[Martin Taylor 2007.08.02.16.38]

[From Bill Powers (2007.08.02.0834 MDT)]

Hello, all. Interesting discussions, but I can't really join right now because my internet connection at home is down. I'm typing this at Mcdonald's.

Finally, I've heard of a legitimate reason for going into a Mcdonald's!

Brief comment. I keep forgetting this, but my initial question on this statistics business was very simple: given a certain correlation, what is the probability that a statement about an individual that is based on group statistics will be incorrect? The answer to that question is not an argument about whether we should do research or apply what is known: it is a number, or a table of numbers. So far only Rick has done any work toward obtaining that number or table. All the rest consists of people telling us what they have concluded on the basis of what they have assumed to be true.

I didn't understand why you said this the first time, and I don't understand why you repeat it now.

The following assumes that the "statement" is of the kind "Since X has value x, Y will be greater than y".

The answer is that there aren't any tables of numbers, and can't be, except for idealized conditions. The answer is not fixed for any given correlation. There are only methods. Richard, in the paper you redistributed a few days ago, demonstrated the equation-based method for the idealized situation in which the distribution is joint Gaussian, and provided numbers for particular levels of the cut. But you can probably use his methods and equations for other choices of the cut, such as "What is the probability that Y will be in the top quartile if X is above the median".

I say "probably" because I didn't reread the paper to check that all the equations are actually there. But I think they are. If not, it's simply a question of taking y' (the value of Y on the regression line when X has the value x), and seeing how many standard deviations y is above or below y', and looking in tables of the Gaussian distribution for your answer.

I gave you the most general method, which you can use for ANY distributional relationship. Remember that the correlation could be zero even when the two variables are functionally related, so that Y is uniquely determined by X. Here's an example: Y = X^2, when the tested X values are symmetrically distributed around zero. The correlation between X an Y is zero for that case, though you know Y exactly when you know X. The answer to your question "what is the probability that a statement about an individual that is based on group statistics will be incorrect" would be "Zero" for that case, even though the correlation is zero.

I'm not at all clear what more you could be asking for.

Martin

The second comment is similar: can anyone lay out the theoretical basis for libertarianism, representative democracy, rule of law, dog-eat-dog, or whatever system concept likes behind the political discussions? We're in a theory-based seminar here, so it seems appropriate to ask what theoretical bases there are for the various points of view.

You mean PCT-theoretic, I imagine. It's not an easy problem to address, though it is worthwhile to attempt.

One real problem that will arise is the same one that comes up in the medical treatment thread -- is it better to improve many people's lives at the cost of making other people's lives worse than they otherwise would have been (casting the question in the Conservative mode), or alternatively, is it better to make a few people's lives better at the cost of making many people's lives a bit worse (casting it in the Liberal mode). Either way you pose the question, the answer is a moral value judgment, not, I think, susceptible to scientific theoretical analysis, PCT or otherwise.

What is potentially susceptible to theory or simulation is whether certain arrangements are evolutionarily stable, and what kinds of dynamic changes are likely given certain starting assumptions. What, for example, might be the far future state of an isolated community with (un)limited resources in which most people choose to cooperate but some don't, and the cooperators allow the defectors to share in the benefits of cooperation without sanction (Samuel Sauders's preferred state, if I understand correctly). It might be possible to analyse and to simulate that oversimplification of a possible society, and likewise with the other concepts.

My replacement DSL model is supposed to be delivered tomorrow afternoon. I may come back to McDonald's tomorrow just to check up on things.

I hope she's pretty when she arrives!

Martin