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Factor Analysis
Presented by:Gurvinder kaur
Under the Guidance of:Dr. Divya Malhan
Multivariate Data Analysis Technique
Dr. Preshth Bhardwaj
Marketing ResearchFactor AnalysisMost popular analysis of interdependence technique.
Basic function is to identify groups of variables that are relatedMain purposes in Marketing & HR:to identify underlying constructs in the data
Using Common Factor Analysisto reduce the number of variables to a more manageable subset
Using Principal Components Analysis
Dr. Preshth Bhardwaj
Marketing ResearchSo Most Common ProceduresCommon factor analysisUsed when we want uncover the underlying dimensions surrounding the original variables (1st purpose)Principal components analysisUsed when the objective is to reduce a large set variables into a smaller number of factors (2nd purpose) Example: Factor analysis of 14 variables to find how similar variables group together to reduce those 14 variables to 4 factors (The reduced variables are known as factors)
Dr. Preshth Bhardwaj
Marketing ResearchFactor, Loading, Eigen valueFactor: A variable (or construct) that is not directly observable but needs to be inferred from the input variablesFactor Loading: A correlation coefficient showing the importance of each variable in defining the factorEigen value: Represents the variance in the original variables that is explained by a factor (standardized so that the avg. variable has an eigenvalue = 1.0)
Dr. Preshth Bhardwaj
Statistics Associated with Factor AnalysisCorrelation matrix. A correlation matrix is a lower triangle matrix showing the simple correlations, r, between all possible pairs of variables included in the analysis. The diagonal elements, which are all 1, are usually omitted. Communality. Communality is the amount of variance a variable shares with all the other variables being considered. This is also the proportion of variance explained by the common factors.
Low communities indicates the variable is independent and cannot be combined with other variables.Scree Plot: Plot of eigen values against the number of factors in order of extraction.
Dr. Preshth Bhardwaj
Marketing ResearchFactor RotationFactor analysis can generate several solutions (loadings & factor scores) for any data seteach solution is called a factor rotationeach time the factors are rotated the pattern of loadings changesgeometrically, rotation simply means that the axes are rotatedVarimax rotation--each factor tends to load high on some variables and low on othersmakes factor interpretation easier
Dr. Preshth Bhardwaj
Statistics Associated with Factor AnalysisBartletts test of Sphericity: used to examine the hypothesis that the variables are uncorrelated in the population. Population correlation matrix is also known as identity matrix. Each variable correlates perfectly (r = 1) but has no correlation with other variables (r = 0).Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy: Index used to examine the appropriateness of Factor Analysis. High values (b/w 0.5 and 1.0) indicates the FA is appropriate.
Dr. Preshth Bhardwaj
Factor Solution: Orthogonal or Oblique
Factor solutions- A factor solution is simply the set of factors that result from a factor analysis experiment. Factor solutions fall into two major types:
Orthogonal factor solutionsyield factors that are statistically independent and can be used with other statistical procedures that must satisfy assumptions of statistical independence.
Oblique factor solutionsare solutions for which factors are correlated. They are usually a better reflection of the underlying reality which the researcher seeks to describe.
Dr. Preshth Bhardwaj
General Steps to FAStep 1: Selecting and Measuring a set of variables in a given domainStep 2: Data screening in order to prepare the correlation matrixStep 3: Factor ExtractionStep 4: Factor Rotation to increase interpretability Step 5: InterpretationFurther Steps: Validation and Reliability of the measures
Dr. Preshth Bhardwaj
Conducting Factor AnalysisTo determine the benefits consumers seek from the purchase of a toothpaste.A sample of 30 respondents were interviewed and asked to indicate their degree of agreement with the following statements using 7 point scale
(1 = Strongly Disagree and 7= Strongly Agree)
Dr. Preshth Bhardwaj
Lifestyle statements constructed for identifying the consumer opinionsV1: It is important to buy a toothpaste that prevents cavities.V2: I like a toothpaste that gives shiny teeth.V3: A toothpaste should strengthen your gums.V4: I prefer a toothpaste that freshens breath.V5: Prevention of tooth decay is not an important benefit offered by a toothpaste brand.V6: The most important consideration in buying a toothpaste is attractive teeth
Dr. Preshth Bhardwaj
Correlation MatrixFactor Analysis of Perceptual Variables
Dr. Preshth Bhardwaj
Sheet1
Toothpaste Attribute Ratings
Respondent No.V1V2V3V5V6
17.003.006.004.002.004.00
21.003.002.004.005.004.00
36.002.007.004.001.003.00
44.005.004.006.002.005.00
51.002.002.003.006.002.00
66.003.006.004.002.004.00
75.003.006.003.004.003.00
86.004.007.004.001.004.00
93.004.002.003.006.003.00
102.006.002.006.007.006.00
116.004.007.003.002.003.00
122.003.001.004.005.004.00
137.002.006.004.001.003.00
144.006.004.005.003.006.00
151.003.002.002.006.004.00
166.004.006.003.003.004.00
175.003.006.003.003.004.00
187.003.007.004.001.004.00
192.004.003.003.006.003.00
203.005.003.006.004.006.00
211.003.002.003.005.003.00
225.004.005.004.002.004.00
232.002.001.005.004.004.00
244.006.004.006.004.007.00
256.005.004.002.001.004.00
263.005.004.006.004.007.00
274.004.007.002.002.005.00
283.007.002.006.004.003.00
294.006.003.007.002.007.00
302.003.002.004.007.002.00
Sheet2
VariablesV1V2V3V4V5V6
V11
V2-0.531
V30.873-0.1551
V4-0.0860.572-0.2481
V5-0.8580.02-0.778-0.0071
V60.0040.64-0.0180.64-0.1361
Rotated Factor Matrix
VariablesFactor 1Factor 2
V10.962-0.027
V2-0.0570.848
V30.934-0.146
V4-0.0980.845
V5-0.933-0.084
V60.0830.885
Sheet3
Sheet1
Toothpaste Attribute Ratings
Respondent No.V1V2V3V5V6
17.003.006.004.002.004.00
21.003.002.004.005.004.00
36.002.007.004.001.003.00
44.005.004.006.002.005.00
51.002.002.003.006.002.00
66.003.006.004.002.004.00
75.003.006.003.004.003.00
86.004.007.004.001.004.00
93.004.002.003.006.003.00
102.006.002.006.007.006.00
116.004.007.003.002.003.00
122.003.001.004.005.004.00
137.002.006.004.001.003.00
144.006.004.005.003.006.00
151.003.002.002.006.004.00
166.004.006.003.003.004.00
175.003.006.003.003.004.00
187.003.007.004.001.004.00
192.004.003.003.006.003.00
203.005.003.006.004.006.00
211.003.002.003.005.003.00
225.004.005.004.002.004.00
232.002.001.005.004.004.00
244.006.004.006.004.007.00
256.005.004.002.001.004.00
263.005.004.006.004.007.00
274.004.007.002.002.005.00
283.007.002.006.004.003.00
294.006.003.007.002.007.00
302.003.002.004.007.002.00
Sheet2
VariablesV1V2V3V4V5V6
V11
V2-0.531
V30.873-0.1551
V4-0.0860.572-0.2481
V5-0.8580.02-0.778-0.0071
V60.0040.64-0.0180.64-0.1361
Rotated Factor Matrix
VariablesFactor 1Factor 2
V10.962-0.0270.926173
V2-0.0570.8480.722353
V30.934-0.1460.893672
V4-0.0980.8450.723629
V5-0.933-0.0840.877545
V60.0830.8850.790114
I =2.6880312.245455
Factor Score Coefficient Matrix
VariablesFactor 1Factor 2
V10.3580.011
V2-0.0010.375
V30.345-0.043
V4-0.0170.377
V5-0.35-0.059
V60.0520.395
Sheet3
Factors representing Variables
Factor 1 has high coefficients for variables: V1(prevention of cavities)and V3 (strong gums). Thus, factor may be labeled as health benefit factor.
Factor 2 has high correlation coefficients for variables: V2(shiny teeth), V4(fresh breath), and V6 (attractive teeth). Thus, factor may be labeled as Social benefit factor.
Dr. Preshth Bhardwaj
Communality. Communality is the amount of variance a variable shares with all the other variables being considered. This is also the proportion of variance explained by the common factors.
h2 = (0.962)2 + (-0.027)2 = 0.926173 and so on.Low communities indicates the variable is independent and cannot be combined with other variables.
Dr. Preshth Bhardwaj
Eigenvalue. The eigen value represents the total variance explained by each factor. Approach to determine the number of factors to be retain in the analysis. If the factor has an eigen value >1, then the factor is retained for further interpretation.
I for Factor 1 = (0.962)2 +(-0.057)2+..+(0.083)2 = 2.688Factor loadings. Factor loadings are simple correlations between the variables and the factors. These are correlation coefficients within the matrix that indicates the importance of the factor. Loading limit is from -1.0 to +1.0
High Correlation between V1, V3 and V5 Low Correlation between V2, V4 and V6.Total Variance: percentage of variance explained i.e. total original variance of 6 variables explained by 2 factors.
C.V. = sum of all h2s x 100 = (0.926173+..+0.790114)/6 number of variables = 4.933486 x 100 = 82.22477%
Dr. Preshth Bhardwaj
Factor Variance:
Amount of variation explained by each factor.= (0.962)2 +(-0.057)2+..+(0.083)2 = 2.688 6 = 0.448005%
Dr. Preshth Bhardwaj
Interpretation of Factor AnalysisConsumers appear to seek two major kinds of benefits from a toothpaste: health benefits and social benefits.Total variance explained about 82.22477 % variance of 6 variables. Therefore, the remaining 17.7752 % of variance indicates that there are some other variables also influencing consumer preferences.
Dr. Preshth Bhardwaj
Factor Analysis in SPSS: Results Here, Null Hypothesis: H0 = Popl. Correlation Matrix is an identity matrix Bartletts Test of Sphericity rejected this hypothesisThe Chi-Square value is 111.314 with 15 degree of freedom (df), which is significant at the 0.05 level.The Value of KMO is 0.660 which is >0.5 level. Therefore factor analysis is an appropriate technique for analysing Correlation Matrix.
Dr. Preshth Bhardwaj
Dr. Preshth Bhardwaj
THANK YOU
Dr. Preshth Bhardwaj
FA vs. PCA conceptuallyFA produces factors; PCA produces componentsFactors cause variables; components are aggregates of the variables
Dr. Preshth Bhardwaj
Conceptual FA and PCA
Dr. Preshth Bhardwaj
Dr. Preshth Bhardwaj
Dr. Preshth Bhardwaj
Dr. Preshth Bhardwaj
Dr. Preshth Bhardwaj
Dr. Preshth Bhardwaj
Dr. Preshth Bhardwaj
Dr. Preshth Bhardwaj
Dr. Preshth Bhardwaj
Dr. Preshth Bhardwaj
Dr. Preshth Bhardwaj
Dr. Preshth Bhardwaj
Dr. Preshth Bhardwaj
Dr. Preshth Bhardwaj
Dr. Preshth Bhardwaj
Dr. Preshth Bhardwaj