I do a measurement where I collect a set of data and fit it to a linear model using ordinary least squares. From that I get a slope, b and the standard error of it, s. Now I repeat the measurement N times and get N slopes and N standard errors. I want to derive the standard deviation of the slope. How do I incorporate the standard errors here?

  • $\begingroup$ This may be relevant: stats.stackexchange.com/questions/88461/… $\endgroup$
    – jon_simon
    Jan 26, 2018 at 3:21
  • $\begingroup$ Thanks, but that question is asking about a derivation of the standard error of the estimate. I am looking for the variance of a collection of such standard errors. $\endgroup$
    – student1
    Jan 26, 2018 at 3:25
  • $\begingroup$ Again, not quite what you're looking for (SE of SD, rather than SD of SE), but may be similar enough to help: stats.stackexchange.com/questions/156518/… $\endgroup$
    – jon_simon
    Jan 26, 2018 at 6:15

1 Answer 1


As I have less than 50 reputations, I'll post this as an answer. A standard way to deal with these kind of problems would be to fit a multilevel model. If $y_{ij}$ is your measurement of your outcome for unit $i$ at the $j$th wave of data collection, the model assumes that

$$ y_{ij} = \alpha_j + \beta_j x_{ij} + \epsilon_{ij}$$

(note the subscripts on the coefficients) where

$$(\alpha_j, \beta_j) \sim \mathcal N(\mu, \Sigma)$$

$$\epsilon_{ij} \sim \mathcal N(0, \Omega),$$ where $\epsilon_{ij}$ and $(\alpha_j,\beta_j)$ are independently distributed. Although the distributions do not have to be Normal, the Normal distribution is the natural choice in many applications. Also it is often assumed that $\Omega = \sigma I$, although this assumption can be easily relaxed. If you have more than one predictor in your regression, $\beta_j$ would be a vector. Note that $\Sigma$ (the covariance matrix of the regression coefficients) will contain the variance of the intercept, slope, and their covariances. You might test whether they are significantly different from zero with likelihood-ratio tests. However, you have to be careful as the null-hypothesis would lie at the boundary of the parameter space.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.