TL;DR: Trump had a 28% chance to win. We shouldn’t be surprised he won.
I’m not going to comment on the political outcome of last week’s US Presidential elections; enough ink—both pen-ink and eye-ink—has been spilled about that. What I am going to comment on though is the growing feeling the polls were wrong, and why an understanding of probability and statistical evidence might lead us to a more positive conclusion (or at least, less negative).
After last week’s “shock” result—read on for why shock is in scare-quotes—many news articles began to ask “Why were the polls wrong?”. (For a list, see the Google search here). This question is largely driven by the fact influential pollsters heavily favoured a Clinton victory. For example, FiveThirtyEight’s polls predicted a 71.4% chance of a Clinton victory. The New York Times predicted an 85% chance of a Clinton victory. Pretty convincing, huh? Something must have gone wrong in these polls, right?
All polls are known to be sub-optimum, but even if we found a way to conduct a perfect poll, and this perfect poll predicted a 71.4% chance of a Clinton victory, could we now state after observing a Trump victory that the polls were wrong? No, and the reason why most of us find this difficult to grasp is that most of us don’t truly appreciate probability.
No poll that I am aware of predicted a 100% chance of a Clinton victory. All polls that I saw had a non-zero chance of a Trump victory. So, even if with our “perfect” poll we see that Trump had a 28.6% chance of winning the election, we should not be surprised with a Trump victory. You can be disgusted, saddened, and /or scared, but you should not be surprised. After all, something with a 28.6% chance of occurring has—you guessed it!—a 28.6% chance of occurring.
28.6% translates to a 1 in 3.5 chance. If you think of a 6-sided die, each number has a 1 in 6 chance of being rolled on a single roll (~16.67% chance). Before you roll the die, you expect to see any other number than a 6. Are you surprised then if when you roll the die you observe a 6? Probably not. It’s not that remarkable. Yet it is expected less than Trump’s 28.6%. Likewise, if the weather-person on TV tells you there is a 28.6% chance of rain today, are you surprised if you get caught in a shower on your lunch break? Again, probably not.
So, the polls weren’t wrong at all. All predicted a non-zero chance of a Trump victory. What was wrong was the conclusion made from the polls.
Richard Royall & “Statistical Evidence”
The above raced through my mind without a second thought when I read numerous articles claiming the polls were wrong, but it was brought into sharper focus today when I was reading Richard Royall’s (excellent) chapter “The Likelihood Paradigm for Statistical Evidence”. In this chapter, he poses the following problem. A patient is given a non-perfect diagnostic test for a disease; this test has a 0.94 probability of detecting the disease if it is present in the patient (and therefore a 0.06 probability of missing the disease when it is present). However, it also has a non-zero probability of 0.02 of producing a “positive” detection even though the disease is not present (i.e., a false-positive).
The table below outlines these probabilities of the test result for a patient who does have the disease (X = 1) and a patient who does not have the disease (X = 0).
Now a patient comes to the clinic and the test is administered. The doctor observes a positive result. What is the correct conclusion the doctor can make based on this positive test result?
- The person probably has the disease.
- The person should be treated for the disease.
- This test result is evidence that this person has the disease.
The person probably has the disease
Intuitively, I think most people would answer this is correct. After all, the test has a 0.94 probability of detecting the disease if present, and we have a positive test result. It’s unlikely that this is a false positive, because this only occurs with a probability of 0.02.
However, this does not take into account the prior probability of the disease being present. (Yes, I have just gone Bayesian on you.) If the disease is incredibly rare, then it turns out that there is a very small probability the patient has the disease even after observing a positive test outcome. For a nice example of how the prior probability of the disease influences the outcome, see here.
The person should be treated for the disease
It should be clear from the above that this conclusion also depends on the prior probability of the disease. If the disease is incredibly rare, the patient doesn’t likely have it (even after a positive test result), so don’t waste resources (and risk potential harm to the patient). Again, the evidence doesn’t allow us to draw this conclusion.
This test result is evidence that this person has the disease
Royall argues that this is the only conclusion one can draw from the evidence. It is subtly different from Conclusion 1, but follows naturally from the “Law of Likelihood”:
If hypothesis A implies that the probability that a random variable X takes the value x is pA(x), while hypothesis B implies that the probability is pB(x), then the observation X = x is evidence supporting A over B if and only if pA(x) is less than pB(x)…
In our “disease” example, the observation of a positive result is evidence that this person has the disease because this outcome (a positive result) is better predicted under the hypothesis of “disease present” than the hypothesis “disease absent”. But it doesn’t mean that the person probably has the disease, or that we should do anything about it.
Back to Trump
Observing a Trump victory after a predicted win of 28.6% isn’t that surprising. The polls weren’t wrong. 28.6% is a non-zero chance. We should interpret this evidence in a similar way to the disease example: These poll results are evidence that Clinton will win. It is a mistake to interpret them as “Clinton probably will win”.