Ignoring the clinic variable, consider a model for the log odds that respiratory status is classified as good, including the main effects of treatment and time (where time is regarded as a categorical variable with five levels), and their interaction.
Use generalized estimating equations (GEE), assuming separate pairwise log odds ratios (or separate pairwise correlations) among the five binary responses.
Construct a test of the null hypothesis of no effect of treatment on changes in the log odds that respiratory status is classified as good based on the empirical standard errors.

clinic id trt y0 y1 y2 y3 y4
1 1 P 0 0 0 0 0
1 2 P 0 0 0 0 0
1 3 A 1 1 1 1 1
1 4 P 1 1 1 1 0
1 5 P 0 0 0 0 0
2 11 P 0 0 0 0 0
2 12 A 0 0 1 1 1
2 13 A 1 1 1 1 1
2 14 P 1 1 0 1 1
2 15 P 1 0 0 1 1
2 16 P 1 1 0 0 0
2 17 P 1 1 1 1 1