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I know that we could use linear regression to find impact of one independent variable on dependent variable and calculate its $r^2$.

What if this event is not continuous and happens only once and want to calculate its impact. How can I do this?

For example: say Uber is giving out $10 worth coupon to every users on Valentine's day. I want to find its impact:

  • did it gather new users?
  • did it gather users who were not recently using uber to use it regularly?

I can get result however there will be a lot of other variables that affect above numbers, how can I measure strength of $10 coupon on valentine's day?

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I believe you are into hypothesis testing. For each of your questions, you can construct two groups: those that got the coupon and those who didn't. Then you can compare the groups, e.g. how frequently they use Uber. An elementary approach would be to use the t-test.

If your samples from the both groups are comparable, the "other variables that affect above numbers" should cancel out. But that's a question of experiment design. E.g. you could take care that the non-Valentine group was also taken in February (weather might influence Uber usage), on a same weekday (people might use Uber differently on weekends), in the same city etc.

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