Upload
melvyn-york
View
214
Download
1
Embed Size (px)
Citation preview
““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 components analysis
(PCA) When using PCA is appropriate Conducting PCA using SPSS 17.0 Interpreting a correlation matrix Interpreting a communality matrix Interpreting a components matrix and the methods used in
determining how many components to retain Analyzing a component to determine which variables to include
and why The concept of reliability and why it is important in 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 (i.e.…20 variables>100 at least 100 observations (i.e.…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 (i.e....regression, used in another GLM technique (i.e....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 2004-collected at a large, metropolitan university between 2004-2006 (IRB approval received)2006 (IRB approval received)
45 questions related to Chickering’s developing purpose stage45 questions related to Chickering’s developing purpose stage Responses on 5 interval scale; 1=”always true” to 5=”never Responses on 5 interval scale; 1=”always true” to 5=”never
true”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 Statistical cases and potential outliers! (APA Task Force on Statistical Inference, 1999).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
OUTPUT COMPONENTSOUTPUT COMPONENTS
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 4 - ARI 1.000 .519
Question 5 - VI 1.000 .539
Question 6 - SL 1.000 .439
Question 7 - ARI 1.000 .605
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
OUTPUT COMPONENTSOUTPUT COMPONENTS
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
OUTPUT COMPONENTSOUTPUT COMPONENTS
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
OUTPUT COMPONENTSOUTPUT COMPONENTS
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
OUTPUT COMPONENTSOUTPUT COMPONENTS
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
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).
OUTPUT COMPONENTSOUTPUT COMPONENTS
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)
OUTPUT COMPONENTSOUTPUT COMPONENTS
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
Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.
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!
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
OUTPUT COMPONENTSOUTPUT COMPONENTS
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
Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.
Component MatrixComponent Matrix
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?
Column values are structure coefficients, or the correlation Column values are structure coefficients, or the correlation between the test question and the synthetic component; between the test question and the synthetic component; REMEMBER: squared structure coefficients inform us of REMEMBER: squared structure coefficients inform us of how well the item can reproduce the effect in the how well the item can reproduce the effect in the component!component!
Rule of thumb, include all items with structure coefficients Rule of thumb, include all items with structure coefficients with an absolute value of .300 or greaterwith an absolute value of .300 or greater
Stevens’ recommends a better way!Stevens’ recommends a better way!
OUTPUT COMPONENTSOUTPUT COMPONENTS
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|.
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
Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.
Sum the interval values for the responses of all Sum the interval values for the responses of all questions included in the retained componentquestions included 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 Anderson-Rubin methodologies…be sure and check “Save as Anderson-Rubin 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
RELIABILITYRELIABILITY
RELIABILITYRELIABILITY
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.
RELIABILITYRELIABILITY
Cronbach’s Alpha Coefficient
Reliability Statistics
Cronbach's Alpha N of Items
.749 45
RELIABILITYRELIABILITY
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.
RELIABILITYRELIABILITY
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