Multivariate Analysis
Latent Class & Mixture Models


Introduction

Cluster analysis techniques are not the only way to find non-observed groupings in your data. In fact, from several perspectives cluster analysis may not be the best way to determine these groupings. There are several latent variable approaches that are available. In this unit we will explore two of them: Latent variable mixture models and latent class analysis.

The advantages of these approaches over cluster analysis are that they are model based, generating probabilities for group membership. It is possible to test these models and to analyze their goodness of fit. The downside to this approach is that it requires specialized software that is more complex to run than general purpose statistical packages. We will demonstrate these techniques using the Mplus from Muthén & Muthén. We will also use Stata for descriptive, subsidiary analyses and for an example of finite mixture modeling.

Latent variable mixture models will use continuous predictors and the latent class analysis will use binary predictor variables. We will the reading, writing, math, science and social studies test scores from the hsb6a dataset. For the binary predictor variables we will be median splits on each of the tests to create hiread, hiwrite, himath, hisci and hiss.

Looking at the data

use http://www.philender.com/courses/data/hsb6a, clear

describe

Contains data from hsb6a.dta
  obs:           600                          highschool and beyond (600
                                                cases)
 vars:            23                          24 Oct 2003 14:18
 size:        31,200 (99.0% of memory free)
-------------------------------------------------------------------------------
              storage  display     value
variable name   type   format      label      variable label
-------------------------------------------------------------------------------
id              int    %9.0g
gender          byte   %9.0g       gl
race            byte   %12.0g      rl
ses             byte   %9.0g       sl
sch             byte   %9.0g       scl
prog            byte   %9.0g       pl
locus           float  %9.0g                  locus of control
concept         float  %9.0g                  self-concept
mot             float  %9.0g                  motivation
career          byte   %14.0g      cl         career choice
read            float  %9.0g                  reading score
write           float  %9.0g                  writing score
math            float  %9.0g                  math score
sci             float  %9.0g                  science score
ss              float  %9.0g                  social studies score
hiread          byte   %9.0g
hiwrite         byte   %9.0g
himath          byte   %9.0g
hisci           byte   %9.0g
hiss            byte   %9.0g

sum read write math sci ss hiread hiwrite himath hisci hiss

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
        read |       600    51.90183    10.10298       28.3         76
       write |       600    52.38483    9.726455       25.5       67.1
        math |       600      51.849    9.414736       31.8       75.5
         sci |       600    51.76333    9.706179         26       74.2
          ss |       600    52.04567    9.879228       25.7       70.5
-------------+--------------------------------------------------------
      hiread |       600        .525    .4997913          0          1
     hiwrite |       600         .54    .4988133          0          1
      himath |       600    .4966667    .5004061          0          1
       hisci |       600    .5266667     .499705          0          1
        hiss |       600    .6483333     .477889          0          1

A 2 Class Latent Variable Mixture Model Using Mplus

Data:
    File is D:\mplus\data\hsb6.dat ;

  Variable:
    Names are
     id gender race ses sch prog locus concept mot career read write math
     sci ss hiread hiwrite himath hisci hiss;
    Usevariables are
       read write math sci ss;
    classes = c(2);
  Analysis:
    Type=mixture;

  MODEL:
    %C#1%
    [read write math sci ss * 30 ];
    %C#2%
    [read write math sci ss * 60 ];


SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                         600

Number of dependent variables                                    5
Number of independent variables                                  0
Number of continuous latent variables                            0
Number of categorical latent variables                           1

Observed dependent variables

  Continuous
   READ        WRITE       MATH        SCI         SS

Categorical latent variables
   C


Estimator                                                      MLR
Information matrix                                        OBSERVED
Optimization Specifications for the Quasi-Newton Algorithm for
Continuous Outcomes
  Maximum number of iterations                                1000
  Convergence criterion                                  0.100D-05
Optimization Specifications for the EM Algorithm
  Maximum number of iterations                                 500
  Convergence criteria
    Loglikelihood change                                 0.100D-06
    Relative loglikelihood change                        0.100D-06
    Derivative                                           0.100D-05
Optimization Specifications for the M step of the EM Algorithm for
Categorical Latent variables
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-05
  Basis for M step termination                           ITERATION
Optimization Specifications for the M step of the EM Algorithm for
Censored, Binary or Ordered Categorical (Ordinal), Unordered
Categorical (Nominal) and Count Outcomes
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-05
  Basis for M step termination                           ITERATION
  Maximum value for logit thresholds                            15
  Minimum value for logit thresholds                           -15
  Minimum expected cell size for chi-square              0.100D-01
Optimization algorithm                                         EMA
Random Starts Specifications
  Number of initial stage starts                                10
  Number of final stage starts                                   1
  Number of initial stage iterations                            10
  Initial stage convergence criterion                    0.100D+01
  Random starts scale                                    0.500D+01
  Random seed for generating random starts                       0

Input data file(s)
  D:\mplus\data\hsb6.dat
Input data format  FREE

Loglikelihood values at local maxima, seeds, and initial stage start numbers:

          -10490.737  285380           1


THE MODEL ESTIMATION TERMINATED NORMALLY


TESTS OF MODEL FIT

Loglikelihood

          H0 Value                      -10490.737

Information Criteria

          Number of Free Parameters             16
          Akaike (AIC)                   21013.474
          Bayesian (BIC)                 21083.825
          Sample-Size Adjusted BIC       21033.029
            (n* = (n + 2) / 24)
          Entropy                            0.853


FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASSES
BASED ON THE ESTIMATED MODEL

    Latent
   Classes
       1        274.08927          0.45682
       2        325.91073          0.54318


FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASS PATTERNS
BASED ON ESTIMATED POSTERIOR PROBABILITIES

    Latent
   Classes
       1        274.08958          0.45682
       2        325.91042          0.54318


CLASSIFICATION OF INDIVIDUALS BASED ON THEIR MOST LIKELY LATENT CLASS MEMBERSHIP

Class Counts and Proportions

    Latent
   Classes
       1              272          0.45333
       2              328          0.54667


Average Latent Class Probabilities for Most Likely Latent Class Membership (Row)
by Latent Class (Column)

          1        2
   1   0.957    0.043
   2   0.042    0.958


MODEL RESULTS

                   Estimates     S.E.  Est./S.E.

Latent Class 1

 Means
    READ              43.783    0.642     68.152
    WRITE             45.068    0.730     61.738
    MATH              44.794    0.469     95.540
    SCI               44.446    0.740     60.051
    SS                45.574    0.658     69.237

 Variances
    READ              46.463    2.785     16.681
    WRITE             49.427    3.011     16.415
    MATH              46.634    3.133     14.884
    SCI               49.022    3.388     14.470
    SS                62.215    4.109     15.141

Latent Class 2

 Means
    READ              58.730    0.605     97.000
    WRITE             58.538    0.497    117.764
    MATH              57.782    0.687     84.120
    SCI               57.917    0.499    116.079
    SS                57.488    0.589     97.629

 Variances
    READ              46.463    2.785     16.681
    WRITE             49.427    3.011     16.415
    MATH              46.634    3.133     14.884
    SCI               49.022    3.388     14.470
    SS                62.215    4.109     15.141

Categorical Latent Variables

 Means
    C#1               -0.173    0.133     -1.298


QUALITY OF NUMERICAL RESULTS

     Condition Number for the Information Matrix              0.383E-02
       (ratio of smallest to largest eigenvalue)

A 3 Class Latent Variable Mixture Model Using Mplus

Data:
    File is D:\mplus\data\hsb6.dat ;

  Variable:
    Names are
     id gender race ses sch prog locus concept mot career read write math
     sci ss hiread hiwrite himath hisci hiss;
    Usevariables are
       read write math sci ss;
    classes = c(3);
  Analysis:
    Type=mixture;

  MODEL:
    %C#1%
    [read write math sci ss * 30 ];
    %C#2%
    [read write math sci ss * 45 ];
    %C#3%
    [read write math sci ss * 60 ];


SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                         600

Number of dependent variables                                    5
Number of independent variables                                  0
Number of continuous latent variables                            0
Number of categorical latent variables                           1

Observed dependent variables

  Continuous
   READ        WRITE       MATH        SCI         SS

Categorical latent variables
   C

Estimator                                                      MLR
Information matrix                                        OBSERVED
Optimization Specifications for the Quasi-Newton Algorithm for
Continuous Outcomes
  Maximum number of iterations                                1000
  Convergence criterion                                  0.100D-05
Optimization Specifications for the EM Algorithm
  Maximum number of iterations                                 500
  Convergence criteria
    Loglikelihood change                                 0.100D-06
    Relative loglikelihood change                        0.100D-06
    Derivative                                           0.100D-05
Optimization Specifications for the M step of the EM Algorithm for
Categorical Latent variables
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-05
  Basis for M step termination                           ITERATION
Optimization Specifications for the M step of the EM Algorithm for
Censored, Binary or Ordered Categorical (Ordinal), Unordered
Categorical (Nominal) and Count Outcomes
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-05
  Basis for M step termination                           ITERATION
  Maximum value for logit thresholds                            15
  Minimum value for logit thresholds                           -15
  Minimum expected cell size for chi-square              0.100D-01
Optimization algorithm                                         EMA
Random Starts Specifications
  Number of initial stage starts                                10
  Number of final stage starts                                   1
  Number of initial stage iterations                            10
  Initial stage convergence criterion                    0.100D+01
  Random starts scale                                    0.500D+01
  Random seed for generating random starts                       0

Input data file(s)
  D:\mplus\data\hsb6.dat
Input data format  FREE

THE MODEL ESTIMATION TERMINATED NORMALLY



TESTS OF MODEL FIT

Loglikelihood

          H0 Value                      -10317.360

Information Criteria

          Number of Free Parameters             22
          Akaike (AIC)                   20678.719
          Bayesian (BIC)                 20775.451
          Sample-Size Adjusted BIC       20705.607
            (n* = (n + 2) / 24)
          Entropy                            0.830

FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASSES
BASED ON THE ESTIMATED MODEL

    Latent
   Classes
       1        194.55844          0.32426
       2        153.04166          0.25507
       3        252.39990          0.42067


FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASS PATTERNS
BASED ON ESTIMATED POSTERIOR PROBABILITIES

    Latent
   Classes
       1        194.55849          0.32426
       2        153.04160          0.25507
       3        252.39991          0.42067


CLASSIFICATION OF INDIVIDUALS BASED ON THEIR MOST LIKELY LATENT CLASS MEMBERSHIP

Class Counts and Proportions

    Latent
   Classes
       1              197          0.32833
       2              154          0.25667
       3              249          0.41500


Average Latent Class Probabilities for Most Likely Latent Class Membership (Row)
by Latent Class (Column)

          1        2        3
   1   0.940    0.000    0.060
   2   0.000    0.913    0.087
   3   0.038    0.050    0.912


MODEL RESULTS

                   Estimates     S.E.  Est./S.E.

Latent Class 1

 Means
    READ              41.735    0.477     87.542
    WRITE             42.703    0.962     44.395
    MATH              43.178    0.516     83.651
    SCI               42.160    0.663     63.627
    SS                43.848    0.695     63.101

 Variances
    READ              32.996    2.820     11.700
    WRITE             42.370    3.775     11.224
    MATH              34.562    2.422     14.269
    SCI               38.395    2.714     14.146
    SS                53.884    3.850     13.996

Latent Class 2

 Means
    READ              63.645    0.948     67.120
    WRITE             61.193    0.453    135.171
    MATH              62.610    0.865     72.405
    SCI               61.648    0.667     92.453
    SS                61.232    0.758     80.762

 Variances
    READ              32.996    2.820     11.700
    WRITE             42.370    3.775     11.224
    MATH              34.562    2.422     14.269
    SCI               38.395    2.714     14.146
    SS                53.884    3.850     13.996

Latent Class 3

 Means
    READ              52.618    0.925     56.872
    WRITE             54.507    0.727     74.942
    MATH              52.008    0.834     62.324
    SCI               53.172    0.835     63.687
    SS                52.794    0.808     65.328

 Variances
    READ              32.996    2.820     11.700
    WRITE             42.370    3.775     11.224
    MATH              34.562    2.422     14.269
    SCI               38.395    2.714     14.146
    SS                53.884    3.850     13.996

Categorical Latent Variables

 Means
    C#1               -0.260    0.130     -2.010
    C#2               -0.500    0.181     -2.767


QUALITY OF NUMERICAL RESULTS

     Condition Number for the Information Matrix              0.707E-02
       (ratio of smallest to largest eigenvalue)

A 2 Class Latent Class Model Using Mplus

Data:
    File is D:\mplus\data\hsb6.dat ;

  Variable:
    Names are
     id gender race ses sch prog locus concept mot career read write math
     sci ss hiread hiwrite himath hisci hiss;
    Usevariables are
       hiread hiwrite himath hisci hiss;
    categorical = hiread hiwrite himath hisci hiss;
    classes = c(2);
  Analysis:
    Type=mixture;

  MODEL:
    %C#1%
    [hiread$1 *2 hiwrite$1 *2 himath$1 *2 hisci$1 *2 hiss$1 *2 ];
    %C#2%
    [hiread$1 *-2 hiwrite$1 *-2 himath$1 *-2 hisci$1 *-2 hiss$1 *-2 ];

INPUT READING TERMINATED NORMALLY

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                         600

Number of dependent variables                                    5
Number of independent variables                                  0
Number of continuous latent variables                            0
Number of categorical latent variables                           1

Observed dependent variables

  Binary and ordered categorical (ordinal)
   HIREAD      HIWRITE     HIMATH      HISCI       HISS

Categorical latent variables
   C

Estimator                                                      MLR
Information matrix                                        OBSERVED
Optimization Specifications for the Quasi-Newton Algorithm for
Continuous Outcomes
  Maximum number of iterations                                1000
  Convergence criterion                                  0.100D-05
Optimization Specifications for the EM Algorithm
  Maximum number of iterations                                 500
  Convergence criteria
    Loglikelihood change                                 0.100D-06
    Relative loglikelihood change                        0.100D-06
    Derivative                                           0.100D-05
Optimization Specifications for the M step of the EM Algorithm for
Categorical Latent variables
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-05
  Basis for M step termination                           ITERATION
Optimization Specifications for the M step of the EM Algorithm for
Censored, Binary or Ordered Categorical (Ordinal), Unordered
Categorical (Nominal) and Count Outcomes
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-05
  Basis for M step termination                           ITERATION
  Maximum value for logit thresholds                            15
  Minimum value for logit thresholds                           -15
  Minimum expected cell size for chi-square              0.100D-01
Optimization algorithm                                         EMA
Random Starts Specifications
  Number of initial stage starts                                10
  Number of final stage starts                                   1
  Number of initial stage iterations                            10
  Initial stage convergence criterion                    0.100D+01
  Random starts scale                                    0.500D+01
  Random seed for generating random starts                       0

Input data file(s)
  D:\mplus\data\hsb6.dat
Input data format  FREE


SUMMARY OF CATEGORICAL DATA PROPORTIONS

    HIREAD
      Category 1    0.475
      Category 2    0.525
    HIWRITE
      Category 1    0.460
      Category 2    0.540
    HIMATH
      Category 1    0.503
      Category 2    0.497
    HISCI
      Category 1    0.473
      Category 2    0.527
    HISS
      Category 1    0.352
      Category 2    0.648


THE MODEL ESTIMATION TERMINATED NORMALLY

TESTS OF MODEL FIT

Loglikelihood

          H0 Value                       -1677.276

Information Criteria

          Number of Free Parameters             11
          Akaike (AIC)                    3376.552
          Bayesian (BIC)                  3424.918
          Sample-Size Adjusted BIC        3389.996
            (n* = (n + 2) / 24)
          Entropy                            0.828

Chi-Square Test of Model Fit for the Binary and Ordered Categorical
(Ordinal) Outcomes

          Pearson Chi-Square

          Value                             54.151
          Degrees of Freedom                    20
          P-Value                           0.0001

          Likelihood Ratio Chi-Square

          Value                             51.429
          Degrees of Freedom                    20
          P-Value                           0.0001


FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASSES
BASED ON THE ESTIMATED MODEL

    Latent
   Classes
       1        331.61601          0.55269
       2        268.38399          0.44731


FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASS PATTERNS
BASED ON ESTIMATED POSTERIOR PROBABILITIES

    Latent
   Classes
       1        331.61545          0.55269
       2        268.38455          0.44731


CLASSIFICATION OF INDIVIDUALS BASED ON THEIR MOST LIKELY LATENT CLASS MEMBERSHIP

Class Counts and Proportions

    Latent
   Classes
       1              334          0.55667
       2              266          0.44333


Average Latent Class Probabilities for Most Likely Latent Class Membership (Row)
by Latent Class (Column)

          1        2
   1   0.958    0.042
   2   0.044    0.956


MODEL RESULTS

                   Estimates     S.E.  Est./S.E.

Latent Class 1

 Thresholds
    HIREAD$1          -1.894    0.215     -8.811
    HIWRITE$1         -1.525    0.182     -8.365
    HIMATH$1          -1.339    0.169     -7.940
    HISCI$1           -1.599    0.168     -9.507
    HISS$1            -2.006    0.199    -10.054

Latent Class 2

 Thresholds
    HIREAD$1           2.201    0.296      7.430
    HIWRITE$1          1.434    0.181      7.901
    HIMATH$1           1.888    0.224      8.436
    HISCI$1            1.738    0.242      7.181
    HISS$1             0.574    0.145      3.953

Categorical Latent Variables

 Means
    C#1                0.212    0.108      1.955

RESULTS IN PROBABILITY SCALE

Latent Class 1

 HIREAD
    Category 1         0.131    0.024      5.351
    Category 2         0.869    0.024     35.574
 HIWRITE
    Category 1         0.179    0.027      6.681
    Category 2         0.821    0.027     30.689
 HIMATH
    Category 1         0.208    0.028      7.487
    Category 2         0.792    0.028     28.554
 HISCI
    Category 1         0.168    0.024      7.150
    Category 2         0.832    0.024     35.363
 HISS
    Category 1         0.119    0.021      5.687
    Category 2         0.881    0.021     42.262

Latent Class 2

 HIREAD
    Category 1         0.900    0.027     33.878
    Category 2         0.100    0.027      3.749
 HIWRITE
    Category 1         0.807    0.028     28.625
    Category 2         0.193    0.028      6.824
 HIMATH
    Category 1         0.869    0.026     33.994
    Category 2         0.131    0.026      5.143
 HISCI
    Category 1         0.850    0.031     27.621
    Category 2         0.150    0.031      4.860
 HISS
    Category 1         0.640    0.033     19.119
    Category 2         0.360    0.033     10.771


ODDS RATIO RESULTS

Latent Class 1 Compared to Latent Class 2

 HIREAD
    Category > 1      60.079   20.083      2.991
 HIWRITE
    Category > 1      19.269    4.750      4.057
 HIMATH
    Category > 1      25.208    6.675      3.776
 HISCI
    Category > 1      28.112    7.882      3.566
 HISS
    Category > 1      13.189    3.164      4.168


QUALITY OF NUMERICAL RESULTS

     Condition Number for the Information Matrix              0.736E-01
       (ratio of smallest to largest eigenvalue)

A 3 Class Latent Class Model Using Mplus

Data:
    File is D:\mplus\data\hsb6.dat ;

  Variable:
    Names are
     id gender race ses sch prog locus concept mot career read write math
     sci ss hiread hiwrite himath hisci hiss;
    Usevariables are
       hiread hiwrite himath hisci hiss;
    categorical = hiread hiwrite himath hisci hiss;
    classes = c(3);
  Analysis:
    Type=mixture;

  MODEL:
    %C#1%
    [hiread$1 *2 hiwrite$1 *2 himath$1 *2 hisci$1 *2 hiss$1 *2 ];
    %C#2%
    [hiread$1 *0 hiwrite$1 *0 himath$1 *0 hisci$1 *0 hiss$1 *0 ];
    %C#3%
    [hiread$1 *-2 hiwrite$1 *-2 himath$1 *-2 hisci$1 *-2 hiss$1 *-2 ];

INPUT READING TERMINATED NORMALLY

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                         600

Number of dependent variables                                    5
Number of independent variables                                  0
Number of continuous latent variables                            0
Number of categorical latent variables                           1

Observed dependent variables

  Binary and ordered categorical (ordinal)
   HIREAD      HIWRITE     HIMATH      HISCI       HISS

Categorical latent variables
   C

Estimator                                                      MLR
Information matrix                                        OBSERVED
Optimization Specifications for the Quasi-Newton Algorithm for
Continuous Outcomes
  Maximum number of iterations                                1000
  Convergence criterion                                  0.100D-05
Optimization Specifications for the EM Algorithm
  Maximum number of iterations                                 500
  Convergence criteria
    Loglikelihood change                                 0.100D-06
    Relative loglikelihood change                        0.100D-06
    Derivative                                           0.100D-05
Optimization Specifications for the M step of the EM Algorithm for
Categorical Latent variables
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-05
  Basis for M step termination                           ITERATION
Optimization Specifications for the M step of the EM Algorithm for
Censored, Binary or Ordered Categorical (Ordinal), Unordered
Categorical (Nominal) and Count Outcomes
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-05
  Basis for M step termination                           ITERATION
  Maximum value for logit thresholds                            15
  Minimum value for logit thresholds                           -15
  Minimum expected cell size for chi-square              0.100D-01
Optimization algorithm                                         EMA
Random Starts Specifications
  Number of initial stage starts                                10
  Number of final stage starts                                   1
  Number of initial stage iterations                            10
  Initial stage convergence criterion                    0.100D+01
  Random starts scale                                    0.500D+01
  Random seed for generating random starts                       0

Input data file(s)
  D:\mplus\data\hsb6.dat
Input data format  FREE

SUMMARY OF CATEGORICAL DATA PROPORTIONS

    HIREAD
      Category 1    0.475
      Category 2    0.525
    HIWRITE
      Category 1    0.460
      Category 2    0.540
    HIMATH
      Category 1    0.503
      Category 2    0.497
    HISCI
      Category 1    0.473
      Category 2    0.527
    HISS
      Category 1    0.352
      Category 2    0.648


     IN THE OPTIMIZATION, ONE OR MORE LOGIT THRESHOLDS APPROACHED AND WERE SET
     AT THE EXTREME VALUES.  EXTREME VALUES ARE -15.000 AND 15.000.
     THE FOLLOWING THRESHOLDS WERE SET AT THESE VALUES:
     * THRESHOLD 1 OF CLASS INDICATOR HIWRITE FOR CLASS 3 AT ITERATION 65

THE MODEL ESTIMATION TERMINATED NORMALLY

TESTS OF MODEL FIT

Loglikelihood

          H0 Value                       -1661.060

Information Criteria

          Number of Free Parameters             17
          Akaike (AIC)                    3356.120
          Bayesian (BIC)                  3430.867
          Sample-Size Adjusted BIC        3376.897
            (n* = (n + 2) / 24)
          Entropy                            0.675

Chi-Square Test of Model Fit for the Binary and Ordered Categorical
(Ordinal) Outcomes

          Pearson Chi-Square

          Value                             17.791
          Degrees of Freedom                    14
          P-Value                           0.2165

          Likelihood Ratio Chi-Square

          Value                             18.996
          Degrees of Freedom                    14
          P-Value                           0.1651


FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASSES
BASED ON THE ESTIMATED MODEL

    Latent
   Classes
       1        220.03955          0.36673
       2        197.52264          0.32920
       3        182.43781          0.30406


FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASS PATTERNS
BASED ON ESTIMATED POSTERIOR PROBABILITIES

    Latent
   Classes
       1        220.03954          0.36673
       2        197.52262          0.32920
       3        182.43784          0.30406


CLASSIFICATION OF INDIVIDUALS BASED ON THEIR MOST LIKELY LATENT CLASS MEMBERSHIP

Class Counts and Proportions

    Latent
   Classes
       1              229          0.38167
       2              175          0.29167
       3              196          0.32667


Average Latent Class Probabilities for Most Likely Latent Class Membership (Row)
by Latent Class (Column)

          1        2        3
   1   0.905    0.094    0.001
   2   0.072    0.818    0.110
   3   0.000    0.168    0.832


MODEL RESULTS

                   Estimates     S.E.  Est./S.E.

Latent Class 1

 Thresholds
    HIREAD$1           3.031    0.554      5.475
    HIWRITE$1          1.596    0.270      5.902
    HIMATH$1           2.202    0.321      6.869
    HISCI$1            2.580    0.488      5.284
    HISS$1             0.754    0.194      3.894

Latent Class 2

 Thresholds
    HIREAD$1          -0.656    0.303     -2.162
    HIWRITE$1         -0.116    0.504     -0.230
    HIMATH$1          -0.105    0.274     -0.384
    HISCI$1           -0.937    0.318     -2.943
    HISS$1            -1.049    0.340     -3.084

Latent Class 3

 Thresholds
    HIREAD$1          -3.134    1.336     -2.346
    HIWRITE$1        -15.000    0.000      0.000
    HIMATH$1          -2.815    1.340     -2.100
    HISCI$1           -1.895    0.488     -3.884
    HISS$1            -2.833    0.625     -4.532

Categorical Latent Variables

 Means
    C#1                0.187    0.256      0.732
    C#2                0.079    0.483      0.165


RESULTS IN PROBABILITY SCALE

Latent Class 1

 HIREAD
    Category 1         0.954    0.024     39.239
    Category 2         0.046    0.024      1.893
 HIWRITE
    Category 1         0.831    0.038     21.940
    Category 2         0.169    0.038      4.449
 HIMATH
    Category 1         0.900    0.029     31.321
    Category 2         0.100    0.029      3.465
 HISCI
    Category 1         0.930    0.032     29.083
    Category 2         0.070    0.032      2.203
 HISS
    Category 1         0.680    0.042     16.146
    Category 2         0.320    0.042      7.600

Latent Class 2

 HIREAD
    Category 1         0.342    0.068      5.005
    Category 2         0.658    0.068      9.646
 HIWRITE
    Category 1         0.471    0.125      3.755
    Category 2         0.529    0.125      4.216
 HIMATH
    Category 1         0.474    0.068      6.938
    Category 2         0.526    0.068      7.708
 HISCI
    Category 1         0.282    0.064      4.375
    Category 2         0.718    0.064     11.160
 HISS
    Category 1         0.259    0.065      3.971
    Category 2         0.741    0.065     11.335

Latent Class 3

 HIREAD
    Category 1         0.042    0.053      0.781
    Category 2         0.958    0.053     17.940
 HIWRITE
    Category 1         0.000    0.000      0.000
    Category 2         1.000    0.000      0.000
 HIMATH
    Category 1         0.057    0.071      0.791
    Category 2         0.943    0.071     13.202
 HISCI
    Category 1         0.131    0.055      2.357
    Category 2         0.869    0.055     15.686
 HISS
    Category 1         0.056    0.033      1.694
    Category 2         0.944    0.033     28.795


ODDS RATIO RESULTS

Latent Class 1 Compared to Latent Class 2

 HIREAD
    Category > 1       0.025    0.015      1.647
 HIWRITE
    Category > 1       0.181    0.121      1.491
 HIMATH
    Category > 1       0.100    0.045      2.206
 HISCI
    Category > 1       0.030    0.017      1.746
 HISS
    Category > 1       0.165    0.071      2.317

Latent Class 1 Compared to Latent Class 3

 HIREAD
    Category > 1       0.002    0.003      0.700
 HIWRITE
    Category > 1       0.000    0.000    999.000
 HIMATH
    Category > 1       0.007    0.009      0.722
 HISCI
    Category > 1       0.011    0.007      1.589
 HISS
    Category > 1       0.028    0.019      1.471

Latent Class 2 Compared to Latent Class 3

 HIREAD
    Category > 1       0.084    0.119      0.706
 HIWRITE
    Category > 1       0.000    0.000    999.000
 HIMATH
    Category > 1       0.067    0.088      0.753
 HISCI
    Category > 1       0.383    0.274      1.399
 HISS
    Category > 1       0.168    0.120      1.395


QUALITY OF NUMERICAL RESULTS

     Condition Number for the Information Matrix              0.314E-02
       (ratio of smallest to largest eigenvalue)


Multivariate Course Page

Phil Ender, 24apr03