Multivariate Research Assignment

8/3/2019 Multivariate Research Assignment

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Business Research Methods

Assignment

On

Multivariate Analysis

Submitted By:

Tariq Mahmood Asghar

Roll No. # 77

MBA (A, B1)


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DATA ANALYSIS

The terms "statistics" and "data analysis" mean the same thing -- the study of how

we describe, combine, and make inferences from numbers. A lot of people are

scared of numbers (quantiphobia), but statistics has got less to do with numbers,

and more to do with rules for arranging them. It even lets you create some of those

rules yourself, so instead of looking at it like a lot of memorization, it's best to see it

as an extension of the research mentality, something researchers do (crunch

numbers) to obtain complete and total power over the numbers. After awhile, the

principles behind the computations become clear, and there's no better way to

accomplish this than by understanding the research purpose of statistics.

MULTIVARIATE DATA ANALYSIS

As the name indicates, multivariate analysis comprises a set of techniquesdedicated to the analysis of data sets with three or more variables. Multivariate

analysis is the simultaneous analysis of three or more variables. It is frequently

done to refine a bivariate analysis, taking into account the possible influence of a

third variable on the original bivariate relationship. Multivariate analysis is also used

to test the joint effects of two or more variables upon a dependent variable. In some

instances, the association between two variables is assessed with a multivariate

rather than a bivariate statistical technique. This situation arises when two or more

variable are needed to express the functional form of the association.

Multivariate Data Analysis refers to any statistical technique used to analyze datathat arises from more than one variable. This essentially models reality where each

situation, product, or decision involves more than a single variable. The information

age has resulted in masses of data in every field. Despite the quantum of data

available, the ability to obtain a clear picture of what is going on and make intelligent

decisions is a challenge. When available information is stored in database tables

containing rows and columns, Multivariate Analysis can be used to process the

information in a meaningful fashion.

Multivariate analysis methods typically used for:

Consumer and market research

Quality control and quality assurance across a range of industries such as

food and beverage, pharmaceuticals, chemicals, energy,

telecommunications, etc.

Process optimization and process control

Research and development


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With Multivariate Analysis you can:

Obtain a summary or an overview of a table. This analysis is often called

Principal Components Analysis or Factor Analysis. In the overview, it is

possible to identify the dominant patterns in the data, such as groups,

outliers, trends, and so on. The patterns are displayed as two plots

Analyze groups in the table, how these groups differ, and to which group

individual table rows belong. This type of analysis is called Classification and

Discriminant Analysis

Find relationships between columns in data tables, for instance relationships

between process operation conditions and product quality. The objective is to

use one set of variables (columns) to predict another, for the purpose of

optimization, and to find out which columns are important in the relationship.

Tools for Multivariate Analysis

The Multiple Correlation Coefficients:

A correlation is a way of measuring the linear association between two variables.

But what is the correlation if Y, the dependent variable, is being predicted from more

than one independent variable? Answer is the multiple correlations are the

correlation between Y and the best linear combination of the independent variables.

The best linear combination of independent variables is the multiple regression

slopes.

In bivariate correlation, the predictor, X, and predicted, Y, were perfectly correlated.

In multiple correlation, the predictors, X1 and X2, are perfectly correlated with the

predicted, Y. You already know how to establish that "best possible predictor" using

multiple regressions described above. You already know how to solve for a

correlation coefficient. In this section, we will show you how these principles are

used to solve for the multiple correlation coefficient.

The multiple correlation coefficients vary in terms of strength; it can vary between .

00 and 1.00. The multiple correlations do not specify direction. If the value of the

dependent variable does not change in any systematic way as any of theindependent variables varies in either direction, then there is no relationship and the

coefficient is 0.00. Thus, if the plane describing the points is flat, indicating no

systematic change in the dependent variable as we increase the values of the

independent variables, the coefficient is 0.


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In formula form, the multiple correlation between X and Y is equal to the covariance

of Y and Y predicted from the best linear combination of the X divided by the

product of the respective standard deviations.

Multiple Regressions (MR):

Multiple Regression is a "general linear model" with a wide range of applications. It

is basically an extension of the bivariate correlation and simple regression analysis.

The primary uses of MR are as follows: Prediction of a continuous Y with several continuous X variables: Unlike

ordinary bivariate regression, MR allows the use of an entire set of variables

to predict another.

Use of categorical variables in prediction: Through the technique of

dummy coding, categorical variables (such as marital status or treatment

group) can be used in addition to continuous variables.

Calculation of the unequal n ANOVA problem: Disproportional cell size in

any factorial ANOVA design produces a correlation among the independent

variables. MR estimates effects and interactions in this situation by the use of

dummy codes.

Model nonlinear relationships between Y and a set of X: By the addition

of "polynomial" terms (e.g. quadratic, cubic, trends) into the equation,

relationships that do not meet the linear assumptions can be analyzed.

Multiple linear regression analysis (MLR):

In MLR, several IV's (which are supposed to be axed or equivalently are measured

without error) are used to predict with a least square approach one DV. If the IV's

are orthogonal, the problem reduces to a set of univariate regressions. When the

IV's are correlated, their importance is estimated from the partial coeffcient of

correlation. An important problem arises when one of the IV's can be predicted from

the other variables because the computations required by MLR can no longer be

performed: This is called multicolinearity.

Non-Linear Regression:

Ordinary multiple regression assumes that each bivariate relationship between X

and Y is linear, and that the relationship between Y and Y' is also linear. How can

we examine how closely we have met this linearity assumption? We could, of

course, always inspect all of the separate scatter plots between all Y and X

relationships. But this does not address the second part of the linearity assumption.


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We can determine the conformity of our data the MR linearity assumption via direct

examination of residual scores. Recall that a residual or "error" score is equal to (Y -

Y') and that we would expect this to be approximately zero once the linear

regression determined. Thus, a plot of residual scores against [Yacute] that shows a

peculiar pattern would suggest the violation of linearity. Two of the most common

type of nonlinear relationships is represented in this way:

These cases can be modeled through polynomial regression. Polynomial regression

simply adds terms to the original equation to account for nonlinear relationships.

Squaring the original variable accounts for a quadratic trend, a cubed term accounts

for the cubic trend, and so on. After the linear component, each term adds another

"bend" in the prediction line. As with traditional MR, polynomial regression can be

interpreted by looking at R Squared and changes in R Squared.

A couple notes of caution: in order to look at any higher order effect (such as the

cubic), all lower order effects (the quadratic and the linear) must be placed into theequation; a degree of freedom is lost for each additional term added into the

equation; and lastly, the N:k Ratio changes for each added term.

Recall that trend tests in ANOVA were accomplished with subjects grouped into

discrete categories. With polynomial regression, the subjects are not grouped, but

rather each individual has a unique score on a continuous variable. Therefore, MR

trend information is much more complete than information available with typical

ANOVA.

Multivariate analysis of variance (MANOVA):

Multivariate analysis of variance (MANOVA) is a generalized form of analysis of

variance (ANOVA) methods to cover cases where there is more than one

(correlated) dependent variable and where the dependent variables cannot simply

be combined. As well as identifying whether changes in the independent variables

have a significant effect on the dependent variables, the technique also seeks to


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identify the interactions among the independent variables and the association

among dependent variables, if any.

MANOVA Procedure

MANOVA procedures are multivariate, significance test analogues of variousunivariate ANOVA experimental designs. MANOVA, as with its univariate

counterparts typically involve random assignment of participants to levels of one or

more nominal independent variables; however, all participants are measured on

several continuous dependent variables.There are three basic variations of MANOVA:

Hotelling's T: This is the MANOVA analogue of the two group T-test

situation; in other words, one dichotomous independent variable, and

multiple dependent variables. One-Way MANOVA: This is the MANOVA analogue of the one-way F

situation; in other words, one multi-level nominal independent variable, and

multiple dependent variables. Factorial MANOVA: This is the MANOVA analogue of the factorial ANOVA

design; in other words, multiple nominal independent variables, and multiple

dependent variables.While all of the above MANOVA variations are used in somewhat different

applications, they all have one feature in common: they form linear combinations of

the dependent variables which best discriminate among the groups in the particular

experimental design. In other words, MANOVA is a test of the significance of group

differences in some m-dimensional space where each dimension is defined by

linear combinations of the original set of dependent variables. This relationship will

be represented for each design into the following sections.

Multivariate Research Assignment

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