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

Question

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?

Answer 1

The standard ANOVA assumes your data are normally distributed within each group. That won't be the case here, because your response variable is an ordinal rating. In addition, difference scores can be a viable option in some cases (where the groups are not necessarily comparable at the start, and where differences are meaningful), but that isn't a good choice in your case: because you randomized your patients, you can consider the distributions to be similar beforehand, and the ordinal nature of the ratings makes the differences nonsensical.

Your best option will be to use ordinal logistic regression. I would include the previous rating as a covariate. Lastly, since your groups are associated with a meaningful number (e.g., 20 mg), I would not think of them as nominal groupings as in an ANOVA, but represent them as numbers on a continuum (20, 30, etc.) using a single continuous variable.

To identify the ideal dose, you could also include the squared dose as an additional variable. That allows the model to fit a curved relationship. The minimum of the curve can be identified by taking the derivative with respect to the dose, setting it equal to 0, and solving for the dose. To do this, you can ignore the other variables that don't include the dose (e.g., the pre- rating, demographic covariates, etc.). You can also work on the scale of the linear predictor—you needn't be concerned about the ordinal nature of the response or trying to figure out how to work with the link function to get actual response categories. If this simplified version of your model is:
$$ \hat \beta_1 {\rm dose} + \hat \beta_2 {\rm dose}^2 $$
Then the derivative would be: $$ \frac{d\ {\rm rating}}{d\ {\rm dose}} = \hat \beta_1 + 2 \hat \beta_2 {\rm dose} $$
And the minimum would be:
$$ {\rm dose} = \frac{-\hat \beta_1}{2 \hat \beta_2} $$
To answer your explicit question, the fact that the doses were given to different numbers of patients does not matter. (It is true that, from the standpoint of experimental design, it is a strange choice; for example, it may leave you with suboptimal power. But that is water over the dam now, and it doesn't affect any of the points discussed above.)