"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.
> >
> > |
> > |
> > |
> > |
> > | *
> > | * *
> > | * *
> > | * *
> > | * *
> > +-------------------------------------------------
> > 0 X²(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]
> > [xxxx] 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 configuration,
> > there are plenty of variations satisfying the same combinatorial
> > distance in terms of the squared deviation - up to a local maximum.
> > My difficulties are, to make this clear in simple words; best in
> > such simple words, as I used, when I explained the rationale of
> > chi-square and significance...
> > Ok, maybe, it's more a subject for news://sci.stat,edu , I guess.
> >
> > Thanks again for your input -
> >
> > Gottfried Helms.
>
For the smoother explanation that you want, point out that a ChiSquared
rv on n degrees of freedom is the sum of (at least) n observations.
(Sometimes n+r observations with r linear constraints.) Your students will
be conscious of this as the way they always calculate this.
If we add n random variables, each of which is positive, the most likely
value of the sum will obviously increase as n increases and the probability
of very small values will decrease. In addition, the Central Limit Theorem
shows that the density of the sum approaches a normal density.
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