# To reduced number of observation by grouping then

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To reduced number of observation by grouping then, into smaller set of cluster. 21.3. CONTENTS 23.3.1. Factor Analysis 23.3.1.1. The Mathematical Basis 23.3.1.2. Important Methods of Factor Analysis 23.3.1.3. Centroid Method 23.3.1.4. Maximum Likelihood method 23.3.1.5. Rotation in Factor Analysis 23.3.1.6. R-Type and Q-Type Factor Analysis 23.3.1.7. Merits and Demerits of Factor Analysis

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216 21.3.2. Cluster Analysis 21.3.2.1. Clustering Algorithms 21.3.2.1.1. Non-hierarchical Clustering 21.3.2.1.2. Hierarchical Clustering 21.3.2.2. Agglomerative Clustering 23.3.1. FACTOR ANALYSIS Factor analysis is by far the most often used multivariate technique of research studies, specially pertaining to social and behavioral sciences. It is a technique applicable when there is a systematic independence among a set of observed or manifest variables and the researcher is interested in finding out something more fundamental or latent which creates this commonality. For ins tance, we might have data, say, about an individual’s income, education, occupation and dwelling area and want to infer from these some factor (such as social class) which summarizes the commonality of all the said four variables. The technique used for such purpose is generally described as factor analysis. Factor analysis, thus, seeks to resolve a large set of measured variables in terms of relatively few categories, known as factors. This technique allows the researcher to group variables into factors (based on correlation between variables) and the factors so derived may be treated as new variables (often termed as latent variables) and their value derived by summing the values of the original variables which have been grouped into the factor. The meaning and name of such new variables is subjectively determined by the researcher. Since the factors happen to be linear combinations of data, the coordinates of each observation or variable is measured to obtain what are called factor loadings. Such factor loadings represent the correlation between the particular variable and the factor, and are usually place in a matrix of correlations between the variable and the factors. 23.3.1.1. The Mathematical Basis The mathematical basis of factor analysis concerns a data matrix (also termed as score matrix), symbolized as S . The matrix contains the scores of N persons of k measures. Thus 1 a is the score of person 1 on measure 2 , a a is the score of person 2 on measure a and n k is the score of person N on measure k . The score matrix then takes the form as shown following: SCORE MATRIX (or Matrix S) Measures (variables) Persons (objects) 1 1 1 1 2 2 2 2 3 3 3 3 1 2 3 . . . . . . . . . . . . . . . . N N N N N a b c k a b c k a b c k a b c k a b c k
217 It is assumed that scores on each measure are standardized 2 [ ., ( ) / ] i i i ie x X X . This being so, the sum of scores in any column of the matrix, S , is zero and the variance of scores in any column is 1,0 . Then factors (a factor is any linear combination of the variables in a data matrix and can be stated in a general way like: ....

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