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MathGroup Archive 2009

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Re: Difference Fit vs. Correlation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg96799] Re: [mg96775] Difference Fit vs. Correlation
  • From: Darren Glosemeyer <darreng at wolfram.com>
  • Date: Wed, 25 Feb 2009 04:03:44 -0500 (EST)
  • References: <200902241048.FAA23721@smc.vnet.net>

Claus wrote:
> Hi,
> in the code below, using Correlation I get 0.501338, using Fit I get 
> 0.514093.
> How come?
>
> Thanks for your help,
> Claus
>
>
> In[47]:= GaltonX = {65.04851`, 63.25094`, 64.95532`, 65.7525`,
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>
> In[59]:= GaltonY = {59.77827`, 63.21404`, 63.34242`, 62.79238`,
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>
> In[60]:= lm = LinearModelFit[GaltonDat, {1, x}, x]
>
> Out[60]= FittedModel[\!\(\*
> PanelBox[
> TagBox[
> RowBox[{"33.88660435407788`", " ", "+",
> RowBox[{"0.5140930386233082`", " ", "x"}]}],
> Short],
> FrameMargins->5]\)]
>
> In[62]:= lm["BestFit"]
>
> Out[62]= 33.8866 + 0.514093 x
>
> In[63]:= r2 = Correlation[GaltonX, GaltonY]
>
> Out[63]= 0.501338
>   

The correlation and slope are two different quantities. Correlation 
gives a measure of how well one variable could predict the other, while 
slope is part of the linear prediction. Note, also that correlation is 
bounded by -1 and 1, while slope could be outside that range. If we take 
the formulas from simple linear regression, we see that the results are 
correct and that correlation and slope will only be equal if the x and y 
sums of squares, sxx and syy, are equal.

In[3]:= xbar = Mean[GaltonX];

In[4]:= ybar = Mean[GaltonY];

In[5]:= n = Length[GaltonX];

In[6]:= sxy = GaltonX.GaltonY - n*xbar*ybar

Out[6]= 4171.58

In[7]:= sxx = GaltonX.GaltonX - n*xbar^2

Out[7]= 8114.44

In[8]:= syy = GaltonY.GaltonY - n*ybar^2

Out[8]= 8532.58

In[9]:= cor = sxy/Sqrt[sxx*syy]

Out[9]= 0.501338

In[10]:= slope = sxy/sxx

Out[10]= 0.514093


Darren Glosemeyer
Wolfram Research


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