** FACTOR ANALYSIS**

**Introduction**

Factor analysis is used when the research problem involves a large number of variables making the analysis and interpretation of the problem difficult. The factor analysis helps the researcher to reduce a number of variables to be analyzed, thereby making the analysis easier. In factor analysis certain variables are combined into specific factors. These factors form the basis of data analysis.

**Methods of conducting Factor Analysis**

The basic purpose of factor analysis is to extract the factors. The objective is to identify the number of factors to be extracted from the data. Theoretically, we can have as many factors as there are original variables. But the main aim is to reduce the variables to a fewer number of factors. The concept of ‘eigen values’ solves this problem. The SPSS default keeps the factor with Eigen values greater than 1.

Once the factors are selected, the next step is to rotate them. The rotated factor matrix gives the loading of each variable on each of the extracted factor. Factor loading varies from (-1) to (+1) and signifies the strength of the loading. The values close to (+1) indicate high loading and the values close to (-1) indicate lower degree of loading of the corresponding variables on the factor. Thus the factor is a linear combination of the variables having high loading on that factor and comparatively low loading on other factors. Then a suitable name is to be given to the factor keeping in mind the essence of the original variables included in that factor.

**INPUT DATA**

The variables should be quantitative at the **interval **or **ratio **level. Categorical data (sex, loyality, etc.) are not suitable for factor analysis.

**CASE ANALYSIS-1**

**PROBLEM**

The problem is to find out the most important factors of choosing a retail store. For this purpose, fifteen attributes such as the location of the store, store ambience, fast check out, attitude of staffs, parking facility, credit card facility etc. are taken into our consideration.

X1 = Location of the store

X2 = Range provided

X3 = Low prices

X4 = Ambience of the store

X5 = Store timings

X6 = Discounts/Promotions/Offers

X7 = Value for money

X8 = Availability of Parking

X9 = AC/Non AC

X10= Staff Attitude

X11= Fast Checkouts

X12= Acceptance of Credit or Debit Cards

X13= Provides Home Delivery

X14= Extends Credit

X15= Exchange Facility

A study has been conducted with 15 attributes of choosing a retail store. A questionnaire on different items related to 15 attributes of choosing a store has been constructed on 5-point likert type scale in all fifteen attributes. The statements are measurable on a Likert scale of 1-5; where 5 indicated strongly disagree and 1 indicated strongly agree. The data so collected from 31 respondents are given below.

**Table-1:** Input Data

**Performing the Analysis with SPSS**

For SPSS Version 11, click on **Analyze ⇒ ****Data Reduction ⇒ Factor**

This will bring up the SPSS screen dialogue box as shown below. The data sheet gives the responses collected from 31 respondents on 15 attributes of choosing a store.

After clicking on **Factor**, the SPSS screen gives the following dialogue box. Select the variables x1, x2 ……..x15 and move them into the variable box.

To do a factor analysis, we need to select an **“Extraction method”** and a **“Rotation method”.** Click the **“Extraction”** button to specify the extraction method. This will bring up the dialogue box shown below.

Select **Principal component method** and **Eigenvalues** more than 1 then** Continue **button. This will return us to the **Factor Analysis **dialogue box. Then click **Rotation**, this will bring up the following dialogue box.

Select **Varimax method** and then** Continue **button. This will bring the following **Factor Analysis **dialogue box.

Now click on the button OK in **Factor Analysis **dialogue box. The output so produced is illustrated on the following pages

**SPSS Output**

**Table-2:** Communalities

* Extraction Method: Principal Component Analysis.*

The first output is a table of communalities which shows how much of the variance in the variables has been accounted for by the extracted factors.

The factor loading for ‘Parking Space’ is comparatively low to the tune of 53.1% of the total variance. However the remaining 14 attributes are explained reasonably with high factor loadings of above 0.6. The attributes like ‘store timings’ and ‘exchange facility’ have the factor loading of 81.1% and 88.4%.

**Table-3: **Total Variance Explained

* Extraction Method: Principal Component Analysis**.*

The table shows six factors extractable from the analysis along with their eigenvalues, the percent of variance attributable to each factor, the cumulative variance of the factor and the previous factors**. **Eigen Value greater than 1, results in six factors being extracted from the data collected. The factors with Eigen values greater than 1 are retained and other factors are not included in the analysis.

There are six factors resulting from the analysis explaining a total of 71.872% of the variance. The first extracted factor accounts for 21.477% of the variance in all 15 variables, the second factor 13.713% of the variance and similarly for other factors.

**Table-4:** Component Matrix

* Extraction Method: Principal Component Analysis.*

* a 6 components extracted.*

Each number represents the correlation between the item and the unrotated factor. The correlation between ‘X5 = Store timings’ and factor 1 is 0.857 and it indicates high loading of ‘Store timings’ on factor 1. These correlations help to formulate an interpretation of the factors or components. This is done by looking for a common thread among the variables that have large loadings for a particular factor or component. It is possible to see items with large loadings on several of the unrotated factors, which can make interpretation difficult. In these cases, it can be helpful to examine a rotated solution.

**Table-5:** Rotated Component Matrix

* Extraction Method: Principal Component Analysis. *

* Rotation Method: Varimax with Kaiser Normalization.*

* a Rotation converged in 7 iterations.*

Rotation is a method used to simplify interpretation of a factor analysis. The idea of rotation is to reduce the number factors on which the variables under investigation have high loadings. Rotation does not actually change anything but makes the interpretation of the analysis easier.

**Component-1** comprises of three variables – Store timings, Value for money, Exchange Facility with factor loading 0.762, 0.771, and 0.843. The factor is named as **Internal factors**.

**Component-2** comprises of three variables – Ambience and Staff attitude with factor loading 0.645, 0.840. The variable ‘low price’ has significant negative loading (-0.643) on second factor. The factor is named as **Transcendence Factors**.

**Component -3** comprises of three variables – Fast Checkouts and AC/Non AC with factor loading 0.679 and 0.605. The variable ‘Range provided’ has negative loading (-0.774) on third factor. The factor is named as **Facilities**.

**Component -4** comprises of two variables – Credit Cards and Home Delivery with factor loading 0.781 and 0.758. The factor is named as **Service.**

**Component -5** comprises of two variables – Location of the store and Extends Credit with factor loading 0.697 and .809. It is named as the factor **Location with credit facility**.

**Component -6** is the variable- Discount /Offers with factor loading 0.823 and it is named as the **Offers.**

**Table-6:** Component Transformation Matrix

* Extraction Method: Principal Component Analysis. *

* Rotation Method: Varimax with Kaiser Normalization.*

Component Transformation Matrix table is a default output of **SPSS** and is of less use in extracting the factors.

The factors so extracted are given in the following table.

**Table-7:**

**CASE ANALYSIS-2**

**PROBLEM**

Adolescence is considered to be a time of great risk. A lot of adolescents are facing pressures to use alcohol, cigarettes, or drugs thereby putting themselves at high health risk. The particular research is intended to find out the factors influencing the adolescents to smoke cigarette or to use alcohol.

The study has been conducted with 23 college students of age group 18 to22. A questionnaire pertaining to ten prominent reasons (X1, X2……..X10) of such habits has been prepared on 5-point likert type scale. The students were asked to rate each reason on a Likert scale of 1-5; where 5 indicates strongly disagree and 1 indicates strongly agree.

X1= to relieve stress

X2= to fit in a group

X3 = to give a cool outlook

X4= to be modern in approach

X5=to grab the attention of the society

X6= to try an experiment

X7 = influenced by role model

X8= experimental attitude

X9= male ego

X10 = attention of the society

The data so collected from 23 respondents are given below.

**Table-1:** Input Data

**Table-2:** Communalities

* Extraction Method: Principal Component Analysis.*

**Table-3:** Total Variance Explained

* Extraction Method: Principal Component Analysis.*

**Table-4:** Component Matrix

* Extraction Method: Principal Component Analysis.*

* a 4 components extracted.*

**Table-5:** Rotated Component Matrix

* Extraction Method: Principal Component Analysis. *

* Rotation Method: Varimax with Kaiser Normalization.*

* a Rotation converged in 4 iterations.*

**Table-6:** Component Transformation Matrix

* Extraction Method: Principal Component Analysis. *

* Rotation Method: Varimax with Kaiser Normalization.*

**Component-1** comprises of three variables – To relieve stress (.749), to give a cool outlook (0.745), to grab the attention of the society (0.638). The factor is named as **Importance in the society and stress release.**

**Component-2** comprises of three variables – To be modern in approach (0.515), to try an experiment (0.786) and not right passage to adulthood (-0.882). The factor is named as **Modernity**.

**Component -3** comprises of three variables – To fit in a group (0.692), not influenced by role model (-0.559), peer group pressure (0.696). The factor is named as **Life style**.

**Component -4** is the variable- Male ego with factor loading 0.881 and it is named as **Ego.**

The factors so extracted are given in the following table.

**Table-7:-**

**SPSS Command**

- Click on ANALYZE at the SPSS menu bar (in older versions of SPSS, click on STATISTICS instead of ANALYZE).
- Click on DATA REDUCTION followed by
- Select the variables and move them to variable box.
- Click on EXTRACTION METHOD and select PRINCIPAL COMPONENT METHOD. Choose EIGEN VALUES more than 1 and then click CONTINUE.
- Click ROTATION and select VARIMAX METHOD and then CONTINUE.
- Now click on the button OK in Factor Analysis dialogue box to get the output.