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Multivariate Analysis. Multivariate Analysis. Multivariate Analysis refers to a group of statistical procedure that simultaneously analyze multiple measurements on each individual being investigated. Some of multivariate methods are straight generalization of univariate analysis .
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Multivariate Analysis • Multivariate Analysis refers to a group of statistical procedure that simultaneously analyze multiple measurements on each individual being investigated. • Some of multivariate methods are straight generalization of univariate analysis. • The mathematical frameworkis relatively complex as compared with the univariate analysis. • These analysis are being used widely around the world.
Why Factor Analysis In many research problem there are large number of variables • Which are difficult to handle • Information provided by the variables are difficult to interpret due to correlation among variables With the help of FA, we can study the combination of the original variables which in some cases provide very useful information regarding any “hidden” underlying structure among the variables
Typical Problem Studied With Factor Analysis FA is typically used to study a complex product or service in order to identify the major characteristics considered to be important by consumers of the product or service. Corporations and their customers are seldom describes on the basis of one dimension. An individual’s decision to visit a fast-food restaurant is often depend on such factors as • Quality, Varity and price of the food • Restaurant’s location • Speed and quality of service When corporations develop databases to better serve their customers the database often includes a vast array of information such as • Demographic • lifestyles • purchasing behavious
Typical Problem Studied With Factor Analysis Example:- A manufacturer of compact automobiles wanted to know which automobile characteristics were considered very important by compact car buyers. To study this topic the company prepared 20 statements that related to all characteristics of automobiles that they believed were important, Six of which are listed below • A compact car should be built to last a long time • Gasoline mileage in a compact car should be at least 30 miles per gallon • A compact car should be easily maintained and serviced by its owner • Four adults should be able to sit comfortably in a compact car • Interior appointments in a compact car should be attractive • A compact car’s brakes are its most critical part
What Factor Analysis Does • Using data for a large sample FA applies an advanced form of correlation analysis to the responses to a number of statement • Determine if the responses to several of the statements are highly correlated • If responses to these statements are highly correlated ,it is believed that the statements measure some FACTOR common to all of them • Researcher use their own judgment to determine what the single “theme” or “factor” is that ties the statement together in the mind of the respondents
Construction of FACTORS FACTORS are identified through the use of extremely complex mathematical calculations
Worry about construction of Factors? I don’t like Mathematics so I am unable to construct FACTORS Don’t worry we will use computer Software like STATISTICA to construct FACTORS for you
Factors Original Variables Height Size Weight Occupation Social Status Education Source of Income
Goals of Factor Analysis • Model correlation patterns in useful way • Suggest new, uncorrelated variables that explain the original correlation structure • Allow for contextual interpretation of the new variables • Evaluate the original data in light of the new variables
Procedure For Factor Analysis • Obtain a random sample of n individuals and measure P traits (variables) and calculate correlation matrix • Initial factor extraction ; Estimate weighted sum of the variables with descending order of importance Methods: • Principal component method • Maximum Likelihood Method
3 Select the suitable number of common factors in the model for which • “suitable proportion “ of the total sample variance has been explained • Where SCREE plot first becomes stable • In case of correlation matrix select the number of common factors equal to the eigenvalues of R that are greater than ONE. • Expert opinion
Rotate the factors in order to simplify the interpretation of factors The main purpose of FA is to define from the data easily interpretable common factors. • The initial factors, however are often difficult to interpret regarding to the method used to extract the initial factor. • It is usual practice to rotate initial factors until a “simpler structure” is achieved Methods: • Orthogonal rotation (e.g Varimax Rotation) • Oblique Rotation (e.g Quartmin Rotation)
Application of FAStep-1&2 V1: Taste V2: Good buy for money V3: Flavor V4: Suitable for snack V5: Provides lots of energy In a consumer-preference study a random sample of customers were asked to rate several attributes of a new product. The responses, on a 7-point semantic differential scale were tabulated and the attribute correlation matrix constructed as
From above matrix entries it is clear that V1 and V3 ( Taste & Flavor ) and V2, V5 ( Good for money & Provides lots of energy) form groups. V4 is close to (V2 ,V5) group than the (V1,V3 ) group. • On the basis of the above results we might expect that the apparent linear relationship between the variables can be explained in terms of at most two or three common factors
Initinal Factor Extraction Variable Factor1 Factor2 Factor3 Factor4 Factor5 Communality Var 1 -0.560 -0.816 -0.045 -0.044 0.129 1.000 Var 2 -0.777 0.524 -0.336 0.090 0.013 1.000 Var 3 -0.645 -0.748 -0.076 -0.037 -0.130 1.000 Var 4 -0.939 0.105 0.272 0.182 0.000 1.000 Var 5 -0.798 0.543 0.100 -0.240 0.002 1.000 Eigen values /Variance 2.8531 1.8063 0.2045 0.1024 0.0337 5.0000 % Var 57 36 4 2 0.7 100 • Factor Loadings: • Correlation between each of the original variable and newly developed factor • Each factor loading is a measure of the importance of the variable in measuring each factor • Factor Loading ( like correlations) can vary from -1 to +1 when constructed from correlation matrix)
Common Factor Analysis Terms 1) FACTOR LOADINGS Correlation between each of the original variable and newly developed factor 2) COMMUNALITY The i–th communalitymeasures the portion of variation of i–th variable explained by the common factors It is obtained by squaring the factor loadings of a variable across all factors and then summing these figures and is denoted by h2 A large value of hi2 indicates that most of variation in i-th variable has been explained by common factors 3) SPECIFIC VARIANCE The i–th specific Variance measures the portion of variation of i–th variable unexplained by factor and is denoted by € In case of correlation matrix =1-h2 Large value of specific variance indicates that common factors fail to explain most of the variation in the i-th factor
Types of Variance Diagonal Value Variance Unity Total Variance Communality Common Specific and error Variance Extracted Variance Lost
Common Factor Analysis Terms 4) TOTAL VARIANCE ( In case of correlation matrix) Total variance is equal to the total number of variable in the data 5) VARIANCE OF A FACTOR: The variance of j-th factor is the sum of the squared factor loadings of that factor and it indicates the variation explained by the j-th factor A large value of the variance of the factor indicates that most of the variation in the data has been explained by that factor % of variation explained by j-th factor= NOTE:- Variance of j-th factor is also the eigen value
Common Factor Analysis Terms 6) TOTAL VARIANCE EXPLAINED Total variance explained by all common factors = % of total variance explained=
STEP-3 Unrotated Factors( One factor Solution) Communality:-The i–th communality; that measures the portion of variation of i–th variable explained by the factors; is given as e.g (-0.560)2= 0.314 Specific Variance:-The i–th specific Variance; that measures the portion of variation of i–th variable unexplained by factor; is given as =1-o.314=0.686 Variance of Factor/Eigen Value=(-0.560)2+. . . + (-0.798)2=2.8531 % of total variance explained=(2.8531/5)x100=57% Total Variance explain by all factors =0.314+0.604+. . . .+0.637=2.8531
Initinal Factor Extraction Variable Factor1 Factor2 Factor3 Factor4 Factor5 Communality Var 1 -0.560 -0.816 -0.045 -0.044 0.129 1.000 Var 2 -0.777 0.524 -0.336 0.090 0.013 1.000 Var 3 -0.645 -0.748 -0.076 -0.037 -0.130 1.000 Var 4 -0.939 0.105 0.272 0.182 0.000 1.000 Var 5 -0.798 0.543 0.100 -0.240 0.002 1.000 Eigen values /Variance 2.8531 1.8063 0.2045 0.1024 0.0337 5.0000 % Var 57 36 4 2 0.7 100
STEP-4 How many of factors • First two factors explain about 93% of the variation • First two eigen vales are greater than 1 So two factors are sufficient to explain correlation structure among five original variables
Factor Rotation • Some times original loadings may not be readily interpretable • It is usual practice to rotate them until a “simpler structure” is achieved. • It is possible to find new factors whose loadings are easier to interpret. These new factors are called the rotated factors, and are selected so that some of the loading are very large (near to +- 1) and the remaining loading are very small (near to zero). • Commonly we would ideally wish for any given variable that it has a large loading on only one factor. In such situation it is easy to give each factor an interpretation arising from the variable with which it is highly correlated. • The rationale is very much similar to sharpening the focus of a microscope in order to see the details more clearly
Key points • The communalities will remain the same before and after any orthogonal rotation. • Although the percentage of the variance explained by the rotated and un rotated factors be different but the cumulative percentage of the variance explained by factors will be same before and after the orthogonal rotation
Interpretation of Factor Var2=Good buy for money Var4=Suitable for Snack Var5=Provides lots of energy Define factor 1= Var1=Taste Var3=Flavor Define Factor 2= Nutritional Factor Taste Factor
EXAMPLE:The marketing manager of a two-wheeler company designed a questionnaire to study the customers feedback about its two-wheeler and in turn he is keen in identifying the factors of his study. He has identified six variables which are as listed below So, the company administered a questionnaire among 50 customers to obtain their opinion on the above characteristics of two-wheeler ( Variables). The range of score for each of the above variables is assumed to be between 0 and 10, both inclusive. The score 0 means LOW RATING and 10 means HIGH RATING. Perform FACTOR ANALYSIS and identify the appropriate number of factors which can represent the variables of the study X1=Fuel Efficiency X2=Life of the two-Wheeler X3=Handling convenience X4=Cost of original spares X5=Breakdown rate X6=Price
Factor Loadings 6-factor Solution
Total Variance Explained Variance of factor/ Eigen Values
How many Number of Factors 3 FACTORS • > 80 % Variance explained by = • Eigen values greater than 1= • Scree plot becomes stable at= 3 FACTORS 3 FACTORS
Example:-2 A telephone industry wanted to study the operations of telephone booths with a view to establish norms for better customers service. A consultant for this task has identified the following variables • Need for deep differential rates • Telephone unit rates • Secrecy of discussion • Influence of external sound on conversation In a pilot study, the questionnaires containing above questions were administered to 50 respondents in different telephone booths and their responses are summarized as
