I conducted a field study where I gathered data about the distribution of visitors over the day in a fitness center. The goal of the study was to implement a treatment that would shift visitors from busy time slots to time slots where the fitness hall was significantly less congested. We wanted to achieve an even distribution of visitors over the day to gain a better yield.

The field experiment is now over and I have 2 data sets of 6 weeks (control and treatment) where I have the average amount of fitness visitors for every hour from 7:00 to 23:00 for every single day of those 6 weeks. I want to measure a few things:

a) Did the distribution change at all, and if so, in what parts of the day did visitor amount increase/decrease

b) Did the distribution of visitors become more equal over the day

c) Over the course of the 6 weeks, did the effect of the treatment lessen

How should I tackle this statistical analysis and what methods should I use?


a. In order to test difference of distributions there are many possible options: Kolmogorov–Smirnov (which "focus" on the maximum difference between distributions, or Von-Misses (which looks on the average difference). If you want to test the difference in specific hours you can average them throughout the weeks (this assumes that at each week the distribution is similar, so be careful if you are willing to assume that), and use a wilcoxon or t-test for each hour between the control and group. Remember to adjust for multiple comparisons, you could also use some hierarchical method such as BB.

b. Same as a but instead of comparing two empirical distributions you can compare it to the theoretical uniform distribution.

c. You could use the statistic of B for each day, and then regress it against the time.


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