From Ted.Harding at manchester.ac.uk  Sun Aug 16 17:26:52 2009
From: Ted.Harding at manchester.ac.uk ( (Ted Harding))
Date: Sun, 16 Aug 2009 16:26:52 +0100 (BST)
Subject: [Rd] Printing the null hypothesis
In-Reply-To: <68b1e2610908160706j5c24c253w7c8fa236ec63e5cc@mail.gmail.com>
Message-ID: <XFMail.090816162652.Ted.Harding@manchester.ac.uk>

On 16-Aug-09 14:06:18, Liviu Andronic wrote:
> Hello,
> On 8/16/09, Ted Harding <Ted.Harding at manchester.ac.uk> wrote:
>> I don't know about *compelling* reasons! But (as a general rule)
>>  if the Alternative Hyptohesis is stated, then the Null Hypothesis
>>  is simply its negation. So, in your example, you can infer
>>
>>   H0: true tau equals 0
>>   Ha: true tau is not equal to 0.
>>
> Oh, I had a slightly different H0 in mind. In the given example,
> cor.test(..., met="kendall") would test "H0: x and y are independent",
> but cor.test(..., met="pearson") would test: "H0: x and y are not
> correlated (or `are linearly independent')" .

Ah, now you are playing with fire! What the Pearson, Kendall and
Spearman coefficients in cor.test measure is *association*. OK, if
the results clearly indicate association, then the variables are
not independent. But it is possible to have two variables x, y
which are definitely not independent (indeed one is a function of
the other) which yield zero association by any of these measures.

Example:
  x <-  (-10:10) ; y <- x^2 - mean(x^2)
  cor.test(x,y,method="pearson")
  #       Pearson's product-moment correlation
  # t = 0, df = 19, p-value = 1
  # alternative hypothesis: true correlation is not equal to 0 
  # sample estimates: cor 0
  cor.test(x,y,method="kendall")
  #       Kendall's rank correlation tau
  # z = 0, p-value = 1
  # alternative hypothesis: true tau is not equal to 0 
  # sample estimates: tau 0
  # cor.test(x,y,method="spearman")
  #      Spearman's rank correlation rho
  # S = 1540, p-value = 1
  # alternative hypothesis: true rho is not equal to 0 
  # sample estimates: rho 0

If you wanted, for instance, that the "method=kendall" should
announce that it is testing "H0: x and y are independent" then
it would seriously mislead the reader!

> To take a different example, a test of normality.
>> shapiro.test(mtcars$wt)
> 
>       Shapiro-Wilk normality test
> 
> data:  mtcars$wt
> W = 0.9433, p-value = 0.09265
> 
> Here both "H0: x is normal" and "Ha: x is not normal" are missing. At
> least to beginners, these things are not always perfectly clear (even
> after reading the documentation), and when interpreting the results it
> can prove useful to have on-screen information about the null.
> 
> Thank you for answering
> Liviu

This is possibly a more discussable point, in that even if you know
what the Shapiro-Wilk statistic is, it is not obvious what it is
sensitive to, and hence what it might be testing for. But I doubt
that someone would be led to try the Shapiro-Wilk test in the
first place unless they were aware that it was a test for normality,
and indeded this is announced in the first line of the response.
The alternative, therefore, is "non-normality".

As to the contrast between absence of an "Ha" statement for the
Shapiro-Wilk, and its presence in cor,test(), this comes back to
the point I made earlier: cot.test() offers you three alternatives
to choose from: "two-sided" (default), "greater", "less". This
distinction can be important, and when cor.test() reports "Ha" it
tells you which one was used.

On the other hand, as far as Shapiro-Wilk is concerned there is
no choice of alternatives (nor of anything else except the data x).
So there is nothing to tell you! And, further, departure from
normality has so many "dimensions" that alternatives like "two
sided", "greater" or "less" would make no sense. One can think of
tests targeted at specific kinds of alternative such as "Distribution
is excessively skew" or "distribution has excessive kurtosis" or
"distribution is bimodal" or "distribution is multimodal", and so on.
But any of these can be detected by Shapiro-Wilk, so it is not
targeted at any specific alternative.

Best wishes,
Ted.

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Date: 16-Aug-09                                       Time: 16:26:48
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[Rd] Printing the null hypothesis
 (Ted Harding