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Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
1
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.
2
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.
3
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.
4
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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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Goals and Objectives
To summarize patterns of correlations (in matrix) among observed variables.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
14
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.
15
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.
16
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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
28
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.
29
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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
40
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.
41
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.
42
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.
43
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.
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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.
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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.
46
Factor Analytic Steps and Procedures
1- Select and measure a set of variables.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
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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.
48
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.
49
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.
50
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.
51
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. 6- Interpret the rotated factor matrix.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
52
Psychological Diagnostic Interview
A case example of a formal psychological / psychiatric intake session will be utilized to display the aforementioned factor analytic steps.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
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Step One
Select and measure a set of variables.
The clinician’s observation of the patient’s mood, affect, behavior, cognitions and their description of the presenting problem or chief complaint during the intake session.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
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Step Two
Compute the matrix of correlations among the variables.
The clinician attempts to understand how the themes that were observed “fit” together; what observations were similar to one another (i.e., physiological or motor disturbances, form or content of thought, perception).
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
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Step Three
Extract a set of unrotated factors.
The clinician makes decisions based on what everything observed had in common. The clinician begins to assess for processes and underlying dimensions (i.e., anxiety & depression).
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
56
Step Four & Five
Determine the number of factors.
Rotate the factors if needed to increase interpretability.
These steps involve the clinician making a decision based on the relative weights, predominance, or importance of each of the aforesaid dimensions.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
57
Step Six
Interpret the rotated factor matrix.
The clinician establishes a formulation or theoretical conceptualization, based on themes of observations, and develops a tentative treatment plan.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
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Standard FA Example Mathematics & Verbal Ability
Suppose you want to study mathematics and verbal ability.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
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Standard FA Example Mathematics & Verbal Ability
Suppose you want to study mathematics and verbal ability.– Research literature to develop test plan.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
60
Standard FA Example Mathematics & Verbal Ability
Suppose you want to study mathematics and verbal ability.– Research literature to develop test plan.– Five items that best measure these abilities.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
61
Standard FA Example Mathematics & Verbal Ability
Suppose you want to study mathematics and verbal ability.– Research literature to develop test plan.– Five items that best measure these abilities.
• Vocabulary, Algebra, Word Analogy, Geometry, Algebra Word Problem.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
62
Standard FA Example Mathematics & Verbal Ability
Suppose you want to study mathematics and verbal ability.– Research literature to develop test plan.– Five items that best measure these abilities.
• Vocabulary, Algebra, Word Analogy, Geometry, Algebra Word Problem.
• Actual research = a vast and nebulous number of items.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
63
Standard FA ExampleMathematics & Verbal Ability
Suppose you want to study mathematics and verbal ability.– Research literature to develop test plan.– Five items that best measure these abilities.
• Vocabulary, Algebra, Word Analogy, Geometry, Algebra Word Problem.
• Actual research = a vast and nebulous number of items.
– Compute matrix of intercorrelations (SPSS).
Cohen, Swerdlik, & Phillips (1996)
64
Mathematics & Verbal AbilityTable 1 The Matrix of Intercorrelations Among the Five Items
1 2 3 4 5
1. Vocab 1.00 .22 .77 .20 .50
2. Algebra .22 .1.00 .21 .65 .48
3. Analogy .77 .21 1.00 .19 .52
4. Geometry .20 .65 .19 1.00 .47
5. Alg-Word .50 .48 .52 .47 1.00
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
65
Matrix of Intercorrelations Mathematics & Verbal Ability
Each entry = correlation coefficient between 2 items.
1 2 3 4 5
1.Vocab 1.0 .22 .77 .20 .50
2.Albegra .22 .1.0 .21 .65 .48
3.Word Analogy
.77 .21 1.0 .19 .52
4.Geometry .20 .65 .19 1.0 .47
5.Algebra-Word
.50 .48 .52 .47 1.0
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
66
Matrix of Intercorrelations Mathematics & Verbal Ability
Each entry = correlation coefficient between 2 items.– Vocab & word analogy
= .77
1 2 3 4 5
1.Vocab 1.0 .22 .77 .20 .50
2.Albegra .22 .1.0 .21 .65 .48
3.Word Analogy
.77 .21 1.0 .19 .52
4.Geometry .20 .65 .19 1.0 .47
5.Algebra-Word
.50 .48 .52 .47 1.0
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
67
Matrix of Intercorrelations Mathematics & Verbal Ability
Each entry = correlation coefficient between 2 items.– Vocab & word analogy
= .77
– Algebra & geometry = .65
1 2 3 4 5
1.Vocab 1.0 .22 .77 .20 .50
2.Albegra .22 .1.0 .21 .65 .48
3.Word Analogy
.77 .21 1.0 .19 .52
4.Geometry .20 .65 .19 1.0 .47
5.Algebra-Word
.50 .48 .52 .47 1.0
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
68
Matrix of Intercorrelations Mathematics & Verbal Ability
Each entry = correlation coefficient between 2 items.– Vocab & word analogy
= .77
– Algebra & geometry = .65
Results suggest…
1 2 3 4 5
1.Vocab 1.0 .22 .77 .20 .50
2.Albegra .22 .1.0 .21 .65 .48
3.Word Analogy
.77 .21 1.0 .19 .52
4.Geometry .20 .65 .19 1.0 .47
5.Algebra-Word
.50 .48 .52 .47 1.0
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
69
Matrix of Intercorrelations Mathematics & Verbal Ability
Each entry = correlation coefficient between 2 items.– Vocab & word analogy
= .77.
– Algebra & geometry = .65.
Results suggest…– 2 underlying factors.
1 2 3 4 5
1.Vocab 1.0 .22 .77 .20 .50
2.Albegra .22 .1.0 .21 .65 .48
3.Word Analogy
.77 .21 1.0 .19 .52
4.Geometry .20 .65 .19 1.0 .47
5.Algebra-Word
.50 .48 .52 .47 1.0
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
70
Matrix of Intercorrelations Mathematics & Verbal Ability
Each entry = correlation coefficient between 2 items.– Vocab & word analogy
= .77.
– Algebra & geometry = .65.
Results suggest…– 2 underlying factors.
– Algebra word may be associate with both.
1 2 3 4 5
1.Vocab 1.0 .22 .77 .20 .50
2.Albegra .22 .1.0 .21 .65 .48
3.Word Analogy
.77 .21 1.0 .19 .52
4.Geometry .20 .65 .19 1.0 .47
5.Algebra-Word
.50 .48 .52 .47 1.0
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
71
Standard FA Example (Cont.)
Mathematics & Verbal Ability
The next step is to factor the intercorrelations matrix (SPSS).
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
72
Standard FA Example (Cont.)
Mathematics & Verbal Ability
The next step is to factor the intercorrelations matrix (SPSS).– Factor loadings can be treated like correlations
between the measure and the underlying factors.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
73
Standard FA Example (Cont.)
Mathematics & Verbal Ability
The next step is to factor the intercorrelations matrix (SPSS).– Factor loadings can be treated like correlations
between the measure and the underlying factors.
– Determine magnitude of factor loading.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
74
Standard FA Example (Cont.)
Mathematics & Verbal Ability
The next step is to factor the intercorrelations matrix (SPSS).– Factor loadings can be treated like correlations
between the measure and the underlying factors.
– Determine magnitude of factor loading.• Look for salient (significant) factor loadings.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
75
Standard FA Example (Cont.)
Mathematics & Verbal Ability
The next step is to factor the intercorrelations matrix (SPSS).– Factor loadings can be treated like correlations
between the measure and the underlying factors.
– Determine magnitude of factor loading.• Look for salient (significant) factor loadings.
– Cattell (1978) proposed .30 for N > 100, and .40 for N < 100.
Cohen, Swerdlik, & Phillips (1996)
76
Mathematics & Verbal AbilityTable 2 Results of Factor Analysis of Five Items
Salient Factor Loadings, N > 100
Factor I Factor II Communality
Vocabulary .917 .101 .851
Algebra .113 .885 .796
Analogy .925 .094 .864
Geometry .086 .891 .801
Algebra-Word .594 .573 .681
Eigenvalue 2.700 1.30
% total Variance
54.00 26.00
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
77
Mathematics & Verbal Ability Vocabulary, Analogy, Algebra
& Geometry are Factorially Simple because they load on only one factor; reflect one dimension.
Factor I Factor II Communality
Vocabulary .917 .101 .851
Algebra .113 .885 .796
Analogy .925 .094 .864
Geometry .086 .891 .801
Algebra-Word
.594 .573 .681
Eigenvalue 2.700 1.30
% total Variance
54.00 26.00
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
78
Mathematics & Verbal Ability Vocabulary, Analogy, Algebra
& Geometry are Factorially Simple because they load on only one factor; reflect one dimension.
Algebra-Word is considered Factorially Complex because it loads on both factors; reflects more than one dimension.
Factor I Factor II Communality
Vocabulary .917 .101 .851
Algebra .113 .885 .796
Analogy .925 .094 .864
Geometry .086 .891 .801
Algebra-Word
.594 .573 .681
Eigenvalue 2.700 1.30
% total Variance
54.00 26.00
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
79
Mathematics & Verbal Ability Two factors named.Verbal
AbilityMathematical
AbilityCommunality
Vocabulary .917 .101 .851
Algebra .113 .885 .796
Analogy .925 .094 .864
Geometry .086 .891 .801
Algebra-Word
.594 .573 .681
Eigenvalue 2.700 1.30
% total Variance
54.00 26.00
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
80
Mathematics & Verbal Ability Two factors named.
Eigenvalue (or characteristic root) indicates the relative strength of each factor.– Range from 0.0 to # of
measures factored.
Verbal Ability
Mathematical Ability
Communality
Vocabulary .917 .101 .851
Algebra .113 .885 .796
Analogy .925 .094 .864
Geometry .086 .891 .801
Algebra-Word
.594 .573 .681
Eigenvalue 2.700 1.30
% total Variance
54.00 26.00
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
81
Mathematics & Verbal Ability Two factors named.
Eigenvalue (or characteristic root) indicates the relative strength of each factor.– Range from 0.0 to # of
measures factored.
% of Total Variance reflects that the Verbal Ability factor (54%) is twice as strong as the Mathematical Ability factor (26%).
Verbal Ability
Mathematical Ability
Communality
Vocabulary .917 .101 .851
Algebra .113 .885 .796
Analogy .925 .094 .864
Geometry .086 .891 .801
Algebra-Word
.594 .573 .681
Eigenvalue 2.700 1.30
% total Variance
54.00 26.00
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
82
Mathematics & Verbal Ability Communality…Verbal
AbilityMathematical
AbilityCommunality
Vocabulary .917 .101 .851
Algebra .113 .885 .796
Analogy .925 .094 .864
Geometry .086 .891 .801
Algebra-Word
.594 .573 .681
Eigenvalue 2.700 1.30
% total Variance
54.00 26.00
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
83
Mathematics & Verbal Ability Communality…
– assesses how well each measure is explained by the common factors.
Verbal Ability
Mathematical Ability
Communality
Vocabulary .917 .101 .851
Algebra .113 .885 .796
Analogy .925 .094 .864
Geometry .086 .891 .801
Algebra-Word
.594 .573 .681
Eigenvalue 2.700 1.30
% total Variance
54.00 26.00
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
84
Mathematics & Verbal Ability Communality…
– assesses how well each measure is explained by the common factors.
– indicates the extent to which variables overlap with factors.
Verbal Ability
Mathematical Ability
Communality
Vocabulary .917 .101 .851
Algebra .113 .885 .796
Analogy .925 .094 .864
Geometry .086 .891 .801
Algebra-Word
.594 .573 .681
Eigenvalue 2.700 1.30
% total Variance
54.00 26.00
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
85
Mathematics & Verbal Ability Communality…
– assesses how well each measure is explained by the common factors.
– indicates the extent to which variables overlap with factors.
– provides proportion of variance in the variables that can be accounted for by the scores in the factors.
Verbal Ability
Mathematical Ability
Communality
Vocabulary .917 .101 .851
Algebra .113 .885 .796
Analogy .925 .094 .864
Geometry .086 .891 .801
Algebra-Word
.594 .573 .681
Eigenvalue 2.700 1.30
% total Variance
54.00 26.00
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
86
Mathematics & Verbal Ability
The results of this factor analysis permits the estimation of not only how many factors or dimensions there were with our five items but also the relative importance or strength of each of the factors.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
87
Problems with Factor Analytic Methods
No criterion variable against which to test the solution.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
88
Problems with Factor Analytic Methods
No criterion variable against which to test the solution.
There is an infinite number of rotations available.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
89
Problems with Factor Analytic Methods
No criterion variable against which to test the solution.
There is an infinite number of rotations available.
Often used to correct improperly conceptualized (sloppy) research.
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
90
Problems with Factor Analytic Methods
[Factor analysis should never be used] as a haphazard method to attempt to make order from chaos; it is totally inappropriate to factor-analyze just any set of measure with the hope of finding meaningful common factors (Cohen, Swerdlik, & Phillips, p. 207, 1996).
Timothy D. Kruse, M.S.Ed. Texas A&M University Commerce.
91
Thank You!
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