Logistic Genetic Effects as Odds Ratios

Why Logistic Coefficients Need Translation

For a binary phenotype, the probability of the outcome must lie between 0 and 1. A linear expression in genotype dosage does not have that restriction, so it is not generally suitable as a model for the probability itself. Logistic regression instead models the log odds of the outcome as a linear function of genotype coding. This makes the coefficient additive on the log-odds scale and multiplicative on the odds scale.

The genetic interpretation depends on the coding. With dosage coding, the coefficient describes the change associated with one additional coded allele copy. With genotype-indicator coding, each coefficient compares one genotype with a specified reference genotype. Exponentiating the relevant coefficient, or coefficient contrast, expresses that comparison as an odds ratio.

Definition: For an outcome with probability \(p\), the odds of the outcome are \(p/(1-p)\), and the logit of \(p\) is \(\log\{p/(1-p)\}\). A logistic model specifies the logit of the conditional outcome probability as a linear predictor. If the linear predictor is \(\eta\), then the odds are \(\exp(\eta)\).

Dosage Coding: One Coefficient, One Allele-Copy Step

Let \(Y=1\) denote the phenotype of interest and \(Y=0\) its absence. For a biallelic locus, let \(G\) be the coded allele dosage, taking values 0, 1, or 2. Under the additive logistic model,

\operatorname{logit}\{P(Y=1\mid G)\}=\beta_0+\beta_1G.

Here \(\beta_0\) is the log odds for the genotype coded \(G=0\), while \(\beta_1\) is the change in log odds for a one-unit increase in dosage. The model assumes that each dosage step has the same log-odds change. It therefore implies the same multiplicative change in odds from 0 to 1 copies as from 1 to 2 copies.

To compare dosage \(G=g+1\) with dosage \(G=g\), subtract the two model expressions. Their log-odds difference is \(\beta_1\); exponentiating that difference gives the odds ratio. For a contrast of \(d\) dosage units, the same subtraction gives \(d\beta_1\).

\begin{aligned} \log\left\{\frac{\operatorname{odds}(Y=1\mid G=g+1)} {\operatorname{odds}(Y=1\mid G=g)}\right\} &=(\beta_0+\beta_1(g+1))-(\beta_0+\beta_1g)\\ &=\beta_1,\\ \operatorname{OR}_{g+1:g}&=\exp(\beta_1),\\ \operatorname{OR}_{g+d:g}&=\exp(d\beta_1)=\{\exp(\beta_1)\}^{d}. \end{aligned}

Thus, \(\exp(\beta_1)\) is the odds ratio for one additional coded allele copy, not for a two-copy difference unless that is the contrast being made. For a 0-versus-2 dosage comparison, the odds ratio is \(\exp(2\beta_1)\). An odds ratio above 1 indicates higher modeled odds in the numerator group; one below 1 indicates lower modeled odds. The odds ratio alone does not state the difference in probabilities.

Worked Example: Interpreting an Additive Dosage Coefficient

Suppose a hypothetical cohort of 900 participants is analyzed for a binary phenotype. The allele of interest is coded as \(G=0,1,2\), and a fitted additive logistic model has estimated coefficient \(\hat{\beta}_1=0.4055\) per allele copy. Interpret the model’s dosage contrasts.

For one additional allele copy, exponentiate the coefficient:

\widehat{\operatorname{OR}}_{\text{one copy}} =\exp(0.4055)\approx 1.500.

Under this model, the odds of the phenotype for participants with one more coded allele copy are estimated to be 1.50 times the odds for participants with one fewer copy. This is a 50.0% increase in odds per copy, not a 50.0 percentage-point increase in phenotype probability.

For the comparison of \(G=2\) with \(G=0\), the dosage difference is 2, so the log-odds contrast is \(2(0.4055)=0.8110\). Therefore,

\widehat{\operatorname{OR}}_{2:0} =\exp\{2(0.4055)\} =\exp(0.8110) \approx 2.250.

The model estimates 2.25 times the phenotype odds at dosage 2 compared with dosage 0. This comparison follows from the additive logit model; it does not establish that the allele causes the phenotype.

Genotype Indicators: Contrasts Against a Reference

A dosage model imposes a constant log-odds step across the three genotypes. When the model instead gives each non-reference genotype its own coefficient, it does not impose that constraint. Let \(I_1\) and \(I_2\) indicate two genotypes, with a third genotype designated as the reference. The model is

\operatorname{logit}\{P(Y=1\mid I_1,I_2)\} =\beta_0+\beta_1I_1+\beta_2I_2.

For the reference genotype, both indicators equal zero, so its log odds are \(\beta_0\). For the genotype indicated by \(I_1=1\), the log odds are \(\beta_0+\beta_1\). Subtracting the reference log odds leaves \(\beta_1\); its exponent is the odds ratio for that genotype versus the reference. The same reasoning applies to \(\beta_2\).

A contrast between two non-reference genotypes uses a difference of coefficients. For example, the log-odds contrast of genotype 2 versus genotype 1 is \(\beta_2-\beta_1\), and the corresponding odds ratio is \(\exp(\beta_2-\beta_1)\). The reference is essential to interpreting the individual coefficients, but it cancels from this direct contrast.

Worked Example: Two Genotype Indicators

In a hypothetical study of 720 participants, the genotype with zero copies of the coded allele is the reference. The other two genotypes have separate indicators. A fitted logistic model gives \(\hat{\beta}_1=0.5878\) for the one-copy genotype and \(\hat{\beta}_2=1.0986\) for the two-copy genotype. Calculate and interpret the odds ratios.

For the one-copy genotype versus the reference, the estimated odds ratio is

\widehat{\operatorname{OR}}_{1:0} =\exp(\hat{\beta}_1) =\exp(0.5878) \approx 1.800.

The modeled phenotype odds for the one-copy genotype are estimated to be 1.80 times the odds for the reference genotype.

For the two-copy genotype versus the reference, the estimated odds ratio is

\widehat{\operatorname{OR}}_{2:0} =\exp(\hat{\beta}_2) =\exp(1.0986) \approx 3.000.

The modeled odds for the two-copy genotype are estimated to be 3.00 times the reference-genotype odds. To compare the two non-reference genotypes directly, subtract the coefficients and exponentiate:

\widehat{\operatorname{OR}}_{2:1} =\exp(\hat{\beta}_2-\hat{\beta}_1) =\exp(1.0986-0.5878) =\exp(0.5108) \approx 1.667.

Thus the two-copy genotype has an estimated 1.667 times the phenotype odds of the one-copy genotype under this model. The indicator model estimates each genotype contrast separately; its coefficients should not automatically be treated as equal dosage steps.

Adjusted Logistic Models

A logistic model can include covariates as well as genotype. For example, with genotype dosage \(G\), age \(A\), and a covariate \(S\), a model might be

\operatorname{logit}\{P(Y=1\mid G,A,S)\} =\beta_0+\beta_GG+\beta_AA+\beta_SS.

In this model, \(\exp(\beta_G)\) compares the modeled phenotype odds for a one-unit dosage difference while holding the included covariates fixed. The interpretation is conditional on the values of those covariates. It is not automatically a causal effect: observational associations can reflect confounding, selection, measurement error, or other features of the study.

Worked Example: A Covariate-Adjusted Dosage Contrast

Suppose a hypothetical analysis of 1,200 participants models a binary phenotype using allele dosage, age in years, and a sex indicator. The estimated dosage coefficient is \(\hat{\beta}_G=-0.2231\). Interpret the one-copy and two-copy contrasts, conditional on age and sex.

For one additional coded allele copy, the adjusted odds ratio is

\widehat{\operatorname{OR}}_{\text{one copy}} =\exp(-0.2231) \approx 0.8000.

At the same modeled age and sex, the phenotype odds are estimated to be 0.8000 times as large for a participant with one additional copy. Equivalently, they are estimated to be 20.0% lower, since \(1-0.8000=0.2000\).

For a two-copy contrast, double the coefficient before exponentiating:

\widehat{\operatorname{OR}}_{2:0} =\exp\{2(-0.2231)\} =\exp(-0.4462) \approx 0.6400.

At the same modeled age and sex, the phenotype odds for dosage 2 are estimated to be 0.6400 times those for dosage 0. This remains an odds comparison, and adjustment for the listed covariates does not by itself justify a causal interpretation.

Assumptions and Interpretation in Practice

Before interpreting a coefficient, check what model it came from. The outcome must be defined consistently as \(Y=1\) for the event whose odds are modeled. The genotype coding must be identified, including which allele dosage counts and which genotype is the reference when indicators are used. The specified logit model must be an appropriate description of the conditional log odds: a linear dosage term assumes a constant log-odds change per copy, whereas separate genotype indicators allow distinct genotype contrasts.

For ordinary logistic-regression inference, the model also treats each outcome as a Bernoulli observation conditional on its predictors and typically assumes independent observations. If participants are clustered or related, that independence assumption may not be appropriate for ordinary model-based standard errors. Such issues affect the validity of uncertainty statements; they do not change the algebra translating a specified coefficient contrast into an odds ratio.

The interpretation is an association in the modeled population and on the odds scale. An odds ratio is not a probability ratio, and it does not reveal an absolute probability change without additional information about baseline risk. The same odds ratio can correspond to different probability changes at different starting risks. Keep the genotype contrast, outcome definition, covariate conditions, and odds scale explicit when reporting a result.

A Reliable Translation Procedure

Identify the outcome and coding.
State which phenotype is coded \(Y=1\), how dosage is defined, or which genotype is the indicator-model reference.

Write the requested contrast.
For dosage, use the difference in coded copies. For indicators, subtract the linear predictors for the two genotypes being compared.

Calculate the log-odds contrast.
Multiply a dosage coefficient by the dosage difference, or form the required coefficient difference for indicator coding.

Exponentiate and describe the comparison.
Report the odds ratio and say which genotype is in the numerator, whether the contrast is conditional on covariates, and that the measure concerns odds.

Common Mistakes

Calling the coefficient itself an odds ratio. A coefficient is on the log-odds scale. Exponentiate it, or the appropriate coefficient contrast, to obtain an odds ratio.
Using the one-copy odds ratio for a two-copy comparison. For a two-unit dosage difference, calculate \(\exp(2\beta_1)\), not \(\exp(\beta_1)\).
Ignoring the reference genotype. An indicator coefficient compares its genotype with the model’s reference, not with every other genotype.
Calling an odds ratio a probability multiplier. An odds ratio compares \(p/(1-p)\), not \(p\) itself.
Claiming causation from a fitted coefficient. A logistic coefficient describes a model-based association unless the study design and assumptions support a causal interpretation.

Key takeaway: Logistic genotype coefficients are changes in log odds. For a specified genotype contrast, calculate its log-odds difference and exponentiate: \(\exp(\text{contrast})\) is the odds ratio for the phenotype odds in the stated numerator genotype relative to the denominator genotype.

Check Your Understanding

For each question, keep the outcome definition and genotype contrast explicit.

A dosage-model coefficient is \(0.30\). What is the estimated odds ratio for one additional allele copy, and what is it for a two-copy difference?
In a genotype-indicator model, what does the coefficient for the heterozygote compare with, and what must be exponentiated to obtain its odds ratio?
Two non-reference genotype indicators have coefficients \(0.4\) and \(0.9\). Write the expression for the odds ratio comparing the second genotype with the first.
Why is an odds ratio of 1.5 not generally a statement that phenotype probability is 50% higher?
In a model adjusted for age, what does a genotype dosage odds ratio compare, and why does adjustment alone not establish causation?