Aligning Phenotypes and Effect Scales

Why Phenotype and Scale Alignment Matter

Before association estimates from different cohorts can be meaningfully compared or combined, they must refer to sufficiently comparable outcomes and effects. A genetic association estimate is not just a number: it describes a change in a particular phenotype, under a particular coding and model, for a specified change in genotype. The value 2.0, for example, might mean a two-unit increase in a quantitative trait, a log odds ratio of 2.0, or an odds ratio of 2.0. Those are not interchangeable interpretations.

Some differences are straightforward to resolve. If one cohort reports a quantitative effect in kilograms and another reports it in pounds, a known linear unit conversion places both estimates on the same scale. Other differences change what is being estimated. If cohorts define disease cases using different diagnostic criteria, the underlying binary outcomes differ; there is generally no single conversion factor that makes their association estimates equivalent. Phenotype alignment therefore requires both a scientific judgment about what the outcomes represent and a mathematical check of the effect scales.

Definition: Phenotype alignment means that cohort outcomes represent sufficiently comparable traits or case definitions for the intended cross-cohort interpretation. Effect-scale alignment means that estimates describe the same type of genetic effect, for the same genotype contrast, in common units or on a common mathematical scale. Alignment does not require identical data collection, but differences must not silently change the estimand.

Specify the Outcome and the Effect

For a quantitative outcome \(Y\), a common association model describes the mean change in \(Y\) per additional copy of an aligned effect allele, conditional on included covariates. In a simple linear model, the coefficient \(\beta\) is that change in the outcome’s units per allele. A coefficient of 0.4 could mean 0.4 kilograms per allele, 0.4 milligrams per deciliter per allele, or 0.4 standard deviations per allele. Units are part of the estimate’s meaning.

For a binary outcome coded \(Y=1\) for cases and \(Y=0\) for controls, logistic regression models the log odds:

\operatorname{logit}\{P(Y=1\mid G,C)\} =\log\left(\frac{P(Y=1\mid G,C)}{1-P(Y=1\mid G,C)}\right) =\alpha+\beta G+\gamma^\mathsf{T}C.

Here \(G\) is the effect-allele dosage, commonly coded from 0 to 2, and \(C\) represents covariates. Under this model, \(\beta\) is the log odds ratio per additional allele, and the odds ratio is \(\exp(\beta)\). The log odds ratio and odds ratio express the same modeled association on different scales, but they are different numerical quantities. An odds ratio is not a log odds ratio.

A comparison also depends on the genotype contrast and outcome coding. Estimates per allele should not be treated as directly comparable to estimates for carrier versus noncarrier without an appropriate model-based conversion. Likewise, if one cohort codes cases as 1 and another codes controls as 1, the direction of the binary association reverses. The earlier material on allele harmonization addresses aligning the allele direction; phenotype coding must be checked as well.

When Does a Difference Permit Conversion?

A unit conversion is justified when the outcome measurements differ by a known linear transformation and otherwise represent the same trait. If \(Y^*=a+bY\), where \(a\) is a fixed shift and \(b\) is a fixed nonzero multiplier, then a linear association coefficient transforms as \(\beta^*=b\beta\). A constant shift changes the outcome’s location but not its per-allele slope. If the standard error is reported on the original scale, it transforms as \(SE^*=|b|SE\); the absolute value is used because a standard error is nonnegative.

\beta^*=b\beta,\qquad SE^*=|b|SE.

For a binary-trait logistic coefficient, converting an odds ratio to a log odds ratio is also exact: take the natural logarithm. If a confidence interval is reported for an odds ratio, its positive endpoints can likewise be logged to obtain endpoints on the log odds ratio scale. This does not turn a change in case definition into a mere unit conversion.

In contrast, different diagnostic criteria, thresholds, measurement instruments with non-linear calibration, or disease subtypes can change the phenotype itself. The resulting estimates may refer to different estimands. Similar labels—such as “asthma” or “hypertension”—do not guarantee equivalent case definitions. The relevant question is whether the underlying outcome and interpretation are sufficiently aligned for the scientific purpose, not whether the cohort names match.

Worked Example: Converting a Quantitative Effect From Kilograms to Pounds

Suppose a hypothetical cohort reports an estimated body-weight effect of \(0.400\) kilograms per additional effect allele, with \(SE=0.120\) kilograms per allele. A second cohort uses pounds for the same weight trait. Use \(1\) kilogram \(=2.20462\) pounds to express the first cohort’s estimate in pounds.

This conversion is appropriate if the measurements represent the same weight outcome and the only difference is the unit. With \(Y^*=2.20462Y\), the coefficient and standard error both multiply by 2.20462:

\beta^*=2.20462(0.400)=0.881848\ \text{lb per allele},

and

SE^*=2.20462(0.120)=0.264554\ \text{lb per allele}.

Rounded to four significant figures, the converted estimate is \(0.8818\) pounds per allele with standard error \(0.2646\) pounds per allele. The conversion changes the numerical values but not the underlying association or its direction. If the second cohort’s phenotype were instead a different measure, such as waist circumference, converting its units would not make it an estimate of the same outcome.

Binary Traits: Align the Definition, Then the Scale

For binary traits, the case definition is especially important. Changing a diagnostic threshold can reclassify individuals and therefore change the association being estimated. It can also change the case-control composition and the relationship between genotype and case status. There is no general multiplier that converts an estimate for one case definition into an estimate for another.

When the binary phenotype and model are considered comparable, report estimates on a consistent scale. If one result is an odds ratio \(OR\), then its log odds ratio is \(\log(OR)\). For example, an odds ratio of 1.5 corresponds to a log odds ratio of \(\log(1.5)\), approximately 0.4055. The log-scale value is used when a common additive scale is needed; it must not be reported as though it were itself an odds ratio.

Worked Example: Different Binary Case Definitions and Log Odds Ratios

Consider two hypothetical case-control analyses of a genetic variant. In each table, “carrier” means carrying one or more copies of the aligned effect allele, and the odds ratio compares carriers with noncarriers. The first analysis uses a strict case definition; the second uses a broader definition. The counts are illustrative and the two analyses need not contain the same individuals.

Strict definition	Carriers	Noncarriers
Cases	60	940
Controls	30	970

For the strict definition, the crude odds ratio is

OR_{\text{strict}} =\frac{60/940}{30/970} =\frac{60\times970}{940\times30} =\frac{97}{47} \approx 2.064.

Its log odds ratio is

\log(OR_{\text{strict}}) =\log(97/47) \approx 0.724563 \approx 0.7246.

For the broader definition, suppose the hypothetical counts are 100 cases who are carriers, 900 cases who are noncarriers, 83 control carriers, and 917 control noncarriers. Then

OR_{\text{broad}} =\frac{100\times917}{900\times83} =\frac{91700}{74700} \approx 1.228,

and \(\log(OR_{\text{broad}})\approx 0.2050\). The two odds ratios are not made equivalent by logging them: the log transformation puts each estimate on the log odds scale, but the estimates still come from different case definitions and data. The difference may reflect changed phenotype classification, different sampled individuals, or other study features. These crude calculations also do not automatically equal covariate-adjusted logistic regression coefficients.

Standardizing Quantitative Effects

A quantitative effect can sometimes be expressed in standard-deviation units. If a cohort’s phenotype has standard deviation \(s_Y\), a raw effect \(\beta\) can be divided by \(s_Y\) to express the per-allele change in phenotype standard deviations. The corresponding standard error is divided by the same positive constant. This is useful when cohorts measure the same construct in different units or with different scales, but it does not make unlike phenotypes equivalent.

\beta_{\text{SD}}=\frac{\beta}{s_Y}, \qquad SE_{\text{SD}}=\frac{SE}{s_Y}.

The interpretation depends on the chosen standard deviation. Using each cohort’s own \(s_Y\) expresses an effect relative to that cohort’s phenotype variability. If cohort variability differs for substantive reasons, the standardized estimates may not represent effects relative to a common reference distribution. A shared external standard deviation, when scientifically appropriate and available, gives a different but explicitly common reference scale. The standardization choice should therefore be documented rather than treated as a mechanical detail.

Worked Example: Comparing Effects in Standard-Deviation Units

Suppose two hypothetical cohorts assess the same quantitative biomarker, with aligned effect-allele dosage and comparable association models. Cohort A reports \(\beta=2.40\) milligrams per deciliter per allele, \(SE=0.720\), and phenotype standard deviation \(s_Y=12.0\) milligrams per deciliter. Cohort B reports \(\beta=3.00\) milligrams per deciliter per allele, \(SE=0.900\), and \(s_Y=15.0\) milligrams per deciliter.

For Cohort A, the standardized coefficient and standard error are

\beta_{\text{SD},A}=\frac{2.40}{12.0}=0.200, \qquad SE_{\text{SD},A}=\frac{0.720}{12.0}=0.0600.

For Cohort B, they are

\beta_{\text{SD},B}=\frac{3.00}{15.0}=0.200, \qquad SE_{\text{SD},B}=\frac{0.900}{15.0}=0.0600.

Both standardized estimates correspond to an increase of 0.200 cohort-specific phenotype standard deviations per additional effect allele. This agreement is informative only if the biomarker definitions, allele contrast, and models are sufficiently comparable. Because each cohort uses its own standard deviation, it does not establish that the raw effects are identical or that each standard deviation represents the same reference scale.

A Practical Alignment Check

A disciplined review separates phenotype alignment from scale conversion. First establish what the outcome means in each cohort; then establish what the reported coefficient measures; only then apply a justified conversion. The following sequence keeps those questions explicit.

Compare phenotype definitions.
Check the measured trait, diagnostic criteria, thresholds, ascertainment, and outcome coding. Decide whether any differences alter the target phenotype rather than merely its units.

Identify each effect estimand.
Record the genotype contrast, effect-allele direction, model, and effect type, including whether the result is a quantitative coefficient, log odds ratio, or odds ratio.

Convert only compatible scales.
Apply an exact linear unit conversion, a clearly specified standardization, or the logarithm of a positive odds ratio when appropriate. Transform the standard error consistently.

Document what remains different.
State unresolved differences in phenotype definition, reference scale, or model. A numerical conversion cannot erase a substantive difference in the quantity estimated.

Why the Distinction Matters

Effect-scale alignment prevents a basic but consequential error: treating numerically similar values as estimates of the same thing. It also avoids the reverse error of rejecting compatible estimates merely because their raw units differ. A genetic association of 0.4 kilograms per allele and one of 0.88 pounds per allele can describe essentially the same change after unit conversion, whereas two odds ratios for different disease definitions may not target a common effect even after both are logged.

Common mistake: “The cohorts report odds ratios, so their estimates are directly comparable; if the numbers differ, I can convert one by taking its logarithm.” The correction is that taking a logarithm converts an odds ratio to a log odds ratio, not one phenotype definition into another. Before comparing scales, confirm the case definition, coding, genotype contrast, and model. Likewise, never label a log odds ratio as an odds ratio: for example, a log odds ratio of 0.7246 corresponds to an odds ratio of approximately 2.064, not 0.7246.

Key takeaway: Align the phenotype and the estimand before aligning the numbers. Exact rescaling is appropriate for known linear unit changes and for odds ratios converted to log odds ratios; it cannot generally repair a change in what counts as the phenotype.

Check Your Understanding

Answer each question by naming both the relevant scale and the reason for your conclusion.

A cohort reports a quantitative effect in centimeters and another in meters for the same measurement. What must happen to the coefficient and standard error to express both in meters?
A binary-trait analysis reports an odds ratio of 1.8. What quantity is obtained by taking its natural logarithm, and is that quantity itself an odds ratio?
Why might estimates from two cohorts with different diagnostic thresholds fail to describe the same phenotype, even if both results are reported as log odds ratios?
Two cohorts report raw effects of 2.4 and 3.0 units per allele, with phenotype standard deviations of 12 and 15 units, respectively. What are their cohort-standardized effects?
What additional interpretation is needed when standardizing each cohort’s effect by its own phenotype standard deviation?