1. 1 Multivariate Analysis

2. 2 Multivariate Analysis refers to a group of statistical procedure that simultaneously analyze multiple measurements on each individual being investigated. 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. Some of multivariate methods are straight generalization of univariate analysis. The mathematical framework is relatively complex as compared with the univariate analysis. The mathematical framework is relatively complex as compared with the univariate analysis. These analysis are being used widely around the world. These analysis are being used widely around the world.

3. 3 Why Factor Analysis In many research problem there are large number of variables Which are difficult to handle Which are difficult to handle Information provided by the variables are difficult to interpret due to correlation among variables 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

4. 4 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 Quality, Varity and price of the food Restaurant’s location Restaurant’s location Speed and quality of service 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

5. 5 Typical Problem Studied With Factor Analysis Example:- 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 1. 1. A compact car should be built to last a long time 2. 2. Gasoline mileage in a compact car should be at least 30 miles per gallon 3. 3. A compact car should be easily maintained and serviced by its owner 4. 4. Four adults should be able to sit comfortably in a compact car 5. 5. Interior appointments in a compact car should be attractive 6. 6. A compact car’s brakes are its most critical part

6. 6 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 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 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 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 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

7. 7 Construction of FACTORS FACTORS are identified through the use of extremely complex mathematical calculations

8. 8 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

9. 9 Height Weight Occupation Education Source of Income Size Social Status Factors Original Variables

10. 10 Origins of Factor Analysis Charles Spearman 1863-1945

11. 11 Origins of Factor Analysis Wanted to estimate intelligence of 24 children in a village school. Wanted to estimate intelligence of 24 children in a village school. Realized way of measuring intelligence was imperfect and that the correlation between any two variables (say, one’s score on a mathematics exam and on a classics exam) would be “wrong”, presumably low, since the variables were not able to be measured perfectly. Realized way of measuring intelligence was imperfect and that the correlation between any two variables (say, one’s score on a mathematics exam and on a classics exam) would be “wrong”, presumably low, since the variables were not able to be measured perfectly. Noticed that the observed correlations between the variables he was interested in were all positive and followed a pattern. Noticed that the observed correlations between the variables he was interested in were all positive and followed a pattern. Spearman wanted to develop a model that would reflect the pattern he saw. Spearman wanted to develop a model that would reflect the pattern he saw.

12. 12 What did Spearman notice? Correlations Between Examination Scores Notice the trend across each row on the upper diagonal

13. 13 ONE Explanation Could model each test score as having two components: one common to all the scores and one specific to the particular test Could model each test score as having two components: one common to all the scores and one specific to the particular test With the “right” assumptions one could generate the correlation structure above With the “right” assumptions one could generate the correlation structure above Factor analysis models reflect this basic idea Factor analysis models reflect this basic idea

14. 14 Goals of Factor Analysis Model correlation patterns in useful way Model correlation patterns in useful way Suggest new, uncorrelated variables that explain the original correlation structure Suggest new, uncorrelated variables that explain the original correlation structure Allow for contextual interpretation of the new variables Allow for contextual interpretation of the new variables Evaluate the original data in light of the new variables Evaluate the original data in light of the new variables

15. 15 Schematically classicsfrenchenglishmathdiscrmusic       f

16. 16 Procedure For Factor Analysis 1. Obtain a random sample of n individuals and measure P traits (variables) and calculate correlation matrix 2. Initial factor extraction ; Estimate weighted sum of the variables with descending order of importance Methods: Principal component method Principal component method Maximum Likelihood Method Maximum Likelihood Method

17. 17 3Select the suitable number of common factors in the model for which “suitable proportion “ of the total sample variance has been explained “suitable proportion “ of the total sample variance has been explained Where SCREE plot first becomes stable 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. In case of correlation matrix select the number of common factors equal to the eigenvalues of R that are greater than ONE. Expert opinion Expert opinion

18. 18 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. 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 It is usual practice to rotate initial factors until a “simpler structure” is achievedMethods: Orthogonal rotation (e.g Varimax Rotation) Orthogonal rotation (e.g Varimax Rotation) Oblique Rotation (e.g Quartmin Rotation) Oblique Rotation (e.g Quartmin Rotation)

19. 19 Application of FA Step-1&2 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 V1: Taste V2: Good buy for money V3: Flavor V4: Suitable for snack V5: Provides lots of energy

20. 20 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

21. 21 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 factorCorrelation 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 factorEach 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)Factor Loading ( like correlations) can vary from -1 to +1 when constructed from correlation matrix)

22. 22 Common Factor Analysis Terms 1) FACTOR LOADINGS Correlation between each of the original variable and newly developed factor 2) COMMUNALITY The i–th communality measures 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 h 2 A large value of hi 2 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 h 2 =1- h 2 Large value of specific variance indicates that common factors fail to explain most of the variation in the i-th factor

23. 23 Types of Variance Variance Total Variance CommonSpecific and error Diagonal Value Unity Communality Variance Extracted Variance Lost

24. 24 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

25. 25 Common Factor Analysis Terms 6) TOTAL VARIANCE EXPLAINED Total variance explained by all common factors = % of total variance explained=

26. 26 Unrotated Factors( One factor Solution) The i–th communality; that measures the portion of variation of i–th variable explained by the factors; is given as Communalit y:- 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 The i–th specific Variance; that measures the portion of variation of i–th variable unexplained by factor; is given as = 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 Variables Factor Loadings Communalities Specific Variance V1-0.5600.3140.686 V2-0.7770.6040.396 V3-0.6450.4160.584 V4-0.9390.8820.118 V5-0.7980.6370.363 Variance2.85312.8531 % variance 5757 STEP-3

27. 27 Unrotated Factors ( two factor Solution) F1F2CommunalitiesSpecific Variance V10.56-0.820.980.02 V20.780.520.880.12 V30.65-0.750.980.02 V40.940.110.890.11 V50.800.540.930.07 Variance2.851.814.66 % Variance57.0436.1293.15

28. 28 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

29. 29 STEP-4 How many of factors First two factors explain about 93% of the variation First two factors explain about 93% of the variation First two eigen vales are greater than 1 First two eigen vales are greater than 1 So two factors are sufficient to explain correlation structure among five original variables

30. 30 Factor Rotation Some times original loadings may not be readily interpretable Some times original loadings may not be readily interpretable It is usual practice to rotate them until a “simpler structure” is achieved. 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). 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. 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 The rationale is very much similar to sharpening the focus of a microscope in order to see the details more clearly

31. 31 Unrotated Factors F1F2Communalities Specific Variance V10.56-0.820.980.02 V20.780.520.880.12 V30.65-0.750.980.02 V40.940.110.890.11 V50.800.540.930.07 Variance2.851.814.66 % Variance57.0436.1293.15

32. 32 Rotated Factors F1F2Communalities Specific Variance V10.0200.9890.9790.021 V20.937-0.0110.8780.122 V30.1290.9760.9690.031 V40.8420.4280.8920.108 V50.965-0.0160.9310.069 Var2.5352.1144.649 % Var50.70442.28592.989

33. 33 Key points The communalities will remain the same before and after any orthogonal rotation. 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 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

34. 34 Interpretation of Factor Var2=Good buy for money Var4=Suitable for Snack Var5=Provides lots of energy Define factor 1= Var1=TasteVar3=Flavor Define Factor 2= Nutritional Factor Taste Factor

35. 35 Factor Analysis Output of the Compact car study F1F2F3CommunalitiesSpecific Variance V10.860.120.040.760.24 V20.840.180.100.750.25 V30.680.240.150.540.46 V40.100.920.050.860.14 V50.060.940.080.890.11 V60.120.140.890.830.17 Var1.941.850.844.63 % Var32311477

36. 36 FACTOR ANALYSIS IN SPSS

37. 37 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 X1=Fuel Efficiency X2=Life of the two-Wheeler X3=Handling convenience X4=Cost of original spares X5=Breakdown rate X6=Price 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

38. 38 Component 123456 Fuel Efficiency.851-.371-.224.082.265.100 Life of the Two-Wheeler.941-.112-.055.088-.283.101 Handling Convenience.549.188.788.190.054-.065 Cost of Original Spares.460.684-.523.175.010-.126 Breakdown Rate.916.087.066-.376.012-.081 Price -.078.972.118-.085.050.158 6-factor Solution Factor Loadings

39. 39 6-factor solution Communalities InitialExtraction Fuel efficiency 1.0001.000 Life of the two-wheeler 1.0001.000 Handling Convenience 1.0001.000 Cost of original spares 1.0001.000 Breakdown rate 1.0001.000 Price 1.0001.000

40. 40 Total Variance Explained Compon Initial Eigenvalues Extraction Sums of Squared Loadings Eigen Value % of Variance Cumulative % Total % of Variance Cumulative % 1 2.97049.49549.4952.97049.49549.495 2 1.60726.78576.2801.60726.78576.280 3.96516.08792.367.96516.08792.367 4.2303.83196.198.2303.83196.198 5.1562.60198.799.1562.60198.799 6.0721.201100.000.0721.201100.000 Variance of factor/ Eigen Values

41. 41 Scree Plot

42. 42 How many Number of Factors  > 80 % Variance explained by =  Eigen values greater than 1=  Scree plot becomes stable at= 3 FACTORS

43. 43 3-FACTOR SOLUTION

44. 44 3-Factor Solution Component Initial Eigenvalues Group Label Extraction Sums of Squared Loadings Total % of Variance Cumulative % Total % of Variance Cumulative % 1 2.97049.49549.4952.97049.49549.495 2 1.60726.78576.2801.60726.78576.280 3.96516.08792.367.96516.08792.367 4.2303.83196.198 5.1562.60198.799 6.0721.201100.000

45. 45 3-Factor Solution Unrotated Factors Component 123 Life of the Two-Wheeler.941 Breakdown Rate.916 Fuel Efficiency.851-.371 Price.972 Cost of Original Spares.460.684-.523 HAndling Convenience.549.788

46. 46 3-Factor Solution Rotated Factors Component 123 Fuel Efficiency.945 Life of the Two-Wheeler.918 Breakdown Rate.812.381 Cost of Original Spares.400.874 Price -.346.868.302 Handling Convenience.945

47. 47 Interpretation of Factors Factors % Variance explained OriginalVariables Factor Labels 149 1. Fuel Efficiency 2. Life 3. Breakdown Rate TECHNIQUAL 227 1. Cost of spares 2. Price COST 316 1. Handling Convenience PERSON AL

48. 48 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 1. Need for deep differential rates 2. Telephone unit rates 3. Secrecy of discussion 4. 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 summarized assummarized as