Paper: Multivariate Analysis

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Subject: Statistics

Paper: Multivariate Analysis

Module: Introduction to Multivariate Analysis

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Development Team

Principal investigator: Dr. Bhaswati Ganguli, Professor, Department of Statistics, University of Calcutta

Paper co-ordinator: Dr. Sugata SenRoy,Professor, Department of Statistics, University of Calcutta

Content writer: Souvik Bandyopadhyay, Senior Lecturer, Indian Institute of Public Health, Hyderabad

Content reviewer: Dr. Kalyan Das,Professor, Department of Statistics, University of Calcutta

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What is Multivariate Analysis?

Multivariate analysis is the simultaneous study of several variables.

I Rather than considering each variable individually, we consider all the variables under study together.

I Formally, in multivariate analysis, we have a set of m related variables

X= (X1, X2, ., Xm)⁰ which are studied together.

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Why Multivariate Analysis?

I A major advantage of multivariate analysis is that, unlike univariate studies, it takes into account the interdependences between the variables under study and hence is more

informative.

I However, this also brings in more complexities in the analyses.

I In spite of it, most often a multivariate analysis of them variables is preferred to mseparate univariate analyses.

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Multivariate Analysis?

I Usually most of the univariate methods can be extended to multivariate methods, with more complexities but better results.

I However, there are several multivariate techniques which are peculiar to themselves. These are

I eithertoo trivialin univariate studies

I ordo not arisein univariate problems

I It is mainly this latter with which we will be concerned here.

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Some Examples

Let us next look at a few examples of multivariate data in different fields :

I Social Science : Characteristics of an individual like (Gender, Age, Nationality)

I Climatology : Weather on a particular day like

(Minimum temperature, Maximum temperature, Rainfall, Humidity)

I Economics : Management of a firm like (Input costs, Production, Profit)

I Socio-Demographic : Profile of a country like (GDP, Life expectancy, Literacy rate)

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Some Examples

There can be several examples from Health Sciences :

I General Health : Health profile of an individual like (Systolic BP, Diastolic BP, Pulse rate)

I Pathology : The pathological profile of a patient like (Blood sugar level, Uric acid concentration, Hemoglobin count)

I Administrative : Day-to-day administration in a hospital (Admissions, Operations, Discharges, Deaths)

I Pharmaceutical : Quantity of sales per day in a pharmacy (Drug A, Drug B, ., Drug Z)

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Visualization

Unlike univariate, bivariate or trivariate data, multivariate data (m≥4)is impossible to plot. Even for trivariate data (as shown in the diagram), the plots are often difficult to understand.

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Example

Problem: In this problem, each face has several features. It is impossible to quantify and then plot these features to analyse their characteristics.

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Different aspects of Multivariate Analysis

A major aspect of multivariate analysis is to extend the univariate results to the multivariate set-up.

This is mostly done to study the inter-relationships between the variables under study, which are lost in univariate studies. Hence emphasis here is onjoint rather thanmarginal studies.

I This may mean the extension of the Binomial or Normal distributions to their multivariate counterparts.

I It can also mean the extensions of the inference techniques to multivariate data. This can relate both to the estimation and hypothesis testing problems.

Different aspects of Multivariate Analysis 10 / 18

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Cause-Effect Relationships

I A major extension is in the study of cause-effect relationships.

The multiple regression model with one response, the ANOVA model and the ANCOVA models can all be extended to their multivariate counterparts where several responses are studied simultaneously.

I The question asked here iswhether a set of variables have an effect on another set of variables, and if so, how ?

I As the model equations are linear in the parameters, the models are broadly referred to as Multivariate Linear Models.

I Special cases of this, depending on the nature of the covariates, areMultivariate Regression,MANOVAand

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Different aspects of Multivariate Analysis

But there are aspects of multivariate analysis which are unique to itself. Two such broad aspects are

I Classification of Individuals

I Dimension Reduction

The problem ofClassification of Individuals is too trivial for univariate data

while theDimension Reductionproblem does not arise for single variable studies.

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Classification of Individuals

At the exploratory stage in the classification problem, the question often asked is

Can a group of individuals or units be separated into smaller subgroups according to similarity or dissimilarity ?

I The answer is often sought to segregate similar units into homogeneous groups and allow the heterogeneity to be treated as between group variability.

I The greater this between group variability, the more distinctly defined are the groups.

I The technique to do this is known as Cluster Analysis.

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Classification of Individuals

Following the formation of such clusters, it is necessary to characterize these clusters. Thus the next natural question is

Why and how are these clusters different ?

I Such answers are generally provided by Discriminant Analysis, which helps to distinguish between the characteristics of the clusters.

I Classification techniques are then used to assign new individuals into one of these well-defined clusters.

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Dimension Reduction

I Very often, a data may contain a large number of variables.

I This may make both the analysis complicated and the interpretations difficult.

I A question thus asked is

Whether it is necessary to look at all the variables,

Or is it possible to capture the same information through a smaller set of variables ?

I The solutions to this problem are obtained throughDimension Reduction methods.

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Dimension Reduction

There are several Dimension Reduction techniques. Four of the important ones are

I Principal Component Analysis

I Factor Analysis

I Canonical Correlation methods

I Multidimensional Scaling

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Summary

In summary, in this course we will be restricting ourselves to

I The extensions of univariate inferential techniques to multivariate data

I The study of Multivariate Linear Models

I The study of Dimension reduction techniques

I The study of different Classification techniques

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Summary

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Summary

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Summary

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Paper: Multivariate Analysis

Subject: Statistics

Paper: Multivariate Analysis

Module: Introduction to Multivariate Analysis

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