I have a linear model with sine and cosine wave as predictor. Y~$\alpha$sine(...)+$\beta$cosine(...). From the parameter estimation, I know the estimation of the coefficients are: $\hat\alpha$ = 0.65(sd = 0.15), $\hat\beta$ =0.54 (sd=0.15). The correlation between $\alpha$ and $\beta$ is 0. Using bivariate normal approximation, I can plot the 0.95 confidence ellipse:

p <- rmvnorm(1000, c(0.65,0.54), matrix(c(0.15^2,0,0,0.15^2),2,2))
plot(p, pch=20, xlab="sine", ylab="cosine")
ellipse(mu=colMeans(p), sigma=cov(p), alpha = .05, npoints = 250, col="red") 

Then I wish to calculate the confidence interval for $\sqrt{\alpha^2+\beta^2}$. Any suggestions is appreciated.

New contributor
Huang Rui is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
  • $\begingroup$ Rewrite your model in the form given in my answer at stats.stackexchange.com/a/77865/919, which incorporates $\sqrt{\alpha^2+\beta^2}$ directly, and compute a confidence interval for it using the usual techniques. $\endgroup$ – whuber Aug 14 at 16:03
  • $\begingroup$ your answer does not include calculating the confidence interval. @whuber $\endgroup$ – Huang Rui Aug 15 at 20:22
  • $\begingroup$ As I indicated, that's a standard textbook calculation--there's little more to say about it. Your software will automatically report the confidence interval, for instance. $\endgroup$ – whuber Aug 15 at 21:13

Your Answer

Huang Rui is a new contributor. Be nice, and check out our Code of Conduct.

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.