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We examined the comparative behavior of subject-specific multivariate and univariate reference regions, using both computer-generated data and serial (semi-annual) measurements of selected analytes in subjects from a large health-maintenance program. Univariate studies under both homeostatic and random-walk time-series models were helpful in defining expected results, but only the homeostatic model was used in multivariate as well as univariate forms. Analysis of the computer-generated data and the real biochemical series produced similar findings, which showed the multivariate subject-specific reference region to be much more conservative than corresponding univariate intervals. That is, a multidimensional point of p correlated observations is quite likely to lie within the individual's multivariate reference region (based on past observation vectors), even when one or more of the observations lie outside their separate reference intervals for that individual. One consequence of this high specificity against univariate false positives in a large surveillance program is a higher than expected proportion of positive multivariate vectors in which none of the values lie outside their univariate ranges. Thus, although the development of multivariate reference regions should be encouraged, they should be used in conjunction with, not instead of, univariate ranges.
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Univariate analysis
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Closure (psychology)
Independence
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ABSTRACT: Multivariate methods of trend analysis offer the potential for higher power in detecting gradual water quality changes as compared to multiple applications of univariate tests. Simulation experiments were used to investigate the power advantages of multivariate methods for both linear model and Mann‐Kendall based approaches. The experiments focused on quarterly observations of three water quality variables with no serial correlation and with several different intervariable correlation structures. The multivariate methods were generally more powerful than the univariate methods, offering the greatest advantage in situations where water quality variables were positively correlated with trends in opposing directions. For illustration, both the univariate and multivariate versions of the Mann‐Kendall based tests were applied to case study data from several lakes in Maine and New York which have been sampled as part of EPA's long term monitoring study of acid precipitation effects.
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This is a brief introductory chapter. Various types of multivariate data are described and illustrated. Some sources of multivariate data are the sciences, humanities, business, medical fields, etc. Most multivariate data sets involve correlated variables, for which there are many useful multivariate exploratory techniques as well as inferential procedures. Many multivariate techniques are extensions of univariate procedures. The univariate procedures are reviewed in the text. Other multivariate techniques have no univariate analog. The objectives for the readers of the book are (1) understand the details of various multivariate techniques, (2) be able to select one or more multivariate procedures for a given multivariate data set, and (3) interpret the results of a computer analysis of a multivariate data set. Various types of data and accompanying analyses are described. Throughout the book there are many examples and problems involving real data sets. Answers to most problems are provided in Appendix B.
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Exploratory data analysis
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Food data are becoming increasing multivariate in nature and response measures can comprise several univariate types. This chapter provides some indication of how to take the univariate methods a stage further and apply them to multivariate situations. The methods illustrated in the chapter require more sophisticated software that is less generally available, although most examples are analysed using SPSS (SPSS Software), which is available as a student version. In the univariate case, independent and dependent variables have been identified. For the multivariate situation, the number of both independent and dependent variables can increase. A selection of commonly used more advanced and multivariate methods are briefly described, with more detail on principal component analysis (PCA). The analysis of sensory and instrumental data and how to combine them in the multivariate context concludes.
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Abstract Brain-Wide Association Studies (BWAS) have become a dominant method for linking mind and brain over the past 30 years. Univariate models test tens to hundreds of thousands of brain voxels individually, whereas multivariate models (‘multivariate BWAS’) integrate signals across brain regions into a predictive model. Numerous problems have been raised with univariate BWAS, including lack of power and reliability and an inability to account for pattern-level information embedded in distributed neural circuits 1–3 . Multivariate predictive models address many of these concerns, and offer substantial promise for delivering brain-based measures of behavioral and clinical states and traits 2,3 . In their recent paper 4 , Marek et al. evaluated the effects of sample size on univariate and multivariate BWAS in three large-scale neuroimaging dataset and came to the general conclusion that “BWAS reproducibility requires samples with thousands of individuals” . We applaud their comprehensive analysis, and we agree that (a) large samples are needed when conducting univariate BWAS of individual differences in trait measures, and (b) multivariate BWAS reveal substantially larger effects and are therefore more highly powered. However, we disagree with Marek et al.’s claims that multivariate BWAS provide “inflated in-sample associations” that often fail to replicate (i.e., are underpowered), and that multivariate BWAS consequently require thousands of participants when predicting trait-level individual differences. Here we substantiate that (i) with appropriate methodology, the reported in-sample effect size inflation in multivariate BWAS can be entirely eliminated, and (ii) in most cases, multivariate BWAS effects are replicable with substantially smaller sample sizes (Figure 1).
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Replicate
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Statistic
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Multivariate statistical analyses are appropriate whenever a study involves two or more outcome variables. Because multiple-outcome models reflect social reality more accurately than do conventional single-outcome or univariate models, multivariate analysis should be studied and practiced more extensively than it is. In this article, several reasons for doing multivariate analysis are presented, and two common errors in statistical analysis are discussed. Examples are presented to show how a single multivariate analysis can produce different results than do separate univariate analyses, and to illustrate the relationship between ANOVA and canonical correlation analysis.
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Canonical correlation
Statistical Analysis
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Sex differences in language-related abilities have been reported. It is generally assumed that these differences stem from a different organization of language in the brains of females and males. However, research in this area has been relatively scarce, methodologically heterogeneous and has yielded conflicting results.
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Univariate analysis
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Current approaches to the issue of interpreting the output of multivariate analyses are examined and two broad classes of approach distinguished: univariate and multivariate. Most researchers opt for interpretations that deal with variables one at a time rather than in combination. This approach is appropriate for univariate but not for multivariate techniques which are essentially tools for investigating combinations of psychological attributes which may determine outcome scores or category membership. Assessing the relative importance of individual variables is also inappropriate. A number of multivariate approaches that deal with combinations of variables are then examined. These include the use of simultaneous test procedures (STPs), rotation, and the simplification of canonical variates.
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Canonical correlation
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The use of multivariate techniques in the analysis of multivariate problems is illustrated by comparing the results of univariate and multivariate techniques applied to the problem of establishing the nutritional requirements of, and the acid tolerance differences between maize cultivars. Forty-eight maize cultivars were statistically separated into three groups, tolerant, intermediate and intolerant, using a univariate approach. A principal components analysis was then carried out to study the grouping at a multivariate level. The variates included were grain yield, plant height and ten leaf chemical analyses: Al, Mg, P, Ca, K, Mn, Zn, Fe, N and Cu. A non-hierarchical classification was applied to classify cultivars into the three tolerance classes. The univariate method resulted in different groupings for each variate under study, while the multivariate approach ensured one single classification of all cultivars into the three groups.
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