I think the issue would be much simpler to understand if we reframe it in terms of estimation than hypothesis testing (https://jamanetwork.com/journals/jama/fullarticle/2813846) and viewing the stated CIs as “compatibility intervals”. “Compatibility” here loosely means “This range of treatment effect estimates is consistent with our data”. There are more detailed discussions one could get into re: the philosophical implications of this definition but we’ll sidestep those for now.
Essentially, the study estimates 2 treatment effects antibiotic use and mortality.
1. For antibiotic use, the trial’s results are compatible with anything from a 0.19 to 1.58 reduction (days) in the duration of antibiotic use with a PCT-guided strategy. The investigators initially hoped (hypothesized) that this compatibility range would exclude 0, which it indeed does. Therefore, they conclude that use of PCT-guided management of Antibiotic use is superior to standard management, insofar as reducing antibiotic use is concerned.
2. For 28-day mortality, the trial’s results are compatible with anything from a 2.18 percentage point decrease to a 5.32 percentage point increase in 28-day mortality with a PCT-guided strategy. The investigators initially hoped (hypothesized) that this compatibility range would exclude a 5.4 percentage point increase in 28-day mortality which it indeed does. Therefore, they conclude that PCT-guided management is “non-inferior” to standard of care, insofar as mortality is concerned.
The only difference between the first and second points is that the value I had hoped to exclude was 0 in the first and 5.4% in the second. Thus, non-inferiority basically boils down to saying “The value I want my compatibility interval to exclude is something other than 0”.
Where does the particular 5.4% threshold come from? It basically demarcates the “unacceptable” increase in risk which would be too much to tolerate from the perspective of the stakeholder (e.g., patient/physician/society). The rationale goes that:
A) As long as we can prove that our intervention improves something (e.g., lessens antibiotic use) by any amount (anything more than 0)
and
B) Does not increase the risk of some unfavorable outcome by an unacceptable amount (in this case, anything exceeding a 5.4% increase in 28-day mortality is considered unacceptable, and anything below that is acceptable)
Then statements like: “Care guided by measurement of PCT reduces antibiotic duration safely compared with standard care” can be made (where “safely” is demonstrated by the second point above).
This could all be analyzed in a Bayesian fashion to frame it in terms of posterior probabilities (which would, I think, be more intuitive to understand). In which case it would go something like: “We want to ensure a >95% probability of reducing antibiotic use by any amount” and a “<5% probability of increasing mortality by 5.4% or more” (for example).