Transforming summary statistics from logistic regression to the liability scale: application to genetic and environmental risk scores

Objective: Stratified medicine requires models of disease risk incorporating genetic and environmental factors. These may combine estimates from different studies, and the models must be easily updatable when new estimates become available. The logit scale is often used in genetic and environmental association studies; however, the liability scale is used for polygenic risk scores and measures of heritability, but combining parameters across studies requires a common scale for the estimates. Methods: We present equations to approximate the relationship between univariate effect size estimates on the logit scale and the liability scale, allowing model parameters to be translated between scales. Results: These equations are used to build a risk score on the liability scale, using effect size estimates originally estimated on the logit scale. Such a score can then be used in a joint effects model to estimate the risk of disease, and this is demonstrated for schizophrenia using a polygenic risk score and environmental risk factors. Conclusion: This straightforward method allows the conversion of model parameters between the logit and liability scales and may be a key tool to integrate risk estimates into a comprehensive risk model, particularly for joint models with environmental and genetic risk factors.