SMDE - (US) Multivariate Analysis

8/10/2019 SMDE - (US) Multivariate Analysis

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MULTIVARIATE ANALYSIS

Pau Fonseca i Casas [email protected]


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Multivariant analisys?

Is the set of statistical methods with the aim to

analyze simultaneously data sets of multivariate

data in the sense that exists different variables

measured for each individual or object to bestudied.

Its reason to exist is to understand better the

phenomenon to be studied, obtaining information

that the univariate or bivariate methods are not

able to obtain.


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Multivariate analysis objectives

Provide methods with the aim of study of

multivariate data sets, that the univariate or

bivariate statistical analysis cannot afford.

To help to the researcher to take optimal decisionsin his context, taking care the available information

for the analyzed dataset.


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Multivariate techniques

Dependency methods Metric Regression analysis

Survival analysis

MANOVA Canonical correlation

No metric Discriminant analysis

Logistic regression

Conjoint analysis Interdependency methods

Structural methods


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Multivariate techniques

Dependency methods

Interdependency methods Metric data

Principal component analysis

Factorial analysis Multidimensional scales

Cluster analysis

No metric data Correspondence analysis

Log-lineal models Multidimensional scales

Cluster analysis

Structural methods


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Regression analysis, Survival analysis, MANOVA,

Canonical regression, Discriminant analysis,

Logistic regression, Cojoint analysis

Dependency methods


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Dependency methods

They assume that the variables analyzed are

divided into two groups: the dependent and

independent variables.

The goal of the methods is dependent on whetherthe set of independent variables affects all

dependent variables and how.


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Dependency methods

They can be classified into two subgroups according

to the variable (s) dependent (s) is (are)

quantitative or qualitative.

If the dependent variable is quantitative sometechniques that can be applied are:

Regression Analysis

Survival Analysis Analysis of variance

Canonical Correlation


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Technique adequate if in the analysis exists one or

several dependent metric variables whose value

depends of one or more independent metrics

variables. For example, trying to predict the annual expenditure

on Christmas of a person from their income level,

education level, gender and age.

Regression analysis


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Survival Analysis

Similar to regression analysis but with the difference

that the independent variable is the time of survival

of an individual or object.

For example, try to predict the time spent inunemployment of an individual from their level of

education and age.


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Analysis of variance

They are used in situations where the total sample is

divided into several groups based on one or more

independent nonmetric variables and the

dependent variables analyzed are metric. It aims tofind out if there are significant differences between

the groups in terms of the dependent variables.

For example, are there differences in the level of

cholesterol by gender? Does it affect also the type

of occupation?.


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Canonical Correlation

Its aim is to connect simultaneously several

independent and dependent metric variables

defining linear combinations of each set of

variables that maximize the correlation between thetwo sets of variables.

For example, analyzing how is related the time

dedicated to work and leisure for a person with an

specific income level, age and education level.


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Dependency methods

If the dependent variable is qualitative some

techniques that can be applied are:

Discriminant Analysis

Logistic regression models

Conjoint Analysis


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Discriminant Analysis

This technique gives optimal classification rules of

new observations where is unknown its source group

based on the information provided by the values

that in it takes the independent variables. For example, determining the financial ratios that

best allow discriminating between profitable and

unprofitable.


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Logistic regression models

Are regression models in which the dependent

variable is not metric. They are used as an

alternative to the discriminant analysis when normal

assumption cannot be assumed.


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Conjoint Analysis

It is a technique that analyzes the effect of independentnon-metric variables on metric or nonmetric variables. Thedifference with the analysis of variance is based on twofacts: the dependent variables can be non-metric and

the values of the independent variables are not set by metricsanalyst. In other disciplines is known as Design of Experiments.

For example, a company wants to design a new product andit needs to specify the shape of the container, its price perpackage content and chemical composition. Presents variouscompositions of these four factors. 100 customers provide aranking of the combinations that are presented. It wants todetermine the optimal values of these four factors.


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Principal Component Analysis, Factorial Analysis,Multidimensional Scales, Cluster Analysis,Correspondence Analysis, Log-Lineal Models,Multidimensional Scales, Cluster Analysis.

Interdependence methods


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These methods do not distinguish between

dependent and independent variables and the

objective is to identify which variables are related,

how they are, and why.


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They can be classified into two groups according to

the type of data to analyze whether metric or non-

metric.

If data are metric can be used, among others, thefollowing techniques:

Factorial Analysis and Principal Component Analysis

Multidimensional Scales Cluster Analysis


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Factor Analysis and Principal

Component Analysis

Is used to analyze interactions between a large number ofvariables such interrelationships explaining metrics in termsof fewer variables called factors (if unobservable) orprincipal components (if they are observable).

For example, if a financial analyst wants to determine whichis the financial health of a company based on theknowledge of a number of financial ratios, building severalnumerical indices that define their situation, the problemwould be resolved by analyzing Principal Components.

If a psychologist wants to determine the factors thatcharacterize an individual's intelligence from their answersto an IQ test, can use to solve this problem a FactorialAnalysis.


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Multidimensional Scales

Is intended to transform judgments of preference or

similarity in distances represented in a multidimensional

space. Consequently a map is constructed in which

positions represents the objects compared. Those whoare similar are closed and far from the dissimilar ones.

For example, look at the soft drinks market, perceptions

that a consumer group has about a list of drinks and

brands in order to study how a consumer uses subjectivefactors when classifying these products.


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If the data are not metric can be used, in addition

to multidimensional scaling and cluster analysis, the

following techniques:

Correspondence Analysis Log-linear models


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Correspondence Analysis

Applies to multidimensional contingency tables and

pursues a similar objective of multidimensional

scales but simultaneously representing the rows and

columns of the contingency tables. For example, unemployment in Aragon analyze

considering the province, sex, age and educational

level of the unemployed


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Log-linear models

They apply to multidimensional contingency tables

and multidimensional dependencies modeling the

observed variables that seek to explain the

observed frequencies.


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Structural Methods


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Structural methods

They assume that the variables are divided into two

groups: the dependent variable and the

independent. The objective of these methods is to

analyze not only as independent variables to thedependent variables affect, but also how variables

relate the two groups together.


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Multivariate analysis steeps

1. Goals of the analysis

2. Design of the analysis

3. Hypotheses of the Analysis

4. Analytical procedure

5. Interpretation of the results

6. Analysis Validation


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The problem is specified defining objectives and

multivariate techniques that will be used.

The investigator must establish the problem

conceptually defining the concepts and relationsthat are fundamental to the investigation. It must

determine whether such relationships will be

relations of dependence or interdependence. With

all these the variables to observe are determined.


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Determine the sample size, the equations to

estimate (if applicable), the distances to calculate

(if applicable) and the estimation techniques

employed. Once this is determined we can proceedto observe the data.


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We evaluate the assumptions underlying the

multivariate technique.

These hypotheses may be of normality, linearity,

independence, homoscedasticity, etc. You must alsodecide what to do with the missing data.


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We estimate the model and we evaluate the fit to

the data.

In this step may appear unusual observations (outliers)

or influential whose influence on the estimates and thegoodness of fit must be analyzed.


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Is to establish the validity of the results obtained by

analyzing whether the results, obtained with the

sample, is generalized to the population from which

it comes. This sample can be divided into several parts in which

the model is re-estimated and the results are

comparared. Other techniques that can be used here

are resampling techniques (jackknife and bootstrap)


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What technique are you going to apply?

Example


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Predicting the amount of money a person spends incinema depending on income level, education level,gender and age which would allow us to betterunderstand what are the patterns of behavior ofthe population.

Variables to consider: Income level

Level of education Sex

Age


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We propose multiple regression analysis in which

the dependent variable would be spending on film

and the other independent variables.


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Are decided how to select the sample, the size of it

and how to measure the variables involved in the

analysis.

Spending on film could be measured as the annualexpenditure on film measured in euros.

The income level could be measured with an ordinal

variable, given the reluctance to give accurate

information on these variables, the level of educationwould be an ordinal variable, a binary sex and age a

quantitative variable measured in years.


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The sample size would be chosen in function of the

power that may be given to the multiple regression.

It should, moreover, that the ratio of the number of

observations to the number of parameters to beestimated is sufficiently broad to estimate model

parameters with the least possible error.


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You have to check the linearity of the relationship,

normality and homoscedasticity. No data are

missing and should study the possible existence of

outliers in each of the variables.


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You can use the least squares estimator, knowing itssampling distribution under normality assumption.

This estimate coincides with the maximum likelihood andis efficient.

You can also use the method of stepwise regression todetermine the independent variables included in theregression.

Once estimated the regression equation examines thegoodness of fit by calculating R2 and residual analysis.

Study the homoscedasticity, independence, possibleomission of variables, existence of outliers and influenceof individual observations.


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5. Interpretation of the results

The value of the coefficients obtained and its sign

are interpreted.


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The sample is divided into two sub-samples of size

50 and re-estimate the regression equation in each

subsample comparing results.

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SMDE - (US) Multivariate Analysis