# Risk-Limiting Audits by Stratified Union-Intersection Tests of Elections (SUITE)

## Abstract

Risk-limiting audits (RLAs) offer a statistical guarantee: if a full manual tally of the paper ballots would show that the reported election outcome is wrong, an RLA has a known minimum chance of leading to a full manual tally. RLAs generally rely on random samples. Stratified sampling—partitioning the population of ballots into disjoint strata and sampling independently from the strata—may simplify logistics or increase efficiency compared to simpler sampling designs, but makes risk calculations harder. We present SUITE, a new method for conducting RLAs using stratified samples. SUITE considers all possible partitions of outcome-changing error across strata. For each partition, it combines *P*-values from stratum-level tests into a combined *P*-value; there is no restriction on the tests used in different strata. SUITE maximizes the combined *P*-value over all partitions of outcome-changing error. The audit can stop if that maximum is less than the risk limit. Voting systems in some Colorado counties (comprising 98.2% of voters) allow auditors to check how the system interpreted each ballot, which allows *ballot-level comparison* RLAs. Other counties use *ballot polling*, which is less efficient. Extant approaches to conducting an RLA of a statewide contest would require major changes to Colorado’s procedures and software, or would sacrifice the efficiency of ballot-level comparison. SUITE does not. It divides ballots into two strata: those cast in counties that can conduct ballot-level comparisons, and the rest. Stratum-level *P*-values are found by methods derived here. The resulting audit is substantially more efficient than statewide ballot polling. SUITE is useful in any state with a mix of voting systems or that uses stratified sampling for other reasons. We provide an open-source reference implementation and exemplar calculations in Jupyter notebooks.

## Keywords

Stratified sampling Nonparametric tests Fisher’s combining function Sequential hypothesis tests Colorado risk-limiting audits Maximizing*P*-values over nuisance parameters Union-intersection test Intersection-union test

## Notes

### Acknowledgements

We are grateful to Ronald L. Rivest and Steven N. Evans for helpful conversations and suggestions.

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