Canonical Redundancy Analysis Research Articles

The prediction of understory vegetation by environmental factors for the purpose of site classification in forestry: an example from northern Ontario using residual ordination analysis

Problems arise in the use of understory vegetation as an indicator of site condition in that impermanent factors such as microclimate, succession, and chance may play significant roles in determining local composition. Residual ordination analysis is a method which facilitates quantification of the sources of variation in understory vegetation over a landscape. Here it is applied to survey data, representing 250 stands upon which the forest ecosystem classification programme for the Clay Belt portion of northeastern Ontario is based, to test the premise that vegetation types will differentiate soil conditions for forestry purposes. Ordination of the data by detrended correspondence analysis yielded a bivariate scatterplot which, through visual appraisal, seemed readily interpretable in terms of site-related nutrient and moisture gradients. Formal exploration, using canonical redundancy analysis, yielded the following predictive model: understory vegetation (detrended correspondence analysis axes 1 and 2) = soils (67%) + canopy (8%) + succession (1%) + error (24%). Extraction of residual ordinations confirmed this general model and demonstrated that although canopy and successional influences are minor in the data, they are significant. Because the nonsite-related, predictable components account for only 9% of the variation at most, the premise of the existing forest ecosystem classification system is judged to be sound insofar as the data upon which it is based adequately describe the range of commercial stand conditions normally encountered. The results are discussed in relation to vegetation survey design and the performance of residual ordination analysis on a large data set is assessed.

The effect of affine transformation on redundancy analysis

Canonical redundancy analysis provides an estimate of the amount of shared variance between two sets of variables and provides an alternative to canonical correlation. The proof that the total redundancy is equal to the average squared multiple correlation coefficient obtained by regressing each variable in the criterion set on all variables in the predictor set is generalized to the case in which there are a larger number of criterion than predictor variables. It is then shown that the redundancy for the criterion set of variables is invariant under affine transformation of the predictor variables, but not invariant under transformation of the criterion variables.

Correspondence between occupational rewards and occupational needs and work satisfaction: A canonical redundancy analysis

The following hypothesis was derived from a general theory of work adjustment: the better the correspondence between occupational rewards and individual occupational needs, the greater will be employee satisfaction at work. Two samples of social workers were studied: one to obtain the occupational rewards levels, and the second to gauge individual occupational needs levels and satisfaction ratings with different work aspects. Then, vectors of difference scores between occupational needs and occupational rewards were computed for each of the subjects in the second sample, as a measure of the above correspondence.The canonical correlation analysis method was employed to ascertain the strength of association between these difference score vectors and satisfaction ratings among the subjects in the second sample. A canonical correlation of 0.84 (P<0.01) was revealed, indicating that a strong association exists. A canonical redundancy analysis was also taken as a precautionary measure. The conceptual implications of this analysis were discussed.

Verbal Interaction of Students and Their Teachers in Junior High Classrooms

Teachers’ and students’ verbal behaviors were investigated using the CVC system of classification in 72 classrooms of grades seven, eight, and nine in a rural and an urban school system. Trained observers observed classes during instruction in language arts, social studies, mathematics, and science twice during 2 weeks. The canonical correlation and redundancy analyses of 12 of the CVC cell percentages of teachers and students produced four significant canonical correlations ranging in value from .76 to .96. The total redundancy coefficients for the teachers’ and students’ factors were, respectively, .59 and .37. When the process and substance marginal categories were analyzed, respectively, one (.64) and two (.88 and .76), significant canonical correlations emerged. Canonical factor structures for the process data sets indicated that teacher-pupil verbal interaction was asymmetrically contingent. The substance level interaction appeared to be reciprocally contingent. Results are extended and related to the theory of instruction and behavior management.

Person-Environment Congruence and Work Satisfaction.

The canonical correlation analysis method was employed to ascertain the strength of association between the occupational rewards - occupational needs congruence and job satisfaction. A canonical coefficient of .89 (p < .01) was revealed, thus indicating that a strong association exists. However, a correction for this finding, using canonical redundancy analysis, although still supportive, suggests the need for precaution in congruence research designs.

Correcting for BIAS in the Canonical Redundancy Statistic

The canonical redundancy statistic, an estimate of the amount of shared variance between two sets of variables, has been proposed as an alternative to the squared canonical correlation coefficient in the interpretation of the results of canonical correlation analysis. The present study was undertaken to investigate the bias of the canonical redundancy statistic and to evaluate methods to correct for any bias. Using Monte Carlo methods, the redundancy statistic was found to exhibit an amount of bias similar to that of the first squared canonical correlation coefficient. Bias is most affected by sample size: bias was observed to decrease by half each time the sample size was doubled. Bias also decreases as the intercorrelations between the two sets of variables increase. The number of predictor and criterion variables, as well as the size of the correlations between variables in each set, have relatively minimal effect on bias. Two formulae, Wherry and Olkin-Pratt, that were developed to estimate the population value of the squared multiple correlation coefficient, were found to correct adequately the bias of the redundancy statistic. As an example of the use and interpretation of canonical redundancy analysis, a set of data relating consumer health values and health behaviors was analyzed. While the results indicate that only a small amount of redundant information is shared between these two sets of measures, the redundancy statistic provides a more conservative interpretation of this overlap than do the squared canonical correlation coefficients.

An alternative method to predict performance: canonical redundancy analysis.

The relationships between predictors of performance and subsequent measures of clinical performance in medical school were examined for two classes of students at Southern Illinois University School of Medicine. A relatively new statistical technique, canonical redundancy analysis, was used to evaluate the association between six academic and three biographical preselection characteristics and four measures clinical competence. While producing results consistent with the univariate methods traditionally used, the technique extends understanding of the total relationship between the two sets of measures. The preselection variable found to be most related to clinical competence was the undergraduate nonscience grade-point average. Overall, 16 percent of the variation in clinical performance was accounted for on the basis of knowledge of the preselection characteristics.