Parametric Linkage Models and the Recombination Fraction

Why Model Linkage and Phenotype Together?

A family can show a pattern of affected and unaffected relatives because a disease-associated allele is inherited through generations. A genetic marker provides another inherited signal: if it lies near the disease locus, the marker allele carried by a relative can tend to travel with the disease-associated allele. Linkage analysis describes this relationship, while a phenotype model describes how often each disease-locus genotype produces the observed disease status.

These are related but distinct parts of a statistical model. The recombination fraction describes the relationship between the disease locus and marker during inheritance. Penetrance describes the relationship between disease-locus genotype and phenotype. A marker can be linked to a disease locus even if penetrance is incomplete, and a phenotype can be modeled without any marker information.

The Recombination Fraction

To define a recombinant transmission, it is necessary to know which disease-locus and marker alleles are on each of a parent's two homologous chromosomes. This arrangement is called the parental phase. A transmitted pair is recombinant when it combines alleles from different parental homologs, rather than preserving one of the parental allele pairings. Thus, whether a transmission is called recombinant depends on the parental arrangement, not just on the allele names.

In the standard two-point model, \(\theta\) is restricted to the interval from 0 to 0.5. At \(\theta=0\), transmissions preserve the parental allele pairings in the model; as \(\theta\) increases, recombinant transmissions become more frequent. At \(\theta=0.5\), the disease-locus and marker transmissions show no two-point linkage. This is the conventional null value for a test of linkage:

The usual linkage alternative is \(\theta<0.5\). A value above 0.5 does not represent stronger linkage in this parameterization; interchanging the phase labels leads to the corresponding value below 0.5. The value 0.5 is also not proof that the loci are physically far apart. Multiple crossovers between loci can make the observed two-point recombination fraction approach 0.5 even when the loci are on the same chromosome. The parameter describes the inheritance relationship captured by the model, not a direct measurement of physical distance.

A useful justification for the null value is to consider the meaning of no linkage in the two-point model. When inheritance at the loci is independent, a marker allele carries no information about which disease-locus homolog was transmitted. Relative to a fixed parental phase, recombinant and nonrecombinant classifications are then equally likely. Their probabilities sum to one, so each is one-half and \(\theta=0.5\). This is why the no-linkage hypothesis is represented by that boundary value rather than by \(\theta=0\).

Phenotype Penetrance

Linkage concerns which alleles are transmitted; it does not by itself determine whether a person is affected. That connection is represented by penetrance. Let \(G_i\) denote individual \(i\)'s disease-locus genotype and let \(Y_i\) denote their recorded phenotype. For a binary phenotype, \(Y_i=1\) can denote affected and \(Y_i=0\) unaffected. The penetrance for genotype \(g\) is the conditional probability that an individual with that genotype is affected.

Penetrance is a probability, not a deterministic rule. A penetrance of 0.80 for genotype \(g\) means that affected status occurs with probability 0.80 among individuals with that genotype under the model; it does not mean that exactly 80% of every small group with genotype \(g\) will be affected. Different genotypes may have different penetrances. Incomplete penetrance allows an individual with a disease-associated genotype to be unaffected, while a nonzero penetrance for other genotypes allows affected individuals who do not carry that genotype.

A parametric linkage model therefore requires a phenotype specification as well as \(\theta\). A simple model might use genotype-specific penetrances, while a more detailed model could account for features such as age. The penetrance assumptions must suit the disease and the recorded phenotype. Otherwise, a mismatch between predicted and observed disease status could be incorrectly attributed to the transmission relationship between the marker and disease locus.

Putting the Model Components in Context

A pedigree linkage model brings together several kinds of information. Disease-locus genotypes connect inheritance to phenotype through penetrance. Marker genotypes provide evidence about inheritance at the marker locus. The parameter \(\theta\) describes how often the transmitted disease-locus and marker-locus alleles are recombinant, relative to phase. These components answer different questions, so their parameters should not be interchanged.

In a basic model, the penetrance probabilities are understood to apply to the population and phenotype definition being analyzed. The analysis also needs a defensible treatment of phase; if phase is uncertain, it must be represented as uncertain rather than silently treated as known. A model using one \(\theta\) assumes that a common recombination fraction is appropriate for the transmissions being analyzed. If sex-specific recombination is relevant, that simplification may need revision. Accurate interpretation also depends on appropriate handling of uncertain genotypes and missing observations.

Assumptions and Interpretation

A parametric model is useful because it makes its biological assumptions explicit. Before interpreting linkage evidence, check that the disease model defines the relevant genotypes and penetrances, that the marker information and phase are handled appropriately, and that the chosen recombination parameterization fits the transmissions under study. If any of these specifications are wrong, the model's predictions can be unreliable even when the arithmetic is correct.

The no-linkage hypothesis concerns the relationship between the two loci, not whether the disease is genetic, whether a major-gene model is correct, or whether penetrance is complete. Conversely, observing that a marker allele and disease status tend to co-occur in a small family does not by itself establish linkage. The pattern must be evaluated against the inheritance and phenotype assumptions of the model.

Common Mistakes

• Confusing \(\theta\) with disease risk. A recombination fraction of 0.08 is not an 8% probability of disease. It is the modeled probability of a recombinant transmission between the two loci.
• Treating penetrance as certainty. A penetrance of 0.80 does not require every person with that genotype to be affected. It assigns an 80% conditional probability under the model.
• Using \(\theta=0\) as the no-linkage null. Zero represents no recombinant transmissions under the model. No two-point linkage is represented by \(\theta=0.5\).
• Interpreting \(\theta=0.5\) as proof that loci are on different chromosomes. It indicates no detectable two-point linkage under this parameterization; multiple crossovers can also produce a recombination fraction near one-half.
• Equating marker association with causation. A marker can track inheritance near a disease locus without causing the disease. The phenotype model concerns the disease genotype, not a causal effect of the marker.