Summary of Analysis
The fit and loss values tell you how well the nonlinear canonical correlation analysis solution fits the optimally quantified data with respect to the association between the sets. The summary of analysis table shows the fit value, loss values, and eigenvalues for the survey example.
Loss is partitioned across dimensions and sets. For each dimension and set, loss represents the proportion of variation in the object scores that cannot be accounted for by the weighted combination of variables in the set. The average loss is labeled Mean. In this example, the average loss over sets is 0.464. Notice that more loss occurs for the second dimension than for the first dimension.
The eigenvalue for each dimension equals 1 minus the average loss for the dimension and indicates how much of the relationship is shown by each dimension. The eigenvalues add up to the total fit. For Verdegaal’s data, 0.801 / 1.536 = 52% of the actual fit is accounted for by the first dimension.
The maximum fit value equals the number of dimensions and, if obtained, indicates that the relationship is perfect. The average loss value over sets and dimensions tells you the difference between the maximum fit and the actual fit. Fit plus the average loss equals the number of dimensions. Perfect similarity rarely happens and usually capitalizes on trivial aspects in the data.
Another popular statistic with two sets of variables is the canonical correlation. Since the canonical correlation is related to the eigenvalue and thus provides no additional information, it is not included in the nonlinear canonical correlation analysis output. For two sets of variables, the canonical correlation per dimension is obtained by the following formula:
ρd = 2 x Ed - 1
where d is the dimension number and E is the eigenvalue.
You can generalize the canonical correlation for more than two sets with the following formula:
ρd = ((K x Ed) - 1)/(K - 1)
where d is the dimension number, K is the number of sets, and E is the eigenvalue. For our example,
ρ1 = ((3 x 0.801) - 1)/2 = 0.702
and
ρ2 = ((3 x 0.735) - 1)/2 = 0.603