Chi-squared test

A chi-squared test rejects a null hypothesis for a statistic whose null distribution is a chi-squared law, exactly or asymptotically. Pearson tests compare observed and expected counts; both goodness-of-fit and chi-squared tests of independence are instances.

Suppose $N_0$ is independently calibrated and counts $y_i$ have predicted means ${\mu }_i=t_i{N}_0{e}^{-b{n}_i}$ under a specified candidate $b$. For independent sufficiently large Poisson counts, the Pearson chi-squared statistic is $Q={\sum }_i(y_i-{\mu }_i{)}^2/{\mu }_i$. With no parameters fitted to these test counts, its approximate reference chi-squared distribution has one degree of freedom per observation. Fitting a continuous $b$ to the same data instead removes one residual degree of freedom. Replacing the denominators by observed counts is a common estimated-error approximation but can behave badly at small counts.

Calibration uncertainty ${\sigma }_0$ also induces correlated prediction errors. First-order propagation of uncertainty givesA generalized quadratic form uses this covariance matrix. For sparse counts use the Poisson deviance or the full count likelihood, with simulation if a calibrated goodness-of-fit probability is required. Choosing a range after inspecting residuals also affects formal test calibration; a physically specified signal-dominated range is preferable when applying a pure attenuation model.

If two independent approximately Gaussian mean lifetime estimates have known standard deviations $s_1,s_2$, equality of their true lifetimes implies that their difference has mean zero and variance $s_1^2+s_2^2$. Its squared standardized difference therefore has a one-degree-of-freedom chi-squared distribution, giving the statistic in the title. A chosen significance threshold converts its upper-tail P-value into a compatibility decision.

Testing a fitted decay curve against a fixed numerical lifetime is a different question: that test does not include the external measurement uncertainty. Comparing the two experiments requires both uncertainties or a joint likelihood with a common lifetime. Correlated estimates replace the denominator by $s_1^2+s_2^2-2\mathrm{Cov}({\hat{\tau }}_1,{\hat{\tau }}_2)$.

For $n$ independent categorical observations cross-classified into an $r$ by $c$ contingency table, let $O_{ij}$ be observed counts and let $O_{i+},O_{+j}$ be their row and column totals. Under the null hypothesis of independence, fitted expected counts are $E_{ij}=O_{i+}{O}_{+j}/n$, and the Pearson statistic isWith fixed $r,c$ and strictly positive underlying cell probabilities, its null distribution tends to ${\chi }_{(r-1)(c-1)}^2$ as $n\to \infty$. The unrestricted cell-probability model has $rc-1$ dimensions; the independence model has $(r-1)+(c-1)$, giving that difference.

For counts $\left(\begin{array}{cc}30& 20\\ 20& 30\end{array}\right)$, all fitted counts are $25$, so $X^2=4$ with one degree of freedom. Sparse counts or structural zeros require different calibration or an appropriately specified exact procedure; the large-sample approximation is not a guarantee for every table.