Competing-Risk Model (VICTOR)

The VICTOR algorithm converts population cancer incidence rates and relative risks into a MECE (Mutually Exclusive, Collectively Exhaustive) penetrance table for use with the segregatr FLB calculator.

Background

Population incidence rates from CI5 reflect the observed rates in a population that is a mixture of non-carriers (NC), heterozygous carriers (Het), and homozygous carriers (Hom). To compute penetrance for genetic variant carriers, we must first back-calculate the carrier-free baseline rate, then scale by the relative risk (RR) for each genotype.

Notation

Symbol	Meaning
\(j\)	Phenotype index
\(a\)	Age in years (0–99)
\(g\)	Genotype: nc, het, hom
\(q\)	CRHF — cumulative risk haplotype frequency
\(\lambda_\text{pop}(j,a)\)	Population hazard rate for phenotype \(j\) at age \(a\)
\(RR_g(j,a)\)	Relative risk for genotype \(g\), phenotype \(j\), age \(a\)
\(\lambda_g(j,a)\)	Genotype-specific hazard for phenotype \(j\)
\(\lambda_{g,\text{all}}(a)\)	All-cause hazard for genotype \(g\) at age \(a\)
\(S_g(a)\)	All-cause survival to age \(a\) for genotype \(g\)
\(F_{j,g}(a)\)	Cause-specific CIF for phenotype \(j\), genotype \(g\), by age \(a\)

Algorithm

Step 1: Hardy-Weinberg correction

The population rate is a weighted mixture of NC, Het, and Hom rates. Under Hardy-Weinberg equilibrium with allele frequency \(q\):

\[D(j,a) = 1 + 2q \cdot (RR_\text{het}(j,a) - 1)\]

\[\lambda_\text{nc}(j,a) = \frac{\lambda_\text{pop}(j,a)}{D(j,a)}\]

\[\lambda_\text{het}(j,a) = RR_\text{het}(j,a) \cdot \lambda_\text{nc}(j,a)\]

\[\lambda_\text{hom}(j,a) = RR_\text{hom}(j,a) \cdot \lambda_\text{nc}(j,a)\]

Critical invariant: \(\lambda_\text{het} / \lambda_\text{nc} = RR_\text{het}\) for every \((j,a,\text{sex})\).

Never use: \(\lambda_\text{het} = RR \cdot \lambda_\text{pop} \cdot (1-q)\) — this is algebraically incorrect.

Step 2: All-cause hazard per genotype

Sum over all tracked phenotypes \(j\) (including OtherTrait with \(RR=1\)):

\[\lambda_{g,\text{all}}(a) = \sum_j \lambda_g(j, a)\]

Step 3: All-cause survival

\[S_g(a) = \exp\!\left(-\sum_{i=0}^{a} \lambda_{g,\text{all}}(i)\right)\]

with \(S_g(-1) = 1\) (boundary condition).

Step 4: Cause-specific Cumulative Incidence Function (CIF)

\[F_{j,g}(a) = \sum_{i=0}^{a} S_g(i-1) \cdot \left(1 - e^{-\lambda_g(j,i)}\right)\]

Step 5: Build MECE rows from age bands

For each age band \([a_0, a_1]\) and each sex:

Affected rows (one per tracked phenotype \(j\)):

\[P_\text{affected}(j, g, [a_0, a_1]) = F_{j,g}(a_1) - F_{j,g}(a_0 - 1)\]

This is the band-specific CIF increment — not the cumulative value \(F_{j,g}(a_1)\).

Unaffected row (one per age band):

\[P_\text{unaffected}(g, [a_0, a_1]) = S_g(a_1)\]

This is the all-cause survival probability at the band end — "P(still event-free through age \(a_1\) | genotype \(g\))". The implementation stores \(1 - S_g(a_1)\); segregatr inverts unaffected penetrances internally, recovering \(S_g(a_1)\) as the likelihood contribution for an unaffected member whose last follow-up was at age \(a_1\).

Competing Risks

Phenotypes not tracked by the phenotype model (e.g. non-HBOPC cancers) are aggregated as OtherTrait with \(RR_g = 1\) for all genotypes. This ensures correct all-cause survival computation without inflating carrier risk.

Non-cancer mortality is excluded — consistent with the COOL3/VICTOR reference implementation.