a test statistic is a large-sample approximation that assumes a regular model, and several common health economic situations breach those conditions. The result is a p-value that looks exact but is not.

• Parameters on a boundary. Chen and colleagues explain that when the null model lies on the boundary of the parameter space, as when testing whether a random-effects variance is zero, the regularity conditions of Wilks' theorem fail and the statistic often follows a mixture of chi-square distributions instead. They list random effects and finite mixture models among the cases where the plain approximation often does not hold.
• Models that are not nested. A log-likelihood difference between a Weibull and a log-logistic curve has no chi-square reference.
• Sparse counts. Pearson's approximation needs adequate expected counts, and the NIST/SEMATECH handbook advises that bins with counts below 5 may need to be combined.
• Significance is not adequacy. A non-significant likelihood ratio or heterogeneity test does not show that the simpler model is right, and, as the Cochrane Handbook notes for heterogeneity, a test with high power can flag differences too small to matter. TSD 14 separates fit to observed trial data from the external validity of an extrapolation, which no chi-square test addresses.
• Degrees of freedom after calibration. When parameters are searched over many candidate sets against the same targets, the number of degrees of freedom used up is unclear, so a chi-square p-value for a calibrated model should be treated as descriptive.

Sources

• Chen Y, Moustaki I, Zhang H. A note on likelihood ratio tests for models with latent variables. Psychometrika. 2020;85(4):996-1012.
• Deeks JJ, Higgins JPT, Altman DG, McKenzie JE, Veroniki AA (editors). Chapter 10: Analysing data and undertaking meta-analyses. In: Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. Section 10.10.2.
• Enns EA, Cipriano LE, Simons CT, Kong CY. Identifying best-fitting inputs in health-economic model calibration: a Pareto frontier approach. Medical Decision Making. 2015;35(2):170-182.
• Higgins JPT, Thompson SG, Deeks JJ, Altman DG. Measuring inconsistency in meta-analyses. BMJ. 2003;327(7414):557-560.
• Karnon J, Vanni T. Calibrating models in economic evaluation: a comparison of alternative measures of goodness of fit, parameter search strategies and convergence criteria. PharmacoEconomics. 2011;29(1):51-62.
• Latimer N. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Decision Support Unit; 2011, last updated March 2013. Sections 2.2, 2.6, 3.3 and 3.5.
• Microsoft. CHISQ.DIST.RT function and CHISQ.INV.RT function. Microsoft Support, Excel documentation. Accessed 2026.
• NIST/SEMATECH. e-Handbook of Statistical Methods. Sections 1.3.5.15 (chi-square goodness-of-fit test), 1.3.6.6.6 (chi-square distribution) and 1.3.6.6.11 (gamma distribution). National Institute of Standards and Technology. Accessed 2026.
• R Core Team. The (non-central) chi-squared distribution: Chisquare, package stats. R Documentation. Accessed 2026.
• SciPy developers. scipy.stats.chi2. SciPy v1.18.0 Manual. 2026.