[e-mail]

[Effect of normalization on correlation matrix]

hello,

To test/demonstrate the effect of normalization on correlations I
generated 3 normally distributed (mean = 10, stddev = 1) random variables
Y, X, Z

Y = target variable
X = normalization variable (not correlated to Y)
Z = correlated to X (r=0,9) but not to Y
Yn = normalized Y (Y/X)

As expected in the correlation matrix Yn is positively correlated to Y and
negatively correlated to the normalization variable X as well as to Z
(although there was no correlation between Y and Z).

As expected z-transforming Y, X, and Z resulted in exactly the same
correlation matrix between zY zX, zZ but now the normalized variable zYn
(zY/zX) was correlated to none of the other variables.

Comparing means and stddev zYn had a much higher variation than zY, zX and
zZ and a mean strongly deviating from zero. In the untransformed variant
mean and stddev of the normalize variable behaved as expected ( mean Yn≈
0, stddev Yn << stddev Y/X/Z).


Does anyone have a theoretical but comprehensible explanation for a non-
statistician why Yn is correlated to X and Z and why corrleations of Ynz
disappear after z-transformation ?


“raw” correlation matrix
  Y  X  Z   Yn
Y  1  .006  .007  .693(**)
X  .006  1  .898(**) -.706(**)
Z  .007  .898(**) 1  -.633(**)
Yn (=Y/X) .693(**) -.706(**) -.633(**) 1

After z-Transformation of Y, X, Z
  zY  zX  zZ  Ynz
zY  1  .006  .007  .018
zX  .006  1  .898(**) .000
zZ  .007  .898(**)  1 -.007
Ynz (zY/zX) .018  .000  -.007  1


Hope someone can help

Greetings Lenz

[statisticsdoc]

Stephen Brand
www.statisticsdoc.com

Lenz,

Consider these cases

X     Y      (X/Y)    zX   zY   (zX/zY)
12   12      1        2    2      1
12    8      1.5      2   -2     -1
 8   12      .67     -2    2     -1
 8    8      1       -2   -2      1

zX is a linear transformation of X, and zY is a linear transformation of Y,
but (zX/zY) is not a linear transformation of (X/Y).  Whenever one, but not
both, of the variables is below the mean, (zX/zY) will be negative.

HTH,

Stephen Brand

P.S. Wonderful username!

For personalized and professional consultation in statistics and research
design, visit
www.statisticsdoc.com


-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]]On Behalf Of
Effect of normalization on correlation matrix
Sent: Sunday, December 10, 2006 9:28 AM
To: [hidden email]
Subject: [hidden email]


hello,

To test/demonstrate the effect of normalization on correlations I
generated 3 normally distributed (mean = 10, stddev = 1) random variables
Y, X, Z

Y = target variable
X = normalization variable (not correlated to Y)
Z = correlated to X (r=0,9) but not to Y
Yn = normalized Y (Y/X)

As expected in the correlation matrix Yn is positively correlated to Y and
negatively correlated to the normalization variable X as well as to Z
(although there was no correlation between Y and Z).

As expected z-transforming Y, X, and Z resulted in exactly the same
correlation matrix between zY zX, zZ but now the normalized variable zYn
(zY/zX) was correlated to none of the other variables.

Comparing means and stddev zYn had a much higher variation than zY, zX and
zZ and a mean strongly deviating from zero. In the untransformed variant
mean and stddev of the normalize variable behaved as expected ( mean Yn≈
0, stddev Yn << stddev Y/X/Z).


Does anyone have a theoretical but comprehensible explanation for a non-
statistician why Yn is correlated to X and Z and why corrleations of Ynz
disappear after z-transformation ?


“raw” correlation matrix
  Y  X  Z   Yn
Y  1  .006  .007  .693(**)
X  .006  1  .898(**) -.706(**)
Z  .007  .898(**) 1  -.633(**)
Yn (=Y/X) .693(**) -.706(**) -.633(**) 1

After z-Transformation of Y, X, Z
  zY  zX  zZ  Ynz
zY  1  .006  .007  .018
zX  .006  1  .898(**) .000
zZ  .007  .898(**)  1 -.007
Ynz (zY/zX) .018  .000  -.007  1


Hope someone can help

Greetings Lenz