Find the confidence interval for the odds ratio, where $OR=e^{\hat{B_1}}=3.5701$, $\hat{B_1}=1.2726$, $\sigma(\hat{B_1})=0.5016$ with 95% confidence.

First I followed the idea from notes that I found $$CI(\hat{B_1},0.95)=\hat{B_1}\pm1.96\sigma(\hat{B_1})=(0.2894;2.2557)$$ then $$CI(OR,0.95)=(e^{0.2894};e^{2.2557})=(1.3356;9.5420)$$ But in some other notes they solve it with delta method.

If $X\sim N(\mu,\sigma^2)$ then $F(X)\sim N(F(\mu),\sigma^2[F'(\mu)]^2)$ , as $$F(\mu)=F(\hat{B_1})=3.5701$$ $$\sigma^2=(0.5016)^2=0.2516$$ $$[F'(\mu)]^2=[e^{\hat{B_1}}]^2=e^{2\hat{B_1}}=12.7458\rightarrow \sigma^2[F'(\mu)]^2=3.2068$$ thus $$e^{\hat{B_1}}\sim N(3.5071;3.2068)$$ $$CI(e^{\hat{B_1}},0.95)=3.5701\pm 1.96\sqrt{3.2068}=(0.0602;7.08)$$

The results are different, what is the right method?

  • $\begingroup$ The delta method is an asymptotic argument, and I don't think it applies here. $\endgroup$ – Greenparker May 15 '16 at 20:31
  • $\begingroup$ @Greenparker The first method is fine? I've seen the delta method for odds ratio in "Categorical Data Analysis, Alan Agresti", but I don't understand it well. $\endgroup$ – user72621 May 15 '16 at 21:00

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