March 21, 2008 | 31 Comments
In part I, we learned that all surveys, and in fact all statistical models, are valid only conditionally on some population (or information). We went into nauseating detail of the conditional information on our own survey of people who wear thinking suppression devices (TSDs; see the original posts), so I’ll skip repeating any of it again.
Today, we look at the data and ignore all other questions. The first matter we have to understand is: what are probability models and statistics for? Although we use the data we just observed to fit these models, they are not for that data. We do not need to ask probability questions of the data we just observed, there is no need to. If we want the probability that all the people in our sample wore TSDs, we just look and see if all wore them or not. The probability is 0 or 1, and is 0 or 1 for any other question we can ask about the observed data (e.g. what is the probability that half or more wore them? again, 0 or 1).
Thus, statistics are useful only for making inferences about unobserved data: usually future data, but really just unknown to you. If you want to make statements or quantify uncertainty in data you have not yet seen, then you need probability models. Some would say statistics are useful for making inferences about unobserved and unobservable parameters, but I’ll try to dissuade you of that opinion in this essay. We have to start, however, with describing what these parameters are and why so much attention is devoted to them.
Before we do, we have to return to our question, which was roughly phrased in English as “How many people wear TSDs?”, and we have to turn it into a mathematical question. We do this by forming a probability model for the English question. If you’ve read some of my earlier posts, you might recall that we have an essentially infinite choice of models which we could use. What we would like is if we could limit our choice to a few or, best of all, to logically deduce the exact model given some set of information that we believe true.
Here is one such statement:
M1 = “The probability that somebody wears a TSD (at the locations and times specified for our for our exactly defined population subset) is fixed, or constant, and knowing whether one person wears a TSDs gives us no information whether any other person wears a TSD.” (Whenever you see
M1, substitute the sentence “The probability…”)
M1 true? Almost certainly not. For example, if two people walk by our observation spot together, say a couple, it might be less likely for either to wear a TSD than it is for two separate people. Again people (not all people, anyway) aren’t going to wear a TSD at all hours equally often, and not equally often at all locations within our subset either.
But let’s suppose that
M1 is true anyway. Why? Because this is what everybody else does in similar situations, which they do because it allows them to write a simple and familiar probability model for the number of people
x out of
n wearing TSDs. Here is the model for the data we just observed:
Pr( x = k | n, θ, M1)
This is actually just a script or shorthand for the model, which is some mathematical equation (binomial distribution), and not of real interest; however it is useful to learn how to read the script. From left to right, it is the probability that the number of people
x equals some number
k given we know
n, something called θ, and
M1 is true. This is the mathematical way of writing the English question.
x is more shorthand meaning “number of people who wore a TSD”. Before we did our experiment, we did not know the value of
x, so we say it was “random.” After we see the data we know
k, the actual number of new people out of the
n people we saw who did wear a TSD. OK so far? We already understand what
M1 is, so all that is left to explain is θ What is it?
It is a parameter, which if you recall previous posts, is an unobservable, unmeasurable number, but which is necessary in order to formulate our probability model. Some people incorrectly call θ “the probability that a single person wears a TSD.” This is false and is an example of the atrocious and confusing terminology so often used in statistics (look in any introductory text and you’ll see what I mean). θ, while giving the appearance of one, is no sort of probability at all. It would be a probability if we knew its value. But we do not: and if we did know, we never would have bothered collecting data in the first place! Now, look carefully. θ is written on the right hand side of the “|”, which is where we put all the stuff that we believe we know, so again it looks as if we are saying we know θ, so it looks like a probability.
But this is because the model is incomplete. Why? Remember that we don’t really need to model the observed data if that is all we are interested in. So the model we have written is only part of a model for future data. There are several pieces that are missing. Those pieces are another probability model for the value of θ, a model for just the observed data, a model for the uncertainty in θ given the observed data, the data model itself again, which are all mathematically manipulated to produce this creature
Pr( xnew = knew | nnew, xold, nold, M1)
which is the true probability model for new data given what we observed with the old data. There is no way that I can even hope to explain this new model without resorting to some heavy mathematics. This is in part why classical statistics just stops with the fragmentary model, because it’s easier. In that tradition, people create a (non-verifiable) point estimate of θ, which means just plugging some value for θ into the probability model fragment, and then call themselves done.
Well, almost done. Good statisticians will give you some measure of uncertainty of the guess of θ, some plus or minus interval. (If you haven’t already, go back and read the post “It depends on what the meaning of mean means.”) The classical estimate used for θ is just the computed mean, the average of the past data. So the plus and minus interval will only be for the guess of the mean. In other words, just as it was in regression models, it will be too narrow and people will be overconfident when predicting new data.
All this is very confusing, so now—finally!—was can return to the data collected by those folks who turned in their homework and work through some examples.
There were 6 separate collections, which I’ll lump together with the clear knowledge that this violates the limits of our population subset (two samples were taken in foreign countries, one China and one New Jersey). This gave
x = 58 and
n = 635.
The traditional estimate of θ is
58/635 = 0.091, with the plus minus interval of
0.07 to 0.12. Well, so what? Remember that our goal is to estimate the number of people who wear TSDs, so this classical estimate of θ is not of much use.
If we just plug in the best estimate of θ to estimate, out of 300 million (the approximate population of the U.S.A.), how many wear TSDs, we get a guess of 27.4 million with a plus-minus window of 27.39 to 27.41 million, which is a pretty tight guess! The length of that interval is only about 20,000 people wide. This is being pretty sure of ourselves, isn’t it?
If we use the modern estimate, we get a guess of 25.5 million, with a plus-minus window of about 19.3 to 31.7 million, which is much wider and hence more realistic. The length of this interval is 12.4 million! Why is this interval so much larger? It’s because we took full account of our uncertainty in the guess of θ, which the classical plug-in guess did not (we essentially recompute a new guess for every possible value of θ and weight them by the probability that θ equals each value: but that takes some math).
Perhaps these numbers are too large to think about easily, so let’s do another example and ask how many people riding a car on the F train wear a TSD. The car at rush hour holds, say, 80 people. The classical guess is 7, with +/- of 3 to 13. The modern guess is also 7 with +/- of 2 to 12. Much closer to each other, right?
Well, how about all the students in a typical college? There might be about 20,000 students. The classical guess is 1750 with +/- 1830 to 1910. The modern is 1700 with +/- 1280 to 2120.
We begin to see a pattern. As the number of new people increases, the modern guess becomes a little lower than the classical one, and the uncertainty in the modern guess is realistically much larger. This begins to explain, however, why so many people are happy enough with the classical guesses: many samples of interest will be somewhat small, so all the extra work that goes into computing the modern estimate doesn’t seem worth it.
Unfortunately, that is only true because we had such a large initial data collection. If, for example, we only had Steve Hempell’s, which was
x = 1 and
n = 41, and we were interested still in the F train, then the classical guess is 2 with +/- 0 to 5; and the modern guess 4 +/- 0 to 13! The difference between the two methods is again large enough to make a difference.
Once again, we have done a huge amount of work for a very, very simple problem. I hope you have read this far, but I would not have blamed you if you hadn’t because, I am very sorry to say, we are not done yet. Everybody who remembers
M1 raise their hands? Not too many. Yes, all these guesses were conditional on
M1 being true. What if it isn’t? At the least, it means that the guesses we made are off a little and that we must widen our plus-minus intervals to take into account our uncertainty in the correctness of our model.
Which I won’t do because I am, and you are probably, too fatigued. This is a very simple problem, like I said. Imagine problems with even more complicated statistics where uncertainty comes at you from every direction. There the differences between the classical and modern way are even more apparent. Here is the second answer for our homework:
- Too many people are far too certain about too many things