DISCRIMINANT ANALYSIS

MULTIVARIATE ANALYSIS

The term “Multivariate analysis” which is a collection of methods for analyzing data in which a number of observations are available for each object.

TECHNIQUES

All statistical techniques which simultaneously analyse more than two variables on a sample of observations can be categorized as multivariate techniques.

DISCRIMINANT ANALYSIS

Through discriminant analysis technique, researcher may classify individuals or objects into one of two or more mutually exclusive and exhaustive groups on the basis of a set of independent variables.

Discriminant analysis requires interval independent variables and a nominal dependent variable.

Conti………

• Regression analysis in such a situation is not suitable because the dependent variable is , not intervally scaled. Thus discriminant analysis is considered an appropriate technique when the single dependent variable happens to be non-metric and is to be classified into two or more groups, depending upon its relationship with several independent variables which all happen to be metric.

EXAMPLES

• Good or bad While grouping investment

alternatives based on rate of return, the criterion of the rate of return will be categorized into ‘good’ or ‘bad’. In this example, ‘investment alternatives’ is an entity and each investment alternative is treated as a member of the entity.

OBJECTIVES

DISCRIMINANT ANALYSIS may be used for two objectives:

(A)either we want to assess the adequacy of the classification, given the group memberships of the objects under study ;

(B) or we wish to assign objects to one of a number of (known) groups of objects.Discriminant analysis may thus have a descriptive or a predictive objective.

TYPES OF ANALYSIS

Multiple discriminant analysis Linear discriminant Fisher’s discriminant Mean square error discriminant Bayesian discriminant Maximum likelihood discrimination K-Nearest neighbours analysis

ASPECTS

• There happens to be a simple scoring system that assigns a score to each individual or object. This score is a weighted average of the individual’s numerical values of his independent variables. On the basis of this score, the individual is assigned to the ‘most likely’ category.

z = bo + b1X1 + b2X2 +………+ bnXn where,

z = the ith individual’s discriminant scorebi = the discriminant coefficiant of the ith variableXi = the ith individual’s value of the jth independent variable

Conti……

• now find the ‘variability between groups’ and ‘variability within groups’

• Now for judging the statistical significance between two groups, we work out the MAHALANOBIS statistic, D² , which happens to be a generalized distance between two groups, where each group is characterized by the same of n variables and where it is assumed that variance-covariance structure is identical for both groups.

Conti…..

D² = (U₁-U₂)v⁻¹(U₁-U₂) where,

U₁ = the mean for group oneU₂ = the mean for group two v = the common variance matrix

Conti…..

• after that, by transformation procedure, this D² statistic becomes an F statistic which can be used to see if the two groups are statistically different from each other.

• Find the table F value and calculated F value.• If the calculated F value is more than the table

F value, reject the null hypothesis, H₀; otherwise, accept the hypothesis.

Conti…….

• Rather than ANOVA categorical independent variable and a continuous dependent variable, discriminant analysis has continuous independent variables and a categorical dependent variable.

• Discriminant analysis is also different from factor analysis in that it is not interdependence technique; a distinction between dependent & independent variables must be made.

APPLICATIONS

(1) Bankruptcy prediction: In bankruptcy prediction, based on

accounting ratios and other financial variables, linear discriminant analysis was the first statistical method applied to systematically explain which firms entered bankruptcy vs. survived.

(2) Marketing : In marketing, discriminant analysis was once often used to determine the factors which distinguish different types of customers and/or products on the basis of surveys or other forms of collected data.

(3) Astronomy : In Astronomy, it helps to define various galaxies and stars.

THANKS