2. Transmission Parameters

Module paptcms assumes Mendelian segregation. Alternatively, module paptctp allows you to estimate the allele transmission probabilities in order to test for major locus inheritance. This test compares the likelihood of the general model with estimated allele transmission probabilities to the likelihood of its submodel, Mendelian segregation; a similar comparison to the likelihood of another submodel, environmental nontransmission, tests an alternative hypothesis. Rejecting environmental nontransmission while failing to reject Mendelian segregation supports major locus inheritance.

Module paptctp defines τ₁, τ₂, τ₃, the allele transmission probabilities, as the probability that a parent of genotype 1, 2, or 3, respectively, transmits allele 1 to an offspring. Module paptctp then applies the Hardy-Weinberg law to compute offspring genotype transmission probabilities for each pair of parental genotypes. For this reason, the frequency module (papfq, selected should also assume Hardy-Weinberg equilibrium and compute the genotype frequencies from the allele frequency p.

Both Mendelian segregation and environmental nontransmission maintain a constant allele frequency across generations. The same restriction on the general model obeys the equation p = p²τ₁ + 2pqτ₂ + q²τ₃. Module paptcet includes only τ₁ and τ₃ as parameters; τ₂ is computed to conform to the constraint.

Modules paptcms, paptctp, and paptcet apply to any Genotype Assignment option. For GA options with sex-specific genotypes, the allele transmission probabilities depend on the sex of both parent and offspring [Demenais & Elston 1981]. Module paptcet restricts the genetic model to 1 locus with 2 alleles; paptcms makes no restrictions. Module paptctp allows any number of loci, but restricts the final locus in the model to 2 alleles and allows estimation of transmission for that locus, assuming Mendelian transmission for all previous loci.

1. Mendelian segregation

Mendelian transmission probabilities of τ₁ = 1, τ ₂ = ½, and τ₃ = 0 restrict a parent of genotype 1 (AA) to transmit exclusively allele A, a parent of genotype 2 (Aa) to transmit allele A with probability ½, and a parent of genotype 3 (aa) to never transmit allele A.

2. Environmental nontransmission

Environmental nontransmission results from making the transmission probabilities independent of the parental genotypes. That is, the transmission probabilities are all the same regardless of the parental or offspring genotypes. The allele frequency may be set equal to the common transmission probability to ensure equilibrium across generations. Otherwise, a different frequency may be estimated for founders than for nonfounders. This is unreasonable in multi-generation pedigrees where founders occur in all generations and at all ages. On the other hand, a frequency difference between founders and nonfounders may accurately represent nuclear families with all founders (parents) older than all nonfounders (children).

Application of the Hardy-Weinberg law in paptctp forces a relationship between the genotype transmission probabilities. When dominance reduces the model to two distinguishable phenotypes, the two proportions are unrestricted. However, when three phenotypes can be distinguished, their occurrence in Hardy-Weinberg proportions places a genetic constraint on a supposedly environmental model.

You can extend the environmental model to any number of alleles and loci by using paptce, which equates the genotype transmission probabilities to the genotype frequencies. You can eliminate the Hardy-Weinberg assumption by using paptce with papfqg. You can perform commingling analysis of a continuous trait by using the environmental model with a transformation.

3. General transmission

Both Mendelian segregation and environmental nontransmission constitute submodels of a general transmission model with each allele transmission probability estimated to a value between 0 and 1. However, τ₁ and τ₃ often estimate to 1 and 0, respectively. Estimation on the boundary complicates determining the degrees of freedom in testing the Mendelian and environmental models.

Estimating p, τ₁, τ₂, and τ₃ ignores the assumption of equilibrium across generations made in the Mendelian and environmental nontransmission models. Alternatively, you can estimate p, τ₁ and τ₃ and compute τ ₂ from p, τ₁ and τ₃ using paptcet [Demenais & Elston 1981].