Genetic Effects on Binary Phenotypes

Why Model a Binary Phenotype on the Probability Scale?

A genetic association question often begins with an outcome such as whether a person developed a disease during a stated follow-up period. The recorded outcome has only two values, but its probability can differ across genotypes. Describing those probabilities directly gives a clear scale for asking how much the chance of the outcome changes between genotype groups.

The probability scale is especially interpretable because a difference such as 0.03 is an absolute change of 3 percentage points. It is not a 3% relative increase. A probability-scale model therefore describes genetic effects in terms of changes in outcome probability, while keeping the outcome itself binary.

The outcome definition matters. “Disease present at the examination” describes the probability of having the disease at that assessment; “new diagnosis within five years” describes risk over five years. A probability is meaningful only when the outcome, its timing, and the population are specified. The conditional probabilities describe how outcome frequency varies with genotype. By themselves, they do not establish that genotype differences cause the outcome differences.

From a Binary Observation to a Probability

For an individual, \(Y\) is still either 0 or 1. The probability model concerns the distribution of that outcome among individuals with a given genotype. If \(P(Y=1\mid G=g)=p_g\), then \(P(Y=0\mid G=g)=1-p_g\). This is a Bernoulli model for the binary outcome conditional on genotype.

The second equality follows directly from the two possible outcomes: \(Y\) contributes 1 with probability \(p_g\) and 0 with probability \(1-p_g\). Thus \(E(Y\mid G=g)=1\cdot p_g+0\cdot(1-p_g)=p_g\). A model for the conditional mean of a binary outcome is therefore also a model for its conditional probability.

Probability-Scale Models and Genotype Coding

A genotype coding represents the genetic categories with a predictor. The earlier material on genotype dosage and indicator coding distinguishes counting alleles from marking particular genotype groups. On the probability scale, the coding specifies which genotype probabilities are constrained to follow the same pattern.

Let \(G\) be the alternate-allele dosage, so \(G\in\{0,1,2\}\). A linear probability model for this dosage is:

Here \(\beta_0\) is the modeled probability for the reference genotype \(G=0\). The coefficient \(\beta_1\) is the change in probability for a one-unit increase in dosage. Under this model, the risk difference is the same at both genotype steps: \(p_1-p_0=\beta_1\) and \(p_2-p_1=\beta_1\). The model therefore imposes equal probability-scale steps; it does not merely describe a general tendency for risk to rise with dosage.

A probability must be between 0 and 1. Consequently, an additive linear probability model is valid for the genotype values being modeled only when \(0\leq\beta_0+\beta_1g\leq1\) for each such value of \(g\). This restriction is important: a straight line is not automatically a sensible probability model for every possible coefficient or extrapolated dosage.

If the genotype-specific probabilities do not have equal steps, a model with a separate parameter for each nonreference genotype can represent them without the additive restriction. Using \(G=0\) as the reference, one such model is:

The indicator \(I(g=k)\) equals 1 when \(g=k\) and 0 otherwise. Thus \(p_0=\beta_0\), \(p_1=\beta_0+\beta_1\), and \(p_2=\beta_0+\beta_2\). Here \(\beta_1\) and \(\beta_2\) are risk differences relative to \(G=0\); the two genotype groups need not have the same difference from the reference. Valid fitted probabilities must still be between 0 and 1.

A dominant coding instead groups the two carrier genotypes together. Define \(D=I(G\geq1)\), and use \(p=\beta_0+\beta_DD\). This model assigns probability \(\beta_0\) to \(G=0\) and probability \(\beta_0+\beta_D\) to both \(G=1\) and \(G=2\). Its coefficient is the carrier-versus-noncarrier difference in modeled probability. The equality of the two carrier probabilities is a modeling constraint, not a fact guaranteed by the coding.

How to Read and Check a Probability-Scale Effect

A careful interpretation names the outcome, the genotype contrast, the scale, and the units. For example, “the modeled probability of disease over five years is 3 percentage points higher per alternate allele” states an absolute difference, its genotype increment, and its time context. It is more informative than saying only that “the genetic effect is 0.03.”

Before interpreting any probability model, check that its coding matches the intended comparison and that each probability it assigns is within \([0,1]\). Also distinguish the model's constraints from the observed genotype pattern. An additive dosage model imposes equal steps; a dominant model imposes equal probabilities for both carrier groups; a genotype-specific model allows those probabilities to differ. These are different descriptions of how probability varies by genotype.

Common Mistake: Confusing Percentage Points With Percent Change

Suppose modeled risk increases from 0.02 to 0.05. The difference is \(0.05-0.02=0.03\), so the absolute increase is 3 percentage points. Calling it “a 3% increase” is ambiguous and can be incorrect: relative to 0.02, the increase is \(0.03/0.02=1.5\), or 150%. For a probability-scale coefficient, use percentage points or state the probability difference explicitly.

Another mistake is interpreting a genotype association as proof that changing genotype would change disease risk. The probability model describes the distribution of the outcome across genotype groups in the population and context under study. A causal interpretation requires additional design and assumptions; it does not follow from the probability-scale representation alone.