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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

770.0. "Are there asymmetric measures of the stat. strength of an assertion ?" by EAGLE1::BEST (R D Best, Systems architecture, I/O) Wed Oct 14 1987 17:58

  I have a question that someone statistically inclined may be able to
answer.

  Suppose that I'm doing a study trying to show a correlation between
two attributes.  Let's say that I'm trying to correlate IQ and some
measure of personal attractiveness (looks).

  I might construct a methodology in which I independently measure
the IQ and looks of a sample of people.  What I get are a bunch
of ordered pairs of the form ( IQ[ n ], looks[ n ] ).

  Now I want to know how to interpret my data.  My questions are:

Can there be a difference between the correlation that I get
when trying to measure the strength of the assertion

(1) higher IQ  --> better looks

and trying to measure the strength of the assertion

(2) better looks --> higher IQ

or are appropriate measures always symmmetric w.r.t. the attributes ?
(i.e. an appropriate measure of assertion (1) will be the same as an
appropriate measure of assertion (2))

What are appropriate measures for what I'm imprecisely referring to
as the 'strength' of such assertions ?

If such 'asymmetric' measures exist, how are they computed ?

I recognise that there is a parameter called 'coefficient of correlation' that
I think is symmetric (in my sense).

I hope this makes some kind of sense.
T.RTitleUserPersonal
Name		DateLines
770.1CLT::GILBERTBuilderThu Oct 15 1987 10:4627
>(1) higher IQ  --> better looks
>(2) better looks --> higher IQ

    You need to be careful here.  A correlation doesn't imply causality.
    I'd intrepret the "-->" to mean "is a good predictor of".

    The coefficient of correlation is symmetric, and can be used with
    either (1) or (2), under the assumption that the predictor is a
    linear function.

    For non-linear relationships, (1) may be better than (2) -- the
    'predictive power' may go one way, but not the other.  Consider
    the following unrealistic sample:

	good looks	|                     
			|         X X         
			|       X     X       
			|     X        X      
			|   X            X    
			| X               X   
	not so good	|X                  X 
			+----------------------
			 low IQ		high IQ

    Notice that "IQ" is a much better predictor of "looks" than "looks"
    is of "IQ".  If we are forced to assume a linear relationship, then
    the predictive powers are necessarily the same.
770.2Asymmetric predictors *do* existGLINKA::GREENEThu Oct 15 1987 11:2631
    re: .0
    
    You are correct that the "correlation coefficient" IS symmetric.
    It is usually written as "r", and r(x,y)=r(y,x)  [note: what is
    in parens is usually subscripted] for two variables X and Y.
    
    There is, however, an asymmetric measure, which is related to the
    correlation coeficient.  That would be the SLOPE of the LINEAR
    REGRESSION LINE of X predicting Y *or* Y predicting X.  These slopes
    are typically referred to by a Beta.  These are asymmetric
    because they are sensitive to the units of measurement (e.g. miles
    vs. inches) whereas the correlation coefficient uses standardized
    units (z scores) for both variables.  
    
    r**2 [r squared] is sometimes used as a measure of the "strength"
    of the relationship between two variables, 0 being 'none' and 1
    representing "perfect" prediction ["perfect" here means that
    information about the value of X gives you complete information
    about the value of Y for any given observed case].
    
    Note .1's comment that "prediction" does NOT equal causality.
    
    Let me know if you have any other questions or if this is not clear.
    
    BTW, there are other measures, such as lambda, for non-continuous
    data when one still wants to determine the predictive power of one
    variable for another.  Or chi-square for un-ordered variables
    (e.g., predicting type of car owned by religious preference).
    
    	Penelope