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
```

•

link
modified 15 months ago
•
written
15 months ago by
mobius • **90**
Related post, but with no answer:

Caclulate effect estimates and SE from Z scores

9.7k