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February 21

phi(n) question

On page 224 of Handbook of Number Theory II, by Sandor and Crstici, it says that C. A Nicol proved that there exist infinitely many numbers n such that phi(n) <= phi(n-k) for all 1 <= k <= n-1. (It references problem E2590 in AMM, vol 83, p 656, with the solution in vol 85, p. 654, but I don't have access to those.) But the statement doesn't make sense to me - when k=n-1, you have phi(n) <= phi(1) = 1, and only phi(1) and phi(2)=1. Is there an error? Bubba73 ^{You talkin' to me?} 03:46, 21 February 2012 (UTC)[reply]

I don't know whether this link is stable but it displays problem E2590 which asks to show that there are infinitely many numbers n such that phi(n) <= phi(k) + phi(n-k) for 1 <= k <= n-1. It sounds like Sandor and Crstici forgot phi(k). PrimeHunter (talk) 04:17, 21 February 2012 (UTC)[reply]

Resolved

Thanks, that makes sense because the next line says that there are infinitely many n such that phi(n) >= phi(k) + phi(n-k) for that range of k. Bubba73 ^{You talkin' to me?} 04:41, 21 February 2012 (UTC)[reply]

Statistics and entailment

On page 173 of Cognitive Strategy Research by McCormick, Miller and Pressley, there is an F-statistic table that evaluates causal inferences, but I've never come across this use of the F-statistic. Part of the table is given here:

Entailment	BETA	F
Importance -> learning	0.13	F(2, 503) = ...
Importance -> attention -> learning	0.12	F(3, 502) = ...

In the experiment (a study of attention-focusing strategies), students had to read a technical passage, then sit a brief test. Those parts of the text that were examined on the test were deemed important by the experimenters (the "importance" part of the table). Students were measured on their ability to focus their attention on these important parts of the text (the "attention" part of the table), as well as their actual performance (the "learning" part). So intuitively, I hope that is clear: the causal chain involving attention (line 2) describes the link between "importance of the text" and "learning of the text", when it is mediated by conscious attention on the part of the reader. The chain without attention (line 1) is blind to this intermediate step.

So I can get the idea but not how the table works. The F-statistic is being used for some kind of test where the numerator steals a degree of freedom from the denominator, whenever an extra link is added to the causal chain. Can anyone explain? IBE (talk) 19:43, 21 February 2012 (UTC)[reply]

F-test is the article you want. Thats got some typical examples of its use.--Salix (talk): 21:22, 21 February 2012 (UTC)[reply]

That doesn't mention anything about entailment, nor does it tell me what Beta might represent. I know how to use an F-test for ANOVA and regression, but I've never seen anything about entailment before. Is it meaningful statistically, or is it just based on assuming causation from correlation? IBE (talk) 15:39, 22 February 2012 (UTC)[reply]

Yes, I realised that its not of much help. I realised you probably knew about F-test as soon as I posted my comment. Wondering if its anything to do with Regression analysis. I've looked at a few papers on Causal inference [1] which seem to involve a lot of regression. The beta could come from the linear regression equation

Y=\beta X+\varepsilon

. But this is a shot in the dark.--Salix (talk): 17:04, 22 February 2012 (UTC)[reply]

I think it's probably better than a shot in the dark - you are probably correct. I just thought Beta might be specific to the application. IBE (talk) 18:51, 22 February 2012 (UTC)[reply]

Could be some form of mediation analysis, but hard to be sure without looking at the source (which I don't have). As explained in that article, in recent years considerable doubt has been cast on simple ways of testing for mediation. Qwfp (talk) 18:09, 22 February 2012 (UTC)[reply]

I think this might be it, but I'll go over it more carefully when I have a moment. More info welcome from anyone, but thanks to you both for the help. IBE (talk) 18:51, 22 February 2012 (UTC)[reply]