survival analysis, event at t=1 Hope someone can clarify this for me. I drew out a timeline that'll hopefully allow me to explain my question clearly.
I'm confused as to how clinical trials record the time of an event. To start from the beginning - let's say patient is recruited at t = 0 (e.g. Aug 2nd at 1pm), if an event is recorded as t = 1, does this mean the event occurred in the first interval [0,1) (Aug 2nd) or the second interval [1,2) (Aug 3rd)? Alternatively, is [0,1) t = 1 ? or is [1,2) t = 1?
and a second follow-up question: if we really consider Aug 3rd day 1, what if event happened on Aug 2nd between 1pm and 11:59pm?? do we exclude this patient from the study??

 A: Generally, the time at which the event is discovered to have occurred is the time that is used for the endpoint. In clinical trials, the screening interval is prespecified for the endpoint so as to improve the timeliness of detection.
To expand: we also don't use "time 0" in analyses. So at any time after randomization (or appropriate time point) up until 48 hours after the fact is considered a "day 1" event. This is simply a "coarsening of time".
Similarly, any time prior to randomization up to 48 hours is generally coded as day -1. Such an endpoint would not be analyzed in the set. Any subsequent event could be captured in analysis provided prior incidence is not an exclusion criterion. For instance, in cancer, progression of disease may occur prior to randomization during screening, but the patient will be followed up again after starting study treatment to see if disease progresses further or the patient dies.
A: Speaking directly to your graphic and question, the person recorded as having experienced an event by time 1 experienced it after $\boldsymbol{t=0}$ but no later than $\boldsymbol{t=1}$ on your timeline.
You can think of time equal $t$ as compassing the interval of possible times $>t-1$ and $\leq t$. So the value of $t=1$ corresponds to events occurring after $t=0$, up to an including $t=1$. In an everyday sense, this is somewhat akin to years of life, as in, someone who is 6 month and 7 days (and $X$ hours, and $Y$ minutes, etc.) old is in their first year of life; after their first birthday, they are in their second year of life, etc.
Your second question is a tad unclear, because you seem to be mixing $t=0$ meaning a point in time, and $t=0$ meaning an interval of time. I think when you land on $t=0$ as a point and consider the period after that but up to $t=1$ the first interval, you will be set.
