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Phenotype normalization

Phenotype normalization is a critical preprocessing step in GWAS to ensure valid statistical inference, numerical stability, and comparability across cohorts.

Raw measures

Definition

The phenotype is analyzed in its original measurement scale:

\[ Y_i = \text{observed phenotype for individual } i \]

When appropriate - Binary traits (case–control) - Approximately normally distributed quantitative traits

Pros - Preserves biological units and interpretability

Cons - Sensitive to skewness and outliers

Residual (covariate and medication adjusted)

Definition

The phenotype is adjusted for covariates and medication effects using regression:

\[Y_i = \alpha + \mathbf{C}_i^\top \gamma + \mathbf{M}_i^\top \delta + \varepsilon_i\]
\[ Y_i^{\text{resid}} = \hat{\varepsilon}_i \]

where

  • \(\mathbf{C}_i\): age, sex, PCs, batch, center
  • \(\mathbf{M}_i\): medication indicators, dosage, or drug class

Medication adjustment strategies - Indicator-based covariate (most common) - Dosage or drug-class covariates - Pre-correction (phenotype shifting, e.g. +10 mmHg for BP) - Exclusion of medicated individuals (not recommended)

Pros - Removes systematic non-genetic effects - Improves power and reduces bias

Cons - Residuals may still be non-normal

Z score

Definition

Standardization to zero mean and unit variance:

\[ Z_i = \frac{Y_i - \mu_Y}{\sigma_Y} \quad \text{or} \quad Z_i = \frac{Y_i^{\text{resid}} - \mu_{\text{resid}}}{\sigma_{\text{resid}}} \]

Pros - Comparable effect sizes across cohorts - Stable regression behavior

Cons - Does not correct skewness - Sensitive to outliers

Rank-based inverse normal transformation (INT)

Definition

Transforms phenotype ranks to a standard normal distribution:

\[ Y_i^{\text{INT}} = \Phi^{-1}\left( \frac{r_i - c}{n + 1 - 2c} \right) \]

where

  • \(r_i\): rank of individual \(i\)
  • \(c = 3/8\) (Blom's transformation, commonly used; \(c = 0.5\) is also used in Rankit transformation)

Pros - Enforces normality - Robust to outliers - Controls type-I error

Cons - Effect sizes lose original scale - Alters genetic architecture

  • Raw → GWAS (binary traits)
  • Residual → Z (well-behaved quantitative traits)
  • Residual → INT (highly skewed traits)
  • Medication correction → Residual → Z / INT (clinical traits)

References