Factor AnalysisN P SinghProfessor

Factor analysis was invented by psychologist Charles Spearman

History

Combination of original variables

What is a factor?

Grades inStudent No

Finance (Y1)

Marketing Y2

Policy (y3)

1 3 6 52 7 3 33 10 9 84 3 9 75 10 6 5

Example

It has been suggested that these grades are functions of two underlying factors, F1 and F2, tentatively

These are as quantitative ability and verbal ability, respectively.

It is assumed that each Y variable is linearly related to the two factors, as given in next slide.

Examples

Factors

The error terms e1, e2, and e3, serve to indicate that the hypothesized relationships are not exact.

the parameters ij are referred to as loadings. For example, 12 is called the loading of variable Y1 on factor F2.

What are these Error Terms

Factor Analysis

A data reduction technique designed to represent a wide range of attributes on a smaller number of dimensions.

In this MBA program, ¯finance is highly quantitative, while marketing and policy have a strong qualitative orientation.

Quantitative skills should help a student in finance, but not in marketing or policy.

Verbal skills should be helpful in marketing or policy but not in finance.

In other words, it is expected that the loadings have roughly the following structure:

Continued ……..

ContinuedIt is expected that the loadings have roughly the following structure:

The Common Factor Model

This model proposes that each observed response (measure 1 through measure 5) is influenced partially by underlying common factors (factor 1 and factor 2) and partially by underlying unique factors (E1through E5).

The strength of the link between each factor and each measure varies, such that a given factor influences some measures more than others.

The Common Factor Model

Factor Analysis

For example, suppose that a bank asked a large number of questions about a given branch. Consider how the following characteristics might be more parsimoniously represented by just a few constructs (factors).

Factor Analysis

Factor Analysis

- Benefits include: (1) a more concise representation of the marketing situation and hence communication may be enhanced; (2) fewer questions may be required on future surveys; and, (3) perceptual maps become feasible. - Ideally, interval data (e.g., a rating on a 7 point scale), regarding the perceptions of consumers are required regarding a number of features, such as those noted above for the bank are gathered.

personality.sav - a set of responses from a personality questionnaire.

SAQ.sav - fictional statistics anxiety questionnaire from Andy Field's textbook resources

Examples of Data

The purpose of PCA is to derive a relatively small number of components that can account for the variability found in a relatively large number of measures.

This procedure, called data reduction, is typically performed when a researcher does

not want to include all of the original measures in analyses but still wants to work with the information that they contain.

Principal Component Analysis

PCA Model

The first difference is that the direction of influence is reversed: EFA assumes that the measured responses are based on the underlying factors while in PCA the principal components are based on the measured responses.

The second difference is that EFA assumes that the variance in the measured variables can be decomposed into that accounted for by common factors and that accounted for by unique factors. The principal components are defined simply as linear combinations of the measurements, and so will contain both common and unique variance.

Difference

FA and PCA (principal components analysis) are methods of data reduction◦Take many variables and explain them

with a few “factors” or “components”◦Correlated variables are grouped together

and separated from other variables with low or no correlation

What is FA & PCA?

FA and PCA are not much different than canonical correlation in terms of generating canonical variates from linear combinations of variables◦ Although there are now no “sides” of the

equation◦ And your not necessarily correlating the

“factors”, “components”, “variates”, etc.

What is FA & PCA?

FA produces factors; PCA produces components

Factors cause variables; components are aggregates of the variables

FA vs. PCA conceptually

FA analyzes only the variance shared among the variables (common variance without error or unique variance); PCA analyzes all of the variance

FA: “What are the underlying processes that could produce these correlations?”; PCA: Just summarize empirical associations, very data driven

FA vs. PCA conceptually

Step 1: Selecting and Measuring a set of variables in a given domain

Step 2: Data screening in order to prepare the correlation matrix

Step 3: Factor Extraction Step 4: Factor Rotation to increase

interpretability Step 5: Interpretation Further Steps: Validation and Reliability of

the measures

General Steps to FA

A good factor: ◦ Makes sense◦ will be easy to interpret ◦ simple structure◦ Lacks complex loadings

“Good Factor”

We are looking for an eigenvalue above 1.0.

Cumulative percent of variance explained.

Expensive

Exciting

Luxury

Distinctive

Not Conservative

Not Family

Not Basic

Appeals to Others

Attractive Looking

Trend Setting

Reliable

Latest Features

Trust

Expensive

Exciting

Luxury

Distinctive

Not Conservative

Not Family

Not Basic

Appeals to Others

Attractive Looking

Trend Setting

Reliable

Latest Features

Trust

What shall these components be called?

Expensive

Exciting

Luxury

Distinctive

Not Conservative

Not Family

Not Basic

Appeals to Others

Attractive Looking

Trend Setting

Reliable

Latest Features

Trust

EXCLUSIVE TRENDY RELIABLE

= (Expensive + Exciting + Luxury + Distinctive – Conservative – Family – Basic)/7

= (Appeals to Others + Attractive Looking + Trend Setting)/3

= (Reliable + Latest Features + Trust)/3

EXCLUSIVE

TRENDY

RELIABLE

Calculate Component Scores

Exclusive Trendy ReliableBeetle 1.4 6.7 6.9Hummer 3.9 6.2 6.7Lotus 4.1 7.3 6.7Minivan -1.67 4.83 6.5Pick-Up -0.43 4.93 6.3

Not much differing on this dimension.

Exclusive Trendy ReliableBeetle 1.4 6.7 6.9Hummer 3.9 6.2 6.7Lotus 4.1 7.3 6.7Minivan -1.67 4.83 6.5Pick-Up -0.43 4.93 6.3

Vehicle by Component

-3 -2 -1 0 1 2 3 4 5 6 7 8

Beetle

Hummer

Lotus

Minivan

Pick-Up

Exclusive Trendy

Exploratory FA◦ Summarizing data by grouping correlated

variables◦ Investigating sets of measured variables related

to theoretical constructs◦ Usually done near the onset of research◦ The type of FA and PCA we are talking here

Types of FA

Confirmatory FA◦ More advanced technique◦ When factor structure is known or at least

theorized◦ Testing generalization of factor structure to new

data, etc.◦ This is tested through SEM

Types of FA

Observed Correlation Matrix Reproduced Correlation Matrix Residual Correlation Matrix

Terminology

Orthogonal Rotation◦ Loading Matrix – correlation between each

variable and the factor Oblique Rotation

◦ Factor Correlation Matrix – correlation between the factors

◦ Structure Matrix – correlation between factors and variables

◦ Pattern Matrix – unique relationship between each factor and variable uncontaminated by overlap between the factors

Terminology

Factor Coefficient matrix – coefficients used to calculate factor scores (like regression coefficients)

Terminology

Three general goals: data reduction, describe relationships and test theories about relationships (next chapter)

How many interpretable factors exist in the data? or How many factors are needed to summarize the pattern of correlations?

Questions

What does each factor mean? Interpretation?

What is the percentage of variance in the data accounted for by the factors?

Questions

Which factors account for the most variance?

How well does the factor structure fit a given theory?

What would each subject’s score be if they could be measured directly on the factors?

Questions

Hypotheses about factors believed to underlie a domain◦ Should have 6 or more for stable solution

Include marker variables◦ Pure variables – correlated with only one factor◦ They define the factor clearly◦ Complex variables load on more than on factor and

muddy the water

Considerations (from Comrey and Lee, 1992)

Make sure the sample chosen is spread out on possible scores on the variables and the factors being measured

Factors are known to change across samples and time points, so samples should be tested before being pooled together

Considerations (from Comrey and Lee, 1992)