Statistical Rarity

With 0 observed quintuples in 18,625 myeloid patients, the Jeffreys non-informative prior Beta(0.5, 0.5) yields a posterior Beta(0.5, 18,625.5). The posterior median (1.22e-5) represents the best point estimate of the true population frequency. The 95% credible interval spans 2.64e-8 to 1.35e-4, providing bounds on the plausible frequency range.

The Rule of Three (Hanley and Lippman-Hand, 1983) provides a quick frequentist upper bound: with 0 events in n trials, the 95% upper confidence limit is approximately 3/n. For 18,625 patients, this gives an upper bound of 1.61e-4. The Clopper-Pearson exact interval (1.61e-4) and Poisson exact 99% bound (2.47e-4) provide alternative frequentist estimates, all consistent with the Bayesian credible interval.

The five genes (DNMT3A, IDH2, SETBP1, PTPN11, EZH2) are individually common in myeloid malignancies. At the gene level, the expected co-occurrence count is 1.94e-2, placing this combination at the 77.8th percentile among all 278,256 possible five-gene combinations from 34 myeloid genes. The rarity is driven entirely by variant specificity: the specific amino acid changes are 7.1 orders of magnitude (1.3e+7x) rarer than the gene-level combination. EZH2 V662A (0 carriers in GENIE) is the primary driver of variant-level rarity.

Zero quintuple matches across two independent, non-overlapping databases totaling 26,642 myeloid patients. GENIE v19.0 provides 18,625 patients with panel-based sequencing from 40+ academic medical centers worldwide. HARMONY Alliance contributes 12,041 AML patients from European institutions with a 41-gene panel (Eurofins-Biomnis). All five target genes are covered by both panels.

Gene-level frequencies in HARMONY (AML-only): DNMT3A 25.2%, IDH2 12.7%, PTPN11 7.5%, EZH2 6%, SETBP1 2%. Under independence, the expected number of quintuples in HARMONY alone is 0.035.

Using the Jeffreys posterior, the probability of observing at least one quintuple match in a future cohort of M patients is computed via the posterior predictive distribution. 55,875 patients are needed for a 50% chance of finding one match. 7,431,474 patients are needed for 95% confidence. These numbers far exceed current database sizes, confirming that the zero-match observation is consistent with expectations and not a sampling artifact.

All 278,256 possible five-gene combinations from the 34 target myeloid genes were enumerated and their expected counts computed under independence. A Generalized Extreme Value (GEV) distribution was fitted to the log-transformed expected counts. The patient quintuple ranks 216,373 out of 278,256 at the gene level (77.8th percentile), confirming that these five genes are individually common enough in myeloid malignancies that their gene-level co-occurrence is unremarkable. The rarity originates from the specific variant combination, not the gene combination.

The GEV fit confirms a heavy-tailed distribution (positive shape parameter 0.23), meaning a small number of gene combinations have disproportionately high expected counts (e.g., DNMT3A+TET2+ASXL1 combinations) while the vast majority are rare. The patient quintuple sits in the body of this distribution at the gene level, but falls in the extreme tail at the variant level.