Consider the following problem: Suppose someone wanted to determine the ideal dosage of an epinephrine shot in order for the shot to be successful 90% of the time. The person is also not willing to use more than 0.5 mg in a dose because it is too expensive. Otherwise, you have no prior knowledge about the proper dosage.
The person is testing the epinephrine shot on mice. The shot is considered to succeed in a test case if the mouse survives the allergic reaction after receiving the shot, and is a failure if the mouse dies. Also, for this example, assume that the dose will never be so high that cause of death to the mouse is always the allergic reaction and not the shot. This is intended to be a simplified problem.
I am dealing with a similar (albeit less morbid) problem at work. Here is the method I am currently using: 1. Choose an initial guess for the ideal dosage. For example- I choose .005 mg. I test the shot on 100 mice to get a sample proportion. Say the shot has a 25% success rate, which is insufficient. 2. I increase my guess, strictly using gut feeling and with no statistical consideration. Now I test a shot with .15 mg on another 100 mice. Say that shot has a 35% success rate. 3. I continue this test process, increasing the dosage when the success rate is too low, and decreasing the dosage when the success rate is too high, until I achieved a 90% success rate.
I think there is a more efficient way to go about the problem of choosing the 'next dosage'. As an added complication, it is unfavorable to test higher 'doses' because those experiments are significantly more expensive than the low dose experiments. In my work application, I notice that the plot of dosage vs. probability success curve is 'S' shaped.
Does anyone have any hints or ideas on how to go about this?