Canonical Analysis Introductory Overview - Sum Scores

A first approach that you might take is simply to add up the responses to the work satisfaction items, and to correlate that sum with the responses to all other satisfaction items. If the correlation between the two sums is statistically significant, we could conclude that work satisfaction is related to satisfaction in other domains.

In a way this is a rather "crude" conclusion. We still know nothing about the particular domains of satisfaction that are related to work satisfaction. In fact, we could potentially have lost important information by simply adding up items. For example, suppose there were two items, one measuring satisfaction with one's relationship with the spouse, the other measuring satisfaction with one's financial situation. Adding the two together is, obviously, like adding "apples to oranges." Doing so implies that a person who is dissatisfied with her finances but happy with her spouse is comparable overall to a person who is satisfied financially but not happy in the relationship with her spouse. Most likely, people's psychological make-up is not that simple...

The problem then with simply correlating two sums is that one might lose important information in the process, and, in the worst case, actually "destroy" important relationships between variables by adding "apples to oranges."

Using a weighted sum. It seems reasonable to correlate some kind of a weighted sum instead, so that the "structure" of the variables in the two sets is reflected in the weights. For example, if satisfaction with one's spouse is only marginally related to work satisfaction, but financial satisfaction is strongly related to work satisfaction, then we could assign a smaller weight to the first item and a greater weight to the second item. We can express this general idea in the following equation:

a1*y1+a2*y2+...+apyp = b1*x1 +b2*x2+...+bqxq

If we have two sets of variables, the first one containing p variables and the second one containing q variables, then we would like to correlate the weighted sums on each side of the equation with each other.

Determining the weights. We have now formulated the general "model equation" for canonical correlation. The only problem that remains is how to determine the weights for the two sets of variables. It seems to make little sense to assign weights so that the two weighted sums do not correlate with each other. A reasonable approach to take is to impose the condition that the two weighted sums shall correlate maximally with each other. This is exactly what canonical correlation does when performing a canonical analysis based on the overall correlation matrix of all variables.