# An Introduction To Multivariate Statistical Ana... PORTABLE

Multivariate meta-analysis is becoming increasingly popular and official routines or self-programmed functions have been included in many statistical software. In this article, we review the statistical methods and the related software for multivariate meta-analysis. Emphasis is placed on Bayesian methods using Markov chain Monte Carlo, and codes in WinBUGS are provided. The various model-fitting options are illustrated in two examples and specific guidance is provided on how to run a multivariate meta-analysis using various software packages.

## An Introduction to Multivariate Statistical Ana...

ANOVA remains one of the most widely used statistical models in academia. Of the several types of ANOVA models, there is one subtype that is frequently used because of the factors involved in the studies. Traditionally, it has found its application in behavioural research, i.e. Psychology, Psychiatry and allied disciplines. This model is called the Multivariate Analysis of Variance (MANOVA). It is widely described as the multivariate analogue of ANOVA, used in interpreting univariate data.

Table 1 shows the effect sizes (given as hazard ratios), 95% confidence intervals (CI), regression coefficients and statistical significance for each of these in relation to overall survival. Each factor is assessed through separate univariate Cox regressions (left-hand columns). However, the aim of the database is to describe how the factors jointly impact on survival, and so all five factors were incorporated into the multivariate model (right-hand columns). It may be seen that higher FIGO stage, higher grade, presence of ascites and increased age impaired survival to varying (and statistically significant) degrees. The histology was also of importance: the figures describe the survival of patients with each histology type in comparison with the serous type. In principle, any type with a reasonable number of patients could be chosen as the baseline of comparison. On multivariate analysis Mucinous and serous were the tumour types with the best prognosis, whereas undifferentiated and mixed mesodermal were the worst. It is possible to present P-values for the comparisons between each type and serous, but we have given an overall likelihood ratio test for the differences between the categories as a whole. The FIGO stage could be modelled as a categorical variable in the same manner as grade and histology, but assuming it is a continuous variable with a linear trend across the four categories performed sufficiently well.

This is a graduate level 3-credit, asynchronous online course. In this course we will examine a variety of statistical methods for multivariate data, including multivariate extensions of t-tests and analysis of variance, dimension reduction techniques such as principal component analysis, factor analysis, canonical correlation analysis, and classification and clustering methods.

Recommended Texts: (i) R. Johnson and D. Wichern, Applied Multivariate Statistical Analysis, 6th ed. Available free as pdf online. This is a popular and good applied book to be used as a source of examples and alternate, intuitive explanations.(ii) We will also refer to topics from two widely referenced statistical-machine-learning books (both free online): C. Bishop (2006), Pattern recognition. Machine learning. T. Hastie, R. Tibshirani and J. Friedman (2009), The Elements of Statistical Learning, 2nd ed. (iii) Anderson, T.W. An Introduction to Multivariate Statistical Analysis, 3rd ed. 2003, Wiley-Interscience. Standard and authoritative, but theoretical and fairly dry, with deeper mathematical treatment than Mardia, Kent and Bibby.(iv) Haerdle, W. and Simar, L. (2007) Applied Multivariate Statistical Analysis, 2nd ed., Springer. Another good applied book, maybe at a slightly higher mathematical level than Johnson-Wichern. Available as free e-book to students through the UMD library.Overview: This course is about statistical models and methods of inference for multivariate observations with dependent coordinates. Much of the theoretical material relates to the multivariate normal distribution and to the statistical sampling behavior of empirical variance-covariance matrices and of various projections and eigen-decompositions of them. Models studied include regression, principal components analysis, factor models, and canonical correlations. In addition, important algorithmic or machine-learning methods like Clustering and Support Vector Machines will also be discussed. All methods will be illustrated using computational data examples in R.

Prerequisite: STAT 420 or STAT 700. Familiarity with some (any) statistical software package would be very helpful, but familiarity with R would be best. The presentation will be geared to second-year Stat grad students. Probability theory material needed throughout this course includes joint probability densities and change-of-variable formulas, law of large numbers and (multivariate) central limit theorem. In addition, the course makes extensive use of linear algebra, especially including eigenvalues and eigenspaces and singular value decompositions.

Multivariate statistical analysis: random vectors and matrices, sample mean and sample covariance, the multivariate normal distribution, the multivariate Central Limit Theorem, assessing normality and outlier detection, the Hotelling's T square, the confidence ellipsoid, simultaneous confidence intervals, Bonferroni methods, the multivariate analysis of variance (MANOVA), the multivariate linear regression.

Izenman, Alan Julian. 2013. Modern multivariate statistical techniques: Regression, classification, and manifold learning. New York: Springer New York. Book Home Page (including R, S-plus and MATLAB code and data sets)

An introduction to the fundamental concepts and methods of statistics. The course will cover topics ranging from descriptive statistics, sampling distributions, confidence intervals, and hypothesis testing. Topics could include simple and multiple linear regression, Analysis of Variance, and Categorical Analysis. Use of statistical software is emphasized. Prerequisite: Graduate standing.

A continuation of SDS 380C: Statistical Methods I. The course presents an overview of advanced statistical modeling topics. Topics may include random and mixed effects models, time series analysis, survival analysis, Bayesian methods, and multivariate analysis of variance. Use of statistical software is emphasized. Prerequisite: Graduate standing, and Statistics and Data Sciences 380C or the equivalent.

Introduction to mathematical concepts and methods essential for multivariate statistical analysis. Topics include basic matrix algebra, eigenvalues and eigenvector, quadratic forms, vector and matrix differentiation, unconstrained optimization, constrained optimization, and applications in multivariate statistical analysis. Prerequisite: Graduate standing and one course in statistics.

In this course, students will learn to describe real-world systems using structured probabilistic models that incorporate multiple layers of uncertainty. Major topics to be covered include: (i) theory of the multivariate normal distribution; (ii) mixture models; (iii) introduction to nonparametric Bayesian analysis; (iv) advanced hierarchical models and latent-variable models; (v) Generalized Linear Models; and (vi) advanced topics in linear and nonlinear regression. Examples will be taken from a wide variety of applied fields in the physical, social, and biological sciences. Prerequisite: Graduate standing.

An introduction to statistical learning methods, exploring both the computational and statistical aspects of data analysis. Topics include numerical linear algebra, convex optimization techniques, basics of stochastic simulation, nonparametric methods, kernel methods, graphical models, decision tress and data re-sampling. Prerequisites: Graduate standing.

Focuses on various mathematical and statistical aspects of data mining. Topics covered include supervised learning (regression, classification, support vector machines) and unsupervised learning (clustering, principal components analysis, dimensionality reduction). The technical tools used in the course draw from linear algebra, multivariate statistics and optimization. Prerequisites: Graduate standing and Mathematics 341 or equivalent.

Multivariate statistics analyzes data on several random variables simultaneously. This course introduces the basic concepts and provides an overview of classical and modern methods of multivariate statistics including visualization, dimension reduction, supervised and unsupervised learning for multivariate data. An emphasis is on applications and solving problems with the statistical software R. 041b061a72