For Matching on a categorical variable with N categories, will it suffice to create (N-1) binary features and match on them? I have data on patients who received different amounts of Occupational therapy (High Dose vs Low Dose) after a stroke. We are investigating if there are differences in recovery between patients from the 2 Dose groups. Because the Therapy Dose (High/Low) was not randomly assigned to patients, we have to account for confounders before concluding that differences between the groups are due to Therapy Dose. The confounders that we consider include Demographic information, medical information (stroke severity/damage), etc. We balanced for confounders by using Genetic Matching.
Current challenge:
Because we are using too many features (>50) to Match on, we are no longer getting a good post-matching balance on the Matching covariates with Genetic Matching. It was working fine when we were using fewer (<20) Matching Covariates. So now we trying to reduce the total number of matching covariates. Which brings me to a more conceptual question:
For Matching on a categorical variable with N categories, will it suffice to create (N-1) binary features and Match on those N-1 features? Or do we lose information in that way?
For example: 
Suppose there was a multicategorical feature indicating region of Brain damaged with the following 4 options:
1) Left hemisphere
2) Right hemisphere
3) Brainstem 
4) Other
In this case, would it suffice to create 3 binary features for 1-3 and then just match on those? I am thinking that 0(s) in the first 3 categories should indicate that the patient belongs to category 4. Would it be redundant to create 4 binary features to match on?
 A: This is a bit too broad of a question because "will it suffice" depends entirely on your goals and assumptions. Broadly, whatever method helps you achieve balance will suffice.
If your question is whether it is sufficient to achieve balance on 3 of the 4 categories, it again depends on what you mean by balance. If you mean exact balance, balancing on 3 categories will automatically balance the 4th. For approximate balancing, though, it is possible to achieve balance up to some threshold on the 3 included categories but not on the 4th. For example, consider the following scenario for the distributions of a 4-category variable in two treatment groups. In the treated group, the proportions might be .25, .25, .25, and .25. In the control group, the proportions might be .2, .2, .2, and .4. If your balance threshold is .05, then the first three categories are indeed balanced and the last category is imbalanced. This would yield bias in an effect estimate. You would need to ensure balance on each category to yield a relatively unbiased effect estimate.
If you're struggling to achieve balance using genetic matching, consider other forms of matching like propensity score matching. You can also combine them by including the propensity score as a covariate in a genetic match that otherwise relies on only a subset of the covariates. You can also try cardinality matching using the designmatch package, which allows you to specify balance constraints for each variable and it will attempt to find the largest matched set that meets those constraints.
If you're using MatchIt or Matching to perform the matching, make sure you assess balance using cobalt, which explicitly assesses balance on all categories, unlike those other packages. 
