Please read the problem till the end. It may appear first that this problem
was answered in earlier posts, but it is not so. I have read all the related
posts.
Problem: Suppose I have two data sets (for two treatments), G and A. I run two logistic regressions
for G and A:
\begin{eqnarray*}
\log \left[ \frac{\Pr (R)}{1-\Pr (R)}\right] _{G} &=&\beta _{0G}+\beta
_{1G}X+\beta _{2G}Y \\
\log \left[ \frac{\Pr (R)}{1-\Pr (R)}\right] _{A} &=&\beta _{0A}+\beta
_{1A}X+\beta _{2A}Y.
\end{eqnarray*}
Based on the estimates of logistic regressions, I have two lines:
\begin{eqnarray*}
x_{G}^{\ast } &=&-\frac{\hat{\beta}_{0G}}{\hat{\beta}_{1G}}-\frac{\hat{\beta}%
_{2G}}{\hat{\beta}_{1G}}Y \\
x_{A}^{\ast } &=&-\frac{\hat{\beta}_{0A}}{\hat{\beta}_{1A}}-\frac{\hat{\beta}%
_{2A}}{\hat{\beta}_{1A}}Y.
\end{eqnarray*}
QUESTION: How do I test that $|\frac{\hat{\beta}_{2G}}{\hat{\beta}_{1G}}|>|%
\frac{\hat{\beta}_{2A}}{\hat{\beta}_{1A}}|$, i.e., slope of $x_{G}^{\ast }$
is greater than the slope of $x_{A}^{\ast }?$
Progress so far (Jan 26, 2016): I came across a document, "Ratios: A short
guide to confidence limits and proper use" by Franz (2007), which mentions
methods such as Fieller, Taylor (or Delta), Bootstrap and Regression.
However, all these methods are based on say, $\rho =\frac{E[Z]}{E[W]}$,
where $Z$ and $W$ are random variables, and a test statistic is derived from
the sample of $N$ paired measurements $(z_{i},w_{i})$, with $i=1,2,…,N$.
Applied to my problem, $Z=\hat{\beta}_{2}$, and $W=\hat{\beta}_{1},$ where $%
\hat{\beta}_{1}\sim N(\beta _{1},s.e.(\hat{\beta}_{1})),\hat{\beta}_{2}\sim
N(\beta _{2},s.e.(\hat{\beta}_{2}))$ (asymptotically; I have large number of
data points). However, I don't have paired measurements such as $\left( \hat{%
\beta}_{11},\hat{\beta}_{21}\right) ,…,\left( \hat{\beta}_{1N},\hat{\beta}%
_{2N}\right) .$ I am kind of stuck here. Will appreciate any help.
Best Answer
If $\beta_{1} \ne 0$, there is a limiting distribution theory. You can find the limiting distribution by using the Delta method.
Let the relevant coefficients be $\beta$, you have:
$\sqrt{n}\left(\hat{\beta} - \beta\right) \overset{d}\rightarrow N\left(0,\Sigma\right)$
For a covariance matrix $\Sigma$ (this is the covariance matrix for all $4$ coefficients in the two ratios).
Define the function $g\left(x_{1},y_{1},x_{2},y_{2}\right) = \frac{x_{1}}{y_{1}} - \frac{x_{2}}{y_{2}}$.
Assuming that $\beta_{1G}$ and $\beta_{1A}$ are non-zero,
$\sqrt{n}\left(g\left(\hat{\beta}_{2G},\hat{\beta}_{1G}, \hat{\beta}_{2A}, \hat{\beta}_{1A}\right) - g\left(\beta_{2G}, \beta_{1G}, \beta_{2A}, \beta_{1A}\right)\right) \overset{d}\rightarrow N\left(0, \nabla g\left(\beta_{2G}, \beta_{1G}, \beta_{2A}, \beta_{1A}\right)^{\top} \Sigma \nabla g\right)$.
Where $\nabla g$ is the gradient of $g$.
Now that you have the asymptotic distribution of the difference of the ratios, you can form hypothesis tests using standard techniques.