Contributed by: Harsha Nimkar

## Introduction

Multivariate means involving multiple dependent variables resulting in one outcome. This explains that the majority of the problems in the real world are Multivariate. For example, we cannot predict the weather of any year based on the season. There are multiple factors like pollution, humidity, precipitation, etc. Here, we will introduce you to multivariate analysis, its history, and its application in different fields.

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## The History of Multivariate analysis

In 1928, Wishart presented his paper. The Precise distribution of the sample covariance matrix of the multivariate normal population, which is the initiation of MVA.

In the 1930s, R.A. Fischer, Hotelling, S.N. Roy, and B.L. Xu et al. made a lot of fundamental theoretical work on multivariate analysis. At that time, it was widely used in the fields of psychology, education, and biology.

In the middle of the 1950s, with the appearance and expansion of computers, multivariate analysis began to play a big role in geological, meteorological. Medical and social and science. From then on, new theories and new methods were proposed and tested constantly by practice and at the same time, more application fields were exploited. With the aids of modern computers, we can apply the methodology of multivariate analysis to do rather complex statistical analyses.

## Multivariate analysis: An overview

Suppose a project has been assigned to you to predict the sales of the company. You cannot simply say that ‘X’ is the factor which will affect the sales.

We know that there are multiple aspects or variables which will impact sales. To analyze the variables that will impact sales majorly, can only be found with multivariate analysis. And in most cases, it will not be just one variable.

Like we know, sales will depend on the category of product, production capacity, geographical location, marketing effort, presence of the brand in the market, competitor analysis, cost of the product, and multiple other variables. Sales is just one example; this study can be implemented in any section of most of the fields.

Multivariate analysis is used widely in many industries, like healthcare. In the recent event of COVID-19, a team of data scientists predicted that Delhi would have more than 5lakh COVID-19 patients by the end of July 2020. This analysis was based on multiple variables like government decision, public behavior, population, occupation, public transport, healthcare services, and overall immunity of the community.

As per the Data Analysis study by Murtaza Haider of Ryerson university on the coast of the apartment and what leads to an increase in cost or decrease in cost, is also based on multivariate analysis. As per that study, one of the major factors was transport infrastructure. People were thinking of buying a home at a location which provides better transport, and as per the analyzing team, this is one of the least thought of variables at the start of the study. But with analysis, this came in few final variables impacting outcome.

Multivariate analysis is part of Exploratory data analysis. Based on MVA, we can visualize the deeper insight of multiple variables.

There are more than 20 different methods to perform multivariate analysis and which method is best depends on the type of data and the problem you are trying to solve.

Multivariate analysis (MVA) is a Statistical procedure for analysis of data involving more than one type of measurement or observation. It may also mean solving problems where more than one dependent variable is analyzed simultaneously with other variables.

• The main advantage of multivariate analysis is that since it considers more than one factor of independent variables that influence the variability of dependent variables, the conclusion drawn is more accurate.
• The conclusions are more realistic and nearer to the real-life situation.

• The main disadvantage of MVA includes that it requires rather complex computations to arrive at a satisfactory conclusion.
• Many observations for a large number of variables need to be collected and tabulated; it is a rather time-consuming process.

## Classification Chart of Multivariate Techniques

Selection of the appropriate multivariate technique depends upon-

a) Are the variables divided into independent and dependent classification?

b) If Yes, how many variables are treated as dependents in a single analysis?

c) How are the variables, both dependent and independent measured?

Multivariate analysis technique can be classified into two broad categories viz., This classification depends upon the question: are the involved variables dependent on each other or not?

If the answer is yes: We have Dependence methods.
If the answer is no: We have Interdependence methods.

Dependence technique:  Dependence Techniques are types of multivariate analysis techniques that are used when one or more of the variables can be identified as dependent variables and the remaining variables can be identified as independent.

Also Read: What is Big Data Analytics?

## Multiple Regression

Multiple Regression Analysis– Multiple regression is an extension of simple linear regression. It is used when we want to predict the value of a variable based on the value of two or more other variables. The variable we want to predict is called the dependent variable (or sometimes, the outcome, target, or criterion variable). Multiple regression uses multiple “x” variables for each independent variable: (x1)1, (x2)1, (x3)1, Y1)

Also Read: Linear Regression in Machine Learning

## Conjoint analysis

Conjoint analysis‘ is a survey-based statistical technique used in market research that helps determine how people value different attributes (feature, function, benefits) that make up an individual product or service. The objective of conjoint analysis is to determine the choices or decisions of the end-user, which drives the policy/product/service. Today it is used in many fields including marketing, product management, operations research, etc.

It is used frequently in testing consumer response to new products, in acceptance of advertisements and in-service design. Conjoint analysis techniques may also be referred to as multi-attribute compositional modeling, discrete choice modeling, or stated preference research, and is part of a broader set of trade-off analysis tools used for systematic analysis of decisions.

There are multiple conjoint techniques, few of them are CBC (Choice-based conjoint) or ACBC (Adaptive CBC).

## Multiple Discriminant Analysis

The objective of discriminant analysis is to determine group membership of samples from a group of predictors by finding linear combinations of the variables which maximize the differences between the variables being studied, to establish a model to sort objects into their appropriate populations with minimal error.

Discriminant analysis derives an equation as a linear combination of the independent variables that will discriminate best between the groups in the dependent variable. This linear combination is known as the discriminant function. The weights assigned to each independent variable are corrected for the interrelationships among all the variables. The weights are referred to as discriminant coefficients.

The discriminant equation:

F = β0 + β1X1 + β2X2 + … + βpXp + ε

where, F is a latent variable formed by the linear combination of the dependent variable, X1, X2,… XP is the p independent variable, ε is the error term and β0, β1, β2,…, βp is the discriminant coefficients.

## A linear probability model

A linear probability model (LPM) is a regression model where the outcome variable is binary, and one or more explanatory variables are used to predict the outcome. Explanatory variables can themselves be binary or be continuous. If the classification involves a binary dependent variable and the independent variables include non-metric ones, it is better to apply linear probability models.

Binary outcomes are everywhere: whether a person died or not, broke a hip, has hypertension or diabetes, etc.

We typically want to understand what the probability of the binary outcome is given explanatory variables.

We could actually use our linear model to do so, it’s very simple to understand why. If Y is an indicator or dummy variable, then E[Y |X] is the proportion of 1s given X, which we interpret as a probability of Y given X.

We can then interpret the parameters as the change in the probability of Y when X changes by one unit or for a small change in X For example, if we model  , we could interpret β1 as the change in the probability of death for an additional year of age

## Multivariate Analysis of Variance and Covariance

Multivariate analysis of variance (MANOVA) is an extension of a common analysis of variance (ANOVA). In ANOVA, differences among various group means on a single-response variable are studied. In MANOVA, the number of response variables is increased to two or more. The hypothesis concerns a comparison of vectors of group means.  A MANOVA has one or more factors (each with two or more levels) and two or more dependent variables. The calculations are extensions of the general linear model approach used for ANOVA.

## Canonical Correlation Analysis

Canonical correlation analysis is the study of the linear relations between two sets of variables. It is the multivariate extension of correlation analysis.

CCA is used for two typical purposes :-

• Data Reduction
• Data Interpretation

You could compute all correlations between variables from the one set (p) to the variables in the second set (q), however interpretation is difficult when pq is large.

Canonical Correlation Analysis allows us to summarize the relationships into a lesser number of statistics while preserving the main facets of the relationships. In a way, the motivation for canonical correlation is very similar to principal component analysis.

## Structural Equation Modelling

Structural equation modeling is a multivariate statistical analysis technique that is used to analyze structural relationships. It is an extremely broad and flexible framework for data analysis, perhaps better thought of as a family of related methods rather than as a single technique.

SEM in a single analysis can assess the assumed causation among a set of dependent and independent constructs i.e. validation of the structural model and the loadings of observed items (measurements) on their expected latent variables (constructs) i.e. validation of the measurement model. The combined analysis of the measurement and the structural model enables the measurement errors of the observed variables to be analyzed as an integral part of the model, and factor analysis combined in one operation with the hypotheses testing.

## Interdependence Technique

Interdependence techniques are a type of relationship that variables cannot be classified as either dependent or independent.

It aims to unravel relationships between variables and/or subjects without explicitly assuming specific distributions for the variables. The idea is to describe the patterns in the data without making (very) strong assumptions about the variables.

## Factor Analysis

Factor analysis is a way to condense the data in many variables into just a few variables. For this reason, it is also sometimes called “dimension reduction”. It makes the grouping of variables with high correlation. Factor analysis includes techniques such as principal component analysis and common factor analysis.

This type of technique is used as a pre-processing step to transform the data before using other models. When the data has too many variables, the performance of multivariate techniques is not at the optimum level, as patterns are more difficult to find. By using factor analysis, the patterns become less diluted and easier to analyze.

## Cluster analysis

Cluster analysis is a class of techniques that are used to classify objects or cases into relative groups called clusters. In cluster analysis, there is no prior information about the group or cluster membership for any of the objects.

• While doing cluster analysis, we first partition the set of data into groups based on data similarity and then assign the labels to the groups.
• The main advantage of clustering over classification is that it is adaptable to changes and helps single out useful features that distinguish different groups.

Cluster Analysis used in outlier detection applications such as detection of credit card fraud. As a data mining function, cluster analysis serves as a tool to gain insight into the distribution of data to observe the characteristics of each cluster.

## Multidimensional Scaling

Multidimensional scaling (MDS) is a technique that creates a map displaying the relative positions of several objects, given only a table of the distances between them. The map may consist of one, two, three, or even more dimensions. The program calculates either the metric or the non-metric solution. The table of distances is known as the proximity matrix. It arises either directly from experiments or indirectly as a correlation matrix.

## Correspondence analysis

Correspondence analysis is a method for visualizing the rows and columns of a table of non-negative data as points in a map, with a specific spatial interpretation. Data are usually counted in a cross-tabulation, although the method has been extended to many other types of data using appropriate data transformations. For cross-tabulations, the method can be considered to explain the association between the rows and columns of the table as measured by the Pearson chi-square statistic. The method has several similarities to principal component analysis, in that it situates the rows or the columns in a high-dimensional space and then finds a best-fitting subspace, usually a plane, in which to approximate the points.

A correspondence table is any rectangular two-way array of non-negative quantities that indicates the strength of association between the row entry and the column entry of the table. The most common example of a correspondence table is a contingency table, in which row and column entries refer to the categories of two categorical variables, and the quantities in the cells of the table are frequencies.

## The Objective of multivariate analysis

(1) Data reduction or structural simplification: This helps data to get simplified as possible without sacrificing valuable information. This will make interpretation easier.

(2) Sorting and grouping: When we have multiple variables, Groups of “similar” objects or variables are created, based upon measured characteristics.

(3) Investigation of dependence among variables: The nature of the relationships among variables is of interest. Are all the variables mutually independent or are one or more variables dependent on the others?

(4) Prediction Relationships between variables: must be determined for the purpose of predicting the values of one or more variables based on observations on the other variables.

(5) Hypothesis construction and testing. Specific statistical hypotheses, formulated in terms of the parameters of multivariate populations, are tested. This may be done to validate assumptions or to reinforce prior convictions.

Also Read: Introduction to Sampling Techniques

## Model Building Process

Model Building–choosing predictors–is one of those skills in statistics that is difficult to tell. It is hard to lay out the steps, because at each step, you must evaluate the situation and make decisions on the next step. But here are some of the steps to keep in mind.

The primary part (stages one to stages three) deals with the analysis objectives, analysis style concerns, and testing for assumptions. The second half deals with the problems referring to model estimation, interpretation and model validation. Below is the general flow chart to building an appropriate model by using any application of the variable techniques-

## Model Assumptions

Prediction of relations between variables is not an easy task. Each model has its assumptions. The most important assumptions underlying multivariate analysis are normality, homoscedasticity, linearity, and the absence of correlated errors. If the dataset does not follow the assumptions, the researcher needs to do some preprocessing. Missing this step can cause incorrect models that produce false and unreliable results.

## Multivariate Statistics Summary

The key to multivariate statistics is understanding conceptually the relationship among techniques with regards to:

• The kinds of problems each technique is suited for.
• The objective(s) of each technique.
• The data structure required for each technique,
• Sampling considerations for each technique.
• Underlying mathematical model, or lack thereof, of each technique.
• Potential for complementary use of techniques

Finally, I would like to conclude that each technique also has certain strengths and weaknesses that should be clearly understood by the analyst before attempting to interpret the results of the technique. Current statistical packages (SAS, SPSS, S-Plus, and others) make it increasingly easy to run a procedure, but the results can be disastrously misinterpreted without adequate care.

One of the best quotes by Albert Einstein which explains the need for Multivariate analysis is, “If you can’t explain it simply, you don’t understand it well enough.”

In short, Multivariate data analysis can help to explore data structures of the investigated samples.

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