What do you mean by test_date[2] ?
As I understand it, this would be the value of the column test_date for the second observation.
This would imply that whatever the calculation in the case_when clause the result period for all observations would be the same.
How could then the value for the first observation be NA ?
Please indicate exactly in words how you would like to calculate period.
And please provide you input as described e.g. in
https://forum.posit.co/t/faq-whats-a-reproducible-example-reprex-and-how-do-i-do-one/5219