From: Andrew.Robinson@anu.edu.au
Newsgroups: sci.math.stat
Subject: Re: Degrees of Freedom Was: Re: Standard Deviation.
Date: 23 Aug 92 05:38:52 GMT
Organization: Australian National University

In article <thompson.714506041@kiyotaki.econ.umn.edu>
thompson@atlas.socsci.umn.edu (T. Scott Thompson) writes:
>Andrew.Robinson@anu.edu.au writes:
>
>>I'm going to go _way_ out on a limb here and say that I think that you have
>>_one_ piece of infomation.  If you don't have X then your data doesn't
consist
>>of a single number, it consists of a single observation from which you have
>>measured a two variable vector - [Y,Z].  You know  that there is an X and you
>>want to use Y and Z to estimate it using a predetermined model, which you
have
>>specified.  This seems to me to be a non-straightforward case of a
multivariate
>>regression, with two dependent variables and one observation.
>
>>Comments?
>
>Well first, I don't have to estimate X since I can calculate it
>exactly from Y and Z.  

Not meaning to nit pick, but this is still estimation, as opposed to
observation.  Your estimation happens to be with a perfect model
(congratulations).

>Secondly, I agree that the single data point X
>contains the exact same information as the vector (Y,Z).  I will even
>agree (although I am not sure that this is so obvious) that the number
>of "pieces of information" in X (hence in (Y,Z) is one.  After all,
>the latter could simply be a normalization of units.
>
>The problem is to relate this to the usual degrees of freedom
>accounting which seems to require that independent "pieces of
>information" add up in some way.  For example, we seem to think of Y
>and Z as independent pieces of information because knowledge of one is
>totally uninformative about the other.  According to the adding up
>rule, then, (Y,Z) should be 2 pieces of information, not one.  Clearly
>we agree that this is not so.  The question is "why not?"

I think I understand our difference.  Correct me if I'm wrong, anyone, but
degrees of freedom are used in the construction and validation of a model.  The
model presented by Scott here has already been constructed (and validated, I
presume), therefore the degres of freedom would seem irrelevant to its further
usage.

Further to the idea expressed above, however, I think that the distinction in
pieces of information lies in the difference between observations and
variables.  Each observation is a report, if you will, of how 'reality' (please
let's not argue about this one :-)) is behaving at that point, irrespective of
how many different independent variables are being measured. So when
considering the number of independent 'pieces of information', we look at the
number of observations with which we are constructing the model, not the number
of variables for each observation.  You seem to have one observation with two
variables, ergo only one piece of information. 

>I think that the definition of "independent pieces of information"
>that is implicitly used in the informal interpretation of "degrees of
>freedom" does not satisfy an adding up rule, and thus fails to provide
>a good interpretation for what "degrees of freedom" really represent.
>Assuming they represent anything.  I am skeptical about this.

I'd be quoting previous posts if I continued on here.

regardless,
Andrew

now hit d
Newsgroups: sci.math.stat