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Principal Component Analysis

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					   SW388R7
Data Analysis &
 Computers II     Principal Component Analysis: Additional Topics
    Slide 1




                               Split Sample Validation

                                  Detecting Outliers

                            Reliability of Summated Scales

                                  Sample Problems
   SW388R7
Data Analysis &
 Computers II                    Split Sample Validation
    Slide 2




                     To test the generalizability of findings from a
                      principal component analysis, we could conduct a
                      second research study to see if our findings are
                      verified.
                     A less costly alternative is to split the sample
                      randomly into two halves, do the principal
                      component analysis on each half and compare the
                      results.
                     If the communalities and the factor loadings are the
                      same on the analysis on each half and the full data
                      set, we have evidence that the findings are
                      generalizable and valid because, in effect, the two
                      analyses represent a study and a replication.
   SW388R7
Data Analysis &
 Computers II             Misleading Results to Watch Out For
    Slide 3



                     When we examine the communalities and factor
                      loadings, we are matching up overall patterns, not
                      exact results: the communalities should all be
                      greater than 0.50 and the pattern of the factor
                      loadings should be the same.
                     Sometimes the variables will switch their
                      components (variables loading on the first
                      component now load on the second and vice versa),
                      but this does not invalidate our findings.
                     Sometimes, all of the signs of the factor loadings will
                      reverse themselves (the plus's become minus's and
                      the minus's become plus's), but this does not
                      invalidate our findings because we interpret the size,
                      not the sign of the loadings.
   SW388R7
Data Analysis &
 Computers II                          When validation fails
    Slide 4




                     If the validation fails, we are warned that the
                      solution found in the analysis of the full data set is
                      not generalizable and should not be reported as valid
                      findings.

                     We do have some options when validation fails:
                         If the problem is limited to one or two variables, we can remove
                          those variables and redo the analysis.
                         Randomly selected samples are not always representative. We
                          might try some different random number seeds and see if our
                          negative finding was a fluke. If we choose this option, we should
                          do a large number of validations to establish a clear pattern, at
                          least 5 to 10. Getting one or two validations to negate the failed
                          validation and support our findings is not sufficient.
   SW388R7
Data Analysis &
 Computers II                              Outliers
    Slide 5



                     SPSS calculates factor scores as standard scores.
                     SPSS suggests that one way to identify outliers is to
                      compute the factors scores and identify those have a
                      value greater than ±3.0 as outliers.
                     If we find outliers in our analysis, we redo the
                      analysis, omitting the cases that were outliers.
                     If there is no change in communality or factor
                      structure in the solution, it implies that there
                      outliers do not have an impact. If our factor solution
                      changes, we will have to study the outlier cases to
                      determine whether or not we should exclude them.
                     After testing outliers, restore full data set before
                      any further calculations
   SW388R7
Data Analysis &
 Computers II                Reliability of Summated Scales
    Slide 6




                     One of the common uses of factor analysis is the
                      formation of summated scales, where we add the
                      scores on all the variables loading on a component to
                      create the score for the component.

                     To verify that the variables for a component are
                      measuring similar entities that are legitimate to add
                      together, we compute Chronbach's alpha.

                     If Chronbach's alpha is 0.70 or greater (0.60 or
                      greater for exploratory research), we have support
                      on the interval consistency of the items justifying
                      their use in a summated scale.
   SW388R7
Data Analysis &
 Computers II                                            Problem 1
    Slide 7
                  In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application
                  of a statistic? Assume that there is no problematic pattern of missing data. Use a level of
                  significance of 0.05. Validate the results of your principal component analysis by splitting
                  the sample in two, using 519447 as the random number seed.
                  Based on the results of a principal component analysis of the 8 variables "highest academic
                  degree" [degree], "father's highest academic degree" [padeg], "mother's highest academic
                  degree" [madeg], "spouse's highest academic degree" [spdeg], "general happiness" [happy],
                  "happiness of marriage" [hapmar], "condition of health" [health], and "attitude toward life"
                  [life], the information in these variables can be represented with 2 components and 3
                  individual variables. Cases that might be considered to be outliers do not have an impact on
                  the factor solution. The internal consistency of the variables included in the components is
                  sufficient to support the creation of a summated scale.
                  Component 1 includes the variables "highest academic degree" [degree], "father's highest
                  academic degree" [padeg], and "mother's highest academic degree" [madeg]. Component 2
                  includes the variables "general happiness" [happy] and "happiness of marriage" [hapmar]. The
                  variables "attitude toward life" [life], "condition of health" [health], and "spouse's highest
                  academic degree" [spdeg] were not included on the components and are retained as individual
                  variables.

                  1.   True
                                                                            The bold text indicates that
                  2.   True with caution                                    parts to the problem that
                  3.   False                                                have been added this week.

                  4.   Inappropriate application of a statistic
   SW388R7
Data Analysis &
 Computers II     Computing a principal component analysis
    Slide 8




                                          To compute a principal
                                          component analysis in SPSS,
                                          select the Data Reduction |
                                          Factor… command from the
                                          Analyze menu.
   SW388R7
Data Analysis &
 Computers II     Add the variables to the analysis
    Slide 9




                                                     First, move the
                                                     variables listed in
                                                     the problem to the
                                                     Variables list box.




                           Second, click on the
                           Descriptives… button to
                           specify statistics to
                           include in the output.
   SW388R7
Data Analysis &
 Computers II                Compete the descriptives dialog box
   Slide 10




                                             First, mark the Univariate
                                             descriptives checkbox to get a
                                             tally of valid cases.

                                                                                 Sixth, click
                                                                                 on the
                                                                                 Continue
               Second, keep the Initial                                          button.
               solution checkbox to get
               the statistics needed to
               determine the number
               of factors to extract.                                   Fifth, mark the Anti-image
                                                                        checkbox to get more
                                                                        outputs used to assess the
                                                                        appropriateness of factor
                                                                        analysis for the variables.


              Third, mark the
              Coefficients checkbox to get
              a correlation matrix, one of
                                                    Fourth, mark the KMO and Bartlett’s test
              the outputs needed to
                                                    of sphericity checkbox to get more outputs
              assess the appropriateness
                                                    used to assess the appropriateness of
              of factor analysis for the
                                                    factor analysis for the variables.
              variables.
   SW388R7
Data Analysis &
 Computers II                 Select the extraction method
   Slide 11




                  First, click on the          The extraction method refers
                  Extraction… button to        to the mathematical method
                  specify statistics to        that SPSS uses to compute the
                  include in the output.       factors or components.
   SW388R7
Data Analysis &
 Computers II     Compete the extraction dialog box
   Slide 12




                                 First, retain the default
                                 method Principal components.




                                                          Second, click
                                                          on the
                                                          Continue
                                                          button.
   SW388R7
Data Analysis &
 Computers II           Select the rotation method
   Slide 13




                                           The rotation method refers to
                  First, click on the      the mathematical method that
                  Rotation… button to      SPSS rotate the axes in
                  specify statistics to    geometric space. This makes
                  include in the output.   it easier to determine which
                                           variables are loaded on which
                                           components.
   SW388R7
Data Analysis &
 Computers II           Compete the rotation dialog box
   Slide 14




                  First, mark the                   Second, click
                  Varimax method                    on the
                  as the type of                    Continue
                  rotation to used                  button.
                  in the analysis.
   SW388R7
Data Analysis &
 Computers II     Complete the request for the analysis
   Slide 15




                                                First, click on the
                                                OK button to
                                                request the output.
   SW388R7
Data Analysis &
 Computers II             Level of measurement requirement
   Slide 16




                  "Highest academic degree" [degree], "father's highest academic
                  degree" [padeg], "mother's highest academic degree" [madeg],
                  "spouse's highest academic degree" [spdeg], "general happiness"
                  [happy], "happiness of marriage" [hapmar], "condition of health"
                  [health], and "attitude toward life" [life] are ordinal level
                  variables. If we follow the convention of treating ordinal level
                  variables as metric variables, the level of measurement
                  requirement for principal component analysis is satisfied. Since
                  some data analysts do not agree with this convention, a note of
                  caution should be included in our interpretation.
   SW388R7
Data Analysis &   Sample size requirement:
                  minimum number of cases
 Computers II

   Slide 17




                                     Descriptiv e Statistics

                                             Mean    Std. Deviation   Analysis N
                  RS HIGHEST DEGREE             1.68         1.085           68
                  FATHERS HIGHEST
                                                 .96          .984           68
                  DEGREE
                  MOTHERS HIGHEST
                                                 .85          .797           68
                  DEGREE
                  SPOUSES HIGHEST
                                                1.97         1.233           68
                  DEGREE
                        The number of valid cases for this
                  GENERAL HAPPINESS             1.65          .617           68
                        set of variables is 68.
                  HAPPINESS OF
                                                1.47          .532           68
                  MARRIAGE
                        While principal component analysis
                  CONDITION OF HEALTH           1.76
                        can be conducted on a sample that .848               68
                        has fewer OR
                  IS LIFE EXCITINGthan 100 cases, but more
                                                1.53          .532           68
                  DULL
                       than 50 cases, we should be
                       cautious about its interpretation.
   SW388R7
Data Analysis &   Sample size requirement:
                  ratio of cases to variables
 Computers II

   Slide 18




                                     Descriptiv e Statistics

                                            Mean     Std. Deviation   Analysis N
                  RS HIGHEST DEGREE            1.68          1.085           68
                  FATHERS HIGHEST
                                                .96           .984           68
                  DEGREE
                  MOTHERS HIGHEST
                                                .85           .797           68
                  DEGREE
                  SPOUSES HIGHEST
                                               1.97          1.233           68
                  DEGREE
                                The ratio of cases to
                  GENERAL HAPPINESS            1.65           .617           68
                                variables in a principal
                  HAPPINESS OF
                                component analysis should .532
                                               1.47                          68
                  MARRIAGE
                                be at least 5 to 1.
                  CONDITION OF HEALTH          1.76           .848           68
                  IS LIFE EXCITING OR and 8 variables,
                                With 68        1.53           .532           68
                  DULL
                                the ratio of cases to
                                variables is 8.5 to 1, which
                                exceeds the requirement
                                for the ratio of cases to
                                variables.
   SW388R7
Data Analysis &                Appropriateness of factor analysis:
                               Presence of substantial correlations
 Computers II

   Slide 19




                                                            Principal components analysis requires that there be
                                                            some correlations greater than 0.30 between the
                                                            variables included in the analysis.

                                                            For this set of variables, there are 7 correlations in
                                                            the matrix greater than 0.30, satisfying this
                                                            requirement. The correlations greater than 0.30 are
                                                            highlighted in yellow.
                                                                          Correlation Matrix

                                                              FATHERS     MOTHERS        SPOUSES                     HAPPINESS                    IS
                                              RS HIGHEST      HIGHEST     HIGHEST        HIGHEST        GENERAL         OF         CONDITION     EX
                                               DEGREE         DEGREE      DEGREE          DEGREE       HAPPINESS     MARRIAGE      OF HEALTH     OR
          Correlation   RS HIGHEST DEGREE           1.000         .490         .410           .595           -.017         -.172         -.246
                        FATHERS HIGHEST
                                                     .490        1.000           .677           .319         -.100         -.131         -.174
                        DEGREE
                        MOTHERS HIGHEST
                                                     .410         .677         1.000            .208          .105         -.046         -.008
                        DEGREE
                        SPOUSES HIGHEST
                                                     .595         .319           .208          1.000         -.053         -.138         -.392
                        DEGREE
                        GENERAL HAPPINESS           -.017         -.100          .105          -.053         1.000          .514          .267
                        HAPPINESS OF
                                                    -.172         -.131         -.046          -.138          .514         1.000          .282
                        MARRIAGE
                        CONDITION OF HEALTH         -.246         -.174         -.008          -.392          .267          .282         1.000
                        IS LIFE EXCITING OR
                                                    -.138         -.012          .151          -.090          .214          .161          .214
                        DULL
   SW388R7
Data Analysis &             Appropriateness of factor analysis:
                         Sampling adequacy of individual variables
 Computers II

   Slide 20




                                                              Anti-image Matrices

                                                        FATHERS         MOTHERS          SPOUSES                          HAPPINESS                            IS LIFE
                                          RS HIGHEST    HIGHEST         HIGHEST          HIGHEST           GENERAL           OF             CONDITION         EXCITING
                                           DEGREE       DEGREE           DEGREE           DEGREE          HAPPINESS       MARRIAGE          OF HEALTH         OR DULL
Anti-image Covariance RS HIGHEST DEGREE          .511       -.101            -.079            -.274             -.058           .067              -.008             .108
                       FATHERS HIGHEST
                                               -.101         .455            -.290            -.024              .103           -.028              .050            .028
                       There are two anti-image
                       DEGREE

                       matrices: the anti-image
                       MOTHERS HIGHEST
                       DEGREE
                                               -.079        -.290             .476             .028             -.102           .043              -.052            -.121
                       covariance matrix and the
                       SPOUSES HIGHEST
                                                                                         Principal component analysis requires
                       anti-image correlation -.274
                       DEGREE
                                                            -.024             .028       that .578 Kaiser-Meyer-Olkin Measure of
                                                                                               the      -.014     -.012        .203                                -.039
                       GENERAL We are
                       matrix.HAPPINESS interested in
                                               -.058         .103            -.102       Sampling Adequacy be greater than 0.50
                                                                                             -.014       .666     -.325       -.085                                -.085
                       the anti-image correlation
                       HAPPINESS OF                                                      for each individual variable as well as the
                       matrix.
                       MARRIAGE
                                                .067        -.028             .043       set of variables.
                                                                                             -.012      -.325      .692       -.099                                -.024
                       CONDITION OF HEALTH     -.008         .050            -.052             .203             -.085           -.099              .749            -.102
                       IS LIFE EXCITING OR                                               On iteration 1, the MSA for all of the
                       DULL
                                                .108         .028            -.121
                                                                                         individual variables included in the-.102
                                                                                             -.039      -.085       -.024                                          .876
Anti-image Correlation RS HIGHEST DEGREE        .701 a      -.210            -.161       analysis was greater than 0.5, supporting
                                                                                             -.503      -.099        .113     -.012                                .162
                       FATHERS HIGHEST
                                               -.210         .640
                                                                    a
                                                                             -.623
                                                                                         their retention .187 the analysis.
                                                                                                          in
                                                                                             -.048                  -.049      .086                                .044
                       DEGREE
                       MOTHERS HIGHEST                                               a
                                               -.161        -.623             .586             .053             -.181           .076              -.087            -.188
                       DEGREE
                       SPOUSES HIGHEST                                                                a
                                               -.503        -.048             .053             .656             -.023           -.018              .309            -.055
                       DEGREE
                       GENERAL HAPPINESS       -.099         .187            -.181            -.023              .549 a         -.478             -.120            -.111
                       HAPPINESS OF                                                                                                     a
                                                .113        -.049             .076            -.018             -.478           .619              -.137            -.030
                       MARRIAGE
                       CONDITION OF HEALTH                                                                                                                a
                                               -.012         .086            -.087             .309             -.120           -.137              .734            -.126
                       IS LIFE EXCITING OR                                                                                                                                 a
                                                .162         .044            -.188            -.055             -.111           -.030             -.126            .638
                       DULL
  a. Measures of Sampling Adequacy(MSA)
   SW388R7
Data Analysis &     Appropriateness of factor analysis:
                  Sampling adequacy for set of variables
 Computers II

   Slide 21




                                    KMO and Bartlett's Test
                   Kaiser-Meyer-Olkin Measure of Sampling
                   Adequacy.                                       .640

                   Bartlett's Test of      Approx. Chi-Square   137.823
                   Sphericity              df                        28
                                           Sig.                    .000
                                                                          In addition, the overall
                                                                          MSA for the set of variables
                                                                          included in the analysis
                                                                          was 0.640, which exceeds
                                                                          the minimum requirement
                                                                          of 0.50 for overall MSA.
   SW388R7
Data Analysis &   Appropriateness of factor analysis:
                      Bartlett test of sphericity
 Computers II

   Slide 22




                                         KMO and Bartlett's Test
                        Kaiser-Meyer-Olkin Measure of Sampling
                        Adequacy.                                       .640

                        Bartlett's Test of      Approx. Chi-Square   137.823
                        Sphericity              df                        28
                                                Sig.                    .000




                              Principal component analysis requires
                              that the probability associated with
                              Bartlett's Test of Sphericity be less
                              than the level of significance.

                              The probability associated with the
                              Bartlett test is <0.001, which satisfies
                              this requirement.
   SW388R7
Data Analysis &                     Number of factors to extract:
                                       Latent root criterion
 Computers II

   Slide 23




                                                                                    Total Variance Explained

                                                      Initial Eigenvalues            Extraction Sums of Squared Loadings
                              Component      Total     % of Variance Cumulative %    Total     % of Variance Cumulative %
                              1               2.600           32.502       32.502     2.600          32.502        32.502
                              2               1.772           22.149       54.651     1.772          22.149        54.651
                              3               1.079           13.486       68.137     1.079          13.486        68.137
                              4                .827           10.332       78.469
                              5                .631             7.888      86.358
                              6                .487             6.087      92.445
                              7                .333             4.161      96.606
                              8                .272             3.394     100.000
                             Extraction Method: Principal Component Analysis.
              Using the output from iteration 1,
              there were 3 eigenvalues greater
              than 1.0.

              The latent root criterion for
              number of factors to derive would
              indicate that there were 3
              components to be extracted for
              these variables.
   SW388R7
Data Analysis &                 Number of factors to extract:
                               Percentage of variance criterion
 Computers II

   Slide 24



                                                                                   Total Variance Explained

                                              Initi al Ei genval ues                 Extracti on Sums of Squared
                  Component         T otal     % of Vari ance     Cumul ati ve %     T otal     % of Vari ance Cu
                  1                   2.600            32.502           32.502         2.600          32.502
                  2                   1.772            22.149           54.651         1.772          22.149
                  3                   1.079            13.486           68.137         1.079          13.486
                  4                    .827            10.332           78.469
                  5                    .631              7.888          86.358
                  6                    .487              6.087          92.445
                  7                    .333              4.161          96.606
                  8                    .272              3.394        100.000
                  Extracti on M ethod: Princi pal Com ponent Anal ysi s.
                                                                       In addition, the cumulative
                                                                       proportion of variance criteria can
                                                                       be met with 3 components to
                                                                       satisfy the criterion of explaining
                                                                       60% or more of the total variance.

                                                                       A 3 components solution would
                                                                       explain 68.137% of the total
                    Since the SPSS default is to extract               variance.
                    the number of components indicated
                    by the latent root criterion, our
                    initial factor solution was based on
                    the extraction of 3 components.
   SW388R7
Data Analysis &
 Computers II                   Evaluating communalities
   Slide 25




                                                                     Communalities

                                                                                Initial   Extraction
                                                     RS HIGHEST DEGREE            1.000        .717
                  Communalities represent the        FATHERS HIGHEST
                                                                                  1.000        .768
                  proportion of the variance in      DEGREE
                  the original variables that is     MOTHERS HIGHEST
                                                                                  1.000        .815
                  accounted for by the factor        DEGREE
                  solution.                          SPOUSES HIGHEST
                                                                                  1.000        .715
                                                     DEGREE
                  The factor solution should         GENERAL HAPPINESS            1.000        .763
                  explain at least half of each      HAPPINESS OF
                                                                                  1.000        .711
                  original variable's variance, so   MARRIAGE
                  the communality value for          CONDITION OF HEALTH          1.000        .548
                  each variable should be 0.50       IS LIFE EXCITING OR
                  or higher.                         DULL
                                                                                  1.000        .415

                                                     Extraction Method: Principal Component Analysis.
   SW388R7
Data Analysis &
 Computers II     Communality requiring variable removal
   Slide 26




                                      Communalities

                                                 Initial   Extraction    On iteration 1, the
                      RS HIGHEST DEGREE            1.000        .717     communality for the
                                                                         variable "attitude toward
                      FATHERS HIGHEST
                                                   1.000        .768     life" [life] was 0.415.
                      DEGREE                                             Since this is less than
                      MOTHERS HIGHEST                                    0.50, the variable should
                                                   1.000        .815
                      DEGREE                                             be removed from the next
                      SPOUSES HIGHEST                                    iteration of the principal
                      DEGREE
                                                   1.000        .715     component analysis.
                      GENERAL HAPPINESS            1.000        .763     The variable was removed
                      HAPPINESS OF                                       and the principal
                                                   1.000        .711     component analysis was
                      MARRIAGE
                      CONDITION OF HEALTH          1.000        .548     computed again.
                      IS LIFE EXCITING OR          1.000        .415
                      DULL
                      Extraction Method: Principal Component Analysis.
   SW388R7
Data Analysis &
 Computers II     Repeating the factor analysis
   Slide 27




                              In the drop down menu,
                              select Factor Analysis to
                              reopen the factor analysis
                              dialog box.
   SW388R7
Data Analysis &
 Computers II     Removing the variable from the list of variables
   Slide 28




                                                         First, highlight
                                                         the life variable.




                             Second, click on the left
                             arrow button to remove
                             the variable from the
                             Variables list box.
   SW388R7
Data Analysis &
 Computers II     Replicating the factor analysis
   Slide 29




                                      The dialog recall command opens
                                      the dialog box with all of the
                                      settings that we had selected the
                                      last time we used factor analysis.

                                      To replicate the analysis without
                                      the variable that we just removed,
                                      click on the OK button.
   SW388R7
Data Analysis &
 Computers II     Communality requiring variable removal
   Slide 30




                                      Communalities

                                                 Initial   Extraction    On iteration 2, the
                      RS HIGHEST DEGREE            1.000        .642     communality for the
                                                                         variable "condition of
                      FATHERS HIGHEST
                                                   1.000        .623     health" [health] was
                      DEGREE                                             0.477. Since this is less
                      MOTHERS HIGHEST                                    than 0.50, the variable
                                                   1.000        .592
                      DEGREE                                             should be removed from
                      SPOUSES HIGHEST                                    the next iteration of the
                      DEGREE
                                                   1.000        .516     principal component
                                                                         analysis.
                      GENERAL HAPPINESS            1.000        .638
                      HAPPINESS OF                                       The variable was removed
                                                   1.000        .594     and the principal
                      MARRIAGE
                      CONDITION OF HEALTH          1.000        .477     component analysis was
                                                                         computed again.
                      Extraction Method: Principal Component Analysis.
   SW388R7
Data Analysis &
 Computers II     Repeating the factor analysis
   Slide 31




                              In the drop down menu,
                              select Factor Analysis to
                              reopen the factor analysis
                              dialog box.
   SW388R7
Data Analysis &
 Computers II     Removing the variable from the list of variables
   Slide 32




                                                         First, highlight
                                                         the health
                                                         variable.




                             Second, click on the left
                             arrow button to remove
                             the variable from the
                             Variables list box.
   SW388R7
Data Analysis &
 Computers II     Replicating the factor analysis
   Slide 33




                                      The dialog recall command opens
                                      the dialog box with all of the
                                      settings that we had selected the
                                      last time we used factor analysis.

                                      To replicate the analysis without
                                      the variable that we just removed,
                                      click on the OK button.
   SW388R7
Data Analysis &
 Computers II     Communality requiring variable removal
   Slide 34




                                                                         On iteration 3, the
                                                                         communality for the
                                                                         variable "spouse's highest
                                                                         academic degree" [spdeg]
                                     Communalities                       was 0.491. Since this is
                                                                         less than 0.50, the
                                                Initial   Extraction
                                                                         variable should be
                       RS HIGHEST DEGREE          1.000        .674      removed from the next
                       FATHERS HIGHEST                                   iteration of the principal
                                                 1.000         .640      component analysis.
                       DEGREE
                       MOTHERS HIGHEST
                       DEGREE
                                                 1.000         .577      The variable was removed
                                                                         and the principal
                       SPOUSES HIGHEST
                                                 1.000         .491      component analysis was
                       DEGREE                                            computed again.
                       GENERAL HAPPINESS         1.000         .719
                       HAPPINESS OF
                                                 1.000         .741
                       MARRIAGE
                      Extraction Method: Principal Component Analysis.
   SW388R7
Data Analysis &
 Computers II     Repeating the factor analysis
   Slide 35




                              In the drop down menu,
                              select Factor Analysis to
                              reopen the factor analysis
                              dialog box.
   SW388R7
Data Analysis &
 Computers II     Removing the variable from the list of variables
   Slide 36




                                                         First, highlight the
                                                         spdeg variable.




                             Second, click on the left
                             arrow button to remove
                             the variable from the
                             Variables list box.
   SW388R7
Data Analysis &
 Computers II     Replicating the factor analysis
   Slide 37




                                      The dialog recall command opens
                                      the dialog box with all of the
                                      settings that we had selected the
                                      last time we used factor analysis.

                                      To replicate the analysis without
                                      the variable that we just removed,
                                      click on the OK button.
   SW388R7
Data Analysis &
 Computers II     Communality satisfactory for all variables
   Slide 38




                                  Communalities
                                                                       Once any variables with
                                             Initial   Extraction      communalities less than
                   RS HIGHEST DEGREE           1.000        .577       0.50 have been removed
                                                                       from the analysis, the
                   FATHERS HIGHEST                                     pattern of factor loadings
                                              1.000         .720
                   DEGREE                                              should be examined to
                   MOTHERS HIGHEST                                     identify variables that
                                              1.000         .684       have complex structure.
                   DEGREE
                   GENERAL HAPPINESS          1.000         .745
                   HAPPINESS OF
                                              1.000         .782
                   MARRIAGE
                   Extraction Method: Principal Component Analysis.

                                                                      Complex structure occurs when
                                                                      one variable has high loadings or
                                                                      correlations (0.40 or greater) on
                                                                      more than one component. If a
                                                                      variable has complex structure, it
                                                                      should be removed from the
                                                                      analysis.

                                                                      Variables are only checked for
                                                                      complex structure if there is more
                                                                      than one component in the
                                                                      solution. Variables that load on
                                                                      only one component are described
                                                                      as having simple structure.
   SW388R7
Data Analysis &
 Computers II           Identifying complex structure
   Slide 39




                                                  a
                           Rotated Component Matrix

                                               Component
                                              1         2
                                                                        On iteration 4, none of the
                  RS HIGHEST DEGREE            .732      -.202
                                                                        variables demonstrated
                                                                        complex structure. It is not
                  FATHERS HIGHEST
                  DEGREE
                                               .848        .031         necessary to remove any
                                                                        additional variables because
                  MOTHERS HIGHEST
                  DEGREE
                                               .810        .169         of complex structure.
                  GENERAL HAPPINESS            .145        .851
                  HAPPINESS OF
                                              -.145        .872
                  MARRIAGE
                  Extraction Method: Principal Component Analysis.
                  Rotation Method: Varimax with Kaiser Normalization.
                    a. Rotation converged in 3 iterations.
   SW388R7
Data Analysis &
 Computers II     Variable loadings on components
   Slide 40




                                                                        On iteration 4, the 2
                                                                        components in the
                                                                        analysis had more than
                                                                        one variable loading on
                                                                        each of them.

                                                  a                     No variables need to be
                           Rotated Component Matrix
                                                                        removed because they
                                               Component                are the only variable
                                              1        2                loading on a component.
                  RS HIGHEST DEGREE         .732        -.202
                  FATHERS HIGHEST
                                                           .031
                  DEGREE                    .848
                  MOTHERS HIGHEST
                                                           .169
                  DEGREE                    .810
                  GENERAL HAPPINESS           .145      .851
                  HAPPINESS OF
                                              -.145
                  MARRIAGE                                  .872
                  Extraction Method: Principal Component Analysis.
                  Rotation Method: Varimax with Kaiser Normalization.
                     a. Rotation converged in 3 iterations.
   SW388R7
Data Analysis &
 Computers II                 Final check of communalities
   Slide 41




                                         Once we have resolved any
                                         problems with complex
                                         structure, we check the
                                         communalities one last time to
                                         make certain that we are
                                         explaining a sufficient portion
                                         of the variance of all of the
                                         original variables.



                                 Communalities

                                            Initial   Extraction
                  RS HIGHEST DEGREE           1.000        .577
                  FATHERS HIGHEST
                                             1.000         .720
                  DEGREE
                  MOTHERS HIGHEST                                      The communalities for all of the
                                             1.000         .684        variables included on the
                  DEGREE
                  GENERAL HAPPINESS          1.000         .745        components were greater than
                  HAPPINESS OF                                         0.50 and all variables had
                  MARRIAGE
                                             1.000         .782        simple structure.
                  Extraction Method: Principal Component Analysis.
                                                                       The principal component
                                                                       analysis has been completed.
   SW388R7
Data Analysis &
 Computers II             Interpreting the principal components
   Slide 42




                  The information in 5 of the
                  variables can be represented
                  by 2 components.
                                                                          Component 1 includes the
                                                                          variables

                                                                               •"highest academic degree"
                                                                               [degree],
                                                  a
                           Rotated Component Matrix                            •"father's highest academic
                                                                               degree" [padeg], and
                                               Component                       •"mother's highest
                                                                               academic degree" [madeg].
                                              1        2
                  RS HIGHEST DEGREE         .732        -.202
                  FATHERS HIGHEST
                                                           .031
                  DEGREE                    .848
                  MOTHERS HIGHEST
                                                           .169
                  DEGREE                    .810                        Component 2 includes the
                  GENERAL HAPPINESS           .145      .851            variables
                  HAPPINESS OF
                  MARRIAGE
                                              -.145                          •"general happiness"
                                                            .872
                                                                             [happy] and
                  Extraction Method: Principal Component Analysis.           •"happiness of marriage"
                  Rotation Method: Varimax with Kaiser Normalization.        [hapmar].
                     a. Rotation converged in 3 iterations.
   SW388R7
Data Analysis &
 Computers II                       Total variance explained
   Slide 43




                                                                         Total Variance Explained

                                           Initial Eigenvalues            Extraction Sums of Squared Loadings     Rotation Sums o
                  Component      Total      % of Variance Cumulative %    Total     % of Variance Cumulative %   Total     % of V
                  1               1.953            39.061       39.061     1.953          39.061        39.061     1.953
                  2               1.555            31.109       70.169     1.555          31.109        70.169     1.556
                  3                .649            12.989       83.158
                  4                .441              8.820      91.977
                  5                .401              8.023     100.000
                  Extraction Method: Principal Component Analysis.
                                                       The 2 components explain
                                                       70.169% of the total
                                                       variance in the variables
                                                       which are included on the
                                                       components.
   SW388R7
Data Analysis &
 Computers II               Split-sample validation
   Slide 44




                  We validate our analysis by conducting an analysis on each half of the
                  sample. We compare the results of these two split sample analyses
                  with the analysis of the full data set.

                  To split the sample into two half, we generate a random variable that
                  indicates which half of the sample each case should be placed in.

                  To compute a random selection of cases, we need to specify the
                  starting value, or random number seed. Otherwise, the random
                  sequence of numbers that you generate will not match mine, and we
                  will get different results.

                  Before we do the do the random selection, you must make certain
                  that your data set is sorted in the original sort order, or the cases in
                  your two half samples will not match mine. To make certain your
                  data set is in the same order as mine, sort your data set in ascending
                  order by case id.
   SW388R7
Data Analysis &
 Computers II     Sorting the data set in original order
   Slide 45




                                     To make certain the data set is
                                     sorted in the original order,
                                     highlight the case id column,
                                     right click on the column header,
                                     and select the Sort Ascending
                                     command from the popup menu.
   SW388R7
Data Analysis &
 Computers II     Setting the random number seed
   Slide 46




                                To set the random number
                                seed, select the Random
                                Number Seed… command
                                from the Transform menu.
   SW388R7
Data Analysis &
 Computers II                   Set the random number seed
   Slide 47




                  First, click on the
                  Set seed to option
                  button to activate
                  the text box.




                                                                Second, type in the
                                                                random seed stated in
                                                                the problem.



                                   Third, click on the OK
                                   button to complete the
                                   dialog box.

                                   Note that SPSS does not
                                   provide you with any
                                   feedback about the change.
   SW388R7
Data Analysis &
 Computers II     Select the compute command
   Slide 48




                              To enter the formula for the
                              variable that will split the
                              sample in two parts, click
                              on the Compute…
                              command.
   SW388R7
Data Analysis &
 Computers II     The formula for the split variable
   Slide 49



                           First, type the name for the
                           new variable, split, into the
                           Target Variable text box.


                                                           Second, the formula for the
                                                           value of split is shown in the
                                                           text box.

                                                           The uniform(1) function
                                                           generates a random decimal
                                                           number between 0 and 1.
                                                           The random number is
                                                           compared to the value 0.50.

                                                           If the random number is less
                                                           than or equal to 0.50, the
                                                           value of the formula will be 1,
                                                           the SPSS numeric equivalent
                                                           to true. If the random
                                                           number is larger than 0.50,
                                                           the formula will return a 0,
                                                           the SPSS numeric equivalent
                   Third, click on the                     to false.
                   OK button to
                   complete the dialog
                   box.
   SW388R7
Data Analysis &
 Computers II     The split variable in the data editor
   Slide 50




                        In the data editor, the
                        split variable shows a
                        random pattern of zero’s
                        and one’s.

                        To select half of the
                        sample for each validation
                        analysis, we will first
                        select the cases where
                        split = 0, then select the
                        cases where split = 1.
   SW388R7
Data Analysis &   Repeating the analysis with the first
                          validation sample
 Computers II

   Slide 51




                                  To repeat the principal
                                  component analysis for the
                                  first validation sample, select
                                  Factor Analysis from the
                                  Dialog Recall tool button.
   SW388R7
Data Analysis &
 Computers II            Using "split" as the selection variable
   Slide 52




                  First, scroll
                  down the list of
                  variables and
                  highlight the
                  variable split.




                                                     Second, click on the
                                                     right arrow button to
                                                     move the split variable
                                                     to the Selection
                                                     Variable text box.
   SW388R7
Data Analysis &
 Computers II      Setting the value of split to select cases
   Slide 53




                  When the variable named
                  split is moved to the
                  Selection Variable text
                  box, SPSS adds "=?" after
                  the name to prompt up to
                  enter a specific value for
                  split.                               Click on the
                                                       Value… button
                                                       to enter a
                                                       value for split.
   SW388R7
Data Analysis &
 Computers II           Completing the value selection
   Slide 54




                  First, type the value         Second, click on the
                  for the first half of the     Continue button to
                  sample, 0, into the           complete the value
                  Value for Selection           entry.
                  Variable text box.
   SW388R7
Data Analysis &   Requesting output for the first validation
                                  sample
 Computers II

   Slide 55




                                                            Click on the OK
                                                            button to
                                                            request the
                                                            output.




                    When the value entry
                    dialog box is closed, SPSS
                    adds the value we entered
                    after the equal sign. This
                    specification now tells      Since the validation analysis
                    SPSS to include in the       requires us to compare the
                    analysis only those cases    results of the analysis using
                    that have a value of 0 for   the two split sample, we will
                    the split variable.          request the output for the
                                                 second sample before doing
                                                 any comparison.
   SW388R7
Data Analysis &   Repeating the analysis with the second
                            validation sample
 Computers II

   Slide 56




                                   To repeat the principal
                                   component analysis for the
                                   second validation sample,
                                   select Factor Analysis from the
                                   Dialog Recall tool button.
   SW388R7
Data Analysis &
 Computers II     Setting the value of split to select cases
   Slide 57




                            Since the split variable is already in the
                            Selection Variable text box, we only need
                            to change its value.

                            Click on the Value… button to enter a
                            different value for split.
   SW388R7
Data Analysis &
 Computers II          Completing the value selection
   Slide 58




                  First, type the value        Second, click on the
                  for the second half of       Continue button to
                  the sample, 1, into the      complete the value
                  Value for Selection          entry.
                  Variable text box.
   SW388R7
Data Analysis &   Requesting output for the second validation
                                   sample
 Computers II

   Slide 59




                                                    Click on the OK
                                                    button to
                                                    request the
                                                    output.




                     When the value entry
                     dialog box is closed, SPSS
                     adds the value we entered
                     after the equal sign. This
                     specification now tells
                     SPSS to include in the
                     analysis only those cases
                     that have a value of 1 for
                     the split variable.
   SW388R7
Data Analysis &
 Computers II                           Comparing communalities
   Slide 60




                        All of the communalities                            All of the communalities
                        for the first split sample                          for the second split sample
                        satisfy the minimum                                 satisfy the minimum
                        requirement of being                                requirement of being
                        larger than 0.50.                                   larger than 0.50.


                                              a                                                  a
                                 Communalities                                      Communalities

                                            Initial   Extraction                               Initial   Extraction
                  RS HIGHEST DEGREE           1.000        .580      RS HIGHEST DEGREE           1.000        .618
                  FATHERS HIGHEST                                    FATHERS HIGHEST
                                             1.000         .647                                 1.000         .802
                  DEGREE                                             DEGREE
                  MOTHERS HIGHEST                                    MOTHERS HIGHEST
                                             1.000         .693                                 1.000         .675
                  DEGREE                                             DEGREE
                  GENERAL HAPPINESS          1.000         .667      GENERAL HAPPINESS          1.000         .807
                  HAPPINESS OF                                       HAPPINESS OF
                                             1.000         .754                                 1.000         .830
                  MARRIAGE                                           MARRIAGE
                  Extraction Method: Principal Component Analysis.   Extraction Method: Principal Component Analysis.
                     a. Only cases for which SPLIT = 0 are used         a. Only cases for which SPLIT = 1 are used
                        in the analysis phase.                             in the analysis phase.


                        Note how SPSS identifies for
                        us which cases we selected
                        for the analysis.
   SW388R7
Data Analysis &
 Computers II                            Comparing factor loadings
   Slide 61




                               The pattern of factor loading for both split samples shows the
                               variables RS HIGHEST DEGREE; FATHERS HIGHEST DEGREE; and
                               MOTHERS HIGHEST DEGREE loading on the first component, and
                               GENERAL HAPPINESS and HAPPINESS OF MARRIAGE loading on the
                               second component.


                                                  a,b                                                   a,b
                           Rotated Component Matrix                              Rotated Component Matrix

                                               Component                                             Component
                                              1         2                                           1         2
                  RS HIGHEST DEGREE            .730      -.215          RS HIGHEST DEGREE            .755      -.219
                  FATHERS HIGHEST                                       FATHERS HIGHEST
                                               .789         .154                                     .895        -.043
                  DEGREE                                                DEGREE
                  MOTHERS HIGHEST                                       MOTHERS HIGHEST
                                               .794         .251                                     .819         .064
                  DEGREE                                                DEGREE
                  GENERAL HAPPINESS            .248         .778        GENERAL HAPPINESS            .049         .897
                  HAPPINESS OF                                          HAPPINESS OF
                                              -.102         .862                                    -.183         .893
                  MARRIAGE                                              MARRIAGE
                  Extraction Method: Principal Component Analysis.      Extraction Method: Principal Component Analysis.
                  Rotation Method: Varimax with Kaiser Normalization.   Rotation Method: Varimax with Kaiser Normalization.
                     a. Rotation converged in 3 iterations.               a. Rotation converged in 3 iterations.
                    b. Only cases for which SPLIT = 0 are used in         b. Only cases for which SPLIT = 1 are used in
                       the analysis phase.                                   the analysis phase.
   SW388R7
Data Analysis &
 Computers II                   Interpreting the validation results
   Slide 62




                               All of the communalities in both validation samples met the criteria.

                               The pattern of loadings for both validation samples is the same, and
                               the same as the pattern for the analysis using the full sample.

                               In effect, we have done the same analysis on two separate sub-
                               samples of cases and obtained the same results.

                               This validation analysis supports a finding that the results of this
                              principal component                                 the population
                           Rotated Component Matrix analysis are generalizable toRotated Component Matrix
                                                  a,b                                                   a,b
                               represented by this data set.
                                               Component                                               Component
                                              1         2                                             1         2
                  RS HIGHEST DEGREE            .730      -.215            RS HIGHEST DEGREE            .755      -.219
                  FATHERS HIGHEST                                         FATHERS HIGHEST
                                               .789         .154                                  .895    -.043
                  DEGREE                                                  DEGREE
                  MOTHERS HIGHEST                                         MOTHERS HIGHEST
                                               .794         .251                                  .819     .064
                  DEGREE                                                  DEGREE
                  GENERAL HAPPINESS            .248         .778             When we are finished with .897
                                                                          GENERAL HAPPINESS       .049
                  HAPPINESS OF
                                              -.102         .862             this analysis, we should select
                                                                          HAPPINESS OF
                                                                                                 -.183     .893
                  MARRIAGE                                                   all cases back into the data
                                                                          MARRIAGE
                  Extraction Method: Principal Component Analysis.            set and remove the variables
                                                                          Extraction Method: Principal Component Analysis.
                  Rotation Method: Varimax with Kaiser Normalization.         we Method: Varimax with Kaiser Normalization.
                                                                          Rotation created.
                     a. Rotation converged in 3 iterations.                 a. Rotation converged in 3 iterations.
                    b. Only cases for which SPLIT = 0 are used in           b. Only cases for which SPLIT = 1 are used in
                       the analysis phase.                                     the analysis phase.
   SW388R7
Data Analysis &
 Computers II     Detecting outliers
   Slide 63




                             To detect outliers, we
                             compute the factor
                             scores in SPSS.

                             Select the Factor
                             Analysis command
                             from the Dialog
                             Recall tool button
   SW388R7
Data Analysis &
 Computers II     Access the Scores Dialog Box
   Slide 64




                        Click on the Scores…
                        button to access the
                        factor scores dialog
                        box.
   SW388R7
Data Analysis &
 Computers II             Specifications for factor scores
   Slide 65




                       First, click on the
                       Save as variables
                       checkbox to create
                       factor variables.




                  Second, accept the
                                                       Third, click on the
                  default method using
                                                       Continue button
                  a Regression equation
                                                       to complete the
                  to calculate the
                                                       specifications.
                  scores.
   SW388R7
Data Analysis &
 Computers II     Compute the factor scores
   Slide 66




                                         Click on the Continue
                                         button to compute
                                         the factor scores.
   SW388R7
Data Analysis &
 Computers II     The factor scores in the data editor
   Slide 67




                                SPSS creates the factor score
                                variables in the data editor window.
                                It names the first factor score
                                “fac1_1,” and the second factor
                                score “fac2_1.”




                                         We need to check to see if we have
                                         any values for either factor score
                                         that are larger than ±3.0. One way
                                         to check for the presence of large
                                         values indicating outliers is to sort
                                         the factor variables and see if any
                                         fall outside the acceptable range.
   SW388R7
Data Analysis &
 Computers II     Sort the data to locate outliers for factor one
   Slide 68




                                          First, select the
                                          fac1_1 column by
                                          clicking on its header.




                                                                    Second, right click
                                                                    on the column
                                                                    header and select
                                                                    the Sort Ascending
                                                                    command from the
                                                                    drop down menu.
   SW388R7
Data Analysis &
 Computers II     Negative outliers for factor one
   Slide 69




                                        Scroll down past the
                                        cases for whom factor
                                        scores could not be
                                        computed. We see that
                                        none of the scores for
                                        factor one are less than
                                        or equal to -3.0.
   SW388R7
Data Analysis &
 Computers II     Positive outliers for factor one
   Slide 70




                                        Scrolling down to the
                                        bottom of the sorted
                                        data set, we see that
                                        none of the scores for
                                        factor one are greater
                                        than or equal to +3.0.

                                        There are no outliers on
                                        factor one.
   SW388R7
Data Analysis &
 Computers II     Sort the data to locate outliers on factor two
   Slide 71




                                         First, select the
                                         fac2_1 column by
                                         clicking on its header.




                                                                   Second, right click
                                                                   on the column
                                                                   header and select
                                                                   the Sort Ascending
                                                                   command from the
                                                                   drop down menu.
   SW388R7
Data Analysis &
 Computers II     Negative outliers for factor two
   Slide 72




                   Scrolling down past the
                   cases for whom factor
                   scores could not be
                   computed, we see that
                   none of the scores for
                   factor two are less than
                   or equal to -3.0.
   SW388R7
Data Analysis &
 Computers II       Positive outliers for factor two
   Slide 73




                  Scrolling down to the bottom
                  of the sorted data set, we
                  see that one of the scores for
                  factor two is greater than or
                  equal to +3.0.

                  We will run the analysis
                  excluding this outlier and see
                  if it changes our
                  interpretation of the analysis.
   SW388R7
Data Analysis &
 Computers II     Removing the outliers
   Slide 74




                                      To see whether or not
                                      outliers are having an
                                      impact on the factor
                                      solution, we will compute
                                      the factor analysis
                                      without the outliers and
                                      compare the results.




                             To remove the outliers, we will
                             include the cases that are not
                             outliers.

                             Choose the Select Cases…
                             command from the Data menu.
   SW388R7
Data Analysis &
 Computers II     Setting the If condition
   Slide 75




                                       Click on the If…
                                       button to enter
                                       the formula for
                                       selecting cases
                                       in or out of the
                                       analysis.
   SW388R7
Data Analysis &
 Computers II     Formula to select cases that are not outliers
   Slide 76




                                                  First, type the
                                                  formula as shown.
                                                  The formula says:
                                                  include cases if the
                                                  absolute value of the
                                                  first and second factor
                                                  scores are less than
                                                  3.0.




                       Second, click on the
                       Continue button to
                       complete the
                       specification.
   SW388R7
Data Analysis &
 Computers II     Complete the select cases command
   Slide 77




                                      Having entered the
                                      formula for including
                                      cases, click on the OK
                                      button to complete the
                                      selection.
   SW388R7
Data Analysis &
 Computers II     The outlier selected out of the analysis
   Slide 78




                        When SPSS selects a case out of the
                        data analysis, it draws a slash
                        through the case number. The case
                        that we identified as an outlier will be
                        excluded.
   SW388R7
Data Analysis &
 Computers II     Repeating the factor analysis
   Slide 79




                                To repeat the factor analysis
                                without the outliers, select the
                                Factor Analysis command from
                                the Dialog Recall tool button
   SW388R7
Data Analysis &   Stopping SPSS from computing factor scores
                                    again
 Computers II

   Slide 80




                  On the last factor analysis,
                  we included the specification
                  to compute factor scores.
                  Since we do not need to do
                  this again, we will remove
                  the specification.




                                                  Click on the Scores…
                                                  button to access the
                                                  factor scores dialog.
   SW388R7
Data Analysis &
 Computers II     Clearing the command to save factor scores
   Slide 81




                   First, clear the Save
                   as variables checkbox.
                   This will deactivate        Second, click on the
                   the Method options.         Continue button to
                                               complete the specification
   SW388R7
Data Analysis &
 Computers II     Computing the factor analysis
   Slide 82




                                           To produce the
                                           output for the factor
                                           analysis excluding
                                           outliers, click on the
                                           OK button.
   SW388R7
Data Analysis &
 Computers II                      Comparing communalities
   Slide 83




                   All of the communalities                               All of the communalities
                   for the factor analysis                                for the factor analysis
                   including all cases satisfy                            excluding outliers satisfy
                   the minimum requirement                                the minimum requirement
                   of being larger than 0.50.                             of being larger than 0.50.



                                 Communalities                                      Communalities

                                            Initial   Extraction                               Initial   Extraction
                  RS HIGHEST DEGREE           1.000        .577      RS HIGHEST DEGREE           1.000        .579
                  FATHERS HIGHEST                                    FATHERS HIGHEST
                                             1.000         .720                                 1.000         .720
                  DEGREE                                             DEGREE
                  MOTHERS HIGHEST                                    MOTHERS HIGHEST
                                             1.000         .684                                 1.000         .681
                  DEGREE                                             DEGREE
                  GENERAL HAPPINESS          1.000         .745      GENERAL HAPPINESS          1.000         .726
                  HAPPINESS OF                                       HAPPINESS OF
                                             1.000         .782                                 1.000         .771
                  MARRIAGE                                           MARRIAGE
                  Extraction Method: Principal Component Analysis.   Extraction Method: Principal Component Analysis.
   SW388R7
Data Analysis &
 Computers II                     Comparing factor loadings
   Slide 84



                     The factor loadings for the                          The factor loadings for the
                     factor analysis including all                        factor analysis excluding
                     cases is shown on the left.                          outliers is shown on the right.

                                                  a
                           Rotated Component Matrix                                                     a
                                                                                 Rotated Component Matrix

                                               Component                                             Component
                                              1        2                                            1        2
                  RS HIGHEST DEGREE            .732     -.202           RS HIGHEST DEGREE            .734     -.201
                  FATHERS HIGHEST                                       FATHERS HIGHEST
                                               .848        .031                                      .846        .060
                  DEGREE                                                DEGREE
                  MOTHERS HIGHEST                                       MOTHERS HIGHEST
                                               .810        .169                                      .810        .157
                  DEGREE                                                DEGREE
                  GENERAL HAPPINESS            .145        .851         GENERAL HAPPINESS            .159        .837
                  HAPPINESS OF                                          HAPPINESS OF
                                              -.145        .872                                     -.143        .866
                  MARRIAGE                                              MARRIAGE
                  Extraction Method: Principal Component Analysis.      Extraction Method: Principal Component Analysis.
                  Rotation Method: Varimax with Kaiser Normalization.   Rotation Method: Varimax with Kaiser Normalization.
                     a. Rotation converged in 3 iterations.                a. Rotation converged in 3 iterations.

                          The pattern of factor loading for both split analyses shows the
                          variables RS HIGHEST DEGREE; FATHERS HIGHEST DEGREE; and
                          MOTHERS HIGHEST DEGREE loading on the first component, and
                          GENERAL HAPPINESS and HAPPINESS OF MARRIAGE loading on the
                          second component.
   SW388R7
Data Analysis &
 Computers II              Interpreting the outlier analysis
   Slide 85



                                   All of the communalities satisfy the criteria of being
                                   greater than 0.50.

                                   The pattern of loadings for both analyses is the same.

                                   Whether we include or exclude outliers, our
                                   interpretation is the same. The outliers do not have an
                                   effect which supports their exclusion from the analysis.

                                   The part of the problem statement that outliers do not
                                   have an impact is true.



                                                  a
                           Rotated Component Matrix                                                     a
                                                                                 Rotated Component Matrix

                                               Component                                             Component
                                              1        2                                            1        2
                  RS HIGHEST DEGREE            .732     -.202           RS HIGHEST DEGREE            .734     -.201
                  FATHERS HIGHEST                                       FATHERS HIGHEST
                                               .848        .031                                  .846            .060
                  DEGREE                                                When we are finished with
                                                                        DEGREE
                  MOTHERS HIGHEST                                       this analysis, we should select
                                                                        MOTHERS HIGHEST
                                               .810        .169                                  .810            .157
                  DEGREE                                                all cases back into the data
                                                                        DEGREE
                  GENERAL HAPPINESS            .145        .851         set and remove the variables
                                                                        GENERAL HAPPINESS        .159            .837
                  HAPPINESS OF                                          we created.
                                                                        HAPPINESS OF
                                              -.145        .872                                 -.143            .866
                  MARRIAGE                                              MARRIAGE
                  Extraction Method: Principal Component Analysis.      Extraction Method: Principal Component Analysis.
                  Rotation Method: Varimax with Kaiser Normalization.   Rotation Method: Varimax with Kaiser Normalization.
                     a. Rotation converged in 3 iterations.                a. Rotation converged in 3 iterations.
   SW388R7
Data Analysis &
 Computers II     Computing Chronbach's Alpha
   Slide 86




                                      To compute
                                      Chronbach's alpha for
                                      each component in
                                      our analysis, we select
                                      Scale | Reliability
                                      Analysis… from the
                                      Analyze menu.
   SW388R7
Data Analysis &
 Computers II     Selecting the variables for the first component
   Slide 87




                           First, move the three
                           variables that loaded
                           on the first component
                           to the Items list box.




                                                    Second, click on the
                                                    Statistics… button to
                                                    select the statistics we
                                                    will need.
   SW388R7
Data Analysis &
 Computers II              Selecting the statistics for the output
   Slide 88




                                                          Second, click on the
              First, mark the                             Continue button.
              checkboxes for Item,
              Scale, and Scale if
              item deleted.
   SW388R7
Data Analysis &
 Computers II     Completing the specifications
   Slide 89




                                                  Second, click on
                                                  the OK button to
                                                  produce the output.




                         First, If Alpha is not
                         selected as the Model
                         in the drop down
                         menu, select it now.
   SW388R7
Data Analysis &
 Computers II     Chronbach's Alpha
   Slide 90




                       Chronbach's Alpha is located at the
                       bottom of the output. An alpha of
                       0.60 or higher is the minimum
                       acceptable level. Preferably, alpha
                       will be 0.70 or higher, as it is in
                       this case.
   SW388R7
Data Analysis &
 Computers II     Chronbach's Alpha
   Slide 91




                  If alpha is too small, this column may
                  suggest which variable should be removed
                  to improve the internal consistency of the
                  scale variables. It tells us what alpha we
                  would get if the variable listed were
                  removed from the scale.
   SW388R7
Data Analysis &
 Computers II     Computing Chronbach's Alpha
   Slide 92




                                      To compute
                                      Chronbach's alpha for
                                      each component in
                                      our analysis, we select
                                      Scale | Reliability
                                      Analysis… from the
                                      Analyze menu.
   SW388R7
Data Analysis &   Selecting the variables for the second
                               component
 Computers II

   Slide 93



                        First, move the three
                        variables that loaded
                        on the second
                        component to the
                        Items list box.




                                                Second, click on the
                                                Statistics… button to
                                                select the statistics we
                                                will need.
   SW388R7
Data Analysis &
 Computers II              Selecting the statistics for the output
   Slide 94




                                                          Second, click on the
              First, mark the                             Continue button.
              checkboxes for Item,
              Scale, and Scale if
              item deleted.
   SW388R7
Data Analysis &
 Computers II     Completing the specifications
   Slide 95




                                                  Second, click on
                                                  the OK button to
                                                  produce the output.




                         First, If Alpha is not
                         selected as the Model
                         in the drop down
                         menu, select it now.
   SW388R7
Data Analysis &
 Computers II     Chronbach's Alpha
   Slide 96




                       Chronbach's Alpha is located at the
                       bottom of the output. An alpha of
                       0.60 or higher is the minimum
                       acceptable level. Preferably, alpha
                                       Second, it is
                       will be 0.70 or higher, asclick in
                       this case.
   SW388R7
Data Analysis &
 Computers II                    Answering the problem question
   Slide 97




                                                                           Total Variance Explained

                                             Initial Eigenvalues               Extraction Sums of Squared Loadings     Ro
                  Component         Total     % of Variance Cumulative %       Total     % of Variance Cumulative %   Tot
                  1                  1.626           40.651         40.651      1.626          40.651        40.651     1.
                           The answer to the original question is true with caution.
                  2                  1.119           27.968         68.619      1.119          27.968        68.619     1.
                  3        Component 1 includes the variables "highest academic degree" [degree],
                                       .694          17.341         85.960
                           "father's highest academic degree" [padeg], and "mother's highest
                  4        academic degree" [madeg]. We can substitute one component variable for
                                       .562          14.040       100.000
                           this combination of Component further
                  Extraction Method: Principal variables in Analysis.analyses.
                         Component 2 includes the variables "general happiness" [happy] and
                         "happiness of marriage" [hapmar]. We can substitute one component
                         variable for this combination of variables in further analyses.

                         The components explain at least 50% of the variance in each of the
                         variables included in the final analysis.

                         The components explain 70.169% of the total variance in the variables
                         which are included on the components.

                         A caution is added to our findings because of the inclusion of ordinal level
                         variables in the analysis.
   SW388R7
Data Analysis &
 Computers II                  Validation with small samples
   Slide 98



                     In the validation example completed above, 105 cases were
                      used in the final principal component analysis model. When we
                      have more than 100 cases available for the validation analysis,
                      an even split should generally results in 50+ cases per validation
                      sample.

                     However, if the number of cases available for the validation is
                      less than 100, then splitting the sample in two may result in a
                      validation samples that are less than the minimum of 50 cases
                      to conduct a factor analysis.

                     When this happens, we draw two random samples of cases that
                      are both larger than the minimum of 50. Since some of the
                      same cases will be in both validation samples, the support for
                      generalizability is not as strong, but it does offer some
                      evidence, especially if we repeat the process a number of
                      times.
   SW388R7
Data Analysis &
 Computers II                  Validation with small samples
   Slide 99



                     We randomly create two split variables which we will call split1
                      and split 2, using a separate random number see for each.

                     In the formula for creating the split variables, we set the
                      proportion of cases sufficient to randomly select fifty cases.

                     To calculate the proportion that we need, we divide 50 by the
                      number of valid cases in the analysis and round up to the next
                      highest 10% increment.

                     For example, if we have 80 valid cases, the proportion we need
                      for validation is 50 / 80 = 0.625, which we would round up to
                      0.70 or 70%. The formulas for the split variables would be:
                           split1 = uniform(1) <= 0.70
                           split2 = uniform(1) <= 0.70
   SW388R7
Data Analysis &
 Computers II             Validation with very small samples
   Slide 100




                     When the number of valid cases in a factor analysis
                      gets close to the lower limit of 50, the results of the
                      validation may appear to support the analysis, but
                      this can be misleading because the validation
                      samples are not really different from the analysis of
                      the full data set.

                     For example, if the number of valid cases were 60, a
                      90% sub-sample of 54 would result in 54 cases being
                      the same in both the full analysis and the validation
                      analysis. The validation may appear to support the
                      full analysis simply because the validation had
                      limited opportunity to be different.
   SW388R7
Data Analysis &
 Computers II                                                   Problem 2
   Slide 101


                  In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic?
                  Assume that there is no problematic pattern of missing data. Use a level of significance of 0.05. Validate the
                  results of your principal component analysis by repeating the principal component analysis on two 70% random
                  samples of the data set, using 743911 and 747454 as the random number seeds.

                  Based on the results of a principal component analysis of the 7 variables "claims about environmental threats
                  are exaggerated" [grnexagg], "danger to the environment from modifying genes in crops" [genegen], "America
                  doing enough to protect environment" [amprogrn], "should be international agreements for environment
                  problems" [grnintl], "poorer countries should be expected to do less for the environment" [ldcgrn], "economic
                  progress in America will slow down without more concern for environment" [econgrn], and "likelihood of
                  nuclear power station damaging environment in next 5 years" [nukeacc], the information in these variables can
                  be represented with 2 components and 3 individual variables. Cases that might be considered to be outliers do
                  not have an impact on the factor solution. The internal consistency of the variables included in the components
                  is sufficient to support the creation of a summated scale.

                  Component 1 includes the variables "danger to the environment from modifying genes in crops" [genegen] and
                  "likelihood of nuclear power station damaging environment in next 5 years" [nukeacc]. Component 2 includes
                  the variables "claims about environmental threats are exaggerated" [grnexagg] and "poorer countries should be
                  expected to do less for the environment" [ldcgrn]. The variables "economic progress in America will slow down
                  without more concern for environment" [econgrn], "should be international agreements for environment
                  problems" [grnintl], and "America doing enough to protect environment" [amprogrn] were not included on the
                  components and are retained as individual variables.

                    1.   True
                    2.   True with caution
                    3.   False
                    4.   Inappropriate application of a statistic
   SW388R7
Data Analysis &
 Computers II                 The principal component solution
   Slide 102




                                             A principal component analysis found a
                                             two-factor solution, with four of the
                                             original seven variables loading on the
                                             components. The communalities and
                                             factor loadings are shown below.

                                                                                                          a
                                 Communalities                                     Rotated Component Matrix

                                             Initial   Extraction                                      Component
                  ENVIRONMENTAL                                                                       1        2
                  THREATS                      1.000        .615         ENVIRONMENTAL
                  EXAGGERATED                                            THREATS                      -.207        .756
                  HOW DANGEROUS                                          EXAGGERATED
                  MODIFYING GENES IN           1.000        .694         HOW DANGEROUS
                  CROPS                                                  MODIFYING GENES IN            .801       -.229
                  POOR COUNTRIES                                         CROPS
                  LESS THAN RICH FOR           1.000        .691         POOR COUNTRIES
                  ENVIRONMENT                                            LESS THAN RICH FOR            .051        .830
                  LIKELIHOOD OF                                          ENVIRONMENT
                  NUCLEAR MELTDOWN             1.000        .744         LIKELIHOOD OF
                  IN 5 YEARS                                             NUCLEAR MELTDOWN              .861        .059
                  Extraction Method: Principal Component Analysis.       IN 5 YEARS
                                                                         Extraction Method: Principal Component Analysis.
                                                                         Rotation Method: Varimax with Kaiser Normalization.
                                                                            a. Rotation converged in 3 iterations.
   SW388R7
Data Analysis &
 Computers II     The size of the validation sample
   Slide 103




                                        Descriptiv e Statistics

                                              Mean    Std. Deviation   Analysis N
                      ENVIRONMENTAL
                      THREATS                   3.28          1.008           75
                      EXAGGERATED
                      HOW DANGEROUS
                      MODIFYING GENES IN        3.11           .953           75
                      CROPS
                      POOR COUNTRIES
                      LESS THAN RICH FOR        3.77           .863           75
                      ENVIRONMENT
                      LIKELIHOOD OF
                      NUCLEAR MELTDOWN          2.47
                         There were 75 valid cases in the final.935           75
                      IN 5 YEARS
                        analysis. The sample is to small to split in
                        half and have enough cases to meet the
                        minimum of 50 cases for factor analysis.

                        We will draw two random samples that
                        each comprise 70% of the full sample. We
                        arrive at 70% by dividing the minimum
                        sample size by the number of valid cases
                        (50 ÷ 75 = 0.667) and rounding up to the
                        next 10% increment, 70%.
   SW388R7
Data Analysis &
 Computers II                               Split-sample validation
   Slide 104




                  The first random
                  number seed stated in
                  the problem is 743911,
                  so we enter this is the
                  SPSS random number
                  seed dialog.
                                                         To set the random number
                                                         seed, select the Random
                                                         Number Seed… command
                                                         from the Transform menu.
   SW388R7
Data Analysis &
 Computers II     Set the random number seed for first sample
   Slide 105




                   First, click on the
                   Set seed to option
                   button to activate
                   the text box.




                                                                 Second, type in the
                                                                 random seed stated in
                                                                 the problem.



                                    Third, click on the OK
                                    button to complete the
                                    dialog box.

                                    Note that SPSS does not
                                    provide you with any
                                    feedback about the change.
   SW388R7
Data Analysis &
 Computers II     Select the compute command
   Slide 106




                              To enter the formula for the
                              variable that will split the
                              sample in two parts, click
                              on the Compute…
                              command.
   SW388R7
Data Analysis &
 Computers II     The formula for the split1 variable
   Slide 107


                           First, type the name for the
                           new variable, split1, into
                           the Target Variable text
                           box.

                                                          Second, the formula for the
                                                          value of split1 is shown in the
                                                          text box.

                                                          The uniform(1) function
                                                          generates a random decimal
                                                          number between 0 and 1.
                                                          The random number is
                                                          compared to the value 0.70.

                                                          If the random number is less
                                                          than or equal to 0.70, the
                                                          value of the formula will be 1,
                                                          the SPSS numeric equivalent
                                                          to true. If the random
                                                          number is larger than 0.70,
                                                          the formula will return a 0,
                                                          the SPSS numeric equivalent
                   Third, click on the                    to false.
                   OK button to
                   complete the dialog
                   box.
   SW388R7
Data Analysis &     Set the random number seed for second
                                   sample
 Computers II

   Slide 108




                  First, click on the
                  Set seed to option
                  button to activate
                  the text box.




                                                                Second, type in the
                                                                random seed stated in
                                                                the problem.



                                   Third, click on the OK
                                   button to complete the
                                   dialog box.

                                   Note that SPSS does not
                                   provide you with any
                                   feedback about the change.
   SW388R7
Data Analysis &
 Computers II     Select the compute command
   Slide 109




                              To enter the formula for the
                              variable that will split the
                              sample in two parts, click
                              on the Compute…
                              command.
   SW388R7
Data Analysis &
 Computers II     The formula for the split2 variable
   Slide 110


                           First, type the name for the
                           new variable, split2, into
                           the Target Variable text
                           box.

                                                          Second, the formula for the
                                                          value of split2 is shown in the
                                                          text box.

                                                          The uniform(1) function
                                                          generates a random decimal
                                                          number between 0 and 1.
                                                          The random number is
                                                          compared to the value 0.70.

                                                          If the random number is less
                                                          than or equal to 0.70, the
                                                          value of the formula will be 1,
                                                          the SPSS numeric equivalent
                                                          to true. If the random
                                                          number is larger than 0.70,
                                                          the formula will return a 0,
                                                          the SPSS numeric equivalent
                   Third, click on the                    to false.
                   OK button to
                   complete the dialog
                   box.
   SW388R7
Data Analysis &   Repeating the analysis with the first
                          validation sample
 Computers II

   Slide 111




                                  To repeat the principal
                                  component analysis for the
                                  first validation sample, select
                                  Factor Analysis from the
                                  Dialog Recall tool button.
   SW388R7
Data Analysis &
 Computers II            Using split1 as the selection variable
   Slide 112




                  First, scroll
                  down the list of
                  variables and
                  highlight the
                  variable split1.




                                                     Second, click on the
                                                     right arrow button to
                                                     move the split1 variable
                                                     to the Selection
                                                     Variable text box.
   SW388R7
Data Analysis &
 Computers II     Setting the value of split1 to select cases
   Slide 113




                  When the variable named
                  split1 is moved to the
                  Selection Variable text
                  box, SPSS adds "=?" after
                  the name to prompt up to
                  enter a specific value for
                  split1.                              Click on the
                                                       Value… button
                                                       to enter a
                                                       value for split1.
   SW388R7
Data Analysis &
 Computers II             Completing the value selection
   Slide 114




                  First, type the value           Second, click on the
                  for the first sample, 1,        Continue button to
                  into the Value for              complete the value
                  Selection Variable text         entry.
                  box.
   SW388R7
Data Analysis &   Requesting output for the first validation
                                  sample
 Computers II

   Slide 115




                                                            Click on the OK
                                                            button to
                                                            request the
                                                            output.




                    When the value entry
                    dialog box is closed, SPSS
                    adds the value we entered
                    after the equal sign. This
                    specification now tells      Since the validation analysis
                    SPSS to include in the       requires us to compare the
                    analysis only those cases    results of the analysis using
                    that have a value of 1 for   the first validation sample,
                    the split1 variable.         we will request the output
                                                 for the second validation
                                                 sample before doing any
                                                 comparison.
   SW388R7
Data Analysis &   Repeating the analysis with the second
                            validation sample
 Computers II

   Slide 116




                                   To repeat the principal
                                   component analysis for the
                                   second validation sample,
                                   select Factor Analysis from the
                                   Dialog Recall tool button.
   SW388R7
Data Analysis &
 Computers II     Removing split1 as the selection variable
   Slide 117




                                                       First, highlight
                                                       the Selection
                                                       Variable text box.




                             Second, click on the
                             left arrow button to
                             move the split1 back to
                             the list of variables.
   SW388R7
Data Analysis &
 Computers II           Using split2 as the selection variable
   Slide 118




                  First, scroll
                  down the list of
                  variables and
                  highlight the
                  variable split2.




                                                    Second, click on the
                                                    right arrow button to
                                                    move the split2 variable
                                                    to the Selection
                                                    Variable text box.
   SW388R7
Data Analysis &
 Computers II     Setting the value of split2 to select cases
   Slide 119




                  When the variable named
                  split2 is moved to the
                  Selection Variable text
                  box, SPSS adds "=?" after
                  the name to prompt up to
                  enter a specific value for
                  split2.                              Click on the
                                                       Value… button
                                                       to enter a
                                                       value for split2.
   SW388R7
Data Analysis &
 Computers II            Completing the value selection
   Slide 120




                  First, type the value          Second, click on the
                  for the second sample,         Continue button to
                  1, into the Value for          complete the value
                  Selection Variable text        entry.
                  box.
   SW388R7
Data Analysis &   Requesting output for the second validation
                                   sample
 Computers II

   Slide 121




                                                     Click on the OK
                                                     button to
                                                     request the
                                                     output.




                    When the value entry
                    dialog box is closed, SPSS
                    adds the value we entered
                    after the equal sign. This
                    specification now tells
                    SPSS to include in the
                    analysis only those cases
                    that have a value of 1 for
                    the split2 variable.
   SW388R7
Data Analysis &        Comparing the communalities for the
                               validation samples
 Computers II

   Slide 122




                     All of the communalities                         All of the communalities
                     for the first validation                         for the second validation
                     sample satisfy the                               sample satisfy the
                     minimum requirement of                           minimum requirement of
                     being larger than 0.50.                          being larger than 0.50.

                                 Communalitiesa                                     Communalitiesa

                                             Initial   Extraction                               Initial   Extraction
                  ENVIRONMENTAL                                      ENVIRONMENTAL
                  THREATS                      1.000        .631     THREATS                      1.000        .672
                  EXAGGERATED                                        EXAGGERATED
                  HOW DANGEROUS                                      HOW DANGEROUS
                  MODIFYING GENES IN           1.000        .648     MODIFYING GENES IN           1.000        .679
                  CROPS                                              CROPS
                  POOR COUNTRIES                                     POOR COUNTRIES
                  LESS THAN RICH FOR           1.000        .773     LESS THAN RICH FOR           1.000        .732
                  ENVIRONMENT                                        ENVIRONMENT
                  LIKELIHOOD OF                                      LIKELIHOOD OF
                  NUCLEAR MELTDOWN             1.000        .691     NUCLEAR MELTDOWN             1.000        .746
                  IN 5 YEARS                                         IN 5 YEARS
                  Extraction Method: Principal Component Analysis.   Extraction Method: Principal Component Analysis.
                     a. Only cases for which SPLIT2 = 1 are used        a. Only cases for which SPLIT1 = 1 are used
                        in the analysis phase.                             in the analysis phase.
   SW388R7
Data Analysis &     Comparing the factor loadings for the
                            validation samples
 Computers II

   Slide 123


                                                                           The factor loadings for the
                  The factor loadings for the first
                                                                           second validation analysis
                  validation analysis including all
                                                                           excluding outliers is shown on the
                  cases is shown on the left.
                                                                           right.
                                                  a,b
                           Rotated Component Matrix                                                     a,b
                                                                                 Rotated Component Matrix

                                                Component                                             Component
                                               1        2                                            1        2
                  ENVIRONMENTAL                                         ENVIRONMENTAL
                  THREATS                       .807        -.147       THREATS                      -.390        .692
                  EXAGGERATED                                           EXAGGERATED
                  HOW DANGEROUS                                         HOW DANGEROUS
                  MODIFYING GENES IN           -.198        .800        MODIFYING GENES IN            .795        -.123
                  CROPS                                                 CROPS
                  POOR COUNTRIES                                        POOR COUNTRIES
                  LESS THAN RICH FOR            .856        .007        LESS THAN RICH FOR            .187        .859
                  ENVIRONMENT                                           ENVIRONMENT
                  LIKELIHOOD OF                                         LIKELIHOOD OF
                  NUCLEAR MELTDOWN              .048        .862        NUCLEAR MELTDOWN              .829        .061
                  IN 5 YEARS                                            IN 5 YEARS
                  Extraction Method: Principal Component Analysis.       Extraction Method: Principal Component Analysis.
                            The pattern of factor loading for both
                  Rotation Method: Varimax with Kaiser Normalization. validation analyses shows theKaiser Normalization.
                                                                         Rotation Method: Varimax with
                            same pattern of iterations.
                     a. Rotation converged in 3variables, though the first and second component
                                                                            a. Rotation converged in 3 iterations.
                          have switched places.
                    b. Only cases for which SPLIT1 = 1 are used in        b. Only cases for which SPLIT2 = 1 are used in
                       the analysis phase.                                   the analysis phase.
                          The communalities and factor loadings of the validation analysis
                          supports the generalizability of the factor model.
   SW388R7
Data Analysis &
 Computers II                     Steps in validation analysis - 1
   Slide 124



                  The following is a guide to the decision process for answering
                  problems about validation analysis:

                                                      Is the number of valid
                                                      cases greater than or
                                 No                   equal to 100?                               Yes


                        •Set the first random seed and
                        compute the split1 variable                             •Set the random seed and
                        •Re-run factor with split1 = 1                          compute the split variable
                        •Set the second random seed                             •Re-run factor with split = 0
                        and compute the split2 variable                         •Re-run factor with split = 1
                        •Re-run factor with split2 = 1




                                                                    Yes


                                                     Are all of the
                                                                                                    No
                                                     communalities in the
                                                                                                                False
                                                     validations greater than
                                                     0.50?


                                                                    Yes
   SW388R7
Data Analysis &
 Computers II     Steps in validation analysis - 2
   Slide 125




                                        Yes


                            Does pattern of factor     No
                          loadings match pattern for        False
                                 full data set?



                                        Yes

                                    True
   SW388R7
Data Analysis &
 Computers II                    Steps in outlier analysis - 1
   Slide 126



                  The following is a guide to the decision process for answering
                  problems about outlier analysis:

                                           Are any of the factor
                                                                          No
                                           scores outliers (larger than
                                           ±3.0)?                              True




                                                           Yes


                                          Re-run factor analysis,
                                          excluding outliers


                                                          Yes


                                           Are all of the                 No
                                           communalities excluding             False
                                           outliers greater than 0.50?



                                                          Yes
   SW388R7
Data Analysis &
 Computers II     Steps in outlier analysis - 2
   Slide 127




                                       Yes


                        Pattern of factor loadings   No
                        excluding outliers match          False
                        pattern for full data set?



                                       Yes


                                   True
   SW388R7
Data Analysis &
 Computers II                    Steps in reliability analysis
   Slide 128



                  The following is a guide to the decision process for answering
                  problems about reliability analysis:

                                           Are Chronbach’s Alpha       No
                                           greater than 0.60 for all        False
                                           factors?



                                                           Yes


                                           Are Chronbach’s Alpha       No
                                           greater than 0.70 for all        True with caution
                                           factors?



                                                           Yes

                                                      True

				
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