# How to identify the ideal dose, when the doses were given to different numbers of patients?

I have six groups of patients. The groups contain different numbers of patients (i.e., 30 in group1, 70 in group2, etc). Each group gets a different dose of a new drug (i.e., group 1 - 20 mg, group 2 - 30 mg, etc.). Each patient is asked to rate the severity of their symptoms (from 1 - very mild, to 6 very bad) before and again one week after the new medicine is administered.

I plan to subtract the post treatment score from the pre-treatment score (i.e., negative numbers mean the patient improved):

Group 1
------------
pre   post    Change
patient 1         5      3        -2
patient 2         6      2        -4
...


Then I would have a table as follows:

                  Group 1     group 2      group 3  group 4  etc.....
Change Patient 1       -2          -5           -4       +1
Change Patient 2       -4          -4           -1       +2


I then need a statistical test to work out which dose of medicine is better at reducing the symptoms.

I have been doing some reading and wondered whether the ANOVA test would do it? But apparently it would just tell me if there is a difference and not specifically which dose is best. How can I deal with that?

• Were the patients randomly assigned to the different dosages? Commented Jul 11, 2015 at 14:12
• thanks for you prompt reply, they were randomly allocated
– JDB
Commented Jul 11, 2015 at 14:17
• Your table layout strongly suggests that each patient participated in multiple groups, with each row describing one individual. If so, you will need to track the temporal sequence of their participation and accommodate that in the modeling. If not (which I suspect), then this is an inappropriate way to tabulate (or even maintain) the data, because it risks confusing distinct individuals and makes it difficult or impossible to associate patient-specific attributes with these results.
– whuber
Commented Jul 15, 2015 at 22:45
• BTW, did you use any control groups, such as groups receiving a (blinded) placebo? If not, you can still identify the best dose--but you will be unable to use these data (by themselves) to support any claims concerning whether it is any better than a zero dose.
– whuber
Commented Jul 15, 2015 at 22:49

$$\hat \beta_1 {\rm dose} + \hat \beta_2 {\rm dose}^2$$
Then the derivative would be: $$\frac{d\ {\rm rating}}{d\ {\rm dose}} = \hat \beta_1 + 2 \hat \beta_2 {\rm dose}$$
$${\rm dose} = \frac{-\hat \beta_1}{2 \hat \beta_2}$$