CHAPTER 3 – FACTOR ANALYSIS
Analyzing patterns of complex, multidimensional relationships of large number of variables and determine
whether information can be condensed or summarized in smaller set of factors or components.
Bartlett: statistical test for overall significance of all correlations within correlation matrix
Common factor analysis: factor model in which factors are based on reduced correlation matrix. Communalities
are inserted in diagonal of correlation matrix, and extracted factors are based only on common variance, with
specific and error variance excluded
Common variance: variance shared with other variables
Communality: total amount of variance an original variable shares with all the other variables
Component analysis: factor model in which factors are based on total variance. Unities are used in diagonal of
correlation matrix. Implies that all variance is common or shared.
Content validity: assessment of degree of correspondence between items selected to constitute summated scale
and its conceptual definition. = face validity
Cronbach’s alpha: measure of reliability. Ranges from 0 to 1, with values of 0.60-0.70 as minimum
Cross-loading: variable has two more factor loadings exceeding threshold value
Dummy variable: binary metric variable used to represent single category of nonmetric variable
Eigenvalue: column sum of squared loadings for factor; latent root. Represents amount of variance accounted
for by factor.
Factor indeterminacy: factor scores are not unique for each individual
Factor loadings: correlation between original variables and factors, and key to understanding the nature of
particular factor. Squared factor loadings indicate what percentage of variance in original variable is explained
by factor
Factor pattern matrix: oblique rotation
Factor matrix: orthogonal rotation
Factor rotation: process of manipulation or adjusting factor axes to achieve simpler and pragmatically more
meaningful factor solution
Factor score: composite measure for each observation on each factor extracted in factor analysis. Factor
weights are used in conjunction with original variable values to calculate each observation’s score. Factor score
can be used to represent factor in subsequent analyses. Standardized to have mean of 0 and standard deviation
of 1.
Factor structure matrix: oblique rotation. Represents simple correlations between variables and factors,
incorporating unique variance and correlations between factors. Factor pattern matrix is preferred
Multicollinearity: extent to which variable can be explained by other variables in analysis
Oblique factor rotation: rotation computed so that extracted factors are correlated.
Orthogonal factor rotation: rotation in which factors are extracted so that their axes are maintained at 90
degrees. Each factor is independent of all other factors. Correlation between factors is 0.
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,Reliability: extent to which variable or set of variables is consistent in what it is intended to measure. If multiple
measurements are taken, reliable measures will all be consistent in their values. Differs from validity in that it
does not relate to what should be measured, but instead to how it is measured. Consistency of measures.
Reverse scoring: process of reversing the scores of variable, while retaining distributional characterstics, to
change relationship between variables. Used in summated scale construction to avoid canceling out between
variables with positive and negative factor loadings on same factor.
Specific variance: variance of each variable unique to that variable and not explained or associated with other
variables. = unique variance
Summated scales: combining several variables that measure same concept into single variable in attempt to
increase reliability of measurement. Most cases, separate variables are summed and then their total or average
score is used.
Surrogate variable: selection of single variable with highest factor loading to represent factor in data reduction
stage
Trace: total amount of variance on which factor solution is based. Is equal to number of variables, based on
assumption that variance in each variable is equal to 1.
Validity: extent to which measure or set of measures correctly represent concept of study – degree to which its
free from any systematic or nonrandom error. How well do the measures define concepts.
VARIMAX: most popular orthogonal factor rotation.
WHAT IS FACTOR ANALYSIS?
Primary purpose: defining underlying structure among variables in analysis.
Employ multivariate techniques and thereby increase number of variables. By more variables, the change of
correlation and thus grouping of variables is higher.
Provides tools for analyzing structure of interrelationships (correlations) among large number of variables by
defining sets of variables that are highly interrelated, known as factors.
Factors represent dimensions within data + reduce number of variables.
If we have conceptual basis for understanding relationships between variables, then dimensions may actually
have meaning for what they collectively represent
Dimensions may correspond to concepts that cannot be adequately described by single measure.
Exploratory or confirmatory perspective.
Many researchers consider it only exploratory: searching for structure among set of variables or as data
reduction method. No set of a priori constraints on estimation of components to be extracted.
Confirmatory: researcher has preconceived thoughts on actual structure of data based on theoretical support of
prior research. Assess degree to which data meet expected structure.
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,FACTOR ANALYSIS DECISION PROCESS
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, Stage 1: objectives
Research problem
General purpose: find way to summarize information contained in number of original variables into smaller set
of new, composite dimensions or variates with minimum loss of information – search for and define
fundamental constructs assumed to underlie original variable.
4 issues:
- Specifying unit of analysis
- Achieving data summarization and/or data reduction
- Variable selection
- Using factor analysis results with other multivariate techniques
Specifying unit of analysis
General model that can identify structure of relationships among either variables or respondents by examining
either correlations between variables or correlations between respondents
- To summarize: factor analysis would be applied to correlation matrix of variables.
R factor analysis: analyze set of variables to identify dimensions that are latent (not easily observed)
- Q factor analysis: factor analysis to correlation matrix of individual respondents. Combines or
condenses large number of people into distinctly different groups within larger population. Not utilized
frequently because of computational difficulties.
Instead, researchers utilize cluster analysis to group individual respondents.
Select unit of factor analysis: variables or respondents
Achieving data summarization and/or data reduction
Summarization: analysis derives underlying dimensions that, when interpreted and understood, describe data
in much smaller number of concepts than original individual variables
Reduction: extends summarization by deriving empirical value (factor score) for each dimension (factor) and
then substituting this value for original values.
Data summarization:
Structure
Through structure, the researcher can view set of variables at various levels of generalization, ranging from
most detailed level (individual variables) to more generalized level (individual variables grouped in collective)
Factor analysis is interdependence technique: all variables are simultaneously considered with no distinction
as to dependent or independent variables. Variates (factors) are formed to maximize explanation of entire
variable set, not to predict dependent variables.
In dependence techniques: one or more variables are explicitly considered the criterion or dependent variables
and all others are predictor or independent variables.
Dependence = prediction
Interdependence = identification of structure. Structure is defined by interrelatedness among variables
allowing for specification of smaller number of dimensions (factors) representing original set of variables
Goal of summarization: defining small number of factors that adequately represent original set of variables
Data reduction:
1. Identifying representative variables from much larger set of variables for use in subsequent
multivariate analyses, or;
2. Creating entirely new set of variables, much smaller in numbers, to partially or completely replace
original set of variables
In both, purpose: retain nature and character of original variables, but reduce their number to simplify
subsequent multiple variables
Identification of underling dimensions or factors ends is made in data summarization.
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