The way to test causation is to introduce ‘interventions’, whereby you change the believed cause and see whether it has an effect. So, for example, if you can find a city where people have switched from cars to PT/bikes in a short period of time and show that health has improved, you could make a stronger claim, or alternatively, a plce where health went down the tubes as a result of a large mode-shift away from PT. I do not know of any strong examples of the former, but perhaps shanghai is a good example of the latter?
Even better if you have a comparable experiment in the opposite direction: a place where obesity increased for other reasons, and a mode shift occurred. Perhaps Samoans, whose weight is a sign of social standing (sitting?) have changed their mode in the last 100 years, but not had a change in average BMI.
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