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