Factor Analysis for Exploratory Studies
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This document presents a factor analysis that was conducted to identify factors related to fitness trainer popularity. It discusses the research problem, domain, and hypotheses. A 13-item questionnaire was administered to 50 fitness trainers. The data was cleaned and factor analysis was performed. Three factors were extracted based on eigenvalues above 1, explaining 72% of the variance. The factors were interpreted as adapting new fitness programs, introducing latest trends to clients, and client view of the trainer. Reliability analysis found the factors to be reliable.
Factor Analysis for Exploratory Studies
- 1. For ExploratoryStudies Presented By- Manohar Kumar Pahan (PhD Scholar) Shrikant (PhD Scholar) February 22nd, 2019 1
- 5. Linear combination of variables 5
- 6. Constructs not Directly Observed 6
- 7. What is Factor Analysis 7
- 10. Types of Factor Analysis • Exploratory Factor Analysis • Confirmatory Factor Analysis 10
- 12. Purpose Theory evaluations of score validity develop theory regarding the nature of constructs Data Reduction 12
- 14. Basics of Analysis 1. Principal Component Analysis 2. Factor Loading 3. Communalities 4. Eigenvalues 5. Kaiser Criterion 6. The Scree Test 7. The Percentage Varimax Explained 8. Rotation a) Varimax 14
- 15. Assumptions • Metric data. (interval or ratio) • Variables normally distributed • Sample size adequate 5- 20 • No outlier • No multicollinearity • No homoscedasticity • Linear variable 15
- 16. Principal Component Analysis • Same as Factor analysis it is also a data reduction technique performed under factor analysis but • In PCA variance- maximizing rotation is performed to account variability in variables • In FA the amount of variability is explained due to common factor 16
- 17. Factor Analysis v/s PCA Regresión Equations Y1 = b1 x F + u1 Y2 = b2 x F + u2 Y3 = b3 x F + u3 Y4 = b4 x F + u4 Simple Equation C = w1(Y1) + w2(Y2) + w3(Y3) + w4(Y4) 17
- 18. Factor Loading Factor Loading Plot • Correlation coefficient between variable and factor 18
- 19. Factor Loading Ɵ = Angle Factor Loading = Cosine of angle between variables Variable = Vector 300 300 1200 600 19
- 20. Factor Loading Cont.. • Factor Loading of Variable 1 on factor 1 = correlation between OA and OC (Factor 1) = cos (30) = 0.886 • Factor Loading of Variable 2 on factor 1 = correlation between OB and OC (Factor 1) = cos (30) = 0.886 • Factor Loading of Variable 1 on Factor 2 = correlation between OA and OD (Factor 2) = cos (60) = 0.500 • Factor Loading of Variable 2 on Factor 2 = correlation between OB and OD (Factor 2) = cos (120) = - 0.500 20
- 22. Factor Loading Cont.. Factor Variable Factor 1 Factor 2 Variable 1 0.866 0.500 Variable 2 0.866 - 0.500 NOTE: Thumb Rule 0.7 & < sufficient variance from factor 22
- 23. KMO and Bartlett’s Test 23
- 24. • Test indicates sample size adequacy for applying Factor analysis. • KMO values should be at least 0.5 to be said adequate. • High values (close to 1.0) generally indicate that a factor analysis may be useful with your data. • If the value is less than 0.50, the results of the factor analysis probably won't be very useful. 24
- 25. • Tests the hypothesis that correlation matrix is an identity matrix, • which indicate variables are unrelated and therefore unsuitable for structure detection. • Small values (less than 0.05) of the significance level indicate that a factor analysis may be useful. 25
- 26. Communality • Sum of a squared factor loading of a variable in all factor • Denoted by h2 • Communality of the variable Adventure = Sum of square of loadings in all factor Factors 1 2 3 Communality (h2) Variable 1 a b c A2+b2+c2 2 d e f D2+e2+f2 3 g h i G2+h2+i2 26
- 27. Communality cont… Factor 1 2 3 Communality Variable 1 .783 .067 .365 .751 2 .788 .186 .332 .766 3 .782 .110 .429 .807 A2 b2 c2 A2+b2+c2 27
- 28. Eigenvalue • Characteristics root • Measures variance in all variable accounted by a factor Factor 1 2 3 Variable 1 a b c 2 d e f 3 g h i a2+d2+g2 b2+e2+h2 c2+f2+i2 28
- 29. Eigenvalue Cont.. Factors 1 2 3 Variable 1 .783 .067 .365 A2 2 .788 .186 .332 B2 3 .782 .110 .429 C2 Eigenva lues 1.845 0.051 0.427 Sum 29
- 30. Kaiser Criterion • Researcher need to decide how much factor to retain • Factors having eigenvalue > 1 should be retained. - Kaiser • Initial eigenvalue is 1 of each factor 30
- 31. The Scree Test • Scree = collection of broken rock fragments at base of mountain • Graphical method to identify important factor • Developed by Raymond B. Cattell (1966) 31
- 32. Percentage of Variance Explained • % of group variability explained by factor. • % variance explained of 1 factor = Sum of square of factor loadings in each factor 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 × 100 32
- 33. Percentage of Variance Explained cont.. Factor 1 2 3 Variable 1 a b c 2 d e f 3 g h i % variance explained of factors 𝐚^𝟐 + 𝐝^𝟐 + 𝐠^𝟐 𝟑 × 𝟏𝟎𝟎 𝐛^𝟐 + 𝐞^𝟐 + 𝐡^𝟐 𝟑 × 𝟏𝟎𝟎 𝐜^𝟐 + 𝐟^𝟐 + 𝐢^𝟐 𝟑 × 𝟏𝟎𝟎 OR 𝐄𝐢𝐠𝐞𝐧𝐯𝐚𝐥𝐮𝐞 𝐨𝐟 𝐞𝐚𝐜𝐡 𝐟𝐚𝐜𝐭𝐨𝐫 𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒗𝒂𝒓𝒊𝒂𝒃𝒍𝒆 × 100 33
- 34. Percentage of Variance Explained cont.. Factors 1 2 3 Variable 1 .783 .067 .365 2 .788 .186 .332 3 .782 .110 .429 Eigenvalues 1.845 0.051 0.427 % variance explained of 1 factor 1.845 3 × 100 0.051 3 × 100 0.427 3 × 100 61.5 1.7 14.23 34
- 35. Rotation Rotation Factors Uncorrelated Orthogonal Varimax Quartimax Equamax Correlated Oblique Direct Oblique Method Promax Varimax Method:- An orthogonal rotation method that minimizes the number of variables that have high loadings on each factor. This method simplifies the interpretation of the factors. 35
- 37. Problem • To construct questionnaire to identify the fitness trainer popularity – Their involvement of self in new fitness programme – Introducing new programme and providing knowledge to clients – The clients liking to the trainer 37
- 38. Domain The questionnaire is constructed so it has three domain (basically for trainers) 1. About the Fitness Trainer (Self) 2. The Fitness Trainer with his client, and 3. The Clients’ view for Fitness Trainer 38
- 39. Hypothesis • Involving new fitness programme for self as well client and client approach to the fitness trainer are highly correlated and linear in nature to know the popularity of fitness trainer. 39
- 41. Information of Participants • Data was collected from the Head Trainers of fitness centres. • Place:- Ranchi • N= 50 • Male & Female 41
- 42. Procedure 1. Data Cleaning a) Descriptive Statistics 2. Performing Factor Analysis 42
- 43. Data Cleaning 43
- 46. Performing Factor Analysis Cont… 46
- 47. SPSS Output 1. KMO value sufficient greater than .6 2. Bartlett’s Test check null hypothesis. A. Significance value less than .05 indicates sufficient correlation. 47
- 48. Unrotated factor solution Factor 1 2 3 Communality (h2) Eigenvalues 5.969 2.131 1.257 % of Variance 45.912 16.391 9.667 Cumulative % 45.912 62.302 71.969 Variable 1 .783 .067 .365 .751 2 .788 .186 .332 .766 3 .782 .110 .429 .807 4 .799 .135 .351 .780 5 .678 -.364 -.028 .593 6 -.525 .553 .244 .641 7 -.031 .181 .372 .172 8 .658 -.608 -.211 .847 9 .723 -.314 -.015 .622 10 .731 -.424 -.212 .758 11 .641 .571 -.410 .905 12 .619 .575 -.435 .902 13 .681 .536 -.246 .812 48
- 49. Procedure Repeated again • Before, • After, 49
- 50. Communality Communalities Initial Extraction q_1.I like introducing new fitness programe and exercise to my clients. 1.000 .710 q_2. I like helping people by providing them with information about many kinds of exercises. 1.000 .763 q_3. People ask me for information about suppliments, fitness and ways of achiving their gym goals. 1.000 .727 q_4. My clients think of me a good source of information when it comes to new fitness programe or exercise. 1.000 .748 q_5. I like to take a chance to new fitness programe or exercise. 1.000 .714 q_8. I like to try new and different fitness programe and exercise. 1.000 .881 q_9. I often try new fitness programe or exercise before others do. 1.000 .515 q_10. I like to experiment with new equipments for doing exercises. 1.000 .834 q_11. When it comes to body shape and fitness, my clients are very likely to ask my opinion. 1.000 .927 q_12. I am often used as a source of advice about body transformation and fitness by my clients. 1.000 .868 q_13. I often tell my friends what I think about fitness and exercise. 1.000 .746 q_6. Introducing a new fitness programe or exercise that has not yet been proven is usually not a waste of time and energy. 1.000 .745 Extraction Method: Principal Component Analysis. 50
- 51. 51
- 52. Rotated Component Matrix Rotated Component Matrixa Component 1 2 3 q_8. .924 q_6. -.835 q_10. .830 q_5. .777 q_9. .636 q_1. .825 q_2. .804 q_3. .744 q_4. .739 q_11. .948 q_12. .907 q_13. .401 .760 Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization. a. Rotation converged in 5 iterations. 52
- 53. • After transformation negative into positive score, • And by modifying question – Introducing a new fitness programme or exercise that has not yet been proven is usually waste of time and energy. • To – Introducing a new fitness programme or exercise that has proven is usually in managing of time and energy. • Factor analysis to be performed again. 53
- 54. Rotated Component Matrix Rotated Component Matrixa Component 1 2 3 q_8. .924 q_6. .835 q_10. .830 q_5. .777 q_9. .636 q_1. .825 q_2. .804 q_3. .744 q_4. .739 q_11. .948 q_12. .907 q_13. .401 .760 Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization. a. Rotation converged in 5 iterations. 54
- 55. Assigning Names to Factors 1. Adapting New Fitness Program 2. Introducing latest trends in fitness to clients 3. Client View for Trainer 55
- 57. Reliability Factor 1 Factor 2 Factor 3 Before changing question and transformation of scores 57
- 58. Thank you 58
- 59. References • Verma. J.P, (2016), Sports Research With Analytical Solution Using SPSS, John Wile & Sons, Hoboken, New Jersey, Chapter 12, pg 319-336. • Kothari. C.R. (2004), Research Methodology Methods & Techniques, New Age International Publishers, New Delhi, Chapter 13, pg 323-334. • Kaushik. N (2014, September, 10), Factor Analsyis-4 (Complete Example), https://youtu.be/2wLn9vS7mA8 59