tricky explanation problem with chi-square on multinomial
Hi , there was a tricky problem, recently, with the chi-square-density of higher dgf's. I discussed thath in sci.stat.consult and in a german newsgroup, got some answers and also think to have understood the real point. But I would like to have a smoother explanation, as I have to deal with it in my seminars. Maybe someone out has an idea or a better shortcut, how to describe it. To illustrate this I just copy&paste an exchange from s.s.consult; hope you forgive my lazyness. On the other hand: maybe the true point comes out better this way. Regards Gottfried 3 postings added: ---(1/3)--- [Gottfried] > Hi - > >im stumbling in the dark... eventually only missing any >simple hint. >I'm trying to explain the concept of significance of the >deviation of an empirical sample from a given, expected >distribution. >If we discuss the chi-square-distribution > | > |* > | * > | * > | * > | * > | * > | * > |* > +- > >then this graph illustrates us very well, that and how a >small deviation is more likely to happen than a high deviation - >thus backing the concept of the 95%tiles etc. in the beginners >literature. >Just cutting it in equal slices this curve gives us expected >frequencies of occurences of samples with individual chi-squared >deviations from the expected occurences. > >If I have more df's, then the curve changes its shape; in this >case a 5 df-curve for samples of thrown dices, where I count >the frequencies of occurences of each number and the deviation >of these frequencies from the uniformity. > > | > | > | > | > |* > | ** > |** > | ** > | * * > +- > 0X²(df=5) > > Now the slices with the highest frequency of occurences > are not the ones with the smallest deviation from the > expected distribution (X²=0) - and even if I accept, that this > is at least so for the cumulative distribution, it is > suddenly no more "self-explaining". It is congruent with > the reality, but our common language is different: > the most likely chisquare-deviation from the uniformity > is now an area which is not at the zero-mark. > So, now: do we EXPECT a deviation from uniformity? > That the count of frequencies of the occurences of the > 6 dices numbers is NOT most likely uniform? HÄH? > Is this suddenly the Nullhypothesis? And do we calculate > the deviation of our empirical sample then from this new > Nullhypothesis??? > > I never thought about that in this way, but since I do > now, I feel a bit confused, maybe I only have to step > aside a bit? > Any good hint appreciated - > > Gottfried. > -- ---(2/3)--- Then one participant answered: > Actually, that corresponds to the notion that if a "random" sequence is > *too* uniform, it isn't really random. For example, if you were to toss a > coin 1000 times, you'd be a little surprised if you got *exactly* 500 > heads and 500 tails. If you think in terms of taking samples from a > multinomial population, the non-monotonicity of the chi-square density > means that a *small* amount of sampling error is more probable than *no* > sampling error, as well as more probable than a *large* sampling error, > which I think corresponds pretty well to our intuition. > --- --(3/3)- I was not really satisfied with this and answered, after I had got some more insight: [Gottfried] > [] wrote: > > Actually, that corresponds to the notion that if a "random" sequence is > > *too* uniform, it isn't really random. For example, if you were to toss a > > coin 1000 times, you'd be a little surprised if you got *exactly* 500 > > heads and 500 tails. If you think in terms of taking samples from a > > > Yes, this is true. But it is the same with each other combination. > No one is more likely to occur (or better: one should say: variation?). > But then, a student would ask, how could you still attribute a near-expected- > variation more likely than a far-away-expected variation in generality? > > The reason is, that we don't argue about a specific variation, > but about properties of a variation, or in this case, of a combination. > We commonly select the property of "having a distance from the > expected variation", measured in terms of squared deviation. > The mess is, that with this criterion, with multinomial con
Re: tricky explanation problem with chi-square on multinomial
Hi Jos, got your msg. Thanks! > You might consider the distribution of chi-square(df) / (df), which > as far as I know has not been given a name; this distribution would be > concentrated around expectation 1 with variance 2/(df). > Seems to be reasonable. Like using Cramer's V instead of Chi-square. The actual problem is that of how to translate this to students, who are used to: the farer away from expectation (i.e. uniformity) the more unlikely is the outcome. Or opposite: the expected is the most likely. If the uniformity is not the most likely, why does it still engaged as the expected, from where we calculate deviations? They have to learn a different slogan, i'm afraid... Gottfried. --- Jos Jansen schrieb: > >(...) > I hope this will clear up the matter a little. > > Jos Jansen = Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at http://jse.stat.ncsu.edu/ =
Re: tricky explanation problem with chi-square on multinomial experiment(dice)
"Gottfried Helms" <[EMAIL PROTECTED]> wrote in message [EMAIL PROTECTED]">news:[EMAIL PROTECTED]... > Hi , > > there was a tricky problem, recently, with the chi-square-density > of higher dgf's. > I discussed thath in sci.stat.consult and in a german newsgroup, > got some answers and also think to have understood the real point. > > But I would like to have a smoother explanation, as I have to > deal with it in my seminars. Maybe someone out has an idea or > a better shortcut, how to describe it. > To illustrate this I just copy&paste an exchange from s.s.consult; > hope you forgive my lazyness. On the other hand: maybe the > true point comes out better this way. > > Regards > Gottfried > > > 3 postings added: > ---(1/3)--- > [Gottfried] > > Hi - > > > >im stumbling in the dark... eventually only missing any > >simple hint. > >I'm trying to explain the concept of significance of the > >deviation of an empirical sample from a given, expected > >distribution. > >If we discuss the chi-square-distribution > > | > > |* > > | * > > | * > > | * > > | * > > | * > > | * > > |* > > +- > > > >then this graph illustrates us very well, that and how a > >small deviation is more likely to happen than a high deviation - > >thus backing the concept of the 95%tiles etc. in the beginners > >literature. > >Just cutting it in equal slices this curve gives us expected > >frequencies of occurences of samples with individual chi-squared > >deviations from the expected occurences. > > > >If I have more df's, then the curve changes its shape; in this > >case a 5 df-curve for samples of thrown dices, where I count > >the frequencies of occurences of each number and the deviation > >of these frequencies from the uniformity. > > > > | > > | > > | > > | > > |* > > | ** > > |** > > | ** > > | * * > > +- > > 0X²(df=5) > > > > Now the slices with the highest frequency of occurences > > are not the ones with the smallest deviation from the > > expected distribution (X²=0) - and even if I accept, that this > > is at least so for the cumulative distribution, it is > > suddenly no more "self-explaining". It is congruent with > > the reality, but our common language is different: > > the most likely chisquare-deviation from the uniformity > > is now an area which is not at the zero-mark. > > So, now: do we EXPECT a deviation from uniformity? > > That the count of frequencies of the occurences of the > > 6 dices numbers is NOT most likely uniform? HÄH? > > Is this suddenly the Nullhypothesis? And do we calculate > > the deviation of our empirical sample then from this new > > Nullhypothesis??? > > > > I never thought about that in this way, but since I do > > now, I feel a bit confused, maybe I only have to step > > aside a bit? > > Any good hint appreciated - > > > > Gottfried. > > > -- > > ---(2/3)--- > Then one participant answered: > > > Actually, that corresponds to the notion that if a "random" sequence is > > *too* uniform, it isn't really random. For example, if you were to toss a > > coin 1000 times, you'd be a little surprised if you got *exactly* 500 > > heads and 500 tails. If you think in terms of taking samples from a > > multinomial population, the non-monotonicity of the chi-square density > > means that a *small* amount of sampling error is more probable than *no* > > sampling error, as well as more probable than a *large* sampling error, > > which I think corresponds pretty well to our intuition. > > > > --- > > --(3/3)- > I was not really satisfied with this and answered, after I had > got some more insight: > > [Gottfried] > > [] wrote: > > > Actually, that corresponds to the notion that if a "random" sequence is > > > *too* uniform, it isn't really random. For example, if you were to toss a > > > coin 1000 times, you'd be a little surprised if you got *exactly* 500 > > > heads and 500 tails. If you think in terms of taking samples from a > > > > > > Yes, this is true. But it is the same with each other combination. > > No one is more likely to occur (or better: one should say: variation?). > > But then, a student would ask, how could you still attribute a near-expected- > > variation more likely than a far-away-expected variation in generality? > > > > The reason is, th
