43955508 factor analysis ppt

7/31/2019 43955508 Factor Analysis Ppt

http://slidepdf.com/reader/full/43955508-factor-analysis-ppt 1/25

Factor Analysis



19-2

What is factor analysis ?

Factor analysis is a general name denoting a class of

Procedures primarily used for data reduction and

summarization.

Variables are not classified as either dependent or

independent. Instead, the whole set of interdependent

relationships among variables is examined in order to

define a set of common dimensions called Factors.



19-3

Purpose of Factor Analysis

To identify underlying dimensions called Factors, that explainthe correlations among a set of variables.

-- lifestyle statements may be used to measure thepsychographic profile of consumers.

To identify a new, smaller set of uncorrelated variables toreplace the original set of correlated variables for subsequentanalysis such as Regression or Discriminant Analysis.

-- psychographic factors may be used as independent

variables to explain the difference between loyal andnon loyal customers.



19-4

Assumptions

Models are usually based on linear relationships

Models assume that the data collected are interval scaled

Multicollinearity in the data is desirable because the objective is to

identify interrelated set of variables.

The data should be amenable for factor analysis. It should not besuch that a variable is only correlated with itself and no correlationexists with any other variables. This is like an Identity Matrix.

Factor analysis cannot be done on such data.



19-5

An Example

A study conducted to determine customers perception and attributesof an airline. A set of 10 statements were constructed and respondentswere asked to rate in a 7 point scale( 1= completely agree, 7 = completely disagree )Statements were as follows:

1. The Airline is always on time2. The seats are very comfortable3. I love the food they provide4. Their air-hostesses are very courteous5. My boss/friend flies with the same airline

6. The airlines have younger aircrafts7. I get the advantage of a frequent flyer program8. It suits my schedule9. My mom feels safe when I fly in this airline10. Flying by this airline compliments my lifestyle and

social standing in the society



19-6

Example Contd..

Do the ten different statements indicate10 different factors which influence acustomer to fly by this airline ?

OR Is there any correlations between these

statements so that we can identify only a

few factors such that some of thesestatements can be associated to thesefactors.



19-7

Factor Analysis – basic ideas

Each of the statement indicated in the example is considered as a Variable. Hence for each respondent there will be a score againsteach variable.Ex: V1 V2 V3 V4 V5 V6 V7 V8 V9 V10respondent 1 2 2 4 3 5 3 5 7 6 2

We can attach suitable weights to each of the variable scores and aWeighted sum of these can be calculated.Ex: weight for V1 = 0.3 , weight for V2 = 0.1 etcHence a score called Factor Score can be calculated as

Factor Score ( Resp 1) = W1x2 + W2x2+ W3x4+w4x3+……….

Similarly factor score can be calculated for each respondent.If there were 20 respondents, we would get a table containing20 factor scores.



19-8

Factor Analysis – basic ideas contd

The weights which are assigned to each of the variables are nottaken arbitrarily but are chosen such that the variance in thefactor scores obtained is the maximum.

Once the first set of weights are obtained, a new set of weights

are obtained so that the new set of factor scores shows themaximum variance but keeping in mind that these set of factorscores are uncorrelated with the first set of factor scores.

This process is repeated till all the variance is explained by thesefactors.

The first set of factor scores obtained is now correlated withthe data for the variable 1 to 10 . This is called factor loadings. Thus factor loading is the correlation between the factor scoresand the variables.



19-9

Factor Analysis – basic ideas contd

An example would clarify what we have discussed so far.

A file in excel data sheet can now be looked at to understandwhat we have just discussed.

The factors thus extracted are done using a technique calledPrincipal – Component Analysis.



19-10

It is possible to extract as many factors as thereare variables but the very purpose of factoranalysis will be defeated and hence a smallernumber of factors need to be found.

Question is --- how many?

Several procedures are available:

-- Determine based on Eigenvalues. An eigenvalue represents the amount of varianceassociated with the factor. Generally only factorswith an Eigenvalue of >1.0 is included.

Determining the number of factors



19-11


Determination based on Scree Plot.

A scree plot is a plot of the eigenvalues against

the number of factors. Typically, the plot has a

distinct break with a gradual trailing off with the

rest of the factors. This trailing off is referred to as

Scree.



19-12

Scree Plot

0.5

2 543 6

Component Number

0.0

2.0

3.0

E i g e n v a l u

e

1.0

1.5

2.5

1

19 13



19-13


Determination based on percentage of Variance.

The number of factors extracted is determined so that the

cumulative percentage of variance reaches a satisfactory level.The amount of variance explained can vary with situation but

above 60% is considered satisfactory.

19 14



19-14

How to check suitability for Factor Analysis

Kaiser-Meyer-Olkin ( KMO ) measure of samplingadequacy . This index compares the magnitude of observed correlation coefficients to the magnitude of partial correlation coefficients. Typically it should be> 0.5 is considered as good enough for conducting

Factor analysis for the data under consideration.

Bartlett test of sphericity : It is a test used toexamine the hypothesis that the variables are

uncorrelated in the population. If the hypothesis canbe rejected then the data is suitable for factoranalysis.

19 15



19-15

Conducting Factor AnalysisRESPONDENT

NUMBER V1 V2 V3 V4 V5 V6

1 7.00 3.00 6.00 4.00 2.00 4.00

2 1.00 3.00 2.00 4.00 5.00 4.00

3 6.00 2.00 7.00 4.00 1.00 3.00

4 4.00 5.00 4.00 6.00 2.00 5.00

5 1.00 2.00 2.00 3.00 6.00 2.00

6 6.00 3.00 6.00 4.00 2.00 4.00

7 5.00 3.00 6.00 3.00 4.00 3.00

8 6.00 4.00 7.00 4.00 1.00 4.00

9 3.00 4.00 2.00 3.00 6.00 3.00

10 2.00 6.00 2.00 6.00 7.00 6.00

11 6.00 4.00 7.00 3.00 2.00 3.00

12 2.00 3.00 1.00 4.00 5.00 4.00

13 7.00 2.00 6.00 4.00 1.00 3.00

14 4.00 6.00 4.00 5.00 3.00 6.00

15 1.00 3.00 2.00 2.00 6.00 4.00

16 6.00 4.00 6.00 3.00 3.00 4.00

17 5.00 3.00 6.00 3.00 3.00 4.00

18 7.00 3.00 7.00 4.00 1.00 4.00

19 2.00 4.00 3.00 3.00 6.00 3.00

20 3.00 5.00 3.00 6.00 4.00 6.00

21 1.00 3.00 2.00 3.00 5.00 3.00

22 5.00 4.00 5.00 4.00 2.00 4.00

23 2.00 2.00 1.00 5.00 4.00 4.00

24 4.00 6.00 4.00 6.00 4.00 7.00

25 6.00 5.00 4.00 2.00 1.00 4.00

26 3.00 5.00 4.00 6.00 4.00 7.00

27 4.00 4.00 7.00 2.00 2.00 5.00

28 3.00 7.00 2.00 6.00 4.00 3.00

29 4.00 6.00 3.00 7.00 2.00 7.0030 2.00 3.00 2.00 4.00 7.00 2.00

19 16



19-16

Correlation Matrix

Variables V1 V2 V3 V4 V5 V6

V1 1.000

V2 -0.530 1.000

V3 0.873 -0.155 1.000 V4 -0.086 0.572 -0.248 1.000

V5 -0.858 0.020 -0.778 -0.007 1.000

V6 0.004 0.640 -0.018 0.640 -0.136 1.000

19 17



19-17

Results of Principal Components Analysis

Communalities

Variables Initial Extraction V1 1.000 0.926 V2 1.000 0.723 V3 1.000 0.894

V4 1.000 0.739 V5 1.000 0.878 V6 1.000 0.790

Initial Eigen values

Factor Eigen value % of variance Cumulat. %1 2.731 45.520 45.5202 2.218 36.969 82.4883 0.442 7.360 89.8484 0.341 5.688 95.5365 0.183 3.044 98.5806 0.085 1.420 100.000

19 18



19-18


Extraction Sums of Squared Loadings

Factor Eigen value % of variance Cumulat. %1 2.731 45.520 45.5202 2.218 36.969 82.488

Factor Matrix

Variables Factor 1 Factor 2

V1 0.928 0.253

V2 -0.301 0.795

V3 0.936 0.131

V4 -0.342 0.789 V5 -0.869 -0.351

V6 -0.177 0.871

19 19

C d i F A l i



19-19

Although the initial or unrotated factor matrixindicates the relationship between the factors andindividual variables, it seldom results in factors thatcan be interpreted, because the factors arecorrelated with many variables. Therefore, throughrotation the factor matrix is transformed into asimpler one that is easier to interpret.

In rotating the factors, we would like each factor tohave nonzero, or significant, loadings or coefficientsfor only some of the variables. Likewise, we would

like each variable to have nonzero or significantloadings with only a few factors, if possible with onlyone.

The rotation is called orthogonal rotation if theaxes are maintained at right angles.

Conducting Factor Analysis Rotate Factors

19-20

C d ti F t A l i



19-20

The most commonly used method for rotation is thevarimax procedure. This is an orthogonal methodof rotation that minimizes the number of variableswith high loadings on a factor, thereby enhancing theinterpretability of the factors. Orthogonal rotation

results in factors that are uncorrelated. The rotation is called oblique rotation when the

axes are not maintained at right angles, and thefactors are correlated. Sometimes, allowing for

correlations among factors can simplify the factorpattern matrix. Oblique rotation should be usedwhen factors in the population are likely to bestrongly correlated.

Conducting Factor Analysis Rotate Factors

19-21



19 21


Rotated Factor Matrix

Variables Factor 1 Factor 2 V1 0.962 -0.027 V2 -0.057 0.848 V3 0.934 -0.146 V4 -0.098 0.845

V5 -0.933 -0.084 V6 0.083 0.885

Rotation Sums of Squared Loadings

Factor Eigenvalue % of variance Cumulat. %1 2.688 44.802 44.8022 2.261 37.687 82.488

19-22

C d ti F t A l i



19 22

A factor can then be interpreted in terms of thevariables that load high on it.

Another useful aid in interpretation is to plot thevariables, using the factor loadings as coordinates. Variables at the end of an axis are those that have

high loadings on only that factor, and hence describethe factor.

Conducting Factor Analysis Interpret Factors

19-23



19 23

Factor Loading Plot

1.0

0.5

0.0

-0.5

-1.0

F a c t o

r

2

Factor 1

factor Variable 1 2

V1 0.962 -2.66E-02

V2 -5.72E-02 0.848 V3 0.934 -0.146

V4 -9.83E-02 0.854

V5 -0.933 -8.40E-02

V6 8.337E-02 0.885

Factor Plot in Rotated Space

-1.0 -0.5 0.0 0.5 1.0

V1

V3

V6

V2

V5

V4

Rotated Component Matrix

19-24



19 24

A few examples

We can now take few examples

with hypothetical data and runfactor analysis using SPSS package.

19-25



19 25

43955508 factor analysis ppt

Documents