Creating a biased sample with two variables matching distributions from another dataset

Question

I have a biased sample from a user activity dataset with known distributions of two variables, the amount of times a user logged in during last week (Poisson distribution) and user's weekly revenue (this one looks normally distributed). I need to create another biased sample from the same dataset and keep the same distributions for both of these variables. Here's my drawing of the situation:

So far I only managed to think of something like splitting all observations into bins, figuring out the number of observations I need in each bin in order for the distribution of the new sample to look like the distribution from the first one (for both variables) and then going through the dataset looking at each user like:

• this one is bin 3 for logins but bin 1 for revenue, check
• this one is bin 2 for logins and bin 1 for revenue, check
• this one is bin 4 for logins and bin 1 for revenue, but bin 1 for revenue is full so it goes nowhere... until all the bins are full.

But this seems tedious, naïve and possibly incorrect. What would be the correct way to do what I want?

Answer 1

If all you are trying to do is create another dataset that has a similar distribution to your existing dataset with similar covariances, variances, etc. then you could simply draw a single bootstrap sample, by sampling with replacement from your original dataset.

Here's a quick example in R. Say your data looks like the data in mydata.

#Generate some fake data for demonstration
logins<-rpois(1000, lambda=2)
revenue<-rnorm(1000, 500, 20000)
mydata<-data.frame(logins, revenue)

#examine distributions of the data with historgrams
hist(mydata$logins)
hist(mydata$revenue)

#Draw 1000 samples with replacement from the existing dataset
simulated.data<-mydata[sample(NROW(mydata), size=1000, replace = TRUE),]

#Verify that the resulting distribution looks similar to the original
hist(simulated.data$logins)
hist(simulated.data$revenue)

When you do this, you'll see that the distributions of both the original data and simulated data look very similar. But I suspect you are trying to do something more complex with your analysis. If you can tell us what you are trying to do, I can update this answer with better guidance.