Question regarding the unit of beta for binary phenotypes

[Chen Lou]

Dear genomicSEM developers,

Thank you very much for your excellent paper on GenomicSEM. I have a question regarding the unit of beta for binary phenotypes.  

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In "Genomic structural equation modelling provides insights into the multivariate genetic architecture of complex traits" (Nature Human Behaviour 2019), the logistic regression coefficient (log-odds ratio) was standardized as follows in GenomicSEM.

The original logistic regression beta (log OR scale) was converted to a standardized SNP effect:

σ2SNP  = 2maf(1-maf) 

v = case/N_total

b_logit, = Z / sqrt( v(1−v) * N_total *σ2SNP)

se(b_logit) = 1 / sqrt( v(1−v) * N_total *  σ2SNP   )

Then the SNP effect was further scaled to the unit-variance liability scale by dividing by the square root of total liability variance:

b_std = b_logit / sqrt( σ2SNP * (b_logit)^2 + π^2/3 )          (Eq. 1)

se(b_std) = se(b_logit) / sqrt(  σ2SNP   * (b_logit)^2 + π^2/3 )

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In "Pervasive Downward Bias in Estimates of Liability-Scale Heritability in GWAS Meta-analysis: A Simple Solution"(Biological Psychiatry， The authors are members of the Genomic SEM research team.), b* is defined instead as the linear regression coefficient of a standardized binary phenotype (assuming balanced case–control design, The 0/1 phenotype was standardized to have mean 0 and variance 1. ):

b* = Z / sqrt( 4 v(1−v)*n_total *σ2SNP)           (Eq. 2)

SE(b*) = 1 / sqrt( 4 v(1−v)*n_total* σ2SNP   )

and approximately:

b_logit ≈ 2 b* → b* ≈ 0.5*b_logit

I am not certain whether this  b*   is on the unit-variance liability scale.

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I have two questions:

1. Between Eqs. (1) and (2), MTAG seems to standardize the binary phenotype to mean 0 and variance 1 and adopts Eq. (2) (4v(1-v)n is input to MTAG as Neff), whereas Genomic SEM uses Eq. (1). Could you clarify the practical distinction between these two formulations?
2. Conceptually, how do Eqs. (1) and (2) differ? They appear numerically close. In Eq. (2), does b* correspond to the unit-variance liability scale? I have not yet conducted a systematic empirical comparison.

Many thanks for your clarification and time!

Best regards,

Lou Chen

Wenzhou Medical University

[Elliot Tucker-Drob]

In the 2019 NHB paper we explain that it is necessary to convert the logistic regression coefficients to the standardized liability scale (using the (pi^2)/3 term) in order to place them on the same scale as the elements in the genetic covariance matrix, which are also scaled relative to phenotypic variances of 1.0. The sumstats function performs this conversion for use with userGWAS.

In the 2023 BP paper, we derive the appropriate N for a GWAS meta-analysis of case-control cohorts. We use the relationship between the logistic and linear probability models as an element within our proof. This does not require converting the logistic coefficient to the standardized liability scale and hence we do not use the (pi^2)/3 term.

Our explication of how Genomic SEM relates to MTAG is in the supplement of the 2019 paper. The notation and formulations are very different even though the models themselves are very close. Therefore, providing a simple answer to how and why two specific terms differ between the two frameworks would be difficult.

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[Chen Lou]

Subject: Clarification on Transforming GWAS Results to Log(OR) Scale

Dear Prof. Tucker-Drob,

Thank you so much for your detailed explanation.

I still have a question regarding the transformation of GWAS results.

For a binary trait, how could we transform the genomicSEM results (on the liability scale of unit 1) back to the log(OR) scale? Similarly, for MTAG (on a linear scale of unit 1), is there a way to convert the results back to the log(OR) scale?

I was wondering if I could estimate log(OR) from the z-score using the following formula:

log(OR) ≈ z / sqrt(v(1 - v) × 2 × n_total × f(1 - f))

The rationale behind this comes from the paper "Causal Associations Between Risk Factors and Common Diseases Inferred from GWAS Summary Data" (GSMR). As per the paper, we know that the sampling variance (SE²) of the log(OR) estimate from a logistic regression is approximately 1 / [v(1-v) × n_var(x)], where v is the proportion of cases, x is the genotype indicator for a SNP (coded as 0, 1, or 2), and n is the sample size. Assuming Hardy-Weinberg equilibrium, var(x) = 2f(1 - f), where f is the allele frequency. Therefore, we can approximate the SE as 1 / sqrt(v(1 - v) × 2nf(1 - f)). With the z-statistic, we can then further estimate log(OR) ≈ z / sqrt(v(1 - v) × 2nf(1 - f)).

The unit being 1 does seem challenging for downstream analyses like Mendelian Randomization, which is why I am reaching out for clarification.

Thank you for your time and assistance.

Best regards,
Chen Lou

[Elliot Tucker-Drob]

Hi Chen,

I can't take the time to check through your equations in detail. However, I will say that it is very common in GWAS to rescale betas and SEs for different applications, back out betas from Z and N, etc... This is something that people working with GWAS sumstats have been doing since long before we ever introduced Genomic SEM. As you have already noted, we commonly perform such transformations in our own work, and document them in the methods and/or supps of various papers. 

Best wishes,

Elliot

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