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.

1. For ExploratoryStudies
Presented By-
Manohar Kumar Pahan (PhD Scholar)
Shrikant (PhD Scholar)
February 22nd, 2019
1

2. What are factors?
2

3. Independent Variable
3

4. Categorical Variable
4

5. Linear combination of variables
5

6. Constructs not Directly Observed
6

7. What is Factor Analysis
7

8. Multivariate Statistical Technique
8

9. Data Reduction
9

10. Types of Factor Analysis
• Exploratory Factor Analysis
• Confirmatory Factor Analysis
10

11. Objectives
Data
Reduction
Structure
Reduction
Obtain
Independent
Factor
11

12. Purpose
Theory
evaluations of score validity
develop theory regarding the nature of constructs
Data Reduction
12

13. Areas of Application
13

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

21. Factor Loading Cont..
21

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
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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
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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

36. Now
An
Example
Through
A
Research Problem
Selected
by
Presenters
36

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
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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

40. Questions in Questionnaire
40

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

44. Data Cleaning cont...
44

45. Performing Factor Analysis
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46. Performing Factor Analysis Cont…
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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

56. Checking Reliability
56

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
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