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## Factor Analysis:

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**Factor Analysis:**A Brief Synopsis of Factor Analytic Methods With an Emphasis on Nonmathematical Aspects. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Factor Analytic Methods**• Factor analysis is a set of mathematical techniques used to identify dimensions underlying a set of empirical measurements. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Factor Analytic Methods**• Factor analysis is a set of mathematical techniques used to identify dimensions underlying a set of empirical measurements. • It is a data reduction method in which several sets of scores (units) and the correlations between them are mathematically considered. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Factor Analytic Methods**• It is an extremely complex procedure that contains numerous, inherent nuances and variety of correlational analyses designed to examine interrelationships among variables; a basic understanding of geometry, algebra, trigonometry and matrix algebra is required. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Fundamental Purposes**• Factor analytic methods can help scientists to define their variables more precisely and decide what variables they should study and relate to each other in the attempt to develop their science to a higher level (Comrey & Lee, 1992) Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Fundamental Purposes**• …the aim is to summarize the interrelationships among the variables in a concise but accurate manner as an aid in conceptualization (Gorsuch, 1983). Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Fundamental Purposes**• …a statistical technique applied to a single set of variables when the researcher is interested in discovering which variables in the set form coherent subsets that are relatively independent of one another (Tabachnick & Fidell, 2001). Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Fundamental Purposes**• …a statistical technique applied to a single set of variables when the researcher is interested in discovering which variable in the set form coherent subsets that are relatively independent of one another (Tabachnick & Fidell, 2001). • …reducing numerous variables down to a few factors (Tabachnick & Fidell, 2001). Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Fundamental Purposes**• All scientists attempt to identify the basic underlying dimensions that can be used to account for the phenomena they study. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Fundamental Purposes**• All scientists attempt to identify the basic underlying dimensions that can be used to account for the phenomena they study. • Scientists analyze the relationships among a set of variables where these relationships are evaluated across a set of individuals under specific conditions. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Fundamental Purposes**• …is to account for the intercorrelations among n variables, by postulating a set of common factors, fewer in number than the number, n, of these variables (Cureton & D’Agostino, 1983). Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Fundamental Purposes**• In other words, factor analytic methods assist the researcher in gaining a more comprehensive understanding and conceptualization of complex and poorly defined interrelationships that exist in a large number of imprecisely measured variables. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Goals and Objectives**• To summarize patterns of correlations (in matrix) among observed variables. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Goals and Objectives**• To summarize patterns of correlations (in matrix) among observed variables. • To reduce a large number of observed variables to a smaller number of factors. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Goals and Objectives**• To summarize patterns of correlations (in matrix) among observed variables. • To reduce a large number of observed variables to a smaller number of factors. • To provide an operational definition (a regression equation) for a process underlying observed variables. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Goals and Objectives**• To summarize patterns of correlations (in matrix) among observed variables. • To reduce a large number of observed variables to a smaller number of factors. • To provide an operational definition (a regression equation) for an underlying process of observed variables. • To test a theory of underlying processes. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Variables – the characteristics being measured and can be anything that can be objectively measured or scored. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Variables – the characteristics being measured and can be anything that can be objectively measured or scored. • Individuals – the units that provide the data by which the relationships among the variables are evaluated (subjects, cases, etc.) Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Conditions – that which pertains to all the data collected and sets the study apart from other similar studies (time, space, treatments, scoring variations, etc.). Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Conditions – that which pertains to all the data collected and sets the study apart from other similar studies (time, space, treatments, scoring variations, etc.). • Observations – a specific variable score of a specific individual under the designated conditions. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Factors • Hypothetical constructs or theories that help interpret the consistency in a data set (Tinsley & Tinsley, 1987). • A dimension or construct that is a condensed statement of the relationship between a set of variables (Kline, 1994). • Hypothesized, unmeasured, and underlying variables (Kim & Meuller, 1978). Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Factors – specific variables that are presumed to influence or explain phenomenon (i.e., test performance); reflect underlying processes or constructs that have created the correlations among variables. • Sometimes referred to as “latent variables.” Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Common Factors • Represent the dimensions that all the measures have in common. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Common Factors • Represent the dimensions that all the measures have in common. • Specific Factors • Are related to a specific variables but are not common to any other variables. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Common Factors • Represent the dimensions that all the variables have in common. • Specific Factors • Are related to a specific variable but are not common to any other variables. • Error Factors • Represent the error of measurement or unreliability of a variable. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Factor Loading – the farther the loading on a factor from zero, the more one can generalize from that factor to the variable; reflects a quantitative relationship. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Factor Loading – the farther the loading on a factor from zero, the more on can generalize from that factor to the variable; reflects a quantitative relationship. • The extent to which the variables are related to the hypothetical factor. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Factor Loading – the farther the loading on a factor from zero, the more on can generalize from that factor to the variable; reflects a quantitative relationship. • The extent to which the variables are related to the hypothetical factor. • May be thought of as correlations between the variables and the factor. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Factor Loading – the farther the loading on a factor from zero, the more on can generalize from that factor to the variable; reflects a quantitative relationship. • The extent to which the variables are related to the hypothetical factor. • May be thought of as correlations between the variables and the factor. • Sometimes referred to as “saturation.” Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Observed Correlation Matrix – matrix of observed variables (i.e., standard test score). Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Observed Correlation Matrix – matrix of observed variables (i.e., standard test score). • Reproduced Correlation Matrix – matrix produced by the factor model. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Observed Correlation Matrix – matrix of observed variables (i.e., standard test score). • Reproduced Correlation Matrix – matrix produced by the factors. • Residual Correlation Matrix – matrix produced by the differences between observed and model matrices. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Rotation – is a process by which the solution is made more interpretable without changing its underlying mathematical properties. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Rotation – is a process by which the solution is made more interpretable without changing its underlying mathematical properties. • Orthogonal rotation – all factors are uncorrelated with each other. • Produces loading & factor-score matrices. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Required Parlance / Lexicon**• Rotation – is a process by which the solution is made more interpretable without changing its underlying mathematical properties. • Orthogonal rotation – all factors are uncorrelated with each other. • Produces loading & factor-score matrices. • Oblique rotation – factors are correlated. • Produces structure, pattern, & factor-score matrices. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Uses of Factor Analysis**• Finding underlying factors of ability tests. • Identify personality dimensions. • Identifying clinical syndromes. • Finding dimensions of satisfaction. • Finding dimensions of social behaviors. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Uses of Factor Analysis**• In psychology - the development of objective tests and assessments for the measurement of personality and intelligence. • Explain inter-correlations. • Test theory about factor constructs. • Determine effect of variation / changes. • Verify previous findings. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Types of Factor Analysis**• Exploratory (EFA) – the researcher attempts to describe and summarize data by grouping together variables that are correlated. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Types of Factor Analysis**• Exploratory (EFA) – the researcher attempts to describe and summarize data by grouping together variables that are correlated. • The variables may or may not have been chosen with potential underlying processes in mind. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Types of Factor Analysis**• Exploratory (EFA) – the researcher attempts to describe and summarize data by grouping together variables that are correlated. • The variables may or may not have been chosen with potential underlying processes in mind. • Used in the early stages of research to consolidate variables and generate hypotheses about possible underlying processes or constructs. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Types of Factor Analysis**• Confirmatory (CFA) – used later in research (advanced stages) to test a theory regarding latent underlying processes / constructs. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Types of Factor Analysis**• Confirmatory (CFA) – used later in research (advanced stages) to test a theory regarding latent underlying processes / constructs. • Variables are specifically chosen to reveal underlying processes / constructs. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Types of Factor Analysis**• Confirmatory (CFA) – used later in research (advanced stages) to test a theory regarding latent underlying processes / constructs. • Variables are specifically chosen to reveal or confirm underlying processes / constructs. • Much more sophisticated technique than EFA. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**The Fundamental Equation of Factor Analysis**• The first step… zjk = aj1F1k + aj2F2k + … + ajmFmk + ajsSjk + jeEjk Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**The Fundamental Equation of Factor Analysis**• Given the limited time available…let’s don’t and say we did. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Factor Analytic Steps and Procedures**• 1- Select and measure a set of variables. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Factor Analytic Steps and Procedures**• 1- Select and measure a set of variables. • 2- Compute the matrix of correlations among the variables. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Factor Analytic Steps and Procedures**• 1- Select and measure a set of variables. • 2- Compute the matrix of correlations among the variables. • 3- Extract a set of unrotated factors. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Factor Analytic Steps and Procedures**• 1- Select and measure a set of variables. • 2- Compute the matrix of correlations among the variables. • 3- Extract a set of unrotated factors. • 4- Determine the number of factors. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.**Factor Analytic Steps and Procedures**• 1- Select and measure a set of variables. • 2- Compute the matrix of correlations among the variables. • 3- Extract a set of unrotated factors. • 4- Determine the number of factors. • 5- Rotate the factors if needed to increase interpretability. Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.