Question: How to calculate effect(beta) and SE from z-score and p-value?
1
anthouliouslive10 wrote:

Hello,

I have a table containing SNPs and their z-score and p-values. I want to calculate their effect and SE.

How can I do this? Is there any code I can use to do this in R?

snp R • 7.0k views
modified 10 months ago by atlas.akhan0 • written 2.4 years ago by anthouliouslive10

Related post, but with no answer:

Caclulate effect estimates and SE from Z scores

1
mobius90 wrote:

Assuming that the model that was fit is from a simple linear regression

The formula is as follows.

Var(Y|X) is the variance of the residual under linear regression and N is the sample size

``````The standardized beta (i.e assuming both Y and X are transformed to have unit variance and mean zero) = Zscore*sqrt(Var(Y|X)/N)

Var(Y|X) = 1/(1 + (Zscore*Zscore)/N)
Var(beta) = Var(Y|X)/N
``````

Here's the code in R that verifies this:

``````re <- lapply(1:1e4, function(u){
x <- rnorm(1e3, 0, 2)
y <-  .6*x + rnorm(1e3)

ft <- summary(lm(scale(y, scale = T)~scale(x, scale = T)))
t_stat <- ft\$coefficients[2,3]
beta_o <- ft\$coefficients[2,1]
se_beta_o <-  ft\$coefficients[2,2]

sigma_sqrd <- 1/(1+(t_stat^2/1e3))
beta_est <- t_stat*sqrt(sigma_sqrd/1e3)
se_beta_est <- sqrt(sigma_sqrd/1e3)

data.table::data.table(beta = beta_o, se_beta = se_beta_o, t_stat = t_stat,
betahat = beta_est, se_betahat = se_beta_est, t_stat_est = beta_est/se_beta_est)
})

re <- do.call(rbind, re)

plot(density(re\$beta), col = "red", lwd = 1)
lines(density(re\$betahat), col = "blue", lwd = 1)

plot(density(re\$se_beta), col = "red", lwd = 1)
lines(density(re\$se_betahat), col = "blue", lwd = 1)

plot(density(re\$t_stat), col = "red", lwd = 1)
lines(density(re\$t_stat_est), col = "blue", lwd = 1)
``````

Here's snapshot based on what I ran beta, se_beta, and t_stat are the truth betahat, se_betahat, and t_stat_est are estimates based on the formula above

``````      beta    se_beta   t_stat   betahat se_betahat t_stat_est
1: 0.7634574 0.02044428 37.34332 0.7631385 0.02043574   37.34332
2: 0.7539762 0.02079386 36.25955 0.7536504 0.02078488   36.25955
3: 0.7716880 0.02013227 38.33090 0.7713754 0.02012411   38.33090
4: 0.7674729 0.02029308 37.81944 0.7671570 0.02028473   37.81944
5: 0.7703690 0.02018282 38.16953 0.7700553 0.02017461   38.16953
``````
0
atlas.akhan0 wrote:

You can use this equation:

Beta = z / sqrt(2p(1− p)(n + z^2)) and

SE =1 / sqrt(2p(1− p)(n + z^2))

Where p is the frequency of the imputed SNP, you could use out reference panel to calculate p. For reference please go to

https://images.nature.com/full/nature-assets/ng/journal/v48/n5/extref/ng.3538-S1.pdf

Link works fine for me. This is the paper:

https://www.nature.com/articles/ng.3538