A second order approximation to the sample influence curve (SIC) in canonical correlation analysis has been derived in the literature. However, it does not seem satisfactory for some cases. In this paper, we present a more accurate second order approximation. As a particular case, the proposed method is exact for the SIC of the squared multiple correlation coefficient. An example is given.

The objective of this paper is to introduce and motivate additional properties and interpretations for the redundancy variables. It is shown that these variables can be derived by application of certain invariance arguments and without reference to the index of redundancy. In addition, an optimality property for the variables is presented which is important whenever one restricts attention in a study to a subset of the redundancy variables. This optimality property pertains to the subset rather than to the individual variables.

This paper extends the biplot technique to canonical correlation analysis and redundancy analysis. The plot of structure correlations is shown to the optimal for displaying the pairwise correlations between the variables of the one set and those of the second. The link between multivariate regression and canonical correlation analysis/redundancy analysis is exploited for producing an optimal biplot that displays a matrix of regression coefficients. This plot can be made from the canonical weights of the predictors and the structure correlations of the criterion variables. An example is used to show how the proposed biplots may be interpreted.

Homogeneity analysis, or multiple correspondence analysis, is usually applied to k separate variables. In this paper we apply it to sets of variables by using sums within sets. The resulting technique is called OVERALS. It uses the notion of optimal scaling, with transformations that can be multiple or single. The single transformations consist of three types: nominal, ordinal, and numerical. The corresponding OVERALS computer program minimizes a least squares loss function by using an alternating least squares algorithm. Many existing linear and nonlinear multivariate analysis techniques are shown to be special cases of OVERALS. An application to data from an epidemiological survey is presented.

The g1- and g2- bipartial canonical correlation analyses are developed as generalizations of the partial, part, and bipartial canonical correlation analysis. Illustrative examples are provided.

To facilitate the interpretation of canonical correlation analysis (CCA) solutions, procedures have been proposed in which CCA solutions are orthogonally rotated to a simple structure. In this paper, we consider oblique rotation for CCA to provide solutions that are much easier to interpret, though only orthogonal rotation is allowed in the existing formulations of CCA. Our task is thus to reformulate CCA so that its solutions have the freedom of oblique rotation. Such a task can be achieved using Yanai’s (Jpn. J. Behaviormetrics 1:46–54, 1974; J. Jpn. Stat. Soc. 11:43–53, 1981) generalized coefficient of determination for the objective function to be maximized in CCA. The resulting solutions are proved to include the existing orthogonal ones as special cases and to be rotated obliquely without affecting the objective function value, where ten Berge’s (Psychometrika 48:519–523, 1983) theorems on suborthonormal matrices are used. A real data example demonstrates that the proposed oblique rotation can provide simple, easily interpreted CCA solutions.

This paper shows essential equivalences among several methods of linearly constrained correspondence analysis. They include Fisher's method of additive scoring, Hayashi's second type of quantification method, ter Braak's canonical correspondence analysis, Nishisato's type of quantification method, ter Braak's canonical correspondence analysis, Nishisato's ANOVA of categorical data, correspondence analysis of manipulated contingency tables, Böckenholt and Böckenholt's least squares canonical analysis with linear constraints, and van der Heijden and Meijerink's zero average restrictions. These methods fall into one of two classes of methods corresponding to two alternative ways of imposing linear constraints, the reparametrization method and the null space method. A connection between the two is established through Khatri's lemma.

Correspondence analysis (CA) is a statistical method for depicting the relationship between two categorical variables, and usually places an emphasis on graphical representations. In this study, we discuss a CA formulation based on canonical correlation analysis (CCA). In CCA-based formulation, the correlations within and between row/column categories in a reduced dimensional space can be expressed by canonical variables. However, in existing CCA-based formulations, only orthogonal rotation is permitted. Herein, we propose an alternative CCA-based formulation that permits oblique rotation. In the proposed formulation, the CA loss function can be defined as maximizing the generalized coefficient of determination, which is a measure of proximity between two variables. Simulation studies and real data examples are presented in order to demonstrate the benefits of the proposed formulation.

Diet, obesity and adipokines play important roles in diabetes and CVD; yet, limited studies have assessed the relationship between diet and multiple adipokines. This cross-sectional study assessed associations between diet, adiposity and adipokines in Mexican Americans. The cohort included 1128 participants (age 34·7±8·2 years, BMI 29·5±5·9 kg/m2, 73·2 % female). Dietary intake was assessed by 12-month food frequency questionnaire. Adiposity was measured by BMI, total percentage body fat and percentage trunk fat using dual-energy X-ray absorptiometry. Adiponectin, apelin, C-reactive protein (CRP), dipeptidyl peptidase-4 (DPP-IV), IL-1β, IL-1ra, IL-6, IL-18, leptin, lipocalin, monocyte chemo-attractant protein-1 (MCP-1), resistin, secreted frizzled protein 4 (SFRP-4), SFRP-5, TNF-α and visfatin were assayed with multiplex kits or ELISA. Joint multivariate associations between diet, adiposity and adipokines were analysed using canonical correlations adjusted for age, sex, energy intake and kinship. The median (interquartile range) energy intake was 9514 (7314, 11912) kJ/d. Overall, 55 % of total intake was accounted for by carbohydrates (24 % from sugar). A total of 66 % of the shared variation between diet and adiposity, and 34 % of diet and adipokines were explained by the top canonical correlation. The diet component was most represented by sugar-sweetened beverages (SSB), fruit and vegetables. Participants consuming a diet high in SSB and low in fruits and vegetables had higher adiposity, CRP, leptin, and MCP-1, but lower SFRP-5 than participants with high fruit and vegetable and low SSB intake. In Mexican Americans, diets high in SSB but low in fruits and vegetables contribute to adiposity and a pro-inflammatory adipokine profile.

Site properties and weed species abundance are known to vary spatially across fields. The extent to which they covary is not well understood. The objective of this research was to assess how canonical correlation analysis could be used to identify associations among site properties and weed species abundance within an agricultural field. A farmer-managed field rotated between Zea mays and Glycine max in Boone County, IA, was grid-sampled for site properties in 1992 and for weed species abundance between 1994 and 1997. Twelve site properties were considered in relation to five weed species that were identified and counted after all weed control operations were completed. Site properties such as total nitrogen, Bray-1 P, percent organic carbon, and texture were spatially variable. Weed species abundance was also spatially variable such that most weeds were found in patches and much of the field was weed-free. Canonical correlation analysis identified one to four significant correlations between linear combinations of site properties and weed species abundance. The first and second pairs of linear combinations explained the majority of variation in the data and were used to identify associations among site properties and weed species abundance. In years with Z. mays, the first pair of linear combinations described an association between herbicide activity and weed presence, and the second described topography and soil texture associations with weed presence. In years with G. max, the single observed association described a link between soil texture and presence of Setaria species and Polygonum coccineum. Several consistent associations were identified across years, indicating that site properties can influence weed abundance. However, annual variation in the associations may be attributed to differences in agronomic and weed management practices for each crop, as well as temporal weather variation influencing weed abundance from year to year. This multivariate technique is an important tool to identify associations between site properties and weed abundance that could help explain observed patchy patterns of weed abundance. These associations are an important first step in the generation of hypotheses to be tested at the whole field scale.

Identification of associations between site properties and weed species abundance led to the generation of hypotheses as to why weed populations occur where they do, or do not, in agricultural fields. The objective of this research was to use a multivariate statistical technique, canonical correlation analysis, to identify the associations. Two continuous Zea mays production fields under center-pivot irrigation in the central Platte River Valley of Nebraska were grid-sampled between 1994 and 1997 for nine site properties and six to seven weed species. Weed species were identified and counted just prior to postemergence weed control in two adjacent quadrats (1 by 0.38 m) at each grid sampling point. These quadrats represented untreated weed populations emerging between crop rows and treated populations that survived preemergence herbicide banded within the crop row. Canonical correlation analysis identified one to five significant correlations between linear combinations of site properties and weed species abundance depending on field site, years, and between- vs. on-crop row weed populations. The first pair of linear combinations consistently described an association that separated weed species across a gradient of topography and soil type. The second pair of linear combinations described associations between weed species and soil fertility. In all cases, it was hypothesized that management practices strongly interacted with site properties to create the observed associations with weed populations. Other hypothesized mechanisms for weed patchiness include patchiness in available soil moisture that would influence weed seed germination, emergence, and seedling growth. Additional variation in plant-available preemergence herbicide concentration across the field site would vary weed control efficacy. Another mechanism would be variation in soil fertility that affects the growth, reproduction, and competitive ability of both the crop and the weed.