Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 29 / 6

The recombination fraction is the probability that a transmitted gamete is recombinant between the two loci in the ordinary two-locus genetic linkage model. Its conventional range isZero corresponds to complete linkage and $1/2$ to independent assortment. Multiple physical crossovers can restore parental phase, so the recombination fraction is not the expected number of crossovers or an unrestricted crossover probability.

The individuals' haplotypes are not identifiable from the supplied source: the pedigree drawings are missing. The original PDF has a blank figure area, and the accompanying TeX contains no family genotypes. Even the two-locus genotypes of individuals 3 and 4 are not stated elsewhere, so numerical allele pairs cannot be supplied honestly.

The method, once the drawings are available, is to use parental or grandparental genotypes to assign the allele inherited at each locus on the same chromosome. A double heterozygote with alleles $A_1,A_2$ and $B_1,B_2$ has the two candidate haplotype pairs $A_1{B}_1/A_2{B}_2$ and $A_1{B}_2/A_2{B}_1$; ancestor and offspring transmissions determine or restrict its phase. These symbols describe the general phasing problem, not invented genotypes for the missing individuals. If pedigree transmissions leave more than one phase possible, retain all compatible phases rather than selecting one without evidence.

For known parental haplotypes, count $r_1$ recombinant transmissions among $m_1$ informative meioses. The conditional pedigree likelihood has the formwhere $C_1$ contains parameter-independent gamete and genotype factors. Differentiating its logarithm gives $r_1/\theta -(m_1-r_1)/(1-\theta )$, so the constrained maximum-likelihood estimate isWhen $r_1=0$ the maximum is at zero; if no meiosis is informative, the likelihood is constant and the parameter is not estimable from that family. This formula presumes the family phase has been resolved; otherwise use phase averaging in a linkage likelihood first.

The family-specific value cannot be calculated because its pedigree, phases and transmission counts are absent from the supplied PDF. The missing data cannot be reconstructed from the requested estimate alone.

Let $s$ index compatible parental phases in family 2, with fixed prior conditional weights ${\pi }_s$. For a given phase, let $r_s$ be the recombinant count and $m_s$ the informative-transmission count, absorbing any fixed genotype factors into $C_s$. The appropriate pedigree likelihood isThis is phase averaging in a linkage likelihood; maximizing over a phase after observing the offspring would not be the same likelihood. In the common special case of two equally likely phases that exchange recombinant and nonrecombinant labels across $m_2$ transmissions, it reduces, up to a constant, toThe constrained maximum-likelihood estimate compares all stationary points and endpoints on $[0,1/2]$.

The actual phase set, weights, counts and numerical maximum for family 2 cannot be determined without its missing pedigree. The special two-phase expression is a conditional example, not an assertion about this unidentified family.

For independent families, multiply their pedigree likelihoods. If family 1 has the known-phase counts above, the combined log-likelihood derivative at an interior point isThus the score equation for a phase-averaged linkage likelihood isFor the general phase sum, let $w_s(\theta )={\pi }_s{C}_s{\theta }^{r_s}(1-\theta {)}^{m_s-r_s}/L_2(\theta )$. Since the weights sum to one, the equivalent interior equation isThe finite likelihood sum is continuous on the compact parameter interval $[0,1/2]$, so a constrained maximum exists. This does not by itself prove an interior score root: complete nonrecombinant data, for example, maximize at zero. To show the requested root lies in the appropriate range for the two particular families, one must insert their transmission counts and evaluate the score or the resulting polynomial at the relevant endpoints.

The source lacks those pedigree counts, so the data-specific equation and its asserted interior-root verification remain undetermined. No universal interior-root claim is substituted for the missing calculation.

The LOD score is the base-ten log likelihood ratio against independent assortment. For the two families,For a known-phase first family this becomeswith endpoint terms interpreted by limits, such as $0\log 0=0$ when the corresponding count is zero. Positive values favour genetic linkage, and the likelihood ratio is $10^{Z_{\max }}$. A conventional large positive threshold such as 3 represents a likelihood ratio of 1000, not automatically a posterior probability or a universal 5% test.

For a test calibrated to this study, use the distribution of the maximized statistic under $\theta =1/2$, for example by simulating transmissions conditional on the same parental genotype and ascertainment scheme. Reject for sufficiently large values with the chosen significance threshold; phase uncertainty and the boundary null prevent assuming an unqualified ordinary interior Wilks reference. The missing pedigrees prevent the numerical value or their explicit likelihood polynomial from being supplied, but the likelihood-ratio definition and testing procedure apply once those data are recovered.