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Applied multivariate statistics is a branch of statistics which encompasses the simultaneous analysis and observation of greater than one outcome variable. One of the applications of Applied multivariate statistics is called as multivariate analysis.
The multivariate statistics relates to the understanding of the different backgrounds and aims of each of the existing different forms of multivariate analysis and the considers the relation between them. The practical implementation of the multivariate analysis which applies to a particular problem may involve different types of multivariate and univariate analysis so that we can understand the existing relationships between variables and consider their relevance in solving actual problem being studied.
In addition to this, multivariate probability distributions is related to multivariate statistics, in both terms:
1. Considering the relation used to represent the distributions of observed data;
2. Their use in statistical inference, particularly where several different quantities are involved in the same analysis.
There exist certain types of problems which involves multivariate data, for example multiple regression and simple linear regression, are not considered as a special cases of multivariate statistics as the analysis is done by considering the univariate conditional distribution counting on a single outcome variable provided the other variables also exist.
The major courses to study in Applied multivariate analysis is Univariate descriptive statistics, Sampling distribution, Estimation, Hypothesis testing, Multivariate descriptive statistics, Multivariate normal distribution, Analysis of variance (ANOVA), Multivariate Inferential statistics, Multivariate analysis of variance (MANOVA), Multiple linear regression (MLR), MLR: Model adequacy tests , Principle component analysis (PCA) etc.

Applied Multivariate Analysis techniques

Applied Multivariate Analysis is focused on many statistical techniques which uses just one or two variables. Multivariate analysis techniques allow more than two variables to be analyzed at once. Multiple regressions is not usually included, but can be considered as a multivariate analysis.

Multivariate is typically more than one measurement is taken on a given experimental unit. It needs to consider all the measurements together so that one can understand how they are related. It needs to consider all the measurements together so that one can extract essential structure. The data include simultaneous measurements on many variables; this body of methodology is called multivariate analysis.

The multivariate analysis methods are the following:-

Analysis of dependence- Where one or more variables are dependent variables which is to be explained or predicted by others. For e.g. Multiple regression, PLS, MDA

Analysis of interdependence- No variables thought of as dependent. For e.g. cluster analysis, factor analysis.

The objectives of multivariate methods include the following:

Data reduction or structural simplification- The event is represented as simply as possible without sacrificing important information. It is hoped that this will make interpretation easier.

Sorting and grouping- Grouping of like objects or variables are created which is based upon measured characteristics.

Investigation of the dependence among variables- The environment of the relationships among variables is very useful. All the variables are mutually independent or are one or more variables dependent on the others.

Prediction- Relationships between variables have to be determined for the reason of predicting the values of one or more variables on the basis of clarification on the other variables.

STAT:6540 (22S:161) APPLIED MULTIVARIATE ANALYSIS

Multivariate descriptive statistics, multivariate normal distribution, T-squared, MANOVA, multivariate regression, principal components, discrimination , classification, cluster analysis, methods of classical applied multivariate statistics, descriptive statistics, test, multivariate regression, MANOVA, principal components, discrimination , classification, modeling continuous longitudinal data, Time permitting, data mining, SAS, R

Graphical methods for Multivariate Data
• Multivariate Normality, Multivariate ANOVA, Principal Components, Multidimensional Scaling, Cluster Analysis, Discriminant Analysis, Factor Analysis, Covariance Structure Models, Canonical Correlation Analysis, basic matrix operations, random vectors.
• Numerical and graphical summaries of multivariate data, Multivariate normal distribution, Inference for multivariate mean, Comparison of two or more mean vectors, Multivariate Linear Regression, Principal Components, Factor Analysis, Discrimination and Classification.
• Cluster Analysis, Canonical Correlation, Characterizing and Displaying Multivariate Data, Multivariate Normal Distribution, Tests on One or Two Mean Vectors, Multivariate Analysis of Variance, Discriminant Analysis, Classiﬁcation Analysis, Logistic Regression, Principal Components, Factor Analysis, Cluster Analysis.

Applied Multivariate Analysis

• Principal components analysis (PCA), Factor analysis and the use of rotations, Cluster analysis for grouping objects, Multidimensional scaling for mapping and relationship to PCA, Canonical variates analysis for highlighting differences between groups.
• Discriminant analysis and allocation rules, Partial least squares regression, Preliminary , Multivariate normal, Multivariate normal , Inference about mean , Principal component analysis , Factor analysis , Canonical correlation analysis , Discrimination and classification, Clustering, matrix theory, univariate normal, chi-squared, F and multivariate normal distributions., multivariate means including Hotelling’s T 2.
• multivariate analysis of variance, covariance structure including principal components, factor analysis and canonical correlation. , Multivariate classification techniques including , discriminant and cluster analyses
• logistic regression
response theory
principal component analysis
confirmatory factor analysis and structural equation modeling, nonparametric regression
nonparametric principal components analysis
quasi-experimental design
multilevel analysis
longitudinal analysis
cluster analysis
mediation and moderation
missing data
• Multivariate Data Analysis
Sample Random Variable Geometry and Vector
Multivariate Normal Distribution
Inference for Vector of Means
Principal Component Analysis
Factor Analysis
Cluster and Discriminant Analysis
Canonical Correlation
• Multivariate Data, Descriptive Statistics, Rows (Subjects) vs. Columns (Variables), Covariances, Correlations and Distances, The Multivariate Normal Distribution, Scatterplots , More than 2 Variable Plots
Assessing Normality
• Multivariate Normal Distribution, MANOVA, & Inference, Details of the Multivariate Normal Distribution, Wishart Distribution, Hotelling T2 Distribution , Multivariate Analysis of Variance (MANOVA), Hypothesis Tests on Covariances, Joint Confidence Intervals
• Multidimensional Scaling & Correspondence Analysis, Principal Components, Correspondence Analysis, Multidimensional Scaling
• Discriminant Analysis, Classification Problem, Population Covariances Known, Population Covariances Estimated, Fisher’s Linear Discriminant Function, Validation
• Introduction to the statistical analysis
• Emphasis is on concepts
• Computer-intensive methods
• Multiple regression
• Multivariate analysis of variance
• Principal components,
• Factor analysis
• Canonical correlations
• Multidimensional scaling
• Clustering.
• Classic multivariate statistics
• Properties of the multivariate normal distribution
• Determinants
• Volumes
• Projections
• Matrix square roots
• The singular value decomposition
• Wishart distributions
• Hotelling's T-square
• Principal components
• Canonical correlations
• Fisher's discriminant
• Cauchy projection formula