Association Effect Scales Across Populations

Why the Effect Scale Matters

A genetic association can be described in more than one mathematically valid way. For a continuous outcome, an effect may be expressed as the expected change in outcome units per additional allele. For a binary outcome, it may be expressed as a change in log odds, an odds ratio, a risk difference, or a risk ratio. These scales answer different questions. In particular, the same odds ratio can correspond to different changes in absolute risk when baseline risks differ across populations.

A cross-population comparison is therefore not complete when it reports only whether two estimated coefficients are numerically equal. First identify the outcome, the genetic contrast, the effect scale, and whether the effect is conditional on covariates or averaged over a population. Throughout, assume the same genetic variant and effect-allele dosage \(G\in\{0,1,2\}\) are being compared. This tutorial concerns interpretation of scales, not allele harmonization or causal interpretation.

Definition: An effect scale specifies the quantity used to describe the contrast in outcome associated with a genetic contrast. For a continuous outcome, a common scale is the mean difference per additional allele. For a binary outcome at a specified time horizon, common scales are the risk difference \(p_1-p_0\), risk ratio \(p_1/p_0\), and odds ratio \(\{p_1/(1-p_1)\}/\{p_0/(1-p_0)\}\). The subscripts 0 and 1 denote the two genetic states being contrasted.

Models and Their Interpretations

For a continuous phenotype \(Y\), a linear association model can be written as

E(Y\mid G,C,A)=\alpha_A+\beta_A G+\gamma_A^\mathsf{T}C,

where \(A\) denotes population, \(C\) is a vector of covariates, and the mean model assumes a constant additive change \(\beta_A\) in the conditional mean per additional allele, at fixed covariates. The coefficient has the units of \(Y\) per allele. A linear model does not require normally distributed outcomes to define this mean contrast; distributional and variance assumptions matter for particular standard errors and tests. If the true association changes with genotype or covariates, the single coefficient is only a model-based summary.

For a binary outcome \(D\), a logistic association model is

\operatorname{logit}P(D=1\mid G,C,A) =\alpha_A+\beta_A G+\gamma_A^\mathsf{T}C, \qquad \operatorname{logit}(p)=\log\left(\frac{p}{1-p}\right).

Under this model, \(\beta_A\) is a conditional log-odds change per allele, and \(\exp(\beta_A)\) is the conditional odds ratio per allele, at fixed covariates. The model assumes a linear genotype dosage term on the log-odds scale; with dosage values 0, 1, and 2, it implies the same log-odds increment from 0 to 1 as from 1 to 2. It does not imply equal increments in risk. In either model, comparisons require compatible outcome definitions, genetic contrasts, covariate conditioning, and population targets.

Risk-scale effects instead compare probabilities. A risk difference is measured in probability units, often reported as percentage points; a risk ratio is dimensionless. Risk must refer to a defined time horizon and population. In a logistic model, the risk for a genotype depends on the intercept and covariates as well as on \(\beta_A\), so the coefficient alone does not specify an absolute risk change.

Worked Example: A Linear Effect in Outcome Units

Suppose a hypothetical continuous biomarker is measured in mg/dL. In population A, the fitted conditional mean is \(100+2.4G\); in population B it is \(112+2.4G\), with the same covariate values held fixed. Interpret the allele effects and calculate the predicted means for \(G=0,1,2\).

In both populations, the slope is \(2.4\) mg/dL per additional allele. For population A the predictions are \(100+2.4(0)=100\), \(100+2.4(1)=102.4\), and \(100+2.4(2)=104.8\) mg/dL. For population B they are \(112\), \(114.4\), and \(116.8\) mg/dL.

Thus the predicted difference from \(G=0\) to \(G=2\) is \(104.8-100=4.8\) mg/dL in A and \(116.8-112=4.8\) mg/dL in B. The baseline means differ by 12 mg/dL, but that difference is not evidence of different additive genetic slopes. These calculations rely on the stated linear mean models and a common covariate setting; they do not establish that the slopes are equal in actual populations.

Why an Odds Ratio Does Not Fix the Risk Difference

For a binary genotype contrast, let \(p_0\) be the risk in the reference group and let \(\theta\) be the odds ratio comparing the other group with it. Since the odds in the reference group are \(p_0/(1-p_0)\), the odds in the comparison group are \(\theta p_0/(1-p_0)\). Converting those odds back to a probability gives

p_1=\frac{\theta p_0}{1-p_0+\theta p_0} =\frac{\theta p_0}{1+(\theta-1)p_0}.

Subtracting \(p_0\) and dividing by \(p_0\), respectively, yields the risk difference and risk ratio:

p_1-p_0 =\frac{(\theta-1)p_0(1-p_0)}{1+(\theta-1)p_0}, \qquad \frac{p_1}{p_0} =\frac{\theta}{1+(\theta-1)p_0}.

These expressions show why a fixed odds ratio does not imply a fixed risk difference or risk ratio: both depend on baseline risk \(p_0\). The formulas assume that the odds ratio and baseline risk refer to the same defined contrast and covariate setting. In practice, estimates of baseline risk are needed to translate an odds ratio into a risk-scale effect.

Worked Example: One Odds Ratio, Two Baseline Risks

Suppose a hypothetical per-allele odds ratio is \(1.5\) in each of two populations. For illustration, compare the risks at genotype states 0 and 1, with baseline risk \(p_0=0.10\) in population A and \(p_0=0.40\) in population B. Assume the odds ratio applies to this contrast at the specified covariate setting.

In A, the risk at \(G=1\) is \(p_1=1.5(0.10)/\{1-0.10+1.5(0.10)\}=0.15/1.05=0.1429\). The risk difference is \(0.1429-0.10=0.0429\), or 4.29 percentage points, and the risk ratio is \(0.1429/0.10=1.429\).

In B, the risk at \(G=1\) is \(p_1=1.5(0.40)/\{1-0.40+1.5(0.40)\}=0.60/1.20=0.5000\). The risk difference is \(0.5000-0.4000=0.1000\), or 10.0 percentage points, and the risk ratio is \(0.5000/0.4000=1.250\).

The conditional odds ratio is 1.5 in both settings, but the absolute risk increase is larger in B. The risk ratio, meanwhile, is larger in A. No scale has changed the data; each scale emphasizes a different contrast.

Conditional and Marginal Effects: Non-Collapsibility

A conditional effect compares genotype groups at fixed covariate values. A marginal effect compares risks or odds averaged over a covariate distribution. For odds ratios, these generally need not be equal, even when genotype is independent of the covariate and there is no confounding. This property is called non-collapsibility: adjusting for a prognostic covariate can change an odds ratio simply because conditional and marginal odds ratios are different estimands.

The reason is that averaging probabilities and then converting the average to odds does not preserve odds ratios. In general, the odds of an average risk are not the average of the stratum-specific odds. This differs from confounding: confounding concerns a distortion in an association caused by a covariate related to both exposure and outcome, whereas non-collapsibility can arise without such a relationship.

Worked Example: Equal Conditional Odds Ratios, Different Marginal Odds Ratios

Consider two hypothetical populations, each with two equally common covariate strata. Within every stratum, genotype is independent of the covariate and the genotype contrast has conditional odds ratio 2. In population A, the reference-genotype risks across the strata are 0.05 and 0.40. In population B, they are 0.20 and 0.60. Calculate each population's marginal odds ratio by averaging risks across strata first.

For any stratum with reference risk \(p\), the risk under odds ratio 2 is \(2p/(1+p)\). In A, the comparison risks are \(2(0.05)/1.05=0.09524\) and \(2(0.40)/1.40=0.5714\). The marginal reference risk is \((0.05+0.40)/2=0.2250\), and the marginal comparison risk is \((0.09524+0.5714)/2=0.3333\). Thus the marginal odds ratio is \((0.3333/0.6667)/(0.2250/0.7750)=1.722\), approximately.

In B, the comparison risks are \(2(0.20)/1.20=0.3333\) and \(2(0.60)/1.60=0.7500\). The marginal reference risk is \(0.4000\), and the marginal comparison risk is \((0.3333+0.7500)/2=0.5417\). The marginal odds ratio is \((0.5417/0.4583)/(0.4000/0.6000)=1.773\), approximately.

The conditional odds ratio is 2 in every stratum and both populations, yet the marginal odds ratios differ from 2 and from each other. Different baseline-risk profiles produce different marginal summaries. Genotype independence from the covariate rules out confounding in this example; the discrepancy is due to non-collapsibility and the differing baseline risks.

Choosing and Comparing Scales in Practice

A comparison should state whether it targets a conditional coefficient or a marginal population contrast. If the scientific question concerns expected changes in a continuous phenotype, report the linear effect in outcome units. For a binary outcome, an odds ratio may be natural for a logistic model, but a risk difference can be more directly informative about absolute burden. A risk ratio describes relative risk; it too can vary with baseline risk even when an odds ratio is fixed.

Marginal risks can be obtained by predicting risk for each genotype under a specified covariate distribution and averaging those predictions. If the same covariate distribution is used for both populations, this targets a standardized comparison; using each population's own covariate distribution instead describes population-specific averages. These are distinct targets, and neither is automatically preferable. The choice should match the question.

Scale	Interpretation	Important comparison issue
Linear mean difference	Change in mean outcome units per allele or genotype contrast	Baseline means can differ while slopes are equal; model form and units must match.
Log odds / odds ratio	Change in log odds or ratio of odds, often conditional on covariates	Non-collapsibility means conditional and marginal odds ratios can differ.
Risk difference	Absolute change in event probability over a stated horizon	Depends on baseline risk and is typically expressed in percentage points.
Risk ratio	Ratio of event probabilities over a stated horizon	Also depends on baseline risk when translated from an odds ratio.

Several checks help reveal when a scale comparison is misleading. Confirm that the outcome definition and time horizon are comparable; inspect whether a single linear dosage term is adequate; and examine whether covariate-specific estimates or predicted risks vary materially. For binary outcomes, check calibration of predicted risks before interpreting risk-scale translations. A case-control sampling design can support estimation of an odds ratio under appropriate sampling assumptions, but the sampled case fraction does not generally estimate population risk; absolute risks require appropriate population information.

A frequent mistake is to see the same logistic coefficient in two populations and conclude that the genetic effect has the same public-health impact. That conclusion does not follow: different baseline risks can imply different risk differences, and conditional and marginal odds ratios may differ because of non-collapsibility. The correction is to name the estimand and scale, then calculate or estimate the corresponding risks for the relevant population and covariate distribution.

Key takeaway: Effect scales are not interchangeable. A coefficient or odds ratio is interpretable only with its model, conditioning set, genetic contrast, and outcome scale stated. Cross-population conclusions should distinguish equality on a model scale from equality of absolute or marginal effects.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A binary outcome has baseline risk 0.20, and the odds ratio for a genotype contrast is 2. What is the comparison-group risk and the risk difference?
Two populations have the same linear effect of 3 mg/dL per allele but different intercepts. Does this imply equal predicted outcome means? Does it imply equal additive slopes under the stated models?
Why can a conditional odds ratio differ from a marginal odds ratio when genotype is independent of a prognostic covariate?
A case-control sample has 40% cases. Can that percentage automatically be used as the population baseline risk to convert an odds ratio to a risk difference? Explain.

Solutions. (1) \(p_1=2(0.20)/\{1-0.20+2(0.20)\}=0.40/1.20=0.3333\). The risk difference is \(0.3333-0.20=0.1333\), or 13.33 percentage points. (2) The intercepts imply different predicted means at a given genotype, but both models have the same additive slope, 3 mg/dL per allele. (3) Averaging stratum-specific risks and then converting to odds does not preserve the conditional odds ratio; this is non-collapsibility, not necessarily confounding. (4) No. Case-control sampling fixes or alters the sample case fraction, so it generally does not estimate population baseline risk. Population risk information is needed for that translation.