= Chi-squared comparison of a calibrated attenuation model
Suppose $N_0$ is independently calibrated and counts $y_i$ have predicted means $\mu_i=t_iN_0e^{-bn_i}$ 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> gives
$$
\operatorname{Cov}(y_i-\mu_i,y_j-\mu_j)\simeq\delta_{ij}\mu_i+t_it_j e^{-b(n_i+n_j)}\sigma_0^2.
$$
A 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.
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