I had originally postedcthis preply on Sep 26, but it failed to make
it, at least as far as Google.. so trying again...
Donald Burrill wrote:
> Them's pretty harsh words, David; and since I cannot see how your
> subsequent words support the assertion, I'd be grateful if you would
> explain. (The more especially as I expect to be addressing
hypothesis
> testing in the class I'm currently teaching, before the term is out;
> and if I've been doing it wrong all this long while, it's probably
> about time I were stopped before another generation of students is
> corrupted.)
Sorry, I didn't mean to upset anyone. And sorry I've been away, hence
delay in reply.
>
> On Wed, 24 Sep 2003, David Jones wrote:
>
>> This is a total misconception of what is going on.
>
> I had written (with rather more context than David quotes, but the
> context was consistent with the quote):
>
>>> The null hypothesis is the one that permits you to define a
>>> sampling distribution for the statistic you will observe. A
>>> distribution cannot be specified for an hypothetical inequality:
>>> so the "null" hypothesis is always the one that contains an equals
>>> sign (=).
>
> David Jones replied as follows, with my responses embedded within
this
> copy of his post:
>
>> This is a total misconception of what is going on. The presence of
an
>> equals sign has nothing to do with choosing the null hypothesis.
>
> There may be a quibble to be had on whether we are discussing the
> choice of a null hypothesis, or the syntax of writing about it after
> the choice has been made. In the context of the original poster,
> there were two hypotheses specified: {Mu < 150 psi} and {Mu >= 150
> psi}. Given these two, which is properly called the null?
>
> (Now, one can argue -- as I would -- that these were not in fact the
> two hypotheses the poster ought to have been discussing. But that
is
> another question -- the one identified as "(2)" in my reply to
jjliu,
> while in the material David quoted I was aiming at "(1)", rephrased
in
> the last sentence of the previous paragraph above ("Given these
> ...").)
>
>> Possibly the "professor" is under the same misapprehension.
>
> Mmm... Just what IS that misapprehension?
My thinking was ....
(i) I don't know of any convention that, if a trained statistician
quotes two mutually exclusive hypotheses, then you can tell which he
wants to treat as the null hypthothesis by whether or not an
equals-sign is included. I think it would be such a "short-hand" way
of specifying things that it veers far too much to brevity while
simultaneously being not particularly clear, easy to make a slip if
trying to use the convention and easy to misinterpret what
someone-else says if they may or may not be using the convention.
Explicit identification of the null hypothesis seems much preferable.
(ii) Since this is supposedly educational, trainee statisticians
should not expect customers to indulge in a particular convention for
stating null hypotheses, or even to use ordinary words in a strict
methematical sense (I believe the phrase in the original question was
similar to "at least as big as"). It is always necessary to understand
the real problem and then respond appropriately, rather than assuming
that a customer's brief note or quick conversation about what he/she
thinks the problem is will do. In this instance, supposing you have a
customer who is scientically trained, and thus apt to use
greater-than's and greater-than-or-equal-to's, the presence of an
equals-sign can't be taken as indicating what should really be taken
as the null hypothesis.
>
>> Where one-sided tests are concerned, the "real" null hypothesis is
a
>> composite one
>
> Yes, that follows from the one-sidedness, I had thought...
>
>> and the critical point of the test statistic is chosen so that the
>> probability of rejection satisfies a certain inequality for all
point
>> "null hypotheses" within the composite null hypothesis region (i.e.
>> any given point null hypothesis is considered to be true and the
>> probability of rejection is determined for this truth)... this
leads
>> to a point null hypothesis which is on the boundary effectively
being
>> the one which determines the critical point.
>
> Yes: and is not the boundary between {Mu < 150 psi} and {Mu > 150
> psi} the hypothesis that {Mu = 150 psi} ? And is not this
hypothesis
> stated with an equals sign?
Yes, but suppose you had started out by stating the null/alternative
hypotheses with the equal-sign included in the "wrong" set ... the
process of imposing the unbiased test requirement implicitly transfers
the equals into the correct (null hypothesis) set if you want to think
of it that way. You get the same test whether or not the
null-hypothesis-reion is open or closed. Thus the presence of an
equals sign doesn't really distinguish the null hypothesis.
>
> I'm sorry, and it is doubtless due to obtuseness on my part, but you
> have not convinced me that my words were either "a total
misconception
> of what is going on" or "a misapprehension".
>
>> As said in another post, the null hypothesis is the one which one
is
>> prepared to accept unless there is strong evidence to the contrary.
>
> Yes. This speaks to choosing the direction of a one-sided
hypothesis,
> or in other terms the locus of the burden of proof. This point had
> been explicitly acknowledged in my response to jjliu, as having been
> dealt with by Duncan (Murdoch).
>
> In case it may help the clarity of discourse, here are the two
> questions as identified at the beginning of my response to jjliu:
>
> ==== DFB ==== DFB ==== DFB ==== DFB ==== DFB ==== DFB ==== DFB ====
> There are two questions lurking here:
> (1) Which is the null hypothesis and which the alternative?
> (2) In which direction ought a one-sided hypothesis to be
specified?
>
> Duncan addressed (2); jjl's professor was dealing with (1):
> ==== DFB ==== DFB ==== DFB ==== DFB ==== DFB ==== DFB ==== DFB ====
>
> -------------------------------------------------------------------
> ---- Donald F. Burrill
> [EMAIL PROTECTED] 56 Sebbins Pond Drive, Bedford, NH 03110
> (603) 626-0816 .
> .
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To go back to ..
>
>>> The null hypothesis is the one that permits you to define a
>>> sampling distribution for the statistic you will observe. A
>>> distribution cannot be specified for an hypothetical inequality:
>>> so the "null" hypothesis is always the one that contains an equals
>>> sign (=).
... the argument for choosing the critical level of the test, looks
not at the "distribution for an hypothetical inequality" but at all
the distributions for which the parameters satisfy an hypothetical
inequality, specically not a single distribution. This gives you the
sampling distributions for the statistic for all parameters contained
in the null hypothesis ... you then choose the critical level for the
test by ensuring that the probability of rejection under each of these
sampling distributions is no more than the nominal probability
associated with the test
David Jones
.
.
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