“A User-Friendly Demonstration of Principal Components Analysis as a Data Reduction Method”

““A User-Friendly A User-Friendly Demonstration of Principal Demonstration of Principal Components Analysis as a Components Analysis as a Data Reduction Method”Data Reduction Method”

R. Michael Haynes, PhDR. Michael Haynes, PhD Keith Lamb, MBAKeith Lamb, MBAAssistant Vice PresidentAssistant Vice President Associate Vice PresidentAssociate Vice PresidentStudent Life StudiesStudent Life Studies Student AffairsStudent AffairsTarleton State UniversityTarleton State University Midwestern State UniversityMidwestern State University

What is Principal Components What is Principal Components Analysis (PCA)? Analysis (PCA)?

A member of the general linear model (GLM) where A member of the general linear model (GLM) where all all analyses are correlational analyses are correlational

Term often used interchangeably with “factor analysis”, Term often used interchangeably with “factor analysis”, however, there are slight differences however, there are slight differences

A method of reducing large data sets into more A method of reducing large data sets into more manageable “factors” or “components” manageable “factors” or “components”

A method of identifying the most A method of identifying the most usefuluseful variables in a variables in a datasetdataset

A method of identifying and classifying variables across A method of identifying and classifying variables across common themes, or constructs that they represent common themes, or constructs that they represent

Before we get started, aBefore we get started, a GLOSSARYGLOSSARY of of terms we’ll be using today:terms we’ll be using today:

Bartletts’s Test of SphericityBartletts’s Test of Sphericity Communality coefficientsCommunality coefficients ConstructConstruct Correlation matrixCorrelation matrix Cronbach’s alpha coefficientCronbach’s alpha coefficient Effect sizes (variance accounted for)Effect sizes (variance accounted for) EigenvaluesEigenvalues ExtractionExtraction Factor or componentFactor or component Kaiser criterion for retaining factorsKaiser criterion for retaining factors Kaiser-Meyer-Olkin Measure of Sampling AdequacyKaiser-Meyer-Olkin Measure of Sampling Adequacy LatentLatent ReliabilityReliability RotationRotation Scree plot Scree plot Split-half reliability Split-half reliability Structure coefficientsStructure coefficients

Desired outcomes fromDesired outcomes fromtoday’s sessiontoday’s session

Understand the terminology associated with principal Understand the terminology associated with principal components analysis (PCA)components analysis (PCA)

Understand when using PCA is appropriateUnderstand when using PCA is appropriate Understand how to conduct PCA using SPSS 17.0Understand how to conduct PCA using SPSS 17.0 Understand how to interpret a correlation matrixUnderstand how to interpret a correlation matrix Understand how to interpret a communality matrixUnderstand how to interpret a communality matrix Understand how to interpret a components matrix and the Understand how to interpret a components matrix and the

methods used in determining how many components to methods used in determining how many components to retain retain

Understand how to analyze a component to determine which Understand how to analyze a component to determine which variables to include and whyvariables to include and why

Understand the concept of reliability and why it is important Understand the concept of reliability and why it is important in survey researchin survey research

LETS GET STARTED!!LETS GET STARTED!!

When is using PCA appropriate?When is using PCA appropriate?

When your data is interval or ratio levelWhen your data is interval or ratio level When you have at least 5 observations per variable and When you have at least 5 observations per variable and

at least 100 observations (ie…20 variables>100 at least 100 observations (ie…20 variables>100 observations)observations)

When trying to reduce the number of variables to be When trying to reduce the number of variables to be used in another GLM technique (ie….regression, used in another GLM technique (ie….regression, MANOVA, etc...)MANOVA, etc...)

When attempting to identify latent constructs that are When attempting to identify latent constructs that are being measured by observed variables being measured by observed variables in the absence of in the absence of a priori theory.a priori theory.

HUERISTIC DATAHUERISTIC DATA

Responses to the Developing Purpose Inventory (DPI) Responses to the Developing Purpose Inventory (DPI) collected at a large, metropolitan university between collected at a large, metropolitan university between 2004-2006 (IRB approval received)2004-2006 (IRB approval received)

45 questions related to Chickering’s developing purpose 45 questions related to Chickering’s developing purpose stagestage

Responses on 5 interval scale; 1=”always true” to Responses on 5 interval scale; 1=”always true” to 5=”never true”5=”never true”

Sample size = 998 participantsSample size = 998 participants SUGGESTION: always visually inspect data for missing SUGGESTION: always visually inspect data for missing

cases and potential outliers! (APA Task Force on cases and potential outliers! (APA Task Force on Statistical Inference, 1999).Statistical Inference, 1999).

Multiple ways of dealing with missing data, but that’s for Multiple ways of dealing with missing data, but that’s for another day! another day!

SPSS 17.0SPSS 17.0

Make sure your set-up in “Variable View” is complete to Make sure your set-up in “Variable View” is complete to accommodate your dataaccommodate your data

Names, labels, possible values of the data, and type of measureNames, labels, possible values of the data, and type of measure

Analyze>Dimension Reduction>Factor Analyze>Dimension Reduction>Factor

SPSS 17.0SPSS 17.0

SPSS 17.0 SYNTAXSPSS 17.0 SYNTAX OrangeOrange indicates sections specific to your analysis! indicates sections specific to your analysis!

DATASET ACTIVATE DataSet1.DATASET ACTIVATE DataSet1.FACTORFACTOR /VARIABLES /VARIABLES question1 question2 question3 question4 question5 question6 question7 question8 question1 question2 question3 question4 question5 question6 question7 question8 question9 question10 question11 question12 question13 question14 question15 question16 question9 question10 question11 question12 question13 question14 question15 question16 question17 question18 question19 question20 question21 question22 question23 question24 question17 question18 question19 question20 question21 question22 question23 question24 question25 question26 question27 question28 question29 question30 question31 question32 question25 question26 question27 question28 question29 question30 question31 question32 question33 question34 question35 question36 question37 question38 question39 question40 question33 question34 question35 question36 question37 question38 question39 question40 question41 question42 question43 question44 question45question41 question42 question43 question44 question45 /MISSING LISTWISE /MISSING LISTWISE /ANALYSIS /ANALYSIS question1 question2 question3 question4 question5 question6 question7 question8 question1 question2 question3 question4 question5 question6 question7 question8 question9 question10 question11 question12 question13 question14 question15 question16 question9 question10 question11 question12 question13 question14 question15 question16 question17 question18 question19 question20 question21 question22 question23 question24 question17 question18 question19 question20 question21 question22 question23 question24 question25 question26 question27 question28 question29 question30 question31 question32 question25 question26 question27 question28 question29 question30 question31 question32 question33 question34 question35 question36 question37 question38 question39 question40 question33 question34 question35 question36 question37 question38 question39 question40 question41 question42 question43 question44 question45question41 question42 question43 question44 question45 /PRINT INITIAL CORRELATION SIG KMO EXTRACTION ROTATION FSCORE/PRINT INITIAL CORRELATION SIG KMO EXTRACTION ROTATION FSCORE /FORMAT SORT BLANK(/FORMAT SORT BLANK(.000.000)) /PLOT EIGEN/PLOT EIGEN /CRITERIA MINEIGEN(1) ITERATE(25)/CRITERIA MINEIGEN(1) ITERATE(25) /EXTRACTION PC/EXTRACTION PC /CRITERIA ITERATE(25)/CRITERIA ITERATE(25) /ROTATION VARIMAX/ROTATION VARIMAX /SAVE AR(ALL)/SAVE AR(ALL) /METHOD=CORRELATION./METHOD=CORRELATION.

OUTPUT COMPONENTSOUTPUT COMPONENTS

Correlation MatrixCorrelation Matrix Pearson R between the individual variablesPearson R between the individual variables Variables range from -1.0 to +1.0; strong, modest, weak; positive, Variables range from -1.0 to +1.0; strong, modest, weak; positive,

negativenegative Correlations of 1.00 on the diagonal; every variable is “perfectly and Correlations of 1.00 on the diagonal; every variable is “perfectly and

positively” correlated with itself!positively” correlated with itself! It is this information that is the basis for PCA! In other words, if you It is this information that is the basis for PCA! In other words, if you

have only a correlation matrix, you can conduct PCAhave only a correlation matrix, you can conduct PCA!!

Question 1 - ARI Question 2 - VI Question 3 - SL Question 4 - ARI Question 5 - VI

Question 1 - ARI 1.000 .157 .077 .165 .069

Question 2 - VI .157 1.000 .261 .109 .211

Question 3 - SL .077 .261 1.000 .157 .017

Question 4 - ARI .165 .109 .157 1.000 .098

Question 5 - VI .069 .211 .017 .098 1.000

KMO Measure of Sampling Adequacy and Bartlett’s Test of SphericityKMO Measure of Sampling Adequacy and Bartlett’s Test of Sphericity KMO values closer to 1.0 are betterKMO values closer to 1.0 are better

Kaiser (1970 & 1975; as cited by Meyers, Gamst, & Guarino, 2006) states that a value of .70 is Kaiser (1970 & 1975; as cited by Meyers, Gamst, & Guarino, 2006) states that a value of .70 is considered adequate.considered adequate.

Bartlett’s Test: you want a statistically significant valueBartlett’s Test: you want a statistically significant value Reject the null hypothesis of a lack of sufficient correlation between the variables.Reject the null hypothesis of a lack of sufficient correlation between the variables.

Kaiser-Meyer-Olkin Measure of Sampling Adequacy. .861

Bartlett's Test of Sphericity

Approx. Chi-Square 9193.879

df 990

Sig. .000


Communality CoefficientsCommunality Coefficients amount of variance in the amount of variance in the

variable accounted for variable accounted for

by the componentsby the components higher coefficientshigher coefficients

=stronger variables =stronger variables lower coefficientslower coefficients

=weaker variables=weaker variables

Initial Extraction

Question 1 - ARI 1.000 .560

Question 2 - VI 1.000 .446

Question 3 - SL 1.000 .773


Question 5 - VI 1.000 .539

Question 6 - SL 1.000 .439


Question 8 - VI 1.000 .527

Question 9 - SL 1.000 .537

Question 10 - ARI 1.000 .775

Question 11 - VI 1.000 .635

Question 12 - SL 1.000 .476

Question 13 - ARI 1.000 .542

Question 14 - VI 1.000 .435

Question 15 - SL 1.000 .426


Total Variance Explained TableTotal Variance Explained Table Lists the individual components (remember, you have as Lists the individual components (remember, you have as

many components as you have variables) by eigenvalue and many components as you have variables) by eigenvalue and variance accounted for variance accounted for

How do we determine how many components to retain?How do we determine how many components to retain?

Component

Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings

Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %

1 7.216 16.035 16.035 7.216 16.035 16.035 3.666 8.147 8.147

2 3.107 6.904 22.938 3.107 6.904 22.938 2.649 5.887 14.034

3 2.455 5.456 28.395 2.455 5.456 28.395 2.597 5.771 19.806

4 1.846 4.103 32.498 1.846 4.103 32.498 2.555 5.677 25.482

5 1.690 3.755 36.253 1.690 3.755 36.253 2.243 4.984 30.466

6 1.458 3.239 39.493 1.458 3.239 39.493 2.189 4.865 35.331

7 1.307 2.906 42.398 1.307 2.906 42.398 1.746 3.880 39.212

8 1.180 2.623 45.021 1.180 2.623 45.021 1.578 3.507 42.719

9 1.107 2.461 47.482 1.107 2.461 47.482 1.555 3.455 46.174

10 1.064 2.364 49.846 1.064 2.364 49.846 1.314 2.919 49.093

11 1.024 2.275 52.121 1.024 2.275 52.121 1.221 2.712 51.805

12 1.014 2.253 54.374 1.014 2.253 54.374 1.156 2.569 54.374

13 .976 2.170 56.544


Component



1 7.216 16.035 16.035 7.216 16.035 16.035 3.666 8.147 8.147

2 3.107 6.904 22.938 3.107 6.904 22.938 2.649 5.887 14.034

3 2.455 5.456 28.395 2.455 5.456 28.395 2.597 5.771 19.806

4 1.846 4.103 32.498 1.846 4.103 32.498 2.555 5.677 25.482

5 1.690 3.755 36.253 1.690 3.755 36.253 2.243 4.984 30.466

6 1.458 3.239 39.493 1.458 3.239 39.493 2.189 4.865 35.331

7 1.307 2.906 42.398 1.307 2.906 42.398 1.746 3.880 39.212

8 1.180 2.623 45.021 1.180 2.623 45.021 1.578 3.507 42.719

9 1.107 2.461 47.482 1.107 2.461 47.482 1.555 3.455 46.174

10 1.064 2.364 49.846 1.064 2.364 49.846 1.314 2.919 49.093

11 1.024 2.275 52.121 1.024 2.275 52.121 1.221 2.712 51.805

12 1.014 2.253 54.374 1.014 2.253 54.374 1.156 2.569 54.374

13 .976 2.170 56.544


Total Variance Explained TableTotal Variance Explained Table Kaiser Criterion (K1 Rule): retain only those components with Kaiser Criterion (K1 Rule): retain only those components with

an eigenvalue of greater than 1; can lead to retaining more an eigenvalue of greater than 1; can lead to retaining more components than necessarycomponents than necessary


Component



1 7.216 16.035 16.035 7.216 16.035 16.035 3.666 8.147 8.147

2 3.107 6.904 22.938 3.107 6.904 22.938 2.649 5.887 14.034

3 2.455 5.456 28.395 2.455 5.456 28.395 2.597 5.771 19.806

4 1.846 4.103 32.498 1.846 4.103 32.498 2.555 5.677 25.482

5 1.690 3.755 36.253 1.690 3.755 36.253 2.243 4.984 30.466

6 1.458 3.239 39.493 1.458 3.239 39.493 2.189 4.865 35.331

7 1.307 2.906 42.398 1.307 2.906 42.398 1.746 3.880 39.212

8 1.180 2.623 45.021 1.180 2.623 45.021 1.578 3.507 42.719

9 1.107 2.461 47.482 1.107 2.461 47.482 1.555 3.455 46.174

10 1.064 2.364 49.846 1.064 2.364 49.846 1.314 2.919 49.093

11 1.024 2.275 52.121 1.024 2.275 52.121 1.221 2.712 51.805

12 1.014 2.253 54.374 1.014 2.253 54.374 1.156 2.569 54.374

13 .976 2.170 56.544

Total Variance Explained TableTotal Variance Explained Table Retain as many factors as will account for a pre-determined Retain as many factors as will account for a pre-determined

amount of variance, say 70%; can lead to retention of amount of variance, say 70%; can lead to retention of components that are variable specific (Stevens, 2002)components that are variable specific (Stevens, 2002)

Scree PlotScree Plot

Plots eigenvalues on Plots eigenvalues on Y Y axis and component number on axis and component number on X X axisaxis

Recommendation is to Recommendation is to

retain all components retain all components

in the descent before in the descent before

the first one on the line the first one on the line

where it levels off where it levels off

(Cattell, 1966; as cited (Cattell, 1966; as cited

by Stevens, 2002).by Stevens, 2002).


Other Retention MethodsOther Retention Methods

Velicer’s Minimum Average Partial (MAP) testVelicer’s Minimum Average Partial (MAP) test

Seeks to determine what components are commonSeeks to determine what components are common Does not seek “cut-off” point, but rather to find a more Does not seek “cut-off” point, but rather to find a more

“comprehensive” solution“comprehensive” solution Components that have high number of highly correlated Components that have high number of highly correlated

variables are retainedvariables are retained However, variable based decisions can result in However, variable based decisions can result in

underestimating the number of components to retainunderestimating the number of components to retain(Ledesma & Valero-Mora, 2007)(Ledesma & Valero-Mora, 2007)

Other Retention MethodsOther Retention Methods

Horn’s Parallel Analysis (PA)Horn’s Parallel Analysis (PA)

Compares observed eigenvalues with “simulated” Compares observed eigenvalues with “simulated” eigenvalueseigenvalues

Retain all components with an eigenvalue greater than the Retain all components with an eigenvalue greater than the “mean” of the simulated eigenvalues“mean” of the simulated eigenvalues

Considered highly accurate and exempt from extraneous Considered highly accurate and exempt from extraneous factorsfactors

(Ledesma & Valero-Mora, 2007)(Ledesma & Valero-Mora, 2007)


Component MatrixComponent Matrix

Column values are structure coefficients, or the Column values are structure coefficients, or the correlation between the test question and the synthetic correlation between the test question and the synthetic component; REMEMBER: squared structure component; REMEMBER: squared structure coefficients inform us of how well the item can coefficients inform us of how well the item can reproduce the effect in the component!reproduce the effect in the component!

Rotated Component Matrixa

Component

1 2 3 4 5 6 7 8 9 10 11 12

Question 42 - SL .781 -.060 .000 .117 .034 .071 .055 -.062 .093 -.002 .032 .025

Question 39 - SL .778 -.132 .107 .109 .008 .024 -.025 .018 .044 -.010 .022 -.025

Question 33 - SL .765 -.042 .115 .098 .034 .090 -.035 -.035 .011 .013 -.012 .020

Question 9 - SL .672 -.103 .127 .092 .050 .126 .005 -.119 -.002 -.063 -.034 -.114

Question 37 - ARI .462 -.173 .193 -.103 .075 .197 .345 -.018 .024 .232 .009 .119

Question 15 - SL .406 -.002 .340 .038 .050 .091 .120 -.007 .067 -.152 -.127 -.273

Question 36 - SL .395 -.067 .212 -.104 .225 .125 .365 -.089 .110 .168 -.037 .221

Question 44 - VI .375 -.033 .360 .128 .175 .091 .221 -.023 .177 -.035 -.027 -.001

Question 26 - VI -.022 .660 -.113 .009 .021 -.063 -.096 .089 .044 .034 -.060 .174

Question 27 - SL -.158 .652 -.088 .032 .069 -.091 .040 .193 -.032 -.150 -.019 .003

Question 38 - VI -.058 .501 -.109 -.171 .032 -.276 -.051 .078 -.042 .255 -.016 -.097

Question 20 - VI -.240 .489 .016 .076 .036 -.092 -.052 .434 -.102 .071 -.079 .056

Question 32 - VI -.101 .488 -.134 .084 -.074 -.415 -.010 .046 .025 -.057 -.050 .020

Question 45 - SL -.144 .443 -.049 -.097 -.105 -.026 -.097 .078 -.031 .057 .421 -.013

Question 29 - VI -.006 .439 .154 -.114 .007 .231 .238 -.196 .145 -.098 .089 -.138

Question 41 - VI -.019 .421 -.087 -.210 .006 -.107 .333 -.005 .125 .091 .300 -.082

Question 24 - SL .129 -.067 .720 .101 .147 .119 -.003 .011 .005 .011 -.012 .203

Question 21 - SL .125 -.164 .676 -.056 .161 .047 .160 -.044 -.012 .137 -.006 .029

Question 23 - VI .313 -.164 .537 .286 .063 .007 .076 -.094 .119 .049 .123 .031

Question 17 - VI .076 -.050 .459 .187 .040 .136 .314 .048 .120 -.212 .083 -.140

Question 30 - SL .120 .114 .420 .287 -.081 .309 -.109 -.165 .061 .328 -.107 .161

Question 22 - ARI .042 .075 .364 .045 .087 -.081 -.135 -.353 .324 .216 .016 -.188

Question 34 - ARI .187 .042 .067 .791 -.002 .075 -.031 -.019 .012 .063 -.050 -.036

Question 1 - ARI -.002 -.062 .082 .722 .055 -.018 .008 -.014 .039 .132 .015 -.075

Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.

Rotated Component Matrixa , continued

Component

Question 35 - VI .113 .077 .015 .569 .030 .439 -.053 -.140 .067 -.089 .095 .105

Question 40 - ARI .194 -.161 .176 .553 .033 .057 -.041 .016 .186 -.086 .216 .147

Question 10 - ARI .029 .016 .144 .033 .860 .036 .010 -.074 .032 .063 -.010 .006

Question 3 - SL .069 -.015 .197 .050 .848 -.029 .025 -.011 .067 -.026 -.003 .004

Question 12 - SL .297 .069 .072 .000 .488 .137 .282 .024 .033 .091 .082 .158

Question 13 - ARI -.046 .058 -.118 .045 .447 -.102 .321 .069 .128 .368 -.222 -.033

Question 11 - VI .151 -.021 .024 .361 .115 .663 .000 -.006 -.124 -.028 .021 .104

Question 5 - VI .154 -.134 .201 .042 -.057 .652 .020 .028 -.019 .124 .039 -.092

Question 8 - VI -.090 .250 -.017 .010 .000 -.623 -.034 .115 -.105 .141 .120 .088

Question 18 - SL .034 .003 .095 -.055 .092 -.039 .686 -.026 .015 .006 -.024 .036

Question 14 - VI .241 -.157 .289 -.007 .132 .221 .418 .061 -.057 -.006 .122 -.080

Question 28 - ARI -.232 .248 .051 .181 -.128 -.237 .357 -.112 .043 .074 -.144 .240

Question 16 - ARI -.069 .213 -.008 .062 -.006 -.075 .033 .678 -.051 -.101 -.103 .023

Question 19 - ARI .001 .054 -.042 -.241 -.033 -.010 -.112 .630 .147 -.010 .127 .036

Question 43 - ARI .138 -.011 .067 .255 .017 .045 -.091 .086 .756 .024 -.074 .075

Question 31 - ARI .062 .045 .069 -.048 .122 -.040 .186 -.053 .721 .140 -.077 .033

Question 4 - ARI .023 -.057 .119 .100 .132 .007 .034 -.131 .184 .643 .020 -.088

Question 6 - SL -.186 .177 -.039 .065 -.051 -.066 .087 .372 -.059 .390 .230 -.080

Question 7 - ARI .024 -.059 .047 .149 .010 .005 .016 -.017 -.133 .008 .736 .126

Question 2 - VI .234 -.198 .246 .175 .233 .094 .203 .086 .179 -.161 .254 -.162

Question 25 - ARI -.048 .063 .119 .021 .073 -.049 .064 .085 .078 -.123 .108 .767




Rule of thumb, include all items with structure Rule of thumb, include all items with structure coefficients with an absolute value of .300 or greatercoefficients with an absolute value of .300 or greater



Component

1 2 3 4 5 6 7 8 9 10 11 12

Question 42 - SL .781 -.060 .000 .117 .034 .071 .055 -.062 .093 -.002 .032 .025

Question 39 - SL .778 -.132 .107 .109 .008 .024 -.025 .018 .044 -.010 .022 -.025

Question 33 - SL .765 -.042 .115 .098 .034 .090 -.035 -.035 .011 .013 -.012 .020

Question 9 - SL .672 -.103 .127 .092 .050 .126 .005 -.119 -.002 -.063 -.034 -.114

Question 37 - ARI .462 -.173 .193 -.103 .075 .197 .345 -.018 .024 .232 .009 .119

Question 15 - SL .406 -.002 .340 .038 .050 .091 .120 -.007 .067 -.152 -.127 -.273

Question 36 - SL .395 -.067 .212 -.104 .225 .125 .365 -.089 .110 .168 -.037 .221

Question 44 - VI .375 -.033 .360 .128 .175 .091 .221 -.023 .177 -.035 -.027 -.001

Question 26 - VI -.022 .660 -.113 .009 .021 -.063 -.096 .089 .044 .034 -.060 .174

Question 27 - SL -.158 .652 -.088 .032 .069 -.091 .040 .193 -.032 -.150 -.019 .003

Question 38 - VI -.058 .501 -.109 -.171 .032 -.276 -.051 .078 -.042 .255 -.016 -.097

Question 20 - VI -.240 .489 .016 .076 .036 -.092 -.052 .434 -.102 .071 -.079 .056

Question 32 - VI -.101 .488 -.134 .084 -.074 -.415 -.010 .046 .025 -.057 -.050 .020

Question 45 - SL -.144 .443 -.049 -.097 -.105 -.026 -.097 .078 -.031 .057 .421 -.013

Question 29 - VI -.006 .439 .154 -.114 .007 .231 .238 -.196 .145 -.098 .089 -.138

Question 41 - VI -.019 .421 -.087 -.210 .006 -.107 .333 -.005 .125 .091 .300 -.082

Question 24 - SL .129 -.067 .720 .101 .147 .119 -.003 .011 .005 .011 -.012 .203

Question 21 - SL .125 -.164 .676 -.056 .161 .047 .160 -.044 -.012 .137 -.006 .029

Question 23 - VI .313 -.164 .537 .286 .063 .007 .076 -.094 .119 .049 .123 .031

Question 17 - VI .076 -.050 .459 .187 .040 .136 .314 .048 .120 -.212 .083 -.140

Question 30 - SL .120 .114 .420 .287 -.081 .309 -.109 -.165 .061 .328 -.107 .161

Question 22 - ARI .042 .075 .364 .045 .087 -.081 -.135 -.353 .324 .216 .016 -.188

Question 34 - ARI .187 .042 .067 .791 -.002 .075 -.031 -.019 .012 .063 -.050 -.036

Question 1 - ARI -.002 -.062 .082 .722 .055 -.018 .008 -.014 .039 .132 .015 -.075



Component

Question 35 - VI .113 .077 .015 .569 .030 .439 -.053 -.140 .067 -.089 .095 .105

Question 40 - ARI .194 -.161 .176 .553 .033 .057 -.041 .016 .186 -.086 .216 .147

Question 10 - ARI .029 .016 .144 .033 .860 .036 .010 -.074 .032 .063 -.010 .006

Question 3 - SL .069 -.015 .197 .050 .848 -.029 .025 -.011 .067 -.026 -.003 .004

Question 12 - SL .297 .069 .072 .000 .488 .137 .282 .024 .033 .091 .082 .158

Question 13 - ARI -.046 .058 -.118 .045 .447 -.102 .321 .069 .128 .368 -.222 -.033

Question 11 - VI .151 -.021 .024 .361 .115 .663 .000 -.006 -.124 -.028 .021 .104

Question 5 - VI .154 -.134 .201 .042 -.057 .652 .020 .028 -.019 .124 .039 -.092

Question 8 - VI -.090 .250 -.017 .010 .000 -.623 -.034 .115 -.105 .141 .120 .088

Question 18 - SL .034 .003 .095 -.055 .092 -.039 .686 -.026 .015 .006 -.024 .036

Question 14 - VI .241 -.157 .289 -.007 .132 .221 .418 .061 -.057 -.006 .122 -.080

Question 28 - ARI -.232 .248 .051 .181 -.128 -.237 .357 -.112 .043 .074 -.144 .240

Question 16 - ARI -.069 .213 -.008 .062 -.006 -.075 .033 .678 -.051 -.101 -.103 .023

Question 19 - ARI .001 .054 -.042 -.241 -.033 -.010 -.112 .630 .147 -.010 .127 .036

Question 43 - ARI .138 -.011 .067 .255 .017 .045 -.091 .086 .756 .024 -.074 .075

Question 31 - ARI .062 .045 .069 -.048 .122 -.040 .186 -.053 .721 .140 -.077 .033

Question 4 - ARI .023 -.057 .119 .100 .132 .007 .034 -.131 .184 .643 .020 -.088

Question 6 - SL -.186 .177 -.039 .065 -.051 -.066 .087 .372 -.059 .390 .230 -.080

Question 7 - ARI .024 -.059 .047 .149 .010 .005 .016 -.017 -.133 .008 .736 .126

Question 2 - VI .234 -.198 .246 .175 .233 .094 .203 .086 .179 -.161 .254 -.162

Question 25 - ARI -.048 .063 .119 .021 .073 -.049 .064 .085 .078 -.123 .108 .767



For heuristic purposes, we’re retaining the first X For heuristic purposes, we’re retaining the first X components; what variables should we include in the components; what variables should we include in the components?components?


Rule of thumb, include all items with structure Rule of thumb, include all items with structure coefficients with an absolute value of .300 or greatercoefficients with an absolute value of .300 or greater

Stevens’ recommends a better way!Stevens’ recommends a better way!


Critical Values for a Correlation Coefficient at α Critical Values for a Correlation Coefficient at α = .01 for a Two-Tailed Test= .01 for a Two-Tailed Test

nn CVCV nn CVCV nn CVCV

5050 .361.361 180180 .192.192 400400 .129.129

8080 .286.286 200200 .182 600.182 600 .105.105

100100 .256.256 250250 .163 800.163 800 .091.091

140140 .217.217 300300 .149.149 10001000 .081.081

(Stevens, 2002, pp. 394)(Stevens, 2002, pp. 394)

Test the structure coefficient for statistical significance against a Test the structure coefficient for statistical significance against a two-tailed table based on sample size and a critical value (CV); for two-tailed table based on sample size and a critical value (CV); for our sample size of 998, the CV would be |.081| doubled (two-tailed), our sample size of 998, the CV would be |.081| doubled (two-tailed), or |.162|.or |.162|.


Component

1 2 3 4 5 6 7 8 9 10 11 12

Question 42 - SL .781 -.060 .000 .117 .034 .071 .055 -.062 .093 -.002 .032 .025

Question 39 - SL .778 -.132 .107 .109 .008 .024 -.025 .018 .044 -.010 .022 -.025

Question 33 - SL .765 -.042 .115 .098 .034 .090 -.035 -.035 .011 .013 -.012 .020

Question 9 - SL .672 -.103 .127 .092 .050 .126 .005 -.119 -.002 -.063 -.034 -.114

Question 37 - ARI .462 -.173 .193 -.103 .075 .197 .345 -.018 .024 .232 .009 .119

Question 15 - SL .406 -.002 .340 .038 .050 .091 .120 -.007 .067 -.152 -.127 -.273

Question 36 - SL .395 -.067 .212 -.104 .225 .125 .365 -.089 .110 .168 -.037 .221

Question 44 - VI .375 -.033 .360 .128 .175 .091 .221 -.023 .177 -.035 -.027 -.001

Question 26 - VI -.022 .660 -.113 .009 .021 -.063 -.096 .089 .044 .034 -.060 .174

Question 27 - SL -.158 .652 -.088 .032 .069 -.091 .040 .193 -.032 -.150 -.019 .003

Question 38 - VI -.058 .501 -.109 -.171 .032 -.276 -.051 .078 -.042 .255 -.016 -.097

Question 20 - VI -.240 .489 .016 .076 .036 -.092 -.052 .434 -.102 .071 -.079 .056

Question 32 - VI -.101 .488 -.134 .084 -.074 -.415 -.010 .046 .025 -.057 -.050 .020

Question 45 - SL -.144 .443 -.049 -.097 -.105 -.026 -.097 .078 -.031 .057 .421 -.013

Question 29 - VI -.006 .439 .154 -.114 .007 .231 .238 -.196 .145 -.098 .089 -.138

Question 41 - VI -.019 .421 -.087 -.210 .006 -.107 .333 -.005 .125 .091 .300 -.082

Question 24 - SL .129 -.067 .720 .101 .147 .119 -.003 .011 .005 .011 -.012 .203

Question 21 - SL .125 -.164 .676 -.056 .161 .047 .160 -.044 -.012 .137 -.006 .029

Question 23 - VI .313 -.164 .537 .286 .063 .007 .076 -.094 .119 .049 .123 .031

Question 17 - VI .076 -.050 .459 .187 .040 .136 .314 .048 .120 -.212 .083 -.140

Question 30 - SL .120 .114 .420 .287 -.081 .309 -.109 -.165 .061 .328 -.107 .161

Question 22 - ARI .042 .075 .364 .045 .087 -.081 -.135 -.353 .324 .216 .016 -.188

Question 34 - ARI .187 .042 .067 .791 -.002 .075 -.031 -.019 .012 .063 -.050 -.036

Question 1 - ARI -.002 -.062 .082 .722 .055 -.018 .008 -.014 .039 .132 .015 -.075



Component

Question 35 - VI .113 .077 .015 .569 .030 .439 -.053 -.140 .067 -.089 .095 .105

Question 40 - ARI .194 -.161 .176 .553 .033 .057 -.041 .016 .186 -.086 .216 .147

Question 10 - ARI .029 .016 .144 .033 .860 .036 .010 -.074 .032 .063 -.010 .006

Question 3 - SL .069 -.015 .197 .050 .848 -.029 .025 -.011 .067 -.026 -.003 .004

Question 12 - SL .297 .069 .072 .000 .488 .137 .282 .024 .033 .091 .082 .158

Question 13 - ARI -.046 .058 -.118 .045 .447 -.102 .321 .069 .128 .368 -.222 -.033

Question 11 - VI .151 -.021 .024 .361 .115 .663 .000 -.006 -.124 -.028 .021 .104

Question 5 - VI .154 -.134 .201 .042 -.057 .652 .020 .028 -.019 .124 .039 -.092

Question 8 - VI -.090 .250 -.017 .010 .000 -.623 -.034 .115 -.105 .141 .120 .088

Question 18 - SL .034 .003 .095 -.055 .092 -.039 .686 -.026 .015 .006 -.024 .036

Question 14 - VI .241 -.157 .289 -.007 .132 .221 .418 .061 -.057 -.006 .122 -.080

Question 28 - ARI -.232 .248 .051 .181 -.128 -.237 .357 -.112 .043 .074 -.144 .240

Question 16 - ARI -.069 .213 -.008 .062 -.006 -.075 .033 .678 -.051 -.101 -.103 .023

Question 19 - ARI .001 .054 -.042 -.241 -.033 -.010 -.112 .630 .147 -.010 .127 .036

Question 43 - ARI .138 -.011 .067 .255 .017 .045 -.091 .086 .756 .024 -.074 .075

Question 31 - ARI .062 .045 .069 -.048 .122 -.040 .186 -.053 .721 .140 -.077 .033

Question 4 - ARI .023 -.057 .119 .100 .132 .007 .034 -.131 .184 .643 .020 -.088

Question 6 - SL -.186 .177 -.039 .065 -.051 -.066 .087 .372 -.059 .390 .230 -.080

Question 7 - ARI .024 -.059 .047 .149 .010 .005 .016 -.017 -.133 .008 .736 .126

Question 2 - VI .234 -.198 .246 .175 .233 .094 .203 .086 .179 -.161 .254 -.162

Question 25 - ARI -.048 .063 .119 .021 .073 -.049 .064 .085 .078 -.123 .108 .767


Sum the interval values for the responses of all questions Sum the interval values for the responses of all questions included in the retained componentincluded in the retained component

Obtain mean values for the responses of all questions Obtain mean values for the responses of all questions included in the retained component…hint…you’ll get the included in the retained component…hint…you’ll get the same same R, RR, R², ß, and structure coefficients as with the ², ß, and structure coefficients as with the sums!sums!

Use SPSS to obtain factor scores for the componentUse SPSS to obtain factor scores for the component

Choose “Scores” button when setting up your PCAChoose “Scores” button when setting up your PCA

Options include calculating scores based on regression, Bartlett, or Options include calculating scores based on regression, Bartlett, or AndersonRubin methodologies…be sure and check “Save as AndersonRubin methodologies…be sure and check “Save as Variables”Variables”

Factor scores will appear in your data set and can be used as Factor scores will appear in your data set and can be used as variables in other GLM analysesvariables in other GLM analyses

Obtaining Continuous Component Values Obtaining Continuous Component Values for Use in Further Analysisfor Use in Further Analysis

RELIABILITYRELIABILITY

The extent to which scores on a test are consistent The extent to which scores on a test are consistent across multiple administrations of the test; the amount of across multiple administrations of the test; the amount of measurement error in the scores yielded by a test (Gall, measurement error in the scores yielded by a test (Gall, Gall, & Borg, 2003).Gall, & Borg, 2003).

While validity is important in ensuring our tests are really While validity is important in ensuring our tests are really measuring what we intended to measure; “You wouldn’t measuring what we intended to measure; “You wouldn’t administer an English literature test to assess math administer an English literature test to assess math competency, would you?”competency, would you?”

Can be measured several ways using SPSS 17.0Can be measured several ways using SPSS 17.0

A Visual Explanation of Reliability and Validity

Cronbach’s Alpha CoefficientRELIABILITY /VARIABLES=question1 question2 question3 question4 question5 question6 question7 question8 question9 question10 question11 question12 question13 question14 question15 question16 question17 question18 question19 question20 question21 question22 question23 question24 question25 question26 question27 question28 question29 question30 question31 question32 question33 question34 question35 question36 question37 question38 question39 question40 question41 question42 question43 question44 question45 /SCALE('ALL VARIABLES') ALL /MODEL=ALPHA.

Split-Half CoefficientRELIABILITY /VARIABLES=question1 question2 question3 question4 question5 question6 question7 question8 question9 question10 question11 question12 question13 question14 question15 question16 question17 question18 question19 question20 question21 question22 question23 question24 question25 question26 question27 question28 question29 question30 question31 question32 question33 question34 question35 question36 question37 question38 question39 question40 question41 question42 question43 question44 question45 /SCALE('ALL VARIABLES') ALL /MODEL=SPLIT.


Cronbach’s Alpha Coefficient

Reliability Statistics

Cronbach's Alpha N of Items

.749 45


Benchmarks for AlphaBenchmarks for Alpha• .9 & up = very good.9 & up = very good• .8 to .9 = good.8 to .9 = good• .7 to .8 = acceptable.7 to .8 = acceptable• .7 & below = suspect..7 & below = suspect.

“… “… don’t refer to the don’t refer to the test as ‘reliable’, but test as ‘reliable’, but scores from this scores from this administration of the administration of the test yielded reliable test yielded reliable results”….Kyle Robertsresults”….Kyle Roberts

Split-Half Coefficient Reliability Statistics

Cronbach's Alpha Part 1 Value .620

N of Items 23a

Part 2 Value .623

N of Items 22b

Total N of Items 45

Correlation Between Forms

.518

Spearman-Brown Coefficient

Equal Length .683

Unequal Length .683

Guttman Split-Half Coefficient

.683

a. The items are: Question 1 - ARI, Question 2 - VI, Question 3 - SL, Question 4 - ARI, Question 5 - VI, Question 6 - SL, Question 7 - ARI, Question 8 - VI, Question 9 - SL, Question 10 - ARI, Question 11 - VI, Question 12 - SL, Question 13 - ARI, Question 14 - VI, Question 15 - SL, Question 16 - ARI, Question 17 - VI, Question 18 - SL, Question 19 - ARI, Question 20 - VI, Question 21 - SL, Question 22 - ARI, Question 23 - VI.

b. The items are: Question 23 - VI, Question 24 - SL, Question 25 - ARI, Question 26 - VI, Question 27 - SL, Question 28 - ARI, Question 29 - VI, Question 30 - SL, Question 31 - ARI, Question 32 - VI, Question 33 - SL, Question 34 - ARI, Question 35 - VI, Question 36 - SL, Question 37 - ARI, Question 38 - VI, Question 39 - SL, Question 40 - ARI, Question 41 - VI, Question 42 - SL, Question 43 - ARI, Question 44 - VI, Questiton 45 - SL.


http://faculty.chass.ncsu.edu/garson/PA765/http://faculty.chass.ncsu.edu/garson/PA765/factor.htmfactor.htm

http://www.uic.edu/classes/epsy/epsy546/Lecturehttp://www.uic.edu/classes/epsy/epsy546/Lecture%204%20---%20notes%20on%20PRINCIPAL%204%20---%20notes%20on%20PRINCIPAL%20COMPONENTS%20ANALYSIS%20AND%20COMPONENTS%20ANALYSIS%20AND%20FACTOR%20ANALYSIS1.pdf%20FACTOR%20ANALYSIS1.pdf

http://www.ats.ucla.edu/stat/Spss/output/factor1.htmhttp://www.ats.ucla.edu/stat/Spss/output/factor1.htm http://www.statsoft.com/textbook/principal-http://www.statsoft.com/textbook/principal-

components-factor-analysis/components-factor-analysis/

RELATED LINKSRELATED LINKS

Gall, M.D., Gall, J.P., & Borg, W.R. (2003). Gall, M.D., Gall, J.P., & Borg, W.R. (2003). Educational research: An introductionEducational research: An introduction 7 7thth ed.). Boson: Allyn and Bacon.ed.). Boson: Allyn and Bacon.

Ledesma, R.D., & Valero-Mora, P. (2007). Determining the number of factors to Ledesma, R.D., & Valero-Mora, P. (2007). Determining the number of factors to retain in EFA: an easy-to-use computer program for carrying out parallel analysis.retain in EFA: an easy-to-use computer program for carrying out parallel analysis. Practical Assessment, Research, & Evaluation,Practical Assessment, Research, & Evaluation, 1212(2).(2).

Meyers, L.S., Gamst, G., & Guarino, A.J. (2006). Meyers, L.S., Gamst, G., & Guarino, A.J. (2006). Applied multivariate research: Applied multivariate research: Design and interpretationDesign and interpretation. Thousand Oaks, CA: Sage.. Thousand Oaks, CA: Sage.

Stevens, J. P. (2002). Stevens, J. P. (2002). Applied multivariate statistics for the social Applied multivariate statistics for the social sciencessciences (4 (4thth ed.). Mahwaw, NJ: Lawrence Erlbaum Associates. ed.). Mahwaw, NJ: Lawrence Erlbaum Associates.

University of California at Los Angeles Academic Technology Services (2009). University of California at Los Angeles Academic Technology Services (2009). Annotated SPSS output: Factor analysis. Retrieved January 11, 2010 from Annotated SPSS output: Factor analysis. Retrieved January 11, 2010 from http://www.ats.ucla.edu/stat/Spss/output/factor1.htmhttp://www.ats.ucla.edu/stat/Spss/output/factor1.htm

University of Illinois at Chicago (2009). Principal components analysis and factor University of Illinois at Chicago (2009). Principal components analysis and factor analysis. Retrieved January 11, 2010 from analysis. Retrieved January 11, 2010 from http://www.uic.edu/classes/epsy/epsy546/Lecture%204%20---%20notes%20onhttp://www.uic.edu/classes/epsy/epsy546/Lecture%204%20---%20notes%20on%20PRINCIPAL%20COMPONENTS%20ANALYSIS%20AND%20FACTOR%20PRINCIPAL%20COMPONENTS%20ANALYSIS%20AND%20FACTOR%20ANALYSIS1.pdf%20ANALYSIS1.pdf

Wilkinson, L. & Task Force on Statistical Inference. (1999). Statistical methods in Wilkinson, L. & Task Force on Statistical Inference. (1999). Statistical methods in psychology journals: Guidelines and psychology journals: Guidelines and explanation. explanation. American Psychologist, 54American Psychologist, 54, , 594-604.594-604.

REFERENCESREFERENCES

“A User-Friendly Demonstration of Principal Components Analysis as a Data Reduction Method”

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