I have a number of thoughts.
First, it is unlikely that there will be enough variation in when (in terms of time of day) a person might have to give up a seat for an elderly person. Public transport is only likely to be that crowded during rush hours and those occur at fairly consistent times.
Second, my feeling is that there will be a much stronger correlation with other factors than with the time of day. I would think that certain kinds of people would always give up their seat to an elderly person regardless of the time of day. You might think that people would be less likely to give up their seats when it is late and they are tired, but it is likely that a person who would give up their seat would realize that the elderly person is tired too. I do not think that many people would say to themselves “I would give my seat to that person, but I’m too tired and I can’t be bothered.
Finally, while this is certainly quantitative, it would be hard to study properly. You would have to control for other factors than time of day. In other words, you would have to have people who were similar in all ways, with time of day being the only variable that was allowed to change. You would also have to figure out a way to determine which people were willing and which were unwilling to give up their seats.