CHAPTER 3: FACTOR ANALYSIS
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CHAPTER 3:
FACTOR ANALYSIS
WHAT IS FACTOR ANALYSIS?
It is an interdependence technique which primarily define the underlying structure among the variables in the analysis
Factor analysis provides the tools for analyzing the structure of the interrelationships among a large number of variables. E.g. (test scores, test items, questionnaire
responses, etc) by defining sets of variables that are highly interrelated known as factors.
WHAT IS FACTOR ANALYSIS?
A statistical approach that can be used to analyze interrelationships among a large number of variables and to explain these variables in terms of their common underlying dimensions (factors).
TYPES OF FACTOR ANALYSIS?
Confirmatory FA It is used to test the pre-specified
relationship. - model is a specified set of dependant
relationships that can be used to test the theory.
Exploratory FA - The primary purpose of exploratory factor
analysis is to define the underlying structure among the variables in the analysis
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HYPOTHETICAL EXAMPLE OF FA
A retail firm identified 80 varied characteristics of retail stores and their service that consumer mentioned as affecting their patronage choice among stores.
--The retailer wants to understand the how the consumer makes decision –
Since the variables are very specific the retailer feels it cannot be evaluated
HYPOTHETICAL EXAMPLE OF FA
Factor analysis result in determining the broader evaluative dimension of the customers
HYPOTHETICAL EXAMPLE OF FA
Factor Analysis result after grouping
OBJECTIVE OF FACTOR ANALYSIS
The general purpose of factor analytic technique is to find a way to condense or summarize the information contained in a large number of original variables into asmaller set of dimension with a minimum loss of information.
Factor analysis is keyed to four issues: 1. Specifying the Unit of Analysis 2. Achieving data summarization and/or data reduction 3. Variable selection 4. Using factor analysis results with other multivariate techniques
DESIGNING A FACTOR ANALYSIS Rule of thumb
Factor analysis is performed most often only on metric variables, although specialized methods exist for the use of
dummy variables
If a study is being designed to reveal factor structure, strive to have at least five variables for each proposed
factor
For sample size:- The sample must have more observations than variables- The minimum absolute sample size should be 50 observations
Maximize the number of observations per variable, with a minimum of 5 and hopefully at least 10 observations per variable
ASSUMPTION IN FACTOR ANALYSIS
The critical assumptions underlying factor analysis are more conceptual than statistical
The conceptual assumptions relate to set of variables selected and the sample chosen
A basic assumptions of factor analysis is that some underlying structure does exist in the set of selected variables
OVER ALL MEASURES OF INTERRELATION 3-2 Rule of Thumb A conceptual foundation needs to support the
assumption that a structure does exist before the factor analysis is performed
A statistically significant Bartlett’s test of sphericity (sig.<.05) indicates that sufficient correlations exist among the variables to proceed
Measure of sampling adequacy (MSA) >.50 for both the overall test and each individual variable; variables with values less than .50 should be omitted from the factor analysis one at a time, lower than 0.50 are being omitted each time
DERIVING FACTORS AND ASSESSING OVERALL FIT
Partitioning the variance of a variable - in order to select a method of extraction
researcher must first understand the variance and how it is divided.
- it is important to understand how much variance a variable shared with other variable in that factor.
3 Types of variance: - 1. Common variance - 2. Unique variance -3. Error variance
METHODS OF EXTRACTION
Component analysis – considers the total variance and derives factors that contain small portions of unique variance and in some error variance.
Common Factor Analysis - considers only the common or shared variance. Assuming that both the unique and error variance are not defining the structure of the variables.
CRITERIA FOR THE NUMBER OF FACTORS TO EXTRACTResearcher decision on the number of factors to be retained should be based on the ff criteria:
Use of several stopping criteria to determine the initial number of factors to retain: - Eigenvalues > 1 - Percentage of Variance Explained ≥ 60%
- A predetermined number of factors based on research objectives and/or prior research - Factors shown by the scree test to have substantial amounts of common variance (i.e., factors before inflection point) - More factors when heterogeneity is present among sample subgroups Consideration of several alternative solutions (one more and one less factor than the initial solution) to ensure the best structure is identified
SCREE TEST CRITERION
It is derived from plotting the eigen value / latent root criterion against the number of factor in their order of extraction.
ASSESSING FACTOR LOADINGS Although factor loadings of ±.30 to ±.40 are minimally
acceptable, values greater than ±.50 are generally considered necessary for practical significance
To be considered significant:- A smaller loading is needed given either a larger sample
size or a larger number of variables being analyzed- A larger loading is needed given a factor solution with a
larger number of factors, especially in evaluating the loadings on later factors
Statistical tests of significance for factor loadings are generally conservative and should be considered only as starting points needed for including a variable for further consideration
INTERPRETING THE FACTORS 3 process of Factor Interpretation 1. Estimate the Factor Matrix 2. Factor Rotation 3. Factor interpretation and re-specification .
An optimal structure exists when all variables have high loadings only on a single factor
Variables that cross-load (load highly on two or more factors and its subtraction <.30) are usually deleted unless theoretically justified or the objective is strictly data reduction
Variables should generally have communalities >.50 to be retained in the analysis
Respecification of a factor analysis can include such options as the following:- Deleting a variable(s)- Changing rotation methods- Increasing or decreasing the number of factors
RELIABILITY Reliability is an assessment of the degree of
consistency between multiple measurement of a variable. The objective is to ensure that responses are not too varied across time periods so that a measurement taken at any point in time is reliable.
Diagnostic measures to assess internal consistency:1. The item-to-total correlation >.50 2. The reliability coefficient with Cronbach’s alpha >.70,
although a .60 level could be used in exploratory research
FACTOR SCORES Factor scores are composite measures of
each factor computed for each subject.
Conceptually the factor scores represents the degree to which each individual scores high on the group of items with high loadings on a factor
Meaning that higher values on the variables with high loadings on a factor will result in a higher factor score.
NAMING THE FACTOR
The final result will be a name or label that represents each of the derived factors as accurately as possible
The label is intuitively developed by the researcher based on its appropriateness for representing the underlying dimensions of a particular factor
Variables with higher loadings are considered more important and have greater influence on the name or label selected to represent a factor
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