Why the Scale of a Trait Matters
Genetic correlation summarizes how genetic effects on two traits covary relative to their genetic variation. But a trait can be recorded in different units, transformed before analysis, or represented as a binary outcome rather than an underlying continuous liability. The resulting number is meaningful only when its scale and genetic estimand are clear.
A change from centimeters to meters is a positive linear rescaling and does not change genetic correlation. Taking a logarithm, squaring values, or thresholding a continuous liability is different: those operations need not simply change the units of genetic effects. They can change which genetic variation and covariance the model describes.
For a transformed trait, the genetic component is the component defined by the model for that transformed outcome. It is not automatically the original genetic component with the same transformation applied to it. That distinction is central: a linear change of units has a simple relationship to the original component, whereas a nonlinear transformation generally does not.
Positive Linear Rescaling Leaves Correlation Unchanged
Suppose each trait is transformed by a positive affine change of units, \(Y_j^*=a_j+b_jY_j\), where \(b_j>0\). In a linear genetic-component model, adding the constant \(a_j\) changes the mean but not the genetic component; multiplying the trait by \(b_j\) multiplies that component by \(b_j\). Thus \(A_j^*=b_jA_j\). The same argument covers standardizing a trait by subtracting its mean and dividing by a positive standard deviation.
Substituting these expressions into the definition of genetic correlation gives a short justification of the invariance result.
If one slope is negative, the correlation changes sign; if both are negative, it retains its sign. The invariance claim is specifically about positive rescalings. The result also assumes that the transformed genetic components really are the original components multiplied by the corresponding constants, as in a linear change of units. It is not a general result for arbitrary transformations or for models whose genetic components are redefined after transformation.
Worked Example: Changing Units for Two Quantitative Traits
Suppose a hypothetical additive genetic model gives a genetic variance of \(4\text{ cm}^2\) for height, a genetic variance of \(9\text{ kg}^2\) for a second trait, and a genetic covariance of \(3\text{ cm}\cdot\text{kg}\). Find the genetic correlation, then express height in meters.
On the original scales, $$ r_G=\frac{3}{\sqrt{4\cdot 9}}=\frac{3}{6}=0.500. $$
Expressing height in meters multiplies its values, and therefore its genetic component, by \(b_1=0.01\); the second trait is unchanged, so \(b_2=1\). The transformed height genetic variance is \(0.01^2(4)=0.0004\text{ m}^2\). The transformed covariance is \(0.01(3)=0.030\text{ m}\cdot\text{kg}\). Hence $$ r_G^*=\frac{0.030}{\sqrt{0.0004\cdot 9}} =\frac{0.030}{0.060}=0.500. $$
The covariance’s numerical value changed because its units changed; the correlation did not. The example assumes a linear genetic-component model and uses the same population and genetic definition before and after changing units.
Nonlinear Transformations Can Change the Estimand
A nonlinear transformation does not generally multiply every genetic value by one constant. For example, taking a logarithm can make differences correspond more closely to relative rather than absolute changes. Squaring can make high values contribute disproportionately and can remove distinctions between positive and negative values. In each case, the genetic covariance and variances for the transformed traits must be considered on the transformed scale.
The following finite example isolates this point. It uses four equally likely inherited genetic backgrounds and treats the listed genetic values as the values whose covariance is being summarized. It illustrates scale dependence across those backgrounds; it is not a claim that a nonlinear transformation preserves a strictly additive genetic model.
Worked Example: Squaring One Genetic Trait Changes the Correlation
Across four equally likely hypothetical genetic backgrounds, let the genetic values for traits \(X\) and \(Y\) be as shown. Both columns have mean zero.
| Background | \(A_X\) | \(A_Y\) | \(A_X^2\) |
|---|---|---|---|
| 1 | -2 | -2 | 4 |
| 2 | -1 | 1 | 1 |
| 3 | 1 | -1 | 1 |
| 4 | 2 | 2 | 4 |
First calculate the original variances and covariance using averages over the four backgrounds: $$ \operatorname{Var}(A_X)=\operatorname{Var}(A_Y)=\frac{4+1+1+4}{4}=2.5, \qquad \operatorname{Cov}(A_X,A_Y)=\frac{4-1-1+4}{4}=1.5. $$
Therefore the original correlation is $$ r_G=\frac{1.5}{\sqrt{2.5\cdot2.5}}=0.600. $$ Now transform \(X\) by squaring. The transformed values have mean \(\overline{A_X^2}=2.5\), so their centered values are \(1.5,-1.5,-1.5,1.5\). Their variance is $$ \operatorname{Var}(A_X^2) =\frac{1.5^2+(-1.5)^2+(-1.5)^2+1.5^2}{4} =2.25. $$ The covariance of the centered transformed values with \(A_Y\) is $$ \operatorname{Cov}(A_X^2,A_Y) =\frac{(1.5)(-2)+(-1.5)(1)+(-1.5)(-1)+(1.5)(2)}{4} =0. $$ Thus the transformed-scale correlation is \(0/\sqrt{2.25\cdot2.5}=0\), rather than \(0.600\).
Squaring changed which contrasts across backgrounds contribute to the genetic component. The original correlation cannot be carried over as though squaring were just a change of units. In an application, the appropriate transformed genetic component depends on the genetic model fitted to the transformed outcome.
A monotone nonlinear transformation may preserve the ordering of individuals but still change variances and covariances. Conversely, a transformation such as squaring can also change order or merge distinct values. Neither preservation of rank nor a high correlation between the original and transformed phenotype establishes that the genetic correlation is unchanged.
Binary Traits and Liability-Scale Conversion
For a binary trait, the observed outcome is often coded \(0\) or \(1\), but a genetic analysis may target a continuous latent liability. A standard liability-threshold model assumes a normally distributed liability with total variance one; an individual is a case when the liability exceeds a threshold. If population prevalence is \(K\), the threshold is \(t=\Phi^{-1}(1-K)\), and the standard normal density at the threshold is \(z=\phi(t)\). The observed case proportion in an ascertained sample is denoted by \(P\).
The multiplier depends on prevalence and ascertainment; it is not generally one. Consequently, an observed-scale heritability and a liability-scale heritability describe genetic variation relative to different trait scales. A liability-scale genetic correlation targets correlation between genetic effects on the latent liabilities, not automatically the correlation between effects on the observed \(0/1\) outcomes.
Worked Example: Converting Heritability at a Ten Percent Prevalence
Consider a hypothetical binary trait with population prevalence \(K=0.10\), sample case proportion \(P=0.10\), and observed-scale heritability \(h^2_{\mathrm{observed}}=0.0150\). For this prevalence, \(t=\Phi^{-1}(0.90)\approx1.2816\) and \(z=\phi(t)\approx0.1755\). The conversion multiplier, using the displayed value of \(z\), is $$ \frac{K^2(1-K)^2}{P(1-P)z^2} = \frac{(0.10)^2(0.90)^2}{(0.10)(0.90)(0.1755)^2} = \frac{0.00810}{(0.1755)^2(0.10)(0.90)} = \frac{0.00810}{0.0027720225} \approx 2.9221. $$
Thus $$ h^2_{\mathrm{liability}}=0.0150(2.9221)=0.04383. $$
Under the stated liability-threshold assumptions, the estimated liability-scale heritability is approximately \(0.0438\), or \(4.38\%\) of the standardized liability variance. This is not the observed-scale estimate of \(1.50\%\) repeated in different units: it is a conversion to a latent-liability estimand. If the sample had been ascertained so that \(P\ne K\), the multiplier would also reflect that difference.
For two binary traits, converting each trait’s variances does not by itself establish how their genetic covariance should be converted. If a model specifies that the liability-scale genetic components are exactly positive multiples of the observed-scale components, then the covariance must be multiplied by the product of those scale factors. In that case, the correlation is invariant by the linear-rescaling result. But thresholding is nonlinear, so that relationship is not guaranteed merely because a univariate liability conversion has been applied to each trait. A bivariate liability model, with its assumptions stated, is needed when the target is the correlation between latent liabilities.
Worked Example: Consistent Covariance Rescaling for Two Binary Traits
Suppose a hypothetical analysis reports observed-scale genetic variances \(0.0200\) and \(0.0100\), and genetic covariance \(0.00600\), for two binary traits. Their observed-scale genetic correlation is $$ r_{G,\mathrm{obs}} = \frac{0.00600}{\sqrt{0.0200\cdot0.0100}} = \frac{0.00600}{0.01414} \approx0.4243. $$
For illustration, suppose a specified model treats both liability-scale genetic components as their observed-scale counterparts multiplied by the positive factor \(\sqrt{c}\), where \(c=2.9221\) is the multiplier in the preceding univariate example. Consistent component scaling gives liability variances \(c(0.0200)=0.05844\) and \(c(0.0100)=0.02922\), and liability covariance \(c(0.00600)=0.01753\). Therefore $$ r_{G,\mathrm{liability}} = \frac{0.01753}{\sqrt{0.05844\cdot0.02922}} \approx0.4243. $$
This equality follows from the assumed common positive scaling, not from a general property of binary outcomes. If the two components are not related by the specified scaling, or if the target is the correlation between latent liabilities under a threshold model, the liability correlation must be estimated or derived under that model rather than inferred from this arithmetic alone.
Common Mistakes and Practical Interpretation
- Calling every transformation a change of units. The reasoning that a correlation is invariant applies to positive linear rescaling of the relevant genetic components. It does not establish invariance after a logarithm, square, rank transformation, or thresholding.
- Reporting a genetic correlation without its scales. State whether the traits are measured in their original units, transformed, or represented on a liability scale, and identify the genetic component being correlated.
- Converting variances but leaving covariance unchanged. A covariance must be adjusted consistently with both component scales. Otherwise, the resulting correlation need not correspond to either the original or intended transformed model.
- Treating the univariate liability correction as a bivariate correlation correction. The univariate formula converts a heritability under specified assumptions. It does not, on its own, identify the covariance between two latent liabilities.
- Interpreting correlation as causation or as identical effects. A genetic correlation summarizes covariance in genetic components in a target population and model. It does not show that one trait causes the other, nor does it imply that every variant has the same effect on both traits.
Check Your Understanding
Use the scale and genetic-component definitions in each question.
- If both traits are multiplied by positive constants, what happens to their genetic correlation? Explain using the covariance and variance scaling.
- Why does a negative rescaling of just one trait reverse the sign of the genetic correlation?
- In the four-background example, why does squaring \(X\) change the covariance with \(Y\), even though the backgrounds themselves have not changed?
- What do \(K\), \(P\), and \(z\) represent in the liability-scale conversion, and why can the multiplier differ from one?
- Under what condition does consistently converting two genetic components preserve their correlation, and why is that condition not guaranteed by thresholding?