Chi-squared comparison of a calibrated attenuation model

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.