- #1
Adel Makram
- 635
- 15
Here are conditions I use to define my problem:
1) I use cumulative distribution of 2 logistic functions g1(x) and g2(x) with g2 is translated to the right of the g1(x) on x-axis.
2) I make a transformation to eliminate both tails of the function which will not have a significant contribution to the result, so my transformation is:
f(x) = g(x) - g(α) / g(1-α) - g(α).
where g(α) can be chosen to be equal to 0.05 and g(1-α)= 0.95.
3) g2(x) is translated with amount r to the right side of g1(x).
The aim of the study is to determine the best odd ratio (OR) between g1(x)/1-g1(x) and g2(x)/1-g2(x). In other words, I am looking for minimizing this ratio by taking the first derivatives and making it equal to zero and then calculate x.
to make the matter more simpler, I assumed that the scale parameters of both g1(x) and g2(x) are equal to each other.
surprisingly, after calculation I got x=-∝ which minimizing the ratio against the intuition that x should equal to 1/2 r. Please look at the attached diagram where the area with pink shade represents the interval where x should lie. On both boundaries of this area, OR will be ∝ and OR should come to a minimum at the required x. I made an initial transformation y=e-x and n=yk where k is the scale parameter which I put it to be the same in 2 functions. So I got n=0 (trivial solution) which means y=0 only at x=-∝. So, how come the calculation came opposite to the right intuition that x should = 1/2 r.[/SUB][/SUB][/SUB][/SUB][/SUB][/SUB][/SUB][/SUB]
1) I use cumulative distribution of 2 logistic functions g1(x) and g2(x) with g2 is translated to the right of the g1(x) on x-axis.
2) I make a transformation to eliminate both tails of the function which will not have a significant contribution to the result, so my transformation is:
f(x) = g(x) - g(α) / g(1-α) - g(α).
where g(α) can be chosen to be equal to 0.05 and g(1-α)= 0.95.
3) g2(x) is translated with amount r to the right side of g1(x).
The aim of the study is to determine the best odd ratio (OR) between g1(x)/1-g1(x) and g2(x)/1-g2(x). In other words, I am looking for minimizing this ratio by taking the first derivatives and making it equal to zero and then calculate x.
to make the matter more simpler, I assumed that the scale parameters of both g1(x) and g2(x) are equal to each other.
surprisingly, after calculation I got x=-∝ which minimizing the ratio against the intuition that x should equal to 1/2 r. Please look at the attached diagram where the area with pink shade represents the interval where x should lie. On both boundaries of this area, OR will be ∝ and OR should come to a minimum at the required x. I made an initial transformation y=e-x and n=yk where k is the scale parameter which I put it to be the same in 2 functions. So I got n=0 (trivial solution) which means y=0 only at x=-∝. So, how come the calculation came opposite to the right intuition that x should = 1/2 r.[/SUB][/SUB][/SUB][/SUB][/SUB][/SUB][/SUB][/SUB]
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