Polygenic Risk Scores: Odds ratio or Beta-coefficient?

Question

[Volka]

Hi all, I have recently started a project on modelling a polygenic risk score model to evaluate its utilitiy in predicting a certain disease. After doing some reading, I have come across various models for unweighted and weighted Genetic Risk Score models.

I am wondering about the use of odds ratio (OR) versus the Beta-coefficient of each SNP variant in a risk score model. For instance, here they used the Beta-coefficient in their model, while here they used the odds ratio. Is there any difference in using the odds ratio versus the Beta-coefficient in a risk score model? Also, I noticed that some papers use log(OR) rather than ln(OR), is there a major difference between both?

Thanks!

Answer 1

[Kevin Blighe]

Hey,

The odds ratio (OR) is the exponent of the beta coefficient. The beta coefficient itself is the per unit increase/decrease in the exposure. A practical example will explain it better:

create random data with a condition A + B ('outcome') and gene1 ('exposure')

modeling <- data.frame(
    condition=factor(c(rep("A",100), rep("B",100)), levels=c("A", "B")),
    gene1=c(runif(100), runif(100)))
head(modeling)
  condition     gene1
1         A 0.3607443
2         A 0.3268301
3         A 0.4237005
4         A 0.7621534
5         A 0.1456797
6         A 0.3201094

Note that we have set A as the reference level.

create a binomial logistic regression model, with gene1's expression 'predicting' the outcome

model <- glm(condition ~ gene1, data=modeling, family=binomial(link='logit'))
summary(model)

Coefficients:
            Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.03767    0.30014  -0.126    0.900
gene1        0.07294    0.51258   0.142    0.887

Here, the beta coefficient for gene1 in relation to condition B versus A is 0.07294. So, gene1 increases in expression in condition B (if it decreased, the beta coefficient would be negative). This is not a statistically significant finding, though, with p=0.887.

We can also test the gene1 via the Wald test on the beta coefficient:

require(aod)
wald.test(b=coef(model), Sigma=vcov(model), Terms=2)
Wald test:
----------
Chi-squared test:
X2 = 0.02, df = 1, P(> X2) = 0.89

obtain the odds ratio and upper / lower confidence intervals (CIs):

exp(cbind(OR=coef(model), confint(model, level = 0.95)))
                   OR     2.5 %   97.5 %
(Intercept) 0.9630288 0.5333548 1.737047
gene1       1.0756700 0.3930383 2.949061

So, odds ratio is just 1.1, which, as you can tell, is not huge and only reflects a slight increase.

log OR

The log OR is just the natural logarithm of the OR. With regard to why we may even want to use log OR over OR, well, there are probably many reasons. One is that we can calculate the Z score from the log OR:

OR <- 1.0756700
lowerCI <- 0.3930383
upperCI <- 2.949061

logOR <- log(OR)
logORlowerCI <- log(lowerCI)
logORSE <- (logOR - logORlowerCI) / 1.96

Then calculate Z:

logOR / logORSE
[1] 0.1420052

Kevin

[dandanli0365]

why can't lnOR be used for z-score?

[Kevin Blighe]

...about what are you talking?

?log
‘log’ computes logarithms, by default natural logarithms...

Answer 2

[Mike]

I'm a student and do research in this area and after a lot of reading, I'm pretty sure you want to use the log odds (Betas) as your weights for your model. The beta is the true weighting even though the OR is more often reported, I believe because it is easier for humans to understand. Also, a lot of the time people say Log(OR) they mean LN(OR). In this field, or generally in bioinformatics, it appears that LN is the default type of logarithmic transformation, so unless you see someone write Log10, they probably mean LN.

Hope this helps!

Mike

[Kevin Blighe]

Indeed

[zhoufeng2ye]

So when you said he should use log odds ratio here, you mean the log10 or ln?

[Kevin Blighe]

We mean the original beta coefficients, referred to as the 'Estimate' from any regression model summary table in R.

library('MASS')
data('menarche')
fit <- glm(cbind(with = Menarche, without = Total - Menarche) ~ Age,
  family = binomial(link = 'logit'), data = menarche)
summary(fit)

Coefficients:
             Estimate Std. Error z value Pr(>|z|)    
(Intercept) -21.22639    0.77068  -27.54   <2e-16 ***
Age           1.63197    0.05895   27.68   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Then we can extract the log odds ratio / beta coefficient / estimate and convert to OR:

coef(fit)['Age']
     Age 
1.631968 

OR <- exp(coef(fit)['Age'])
OR
     Age 
5.113931

Then, if you already have the OR and want to convert back to a log odds ratio / beta coefficient/ estimate:

log(OR, base = exp(1))
     Age 
1.631968

[Mike]

I mean ln, which if you use R, is the default for the log function.

Answer 3

[willebaldo.garcia.m]

Also, your parameter of choice, Beta/OR, will affect the units of the PRS on the individuals. The association input dataset and its units will affect both the ranges of the PRS distribution and risk units. The units are also affected by binary or quantitative nature of the studied traits.

https://doi.org/10.1038/s41596-020-0353-1