Genetic association

A genetic association is a statistical dependence between a genetic variant and a phenotype. An associated marker need not be causal: linkage disequilibrium can correlate it with a causal locus, and population stratification can produce an association without such a link.

A study testing associations between genomic variants and a measured outcome across many genomic positions. Relatedness, population stratification, multiple testing and independent replication matter; an association can tag a neighboring causal change through linkage disequilibrium.

The genotype relative risk compares the penetrance for genotype $g$ with that of a chosen baseline $g_0$. The common baseline factor cancels when computing conditional probabilities of transmitted genotypes given an affected offspring. It is a ratio of disease probabilities, unlike an odds ratio.

Let $q_g=P(g\mid g_f,g_m)$ be the Mendelian segregation probability, and assume the child's disease status depends on the parental genotypes only through its own genotype, with penetrance proportional to $R_g$. Bayes' theorem gives the displayed conditional law. For a diallelic parent pair there are four equally likely labelled transmission outcomes. If an unordered genotype occurs in $m_g$ of them, $q_g=m_g/4$, so its probability is $m_g{R}_g/{\sum }_h{m}_h{R}_h$. Dropping multiplicities would, for example, misweight a heterozygous child of two heterozygotes.

The haplotype-relative-risk design compares transmitted alleles to the corresponding untransmitted parental copies, used as internal controls for an affected offspring. If allele-1/allele-2 transmitted counts are $w,x$ and untransmitted counts are $y,z$, the allele-1 odds ratio estimate is $wz/(xy)$. An unmatched Pearson chi-squared test of independence treats this as a two-by-two table. The transmission disequilibrium test instead keeps parent-specific pairing and uses only discordant transmissions. The sampling assumptions of the unmatched calculation and any missing-parent reconstruction must be justified separately.

Association mapping localizes genetic influences on a phenotype by testing population genetic association with markers. Linkage disequilibrium allows a typed marker to tag a nearby causal variant; historical genetic recombination can narrow the associated interval. Population stratification can produce association without a local causal relation.

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An observed genetic marker tags an unobserved causal genetic variant when linkage disequilibrium correlates their allele indicators. Their squared allelic correlation $r^2$ measures prediction strength. Under standard local-alternative score-test approximations, using the marker rather than the causal allele reduces the noncentrality parameter by roughly $r^2$, suggesting a sample-size multiplier about $1/r^2$. This is an approximation requiring the same disease model and sampling design, not a universal power identity.

For affected offspring and heterozygous parents at a diallelic locus, let $b$ count transmissions of allele 1 with allele 2 untransmitted and $c$ count the reverse. Under the fair-transmission null, conditional on $b+c$, $b$ is binomial with probability $1/2$. The asymptotic statistic is $(b-c{)}^2/(b+c)$, with a one-degree-of-freedom chi-squared null distribution. Homozygous-parent transmissions give no contrast. Conditioning within families protects this test from ordinary population stratification.

If one parent is heterozygous and the other parent's genotype is missing, a homozygous child reveals the observed parent's transmitted allele, whereas a heterozygous child need not. Selecting only revealed transmissions can therefore bias the transmission disequilibrium test. For allele-1 frequency $p$ under random mating and fair transmission, the probabilities of an allele-1 or allele-2 homozygous child from an observed heterozygous parent are $p/2$ and $(1-p)/2$. Among selected homozygous children, the observed allele-1 transmission probability is $p$, not generally $1/2$. A complete-trio analysis avoids this reconstruction rule; methods retaining missing-parent families need an explicit likelihood function or a valid conditional adjustment. Family-based association tests using genotype data with uncertainty discusses this bias.

Combine the untransmitted maternal and paternal alleles into a hypothetical control genotype, paired with the child's transmitted genotype. Even when two heterozygous parents have a heterozygous child and parent-specific phase is ambiguous, the combined untransmitted genotype is unambiguously heterozygous. Such pseudo-controls are matched family controls, not independent random individuals; Hardy-Weinberg principle is therefore not guaranteed merely by this construction.

Under random mating and Hardy-Weinberg equilibrium, label each parental allele by whether it is transmitted. Let $(A,B)$ and $(C,D)$ be the transmitted/untransmitted pairs of two parents. Before ascertainment all four draws are independent with population frequencies $\pi$. If offspring disease weights the transmitted pair by ${\psi }_A{\psi }_C$, thenThe transmitted pair is an affected-case genotype and the untransmitted pair a population family pseudo-control genotype under this homogeneous model. For unordered genotypes, compatible labelled transmissions must be summed. Conditioning on parental genotypes instead removes nuisance population frequencies and gives the usual transmission disequilibrium test.

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