1 Peter Fox Data Analytics – ITWS-4963/ITWS-6965 Week 10a, April 7, 2015 Factor Analysis (v. PCA),...

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Peter Fox

Data Analytics – ITWS-4963/ITWS-6965

Week 10a, April 7, 2015

Factor Analysis (v. PCA), Fischer Linear Discriminant

Factor Analysis• Exploratory factor analysis (EFA) is a

common technique in qualitative sciences for explaining the (shared) variance among several measured variables as a smaller set of latent (hidden/not observed) variables.

• EFA is often used to consolidate survey data by revealing the groupings (factors) that underlie individual questions.

• A large number of observable variables can be aggregated into a model to represent an underlying concept, making it easier to understand the data.

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Examples• E.g. business confidence, morale, happiness

and conservatism - variables which cannot be measured directly.

• E.g. Quality of life. Variables from which to “infer” quality of life might include wealth, employment, environment, physical and mental health, education, recreation and leisure time, and social belonging. Others?

• Tests, questionnaires, visual imagery…

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Relation among factors• “correlated” (oblique) or “orthogonal” factors?

• E.g. wealth, employment, environment, physical and mental health, education, recreation and leisure time, and social belonging

• Relations?

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Factor Analysis

PCA and FA?• CFA analyzes only the reliable common variance of

data, while PCA analyzes all the variance of data. • An underlying hypothetical process or construct is

involved in CFA but not in PCA. • PCA tends to increase factor loadings especially in

a study with a small number of variables and/or low estimated communality.

• Thus, PCA is not appropriate for examining the structure of data.

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FA vs. PCA conceptually

• FA produces factors; PCA produces components

• Factors cause variables; components are aggregates of the variables

FA

I1 I3I2

PCA

I1 I3I2

Conceptual FA and PCA

PCA and FA?• If the study purpose is to explain correlations

among variables and to examine the structure of the data, FA provides a more accurate result.

• If the purpose of a study is to summarize data with a smaller number of variables, PCA is the choice.

• PCA can also be used as an initial step in FA because it provides information regarding the maximum number and nature of factors. – Scree plots (Friday)

• More on this later…

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The Relationship between Variables

• Multiple Regression

– Describes the relationship between several variables, expressing one variable as a function of several others, enabling us to predict this variable on the basis of the combination of the other variables

• Factor Analysis

– Also a tool used to investigate the relationship between several variables

– Investigates whether the pattern of correlations between a number of variables can be explained by any underlying dimensions, known as ‘factors’

From: Laura McAvinue School of PsychologyTrinity College Dublin

Uses of Factor Analysis

Test / questionnaire constructiono For example, you wish to design an anxiety questionnaire…o Create 50 items, which you think measure anxietyo Give your questionnaire to a large sample of peopleo Calculate correlations between the 50 items & run a factor

analysis on the correlation matrixo If all 50 items are indeed measuring anxiety…

• All correlations will be high• One underlying factor, ‘anxiety’

Verification of test / questionnaire structureo Hospital Anxiety & Depression Scaleo Expect two factors, ‘anxiety’ & ‘depression’

How does it work?

• Correlation Matrix

– Analyses the pattern of correlations between variables in the correlation matrix

– Which variables tend to correlate highly together?

– If variables are highly correlated, likely that they represent the same underlying dimension

• Factor analysis pinpoints the clusters of high correlations between variables and for each cluster, it will assign a factor

Correlation Matrix

• Q1-3 correlate strongly with each other and hardly at all with 4-6• Q4-6 correlate strongly with each other and hardly at all with 1-3• Two factors!

Q1 Q2 Q3 Q4 Q5 Q6

Q1 1

Q2 .987 1

Q3 .801 .765 1

Q4 -.003 -.088 0 1

Q5 -.051 .044 .213 .968 1

Q6 -.190 -.111 0.102 .789 .864 1

Factor Analysis

• Two main things you want to know…

– How many factors underlie the correlations between the variables?

– What do these factors represent?• Which variables belong to which factors?

Steps of Factor Analysis

1. Suitability of the Dataset

2. Choosing the method of extraction

3. Choosing the number of factors to extract

4. Interpreting the factor solution

1. Suitability of Dataset

Selection of Variables

Sample Characteristics

Statistical Considerations

Selection of Variables

Are the variables meaningful?• Factor analysis can be run on any dataset• ‘Garbage in, garbage out’ (Cooper, 2002)

Psychometrics • The field of measurement of psychological constructs• Good measurement is crucial in Psychology• Indicator approach

• Measurement is often indirect• Can’t measure ‘depression’ directly, infer on the basis of an

indicator, such as questionnaire

Based on some theoretical / conceptual framework, what are these variables measuring?

Selection of Variables, Example

How would you group these faces?

Sample Characteristics

Size At least 100 participants

Participant : Variable Ratio Estimates vary Minimum of 5 : 1, ideal of 10 : 1

Characteristics Representative of the population of interest? Contains different subgroups?


Assumptions of factor analysis regarding data Continuous Normally distributed Linear relationships

These properties affect the correlations between variables

Independence of variables Variables should not be calculated from each other

e.g. Item 4 = Item 1 + 2 + 3


Are there enough significant correlations (> .3) between the variables to merit factor analysis?

Bartlett Test of Sphericity

Tests Ho that all correlations between variables = 0

If p < .05, reject Ho and conclude there are significant correlations between variables so factor analysis is possible


Are there enough significant correlations (> .3) between the variables to merit factor analysis?

Kaiser-Meyer-Olkin Measure of Sampling Adequacy

Quantifies the degree of inter-correlations among variables Value from 0 – 1, 1 meaning that each variable is perfectly

predicted by the others Closer to 1 the better If KMO > .6, conclude there is a sufficient number of

correlations in the matrix to merit factor analysis

Statistical Considerations, Example

• All variables• Continuous• Normally Distributed• Linear relationships• Independent

• Enough correlations?• Bartlett Test of Sphericity (χ2; df ; p < .05)• KMO

2. Choosing the method of extraction

Two methods

Factor AnalysisPrincipal Components AnalysisDiffer in how they analyse the variance in the

correlation matrix

Variable

Specific

Variance

Error

Variance

Common Variance

Variance unique to the variable itself

Variance due to measurement error or some

random, unknown source

Variance that a variable shares

with other variables in a

matrix

When searching for the factors underlying the relationships between a set of variables, we are interested in detecting and explaining the common variance

Principal Components Analysis

• Ignores the distinction between the different sources of variance

• Analyses total variance in the correlation matrix, assuming the components derived can explain all variance

• Result: Any component extracted will include a certain amount of error & specific variance

Factor Analysis

• Separates specific & error variance from common variance

• Attempts to estimate common variance and identify the factors underlying this

Which to choose?

• Different opinions

• Theoretically, factor analysis is more sophisticated but statistical calculations are more complicated, often leading to impossible results

• Often, both techniques yield similar solutions

V


• Statistical Modelling– You can create many solutions using different

numbers of factors

• An important decision– Aim is to determine the smallest number of factors

that adequately explain the variance in the matrix– Too few factors

• Second-order factors – Too many factors

• Factors that explain little variance & may be meaningless

Criteria for determining Extraction

Theory / past experience

Latent Root Criterion

Scree Test

Percentage of Variance Explained by the factors

Latent Root Criterion (Kaiser-Guttman)

• Eigenvalues

– Expression of the amount of variance in the matrix that is explained by the factor

– Factors with eigenvalues > 1 are extracted– Limitations

• Sensitive to the number of variables in the matrix• More variables… eigenvalues inflated… overestimation of

number of underlying factors

Scree Test (Cattell, 1966)

• Scree Plot

– Based on the relative values of the eigenvalues– Plot the eigenvalues of the factors– Cut-off point

• The last component before the slope of the line becomes flat (before the scree)

Elbow in the graph

Take the components above the elbow

Percentage of Variance

• Percentage of variance explained by the factors

– Convention– Components should explain at least 60% of the

variance in the matrix (Hair et al., 1995)


• Three components with eigenvalues > 1

• Explained 67.26% of the variance

Scree Plot

Co mp o n e n t Nu mb e r

987654321

Eig

en

va

lue

4. 0

3. 5

3. 0

2. 5

2. 0

1. 5

1. 0

. 5

0. 0

BFI data in psych (R)

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4. Interpreting the Factor Solution

• Factor Matrix– Shows the loadings of each of the variables on the

factors that you extracted– Loadings are the correlations between the variables

and the factors– Loadings allow you to interpret the factors

• Sign indicates whether the variable has a positive or negative correlation with the factor

• Size of loading indicates whether a variable makes a significant contribution to a factor

– ≥ .3

Component 1 – Visual imagery tests

Component 2 – Visual imagery questionnaires

Component 3 – ?

Variables Component 1 Component 2 Component 3

Vividness Qu -.198 -.805 .061

Control Qu .173 .751 .306

Preference Qu .353 .577 -.549

Generate Test -.444 .251 .543

Inspect Test -.773 .051 -.051

Maintain .734 -.003 .384

Transform (P&P) Test

.759 -.155 .188

Transform (Comp) Test

-.792 .179 .304

Visual STM Test .792 -.102 .215

Factor Matrix

• Interpret the factors

• Communality of the variables– Percentage of variance in each variable that can be

explained by the factors

• Eigenvalues of the factors– Helps us work out the percentage of variance in the

correlation matrix that the factor explains

Communality of Variable 1 (Vividness Qu) = (-.198)2 + (-.805)2 + (.061)2 = . 69 or 69%

Eigenvalue of Comp 1 = ( [-.198]2 + [.173]2 + [.353]2 + [-.444]2 + [-.773]2 +[.734]2 + [.759]2 + [-.792]2 + [.792]2 ) = 3.36

3.36 / 9 = 37.3%

Variables Component 1 Component 2 Component 3 Communality

Vividness Qu -.198 -.805 .061 69%

Control Qu .173 .751 .306 69%

Preference Qu .353 .577 -.549 76%

Generate Test -.444 .251 .543 55%

Inspect Test -.773 .051 -.051 60%

Maintain .734 -.003 .384 69%


.759 -.155 .188 64%


-.792 .179 .304 75%

Visual STM Test .792 -.102 .215 69%

Eigenvalues 3.36 1.677 1.018 /

% Variance 37.3% 18.6% 11.3% /

Factor Matrix

• Unrotated Solution– Initial solution– Can be difficult to interpret– Factor axes are arbitrarily aligned with the variables

• Rotated Solution– Easier to interpret– Simple structure– Maximises the number of high and low loadings on

each factor

Factor Analysis through Geometry

• It is possible to represent correlation matrices geometrically

• Variables– Represented by straight lines of equal length– All start from the same point– High correlation between variables, lines positioned

close together– Low correlation between variables, lines positioned

further apart– Correlation = Cosine of the angle between the lines

60º

30º

V1

V2

V3

The smaller the angle, the bigger the cosine and the bigger the correlation

V1 & V3

90º angle

Cosine = 0

No relationship

V1 & V230º angleCosine = .867r = .867

V2 & V360º angleCosine = .5R = .5

V1

V5

V4

Factor Loading

Cosine of the angle between each factor and the variable

Factor Analysis

Fits a factor to each cluster of

variables

Passes a factor line through

the groups of variables

V2 V3

V6

F1

F2

V1

V5

V4

V2 V3

V6

F1

F2

V1

V5

V4

V2 V3

V6

F1

F2

Two Methods of fitting Factors

Orthogonal Solution

Factors are at right angles

Uncorrelated

Oblique Solution

Factors are not at right angles

Correlated

V1

V5

V4

V2 V3

V6

F1

F2

Two Step Process

Factors are fit arbitrarily

Factors are rotated to fit the clusters of variables better

V1

V5

V4

V2 V3

V6

F1

F2

Variables C1 C2 C3

Vividness Qu -.198 -.805 .061

Control Qu .173 .751 .306

Preference Qu .353 .577 -.549


Inspect Test -.773 .051 -.051

Maintain Test .734 -.003 .384


.759 -.155 .188

Transform(Comp) Test

-.792 .179 .304

Visual STM Test .792 -.102 .215

Variables C1 C2 C3

Vividness Qu -.029 -.831 .008

Control Qu .174 .744 .323

Preference Qu -.010 .679 -.547


Inspect Test -.717 -.103 .279

Maintain Test .819 .116 .043


.779 -.013 -.166


-.599 -.01 .626

VisualSTM Test .813 .045 -.147

Unrotated Solution Solution following Orthogonal Rotation

For example…

Factor Rotation

• Changes the position of the factors so that the solution is easier to interpret

• Achieves simple structure

– Factor matrix where variables have either high or low loadings on factors rather than lots of moderate loadings

Evaluating your Factor Solution

• Is the solution interpretable?– Should you re-run and extract a bigger or smaller number of

factors?

• What percentage of variance is explained by the factors?– >60%?

• Are all variables represented by the factors?– If the communality of one variable is very low, suggests it is not

related to the other variables, should re-run and exclude

Variables C1 C2 C3

Vividness Qu -.029 -.831 .008

Control Qu .174 .744 .323

Preference Qu -.010 .679 -.547


Inspect Test -.717 -.103 .279

Maintain Test .819 .116 .043


.779 -.013 -.166


-.599 -.01 .626

VisualSTM Test .813 .045 -.147

First Solution Second Solution

For example…

Variables Component 1 Component 2

Vividness Qu .013 -.829

Control Qu -.023 .770

Preference Qu .195 .648

Generate Test -.493 .130

Inspect Test -.760 -.146

Maintain Test .711 .183

Transform (P&P) Test .773 .042


-.811 -.028

Visual STM Test .792 .103

Component 3 = ?C1 = Efficiency of objective visual imagery

C2 = Self-reported imagery efficacy

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

References

• Some…• Cooper, C. (1998). Individual differences.

London: Arnold.

• Kline, P. (1994). An easy guide to factor analysis. London: Routledge.

Assignment to come…

• Assignment 7: Predictive and Prescriptive Analytics. Due ~ week ~12. 20%..

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1 Peter Fox Data Analytics – ITWS-4963/ITWS-6965 Week 10a, April 7, 2015 Factor Analysis (v. PCA),...

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