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In biological sciences, ($CI$) stands for the Combination Index which is a conventional method for dose-response assessment and drug interaction analysis. It can be defined as:

$$CI =

\frac{\bar{D}{1} \hat{\beta}{1}}{ln(\bar{SF}{1}) - \hat\alpha{1}} + \frac{\bar{SF}{1} - \hat{\alpha}{2}}{{0.5} - \hat\alpha_{2}}$$

Notes:

  1. $\bar{D}_{1}$ is independent of $(\hat\alpha_{1}, \hat\beta_{1})$.
  2. $(\hat\alpha_{1}, \hat\beta_{1})$ are independent of $(\hat\alpha_{2})$.
  3. $log(SF_{1}) = \alpha_{1} + \beta_{1} x + \epsilon$
  4. $E(\bar{D}_{1}) = \mu_{D}$, and $Var(\bar{D}_{1}) = \sigma^{2}_{D}$

I'm trying to find $E({log(CI))}$ and $Var(log(CI))$. I know this can be done using delta method, but I'm not sure how... your help is appreciated.

Thanks!

Michael Bishop
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user9292
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  • and $\epsilon$? – conjugateprior Apr 18 '12 at 12:36
  • $\epsilon$ is the random term for the linear regression model which assumed to be normally distributed with mean zero and fixed variance. – user9292 Apr 18 '12 at 15:40
  • OK, then personally I'd put away the pencil and paper and simulate a lot of values of $CI$ using the distributions of $\epsilon$ and $D_1$ with the parameter estimates you've got, then log 'em all and compute your quantities of interest. Although I appreciate that's not your question, it seems like it would solve the practical problem of knowing how certain to be about $\log CI$. – conjugateprior Apr 18 '12 at 16:01
  • I did find both the mean and variance of $log(CI)$ using bootstrapping, and now I'm trying to find an analytical form... – user9292 Apr 18 '12 at 16:59

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