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# Factorial Analysis

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Rosie M.

Jade S. Factor Analysis Factor Analysis

Brief overview

Statistical Tests Measures

Assumptions

Hypothesis/research questions

DV/IV explained

Journal articles

Demonstration

Quiz Factor Analysis

Overview Contents Two Main Types Exploratory Factor Analysis Steps Step 1 Assessing the Suitability Step 2 Factor Extraction 1. Assessing the Suitability of the Data

2. Factor Extraction

3. Factor Rotation

4. Loadings and interpretation Main uses... Exploratory (EFA)

Used in the early stages to discover underlying structure of a large set of variables and to examine its' internal reliability.

Confirmatory (CFA)

More complex technique.

Used to decide on the number of factors and theories. Large sample size

At least 300 or 10:1 items Interrcorrelation strength r=.3

KMO and Bartlett's Tests of Sphericity Extracting the smallest number of variable to account for the largest amount of variability in the data. Not designed to test hypothesis

Designed to reduce the data into smaller set of factors

Looks for intercorrelations between the variables Factor Extraction •Based on its own particular statistical assumptions

•Results from different extraction methods are either identical, or their differences trivial.

•The most common method used is the Least Square extraction of Charactristic Roots and Vectors (Eigen values and vectors). Used for understanding the structure of a 'latent variable' such as Intelligence.

Constructing and developing set of scales for testing.

Designed to reduce a large amount of data into more meaningful data. 3 Main methods Principal component analysis (PCA)

•Seeks a linear combination of variables so maximum variance is extracted

•Removes variance and seeks a second linear combination

Common factor analysis (PFA)

•Seeks the least number of factors which can account for the common variance (correlation) of a set of variables.

Image factoring

•Based on the correlation matrix of predicted variables rather than actual variables

We will focus on the Principal component method Principal components

Step 2 Factor rotation and interpretation

Step 3

Once the number of factors has been determined you need to interpret these factors

SPSS does not label or interpret factors for you. You need to do this using past research and theory

Rotation does not change the underlying solution rather it presents the pattern of loadings in a manner that is easier to interpret Two types of rotation Orthogonal methods give factors which are not correlated with each other

Easier to interpret and report but assume independence of factors

Oblique methods allow correlation between factors

The two approach often result in very similar solutions particularly when the pattern of correlations among the items is clear

Researchers usually do both and report the simplest Researcher to determine the number of factors that best describe the underlying relationship

Few factors as possible to explain majority of variance

Used to reduce large multi-variate data sets so that they are easier to analysis (Jolliffe, 2005)

Can be used to recreate the original data

Is used to explore the relationship between the variables.

To do this the final residual matrix must have no correlation left

Issues;

Too many factors and some of the correlations would be exhausted before others

The 1s in the diagonal is replaced by an estimate of the communality Assumptions •Factor analysis is designed for interval data

•The variables used in factor analysis should be reliable and linearly related to each other

•The variables must also be at least moderately correlated to each other.

•Highly affected by missing data and outlying cases

•Need enough subjects for stable estimates; approx. 300

•Univariate - normally distributed variables make the solution stronger

•KMO and Barlette’s test of Sphericity Hypothesis, research questions, IV and DV

•Factor Analysis is not designed to test hypothesis

•Data reduction procedure.

•Searches for intercorrelations of a set of variables

•Is used in the development and evaluation of tests and scales. Assessment of the suitability of the data What do you notice

about 'Analysis N'? Intercorrelations between items: Step 1

What do you notice about the values? KMO AND BARTLETT'S TEST OF SPHERICITY

KMO: Tests amount of variance that could be explained by data.

Factorability of the data

varies between 0 and 1, and values closer to 1 are better.

value of .6 is a suggested minimum.

Bartlett's Test of Sphericity - Tests the null hypothesis that the data is factorable

Aim to reject this null hypothesis.

Value < .05 Extraction Step 2

Eigenvalues Most commonly used in Factor Analysis

How many factors should be extracted in Factor Analysis

Explains how much of the variance in the data set is explained by each factor. SCREE PLOT

Eigenvalues plotted: against the factor number

In decreasing order

Dotted line: indicates where the scree plot breaks into steep and shallow parts:

Less variance accounted for in the shallow parts (values <1)

Components > 1 chosen (more variance accounted). Rotation and Sum of square loadings:

Factor loading before the rotation. Performed after factors have been extracted in order to show the value or the “loading” of each factor onto the corresponding item.

Table

3 underlying extracted factors presented.

Columns showing loading of factor on the item.

Correlation between the factor and the item Factor loading after rotation Table that is reported when interpreting SPSS data.

Provides us with the unique relationship between the individual item and a factor, as it excludes the overlap of the factors.

We look for the items that load highly

Loading of <.3 is too low, therefore not considered. Questions? Bruin, J. 2006. newtest: command to compute new test. UCLA: Statistical Consulting Group. http://www.ats.ucla.edu/stat/stata/ado/analysis/.

Field, A. (2009). Discovering Statistics using SPSS. Singapore: Sage Publications.

Introduction to SAS. UCLA: Statistical Consulting Group. from http://www.ats.ucla.edu/stat/sas/notes2/ (accessed November 24, 2007).

James Dean BrownShiken Statistics Corner: Questions and answers about language testing statistics:What iis an eigenvallue?: JALT Testing & Evaluation SIG Newsletter, 5 (1) April 200p, 15-19.

Jollife I (2005), Encyclopedia of Statistics in Behavioral Science’, Principal Component Analysis, pp. 30.

Ramli, M., Salmiah, M. A., Nurul, A. M. (2009). Validation and psychometric properties of Bahasa Malaysia version of the Depression Anxiety and Stress Scales (DASS) among diabetic patients. ASEAN Journal of Psychiatry, 8 (2), 82-89.

Zyl, C. J., & Bruin, K. (2012). The relationship between mixed model emotional intelligence and personality. Psychological Society of South Africa, 42 (4), pp.532-542. ...Chocolate? QUIZ

TIME!!! Q Q Q JOURNAL EXAMPLE Aim: to validate the Depression Anxiety Stress Scales 21-item (DASS-21) Bahasa Malaysia (BM) version among clinical subjects who were diabetic patients.

Determining reliability and construct validity of the BM DASS by looking at internal consistency and exploratory factor analysis respectively.

153 diabetic patients.

The BM DASS-21 had very good Cronbach’s alpha values of 0.75, 0.74 and 0.79, respectively for depression, anxiety and stress subscales.

For construct validity, it also had good factor loading values for 17 out of 21 items (.31 to .75).

The results of this study entrenched the evidence that the BM DASS-21 had excellent sychometric properties.

Validity was determined by Confirmatory Factor Analysis (CFA)

Proved that BM DASS-21 managed to delineate its items into 3 main categories (depression, anxiety and stress).

All items except four had factor loadings of more than .30, good! Factor Analysis Application in Research Zyl and Bruin (2012) investigated the relationship between the mixed model emotional intelligence and personality using the factor analysis.

Trait personality questionnaires, the Hogan Personality inventory and the Work Personality index were assessed together with the Emotional Quotient Inventory.

Some similar factors emerged indicating the overlap between the models

Therefore, the application of factor analysis In this study has uncovered common factors between the two variable sets ( EI and Personality). ReferenceFull transcript

by

Tweet## Rosie Milojevic

on 20 February 2014#### Transcript of Factorial Analysis

Rosie M.

Jade S. Factor Analysis Factor Analysis

Brief overview

Statistical Tests Measures

Assumptions

Hypothesis/research questions

DV/IV explained

Journal articles

Demonstration

Quiz Factor Analysis

Overview Contents Two Main Types Exploratory Factor Analysis Steps Step 1 Assessing the Suitability Step 2 Factor Extraction 1. Assessing the Suitability of the Data

2. Factor Extraction

3. Factor Rotation

4. Loadings and interpretation Main uses... Exploratory (EFA)

Used in the early stages to discover underlying structure of a large set of variables and to examine its' internal reliability.

Confirmatory (CFA)

More complex technique.

Used to decide on the number of factors and theories. Large sample size

At least 300 or 10:1 items Interrcorrelation strength r=.3

KMO and Bartlett's Tests of Sphericity Extracting the smallest number of variable to account for the largest amount of variability in the data. Not designed to test hypothesis

Designed to reduce the data into smaller set of factors

Looks for intercorrelations between the variables Factor Extraction •Based on its own particular statistical assumptions

•Results from different extraction methods are either identical, or their differences trivial.

•The most common method used is the Least Square extraction of Charactristic Roots and Vectors (Eigen values and vectors). Used for understanding the structure of a 'latent variable' such as Intelligence.

Constructing and developing set of scales for testing.

Designed to reduce a large amount of data into more meaningful data. 3 Main methods Principal component analysis (PCA)

•Seeks a linear combination of variables so maximum variance is extracted

•Removes variance and seeks a second linear combination

Common factor analysis (PFA)

•Seeks the least number of factors which can account for the common variance (correlation) of a set of variables.

Image factoring

•Based on the correlation matrix of predicted variables rather than actual variables

We will focus on the Principal component method Principal components

Step 2 Factor rotation and interpretation

Step 3

Once the number of factors has been determined you need to interpret these factors

SPSS does not label or interpret factors for you. You need to do this using past research and theory

Rotation does not change the underlying solution rather it presents the pattern of loadings in a manner that is easier to interpret Two types of rotation Orthogonal methods give factors which are not correlated with each other

Easier to interpret and report but assume independence of factors

Oblique methods allow correlation between factors

The two approach often result in very similar solutions particularly when the pattern of correlations among the items is clear

Researchers usually do both and report the simplest Researcher to determine the number of factors that best describe the underlying relationship

Few factors as possible to explain majority of variance

Used to reduce large multi-variate data sets so that they are easier to analysis (Jolliffe, 2005)

Can be used to recreate the original data

Is used to explore the relationship between the variables.

To do this the final residual matrix must have no correlation left

Issues;

Too many factors and some of the correlations would be exhausted before others

The 1s in the diagonal is replaced by an estimate of the communality Assumptions •Factor analysis is designed for interval data

•The variables used in factor analysis should be reliable and linearly related to each other

•The variables must also be at least moderately correlated to each other.

•Highly affected by missing data and outlying cases

•Need enough subjects for stable estimates; approx. 300

•Univariate - normally distributed variables make the solution stronger

•KMO and Barlette’s test of Sphericity Hypothesis, research questions, IV and DV

•Factor Analysis is not designed to test hypothesis

•Data reduction procedure.

•Searches for intercorrelations of a set of variables

•Is used in the development and evaluation of tests and scales. Assessment of the suitability of the data What do you notice

about 'Analysis N'? Intercorrelations between items: Step 1

What do you notice about the values? KMO AND BARTLETT'S TEST OF SPHERICITY

KMO: Tests amount of variance that could be explained by data.

Factorability of the data

varies between 0 and 1, and values closer to 1 are better.

value of .6 is a suggested minimum.

Bartlett's Test of Sphericity - Tests the null hypothesis that the data is factorable

Aim to reject this null hypothesis.

Value < .05 Extraction Step 2

Eigenvalues Most commonly used in Factor Analysis

How many factors should be extracted in Factor Analysis

Explains how much of the variance in the data set is explained by each factor. SCREE PLOT

Eigenvalues plotted: against the factor number

In decreasing order

Dotted line: indicates where the scree plot breaks into steep and shallow parts:

Less variance accounted for in the shallow parts (values <1)

Components > 1 chosen (more variance accounted). Rotation and Sum of square loadings:

Factor loading before the rotation. Performed after factors have been extracted in order to show the value or the “loading” of each factor onto the corresponding item.

Table

3 underlying extracted factors presented.

Columns showing loading of factor on the item.

Correlation between the factor and the item Factor loading after rotation Table that is reported when interpreting SPSS data.

Provides us with the unique relationship between the individual item and a factor, as it excludes the overlap of the factors.

We look for the items that load highly

Loading of <.3 is too low, therefore not considered. Questions? Bruin, J. 2006. newtest: command to compute new test. UCLA: Statistical Consulting Group. http://www.ats.ucla.edu/stat/stata/ado/analysis/.

Field, A. (2009). Discovering Statistics using SPSS. Singapore: Sage Publications.

Introduction to SAS. UCLA: Statistical Consulting Group. from http://www.ats.ucla.edu/stat/sas/notes2/ (accessed November 24, 2007).

James Dean BrownShiken Statistics Corner: Questions and answers about language testing statistics:What iis an eigenvallue?: JALT Testing & Evaluation SIG Newsletter, 5 (1) April 200p, 15-19.

Jollife I (2005), Encyclopedia of Statistics in Behavioral Science’, Principal Component Analysis, pp. 30.

Ramli, M., Salmiah, M. A., Nurul, A. M. (2009). Validation and psychometric properties of Bahasa Malaysia version of the Depression Anxiety and Stress Scales (DASS) among diabetic patients. ASEAN Journal of Psychiatry, 8 (2), 82-89.

Zyl, C. J., & Bruin, K. (2012). The relationship between mixed model emotional intelligence and personality. Psychological Society of South Africa, 42 (4), pp.532-542. ...Chocolate? QUIZ

TIME!!! Q Q Q JOURNAL EXAMPLE Aim: to validate the Depression Anxiety Stress Scales 21-item (DASS-21) Bahasa Malaysia (BM) version among clinical subjects who were diabetic patients.

Determining reliability and construct validity of the BM DASS by looking at internal consistency and exploratory factor analysis respectively.

153 diabetic patients.

The BM DASS-21 had very good Cronbach’s alpha values of 0.75, 0.74 and 0.79, respectively for depression, anxiety and stress subscales.

For construct validity, it also had good factor loading values for 17 out of 21 items (.31 to .75).

The results of this study entrenched the evidence that the BM DASS-21 had excellent sychometric properties.

Validity was determined by Confirmatory Factor Analysis (CFA)

Proved that BM DASS-21 managed to delineate its items into 3 main categories (depression, anxiety and stress).

All items except four had factor loadings of more than .30, good! Factor Analysis Application in Research Zyl and Bruin (2012) investigated the relationship between the mixed model emotional intelligence and personality using the factor analysis.

Trait personality questionnaires, the Hogan Personality inventory and the Work Personality index were assessed together with the Emotional Quotient Inventory.

Some similar factors emerged indicating the overlap between the models

Therefore, the application of factor analysis In this study has uncovered common factors between the two variable sets ( EI and Personality). Reference