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Some statistical formulas for calculating the 95% confidence intervals for estimators such as Relative Risk (RR) and Hazard Ratio (HR):

95% CI for the log RR = logRR ± 1.96 × SE

95% CI for the log HR = logHR ± 1.96 × SE

They tell me one fact that log(RR) or log(HR) follows normal distribution, but how and why?

I recalled my statistic teacher told me that, take simple linear regression as an example, the fact that $Y$ data independent with normal distribution leads to normally distributed $\hat{\beta_0}$ and $\hat{\beta_1}$.

I read this from an article by David C. Howell

Perhaps when Hogg and Craig (and many other people) say that b is normally distributed, what they really mean is that b is asymptotically normally distributed. In other words if each sample were infinitely large the distribution would be normal.

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The estimators for parameters in a regression model are functions of your data. If your dataset is extremely large then these estimators follow a normal distribution. However for sufficiently sized datasets ($n=30$ is a rule of thumb) the central limit theorem kicks in and your estimators are sufficiently close to normal to construct confidence intervals etc. based on the normal distribution. This behavior is independent of the distributions of $X$ and $Y$.

This means that if you repeat the experiment over and over again, the parameter estimates over all these experiments will be seen to follow an normal distribution around the true parameter. In practice of course you do the experiment only once and rely on this asymptotic result.

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