Canonical Correlation Analysis as a General Linear Model: A Heuristic Lesson for Teachers and Students

The Catholic Education Office (CEO) Sydney is a large non-government education authority which administers the systemic, Catholic schools of the Archdiocese of Sydney, Australia. The system consists of 148 primary and secondary schools with an enrolment of some 62,000 students. The major research question was: What characteristics of a learning organization can be identified in the Catholic Education Office (CEO) Sydney and are these perceived to raise standards in systemic schools of the Archdiocese of Sydney? Like all western education systems the CEO Sydney is immersed in constant change and is expected to account for improving educational standards within the system. The learning organization with its emphasis on adaptability and continuous improvement was considered an appropriate framework within which to conduct this research. The study consisted of two main parts the first investigated the CEO Sydney as a learning organization using a survey questionnaire distributed, using a dedicated web site, to a sample of primary and secondary principals in the system and a smaller number of senior CEO Sydney personnel. The response rate was 91%. This was complemented by examination of relevant CEO Sydney documentation and policies. The definition of the learning organization adopted for the study consisted of eight characteristics each of which formed a scale in the questionnaire. The eight characteristics adopted were: 'Systemic Thinking and Mental Models', 'Continuous Improvement of Work', 'Taking Initiatives and Risks', 'Ongoing Professional Development', 'Trusting and Collaborative Climate', 'Shared and Monitored Vision/Mission', 'Effective Communication Channels' and 'Team Work and Team Learning'. This part of the study was essentially a quantitative one, with the data subjected to descriptive, statistical analysis complemented by some clarifying and contextualising qualitative data. The second part of the study investigated the perceived relationship between the CEO Sydney and its learning organization characteristics and the standards in three curriculum outcome areas (religious education, literacy and numeracy). This part of the study was also quantitative using descriptive statistics complemented by Pearson correlation, multiple regression and canonical correlational analyses. Once again some relevant contextualising qualitative data was gathered. Five demographic groups (gender, role, region (principals only), years of experience as a principal and age) were examined to see if there were any differences in the extent to which the various learning organization characteristics and curriculum outcomes were identified by each group. The results of this study indicated that the CEO Sydney exhibited many of the characteristics of a learning organization with particular strengths in 'Continuous Improvement of Work', 'Systemic Thinking and Mental Models' and 'Shared and Monitored Vision/Mission'. The weakest characteristic was 'Taking Initiatives and Risks'. Demographic group analysis of this data revealed that there were no statistically significant differences in the responses of the different demographic groups. The results also indicated that there were correlations between the CEO Sydney as a learning organization and raising standards particularly in religious education and literacy and less so in numeracy. Finally, the study made a number of recommendations for the further development of the CEO Sydney as a learning organization and ways that it can further raise standards in the schools of the system.

Chapter 11 - Canonical analysis

Data integration through canonical correlation analysis and its application to OMICs research

This study explores various options available for choosing the number of principal coordinates m in the canonical analysis of principal coordinates ‘CAP’, a useful procedure that has wide-ranging application wherever multivariate data sets are collected or generated. Choosing too few coordinates (small m) in this constrained (i.e. hypothesis-based) ordination procedure may lead to inadequate separation of the groups (when used as a canonical discriminant analysis) or to inadequate correlation between explanatory and response variables (when used as a canonical correlations analysis), whereas choosing too many (large m) may lead to overparameterization, resulting in overfitting of the data and spurious relationships. It is shown here that the optimum number of principal coordinates is simply the one that results in the smallest P value in the canonical analysis carried out using permutations. For data in which more than one m value results in the same minimum P value, m should be chosen from that set to be the number of principal coordinates that minimizes the leave-one-out residual sum of squares. This choice of m provides suitable solutions for each of the 17 case studies investigated here (which yielded 17 canonical discriminant analyses and 7 canonical correlation analyses).

The application of three multivariate analysis techniques (canonical discriminant analysis (CDA), principal component analysis (PCA), and canonical correlation analysis (CCA)) for evaluation of pasture botanical composition data is illustrated and discussed. CDA and PCA were used to describe differences in pasture botanical composition for different microsites within a pasture near Palmerston North, New Zealand. CCA could not be validly applied to this data set because a sampling strategy inappropriate for CCA had been used to collect the data. However, CCA is conceptually ideal for determining association between two groups of variables and CCA was used for a second data set from the Hawkes Bay region to establish association between differences in pasture botanical composition and differences in environmental variables. CCA identified a transition from white clover (Trifolium repens L.) to subterranean clover {Trifolium subterraneum L.) presence associated with decreasing rainfall, and a similar transition from ryegrass presence to browntop presence associated with altitude. There are practical difficulties in obtaining a suitable data set for canonical correlation, but with attention to sampling strategy a more precise definition of effect of environmental factors on pasture botanical composition would be expected.

We have endeavored in the preceding chapters to draw attention to the richness and flexibility of canonical analysis. The flexibility of the underlying model, in particular, is expressed in part by the number of apparently distinct methods which prove on examination to be special cases of canonical correlation analysis. Canonical variate analysis and dual scaling were both shown abstractly to be equivalent to canonical correlation analysis. The model’s flexibility is complemented by a correspondingly diverse range of application. While, in relation to the worked examples of Chaps. 6–12, it may be that the ability of canonical analysis to have parsimoniously summarized the dependence structures in question is to be attributed to the small scale of these investigations, within the class of small-scale investigations the applicability of canonical analysis was shown to be very wide. Altogether, it seems justifiable to conclude that canonical analysis offers an unusually versatile structure by means of which statistical theory, ecological theory and empirical data can be harnessed to provide insight into the organization of ecological systems.

Canonical correlation analysis (CCA) is a powerful technique for assessing the relationship between two sets of variables. However, classical CCA does not distinguish between treatment and outcome variables, and it fails to account for potential confounding effects that may bias the estimated results. This study proposes a novel method called canonical causal analysis (Causal‐CCA) that extends CCA to estimate the canonical causal effects between multivariate continuous treatments and outcomes while adjusting for confounding variables. Causal effects are estimated within a framework where treatments and outcomes are prespecified. The proposed approach uses Gram–Schmidt orthogonalization and entropy balancing weights for multivariate treatments (EBMT) to extract unconfounded canonical causal variates and compute the canonical causal effects. Extensive simulations across various scenarios demonstrate that Causal‐CCA consistently outperforms classical CCA, weighted CCA with multivariate generalized propensity score weighting (MVGPS) and CCA with EBMT in terms of bias and mean squared error. The proposed method is applied to a real‐world dataset to study the causal relationship between psychological factors (self‐concept and motivation) and academic achievement (math and science scores), showing a significant positive causal effect of the psychological variables on academic performance.

The data of 60 male Chios lambs raised in western Turkey on a farm registered with the Sheep and Goat Breeders' Association were used. Two datasets were generated to determine the correlations between weaning (90 d) and six-month traits in lambs. A canonical analysis was performed to determine the correlation between body weight measured at the weaning period and six morphological traits —dataset X-weaning weight (WW), body length (BL), wither height (WH), heart girth (HG), chest width (CW), rump width (RW), and rump length (RL)— and body weight measured at six months and six morphological traits —dataset Y-body weight (LW_6), body length (BL_6), wither height (WH_6), heart girth (HG_6), chest width (CW_6), rump width (RW_6), and rump length (RL_6). Only the first (U1V1) of the seven pairs of canonical dimensions was statistically significant. The contribution of body weight, WH, and RW to the explanatory effect of canonical variables estimated from morphological traits of Chios lambs was greater at weaning age than other body measurements. Canonical correlation analysis and variables (U1 and V1) can be used as a generic measure of variables, X and Y, respectively. The cross-loadings indicated that body weight, RW, and WH in the weaning period were more effective in V1 determination than other traits; WH, RL, and LW_6 were more influential in U1 determination. This study is the first to use canonical correlation analysis to determine the correlation between live weight and body measurements in Chios lambs, presenting early selection criteria for Chios lamb breeding studies.

Parametric statistical tests, as opposed to nonparametric or distribution‐free tests, are based on various assumptions regarding the characteristics, properties, and forms of the distributions of populations from which the data are drawn. A large number of statistical tests are included among the parametric tests, primarily hypothesis testing procedures derived from the general linear model. These include both univariate and multivariate statistical tests: the t test, univariate and multivariate analysis of variance (ANOVA) and covariance (including repeated measures), Pearson product‐moment correlation, simple and multiple regression (and variants including logistic regression), Hotelling's T 2 , discriminant function analysis, canonical correlation, and multivariate set correlation.

In the face of multicollinearity, researchers face challenges interpreting canonical correlation analysis (CCA) results. Although standardized function and structure coefficients provide insight into the canonical variates produced, they fall short when researchers want to fully report canonical effects. This article revisits the interpretation of CCA results, providing a tutorial and demonstrating canonical commonalty analysis. Commonality analysis fully explains the canonical effects produced by using the variables in a given canonical set to partition the variance of canonical variates produced from the other canonical set. Conducting canonical commonality analysis without the aid of software is laborious and may be untenable, depending on the number of noteworthy canonical functions and variables in either canonical set. Commonality analysis software is identified for the canonical correlation case and we demonstrate its use in facilitating model interpretation. Data from Holzinger and Swineford (1939) are employed to test a hypothetical theory that problem-solving skills are predicted by fundamental math ability.

Larval diet quality and rearing conditions have a direct and irreversible effect on adult traits. Therefore, the current study was carried out to optimize the larval diet for mass rearing of Aedes aegypti, for Sterile Insect Technique (SIT)-based applications in Sri Lanka. Five batches of 750 first instar larvae (L1) of Ae. aegypti were exposed to five different concentrations (2-10%) of International Atomic Energy Agency (IAEA) recommended the larval diet. Morphological development parameters of larva, pupa, and adult were detected at 24 h intervals along with selected growth parameters. Each experiment was replicated five times. General Linear Modeling along with Pearson's correlation analysis were used for statistical treatments. Significant differences (P < 0.05) among the larvae treated with different concentrations were found using General Linear Modeling in all the stages namely: total body length and the thoracic length of larvae; cephalothoracic length and width of pupae; thoracic length, thoracic width, abdominal length and the wing length of adults; along with pupation rate and success, sex ratio, adult success, fecundity and hatching rate of Ae. aegypti. The best quality adults can be produced at larval diet concentration of 10%. However, the 8% larval diet concentration was most suitable for adult male survival.

A number of measures of canonical correlation coefficient are now used in pattern recognition in the different literature. Some robust forms of classical canonical correlation coefficient are introduced recently to address the robustness issue of the canonical coefficient in the presence of outliers and departure from normality. Also a few number of kernels are used in canonical analysis to capture nonlinear relationship in data space, which is linear in some higher dimensional feature space. But not much work has been done to investigate their relative performances through simulation and also from the view point of sensitivity. In this paper an attempt has been made to compare performances of kernel canonical correlation coefficients (Gaussian, Laplacian and Polynomial) with that of classical and robust canonical correlation coefficient measures using simulation and influence function. We investigate the bias, standard error, MSE, qualitative robustness index, sensitivity curve of each estimator under a variety of situations and also employ boxplots and scatter plots of canonical variates to judge their performances. We observe that the class of kernel estimators perform better than the class of classical and robust estimators in general and the kernel estimator with Laplacian function has shown the best performance for large sample size.

This paper gives results from joint analyses of dual polarimety synthetic aperture radar data from the Sentinel-1 mission and optical data from the Sentinel-2 mission. The analyses are carried out by means of traditional canonical correlation analysis (CCA) and canonical information analysis (CIA). Where CCA is based on maximising correlation between linear combinations of the two data sets, CIA maximises mutual information between the two. CIA is a conceptually more pleasing method for the analysis of data with very different modalities such as radar and optical data. Although a little inconclusive as far as the change detection aspect is concerned, results show that CIA analysis gives conspicuously less noisy appearing images of canonical variates (CVs) than CCA. Also, the 2D histogram of the mutual information based leading CVs clearly reveals much more structure than the correlation based one. This gives promise for potentially better change detection results with CIA than can be obtained by means of CCA.

BackgroundAlthough hypercaloric interventions are associated with nutritional, endocrine, metabolic, and cardiovascular disorders in obesity experiments, a rational distinction between the effects of excess adiposity and the individual roles of dietary macronutrients in relation to these disturbances has not previously been studied. This investigation analyzed the correlation between ingested macronutrients (including sucrose and saturated and unsaturated fatty acids) plus body adiposity and metabolic, hormonal, and cardiovascular effects in rats with diet-induced obesity.MethodsNormotensive Wistar-Kyoto rats were submitted to Control (CD; 3.2 Kcal/g) and Hypercaloric (HD; 4.6 Kcal/g) diets for 20 weeks followed by nutritional evaluation involving body weight and adiposity measurement. Metabolic and hormonal parameters included glycemia, insulin, insulin resistance, and leptin. Cardiovascular analysis included systolic blood pressure profile, echocardiography, morphometric study of myocardial morphology, and myosin heavy chain (MHC) protein expression. Canonical correlation analysis was used to evaluate the relationships between dietary macronutrients plus adiposity and metabolic, hormonal, and cardiovascular parameters.ResultsAlthough final group body weights did not differ, HD presented higher adiposity than CD. Diet induced hyperglycemia while insulin and leptin levels remained unchanged. In a cardiovascular context, systolic blood pressure increased with time only in HD. Additionally, in vivo echocardiography revealed cardiac hypertrophy and improved systolic performance in HD compared to CD; and while cardiomyocyte size was unchanged by diet, nuclear volume and collagen interstitial fraction both increased in HD. Also HD exhibited higher relative β-MHC content and β/α-MHC ratio than their Control counterparts. Importantly, body adiposity was weakly associated with cardiovascular effects, as saturated fatty acid intake was directly associated with most cardiac remodeling measurements while unsaturated lipid consumption was inversely correlated with these effects.ConclusionHypercaloric diet was associated with glycemic metabolism and systolic blood pressure disorders and cardiac remodeling. These effects directly and inversely correlated with saturated and unsaturated lipid consumption, respectively.

The paper deals with the analysis of the relationships between the visual characteristics of mountain ecosystems as they appear on spatial images, and the characteristics of the abiotic conditions as being governed and (or) indicated by the terrain morphometric parameters, in the study area in Ukrainian Carpathians. LANDSAT 7 ETM+ spatial imagery with 30 m resolution, and SRTM v. 4.1 DEM with 90 m ground resolution were the main data sources used in the study. Based on the analysis of LANDSAT imagery, three principal components have been picked out that describe the different characteristics of vegetation and land cover spatial differentiation. Terrain parameters accounted for in the study included raw elevation values, flow energy index, slope values, topographic wetness index, and insolation index. The canonical correlation analysis has then been applied to analyze the relationships between the landscape reflectance characteristics, and the terrain morphometric parameters, allowing deriving two statistical significant canonical roots and determining their structure. The relationships between the structure and properties of vegetation cover (as conditioned by landscape abiotic factors and the land use structure), and the terrain morphometric parameters have thus been revealed and statistically substantiated. Key words: LANDSAT imagery, digital elevation models, terrain morphometric parameters, factor analysis, canonical analysis.