Truths, Damned Truths, and Statistics
Would you answer truthfully if you were asked in a survey whether you consume cannabis? Maybe you do like to wind down with an edible after a long day. But the legal contradictions between federal and state laws make you wary. Besides, what will your kids' friends' parents think? The glare of judgment is blinding. Or, what if someone asked you whether you voted for Donald Trump in 2016? What about in 2020? The systematic polling errors in each of those US presidential elections live in infamy. I wonder if the inaccuracies can be attributed, at least in part, to the reluctance of some closeted Trump voters to admit their support.
How could we ever get at the truth of what people think or do if they are prone to lie about it, especially on sensitive topics? Well, what if we could provide statistical cover to the respondents? That is the key behind a well-known surveying technique called randomized response. Here’s the simplest version. You, the surveyee, privately flip a coin. If it lands tails, you admit “yes” to the sensitive question, regardless of the truth. Any stigma associated with this response is blunted by the fact that the surveyor does not know whether you were forced into a “yes” due to the coin toss. If the toss results in heads, you answer “yes” or “no” truthfully. This statistical cover obviates the incentive to lie. But how can we sift through the responses and separate the truthful yeses from the forced ones?
Let 'N' be the number of surveyees.
Let 'f' be the fraction of surveyees that truthfully answered "yes".
Let 'Y' be the actual number of surveyees who answered "yes".Y = (prob. of tails) x (N) + (prob. of heads) x f x N
=> Y = 1/2 x N + 1/2 x f x N
┌--------------┐
=> | f = 2Y/N - 1 |
└--------------┘The fraction of people who truthfully answered “yes” in the coin-toss randomized-response survey:
2 x (# yes responses) / (# total responses) - 1
So teachers, if you really want to know what percentage of your students like you, go ahead and ask them with a randomized response survey. Corporate leaders, if you’re truly interested in finding out what your employees honestly think about the company, now you know how to find out. And yes, political pollsters, go ahead and provide statistical cover to people to uncover the truth of how they will really vote next time.
