The standardized and unstandardized canonical correlation coefficients

In summary, the output of SPSS 27 Canonical Correlation provides both the standardized and unstandardized canonical correlation coefficients, with the difference being that the standardized coefficients are adjusted for scale differences between the two variables, while the unstandardized coefficients are not.
  • #1
Ad VanderVen
169
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TL;DR Summary
What exactly are the standardized and unstandardized canonical correlation coefficients and what is the difference between them?
The output of SPSS 27 Canonical Correlation gives the standardized and unstandardized canonical correlation coefficients.

What exactly are the standardized and unstandardized canonical correlation coefficients and what is the difference between them?
 
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  • #2
standardized:
$$
\operatorname{cov}\left(\dfrac{X-\mu_X}{\sigma_X}\, , \,\dfrac{Y-\mu_Y}{\sigma_Y}\right)=\dfrac{1}{\sigma_X\,\sigma_Y}\,\operatorname{cov}(X,Y)
$$

unstandardized:
##\operatorname{cov}(X,Y)##
 
  • #3
fresh_42 said:
standardized:
$$
\operatorname{cov}\left(\dfrac{X-\mu_X}{\sigma_X}\, , \,\dfrac{Y-\mu_Y}{\sigma_Y}\right)=\dfrac{1}{\sigma_X\,\sigma_Y}\,\operatorname{cov}(X,Y)
$$

unstandardized:
##\operatorname{cov}(X,Y)##

You simply define the standardized and unstandardized correlation coefficient, but we are talking about the canonical correlation coefficient here.
 

FAQ: The standardized and unstandardized canonical correlation coefficients

What is the difference between standardized and unstandardized canonical correlation coefficients?

Standardized canonical correlation coefficients are computed using standardized variables, meaning that the variables have been transformed to have a mean of zero and a standard deviation of one. This allows for direct comparison between variables with different units or scales. Unstandardized canonical correlation coefficients, on the other hand, are computed using the original variables without any transformation, so they retain the units and scales of the original data.

When should I use standardized canonical correlation coefficients?

Standardized canonical correlation coefficients should be used when you need to compare the relative importance of variables that are measured on different scales or units. This is often the case in psychological, social, and educational research where variables may be measured in different ways (e.g., test scores, survey responses, physiological measurements).

How do I interpret unstandardized canonical correlation coefficients?

Unstandardized canonical correlation coefficients are interpreted in the context of the original units of the variables. They indicate the strength and direction of the relationship between the original variables in each canonical function. However, because they retain the original units, they are not directly comparable across different variables unless the variables are on the same scale.

Can standardized and unstandardized canonical correlation coefficients yield different results?

Yes, standardized and unstandardized canonical correlation coefficients can yield different numerical results because they are based on different scales of the variables. However, the overall pattern of relationships between the sets of variables (i.e., which variables are most strongly related to each other) should be similar. The choice between standardized and unstandardized coefficients depends on the research question and the need for comparability across variables.

How are canonical correlation coefficients calculated?

Canonical correlation coefficients are calculated by finding linear combinations of the variables in each set that maximize the correlation between the sets. This involves solving an eigenvalue problem for the cross-covariance matrices of the two sets of variables. The resulting canonical correlation coefficients represent the strength of the relationship between the linear combinations of variables in each set. Standardized coefficients are obtained by standardizing the variables before performing the analysis.

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