Factor Analysis Gurvinder

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


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)


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)


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.


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


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.


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.


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


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)


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


Correlation MatrixFactor Analysis of Perceptual Variables


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


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


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.


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.


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%


Factor Variance:

Amount of variation explained by each factor.= (0.962)2 +(-0.057)2+..+(0.083)2 = 2.688 6 = 0.448005%


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.


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.


THANK YOU


FA vs. PCA conceptuallyFA produces factors; PCA produces componentsFactors cause variables; components are aggregates of the variables


Conceptual FA and PCA


Factor Analysis Gurvinder

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