Theory¶

Population Resemblance Statistic¶

Let

\[ p_0=(p_{01},\ldots,p_{0B})^\top \]

be a fixed reference distribution with \(p_{0j}>0\), and let

\[ \hat p=(\hat p_1,\ldots,\hat p_B)^\top \]

be the empirical current distribution.

The Population Resemblance Statistic is

\[ \mathrm{PRS} = \sum_{j=1}^{B} \frac{(\hat p_j-p_{0j})^2}{p_{0j}}. \]

The sample-size-scaled statistic is

\[ Q_n=n\mathrm{PRS}. \]

Under local alternatives, \(Q_n\) is asymptotically non-central chi-square with \(B-1\) degrees of freedom.

\(\delta\)-resemblance¶

The framework defines a current population \(p\) as \(\delta\)-resemblant to \(p_0\) when

\[ \max_j |p_j-p_{0j}| \le \delta. \]

This changes the monitoring question from exact equality to whether the population shift exceeds a pre-specified tolerable amount.

The recommended automatic tolerance is

\[ \delta = c\min_j \sqrt{ \frac{p_{0j}(1-p_{0j})}{n} }, \]

where \(c>0\) scales the acceptable shift relative to sampling variability.

Least-favourable non-centrality¶

The non-centrality parameter is

\[ \lambda = n\sum_{j=1}^{B} \frac{(p_j-p_{0j})^2}{p_{0j}}. \]

Because the true current probabilities are unknown, the framework uses the maximal non-centrality over the \(\delta\)-resemblance region.

For even \(B\),

\[ \lambda_{\sup} = n\delta^2\sum_{j=1}^{B}p_{0j}^{-1}. \]

For odd \(B\),

\[ \lambda_{\sup} = n\delta^2 \left( \sum_{j=1}^{B}p_{0j}^{-1} - p_*^{-1} \right), \]

where \(p_*=\max_j p_{0j}\).

Three decision regions¶

The package implements two nested resemblance hypotheses with tolerances \(\delta\) and \(M\delta\), where \(M>1\).

The PRS decision regions are

\[ R_1=\{\mathrm{PRS}\le \tau_1\}, \]

\[ R_2=\{\tau_1<\mathrm{PRS}\le\tau_2\}, \]

and

\[ R_3=\{\mathrm{PRS}>\tau_2\}. \]

The critical values are

\[ \tau_1 = \frac{ F^{-1}_{B-1}(\alpha_2,M^2\lambda_{\sup}) }{n}, \]

\[ \tau_2 = \frac{ F^{-1}_{B-1}(1-\alpha_1,\lambda_{\sup}) }{n}, \]

where \(F^{-1}_{\nu}(\cdot,\lambda)\) is the non-central chi-square quantile function.

The package labels the regions as:

Region	Package label	Interpretation
\(R_1\)	`acceptable`	Continue using the model or process
\(R_2\)	`partially discrepant`	Enhanced monitoring
\(R_3\)	`fully discrepant`	Material discrepancy requiring action

Structural constraint¶

The widened tolerance must satisfy

\[ M\delta \le \min_j p_{0j}. \]

This prevents the resemblance region from implying invalid negative probabilities.

Structural sample-size planning¶

For the recommended tolerance,

\[ \delta = c\min_j\sqrt{\frac{p_{0j}(1-p_{0j})}{n}}, \]

the structural requirement

\[ M\delta \le \min_j p_{0j} \]

can be solved directly for the smallest positive integer sample size.

Use minimum_structural_sample_size() to compute that bound.

What this bound means

The returned value guarantees only that the recommended tolerance satisfies the probability-domain constraint. It is not a power calculation and does not guarantee good finite-sample calibration, sufficient expected cell counts, or asymptotic accuracy.

Calibration¶

Use evaluate_calibration() to inspect the derived tolerance, non-centrality, critical values, and margin to the structural bound before running an assessment.

Use sweep_calibration_parameters() to examine sensitivity across combinations of \(c\), \(M\), \(\alpha_1\), and \(\alpha_2\).

Statistical scope¶

The implementation follows the fixed-reference, one-sample formulation. An empirical reference sample can be converted into a fixed probability vector conditionally, but that does not make the procedure a two-sample test.

This distinction is deliberate. Reference-sample uncertainty requires a different statistical formulation.