Mplus VERSION 8
MUTHEN & MUTHEN
04/10/2017   5:06 AM

INPUT INSTRUCTIONS

  TITLE:	this is an example of a two-level time
  	series analysis with a univariate first-order autoregressive AR(1) model
      for a continuous dependent variable with a covariate, random intercept,
      random AR(1) slope, random slope, and random residual variance
  DATA:	FILE = ex9.31.dat;
  VARIABLE:	NAMES = y x w xm subject;
  	WITHIN = x;
  	BETWEEN = w xm;
  	CLUSTER = subject;
  	LAGGED = y(1);
  DEFINE:	CENTER X (GROUPMEAN);
  ANALYSIS:	TYPE = TWOLEVEL RANDOM;
  	ESTIMATOR = BAYES;
  	PROCESSORS = 2;
  	BITERATIONS = (2000);
  MODEL:	%WITHIN%
  	sy | y ON y&1;
  	sx | y ON x;
  	logv | y;
  	%BETWEEN%
  	y ON w xm;
  	sy ON w xm;
  	sx ON w xm;
  	logv ON w xm;
  	y sy sx logv WITH y sy sx logv;
  OUTPUT:	TECH1 TECH8;
  PLOT:	TYPE= PLOT3;



INPUT READING TERMINATED NORMALLY



this is an example of a two-level time
series analysis with a univariate first-order autoregressive AR(1) model
for a continuous dependent variable with a covariate, random intercept,
random AR(1) slope, random slope, and random residual variance

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                        5000

Number of dependent variables                                    1
Number of independent variables                                  4
Number of continuous latent variables                            3

Observed dependent variables

  Continuous
   Y

Observed independent variables
   X           W           XM          Y&1

Continuous latent variables
   SY          SX          LOGV

Variables with special functions

  Cluster variable      SUBJECT

  Within variables
   X           Y&1

  Between variables
   W           XM

  Centering (GROUPMEAN)
   X


Estimator                                                    BAYES
Specifications for Bayesian Estimation
  Point estimate                                            MEDIAN
  Number of Markov chain Monte Carlo (MCMC) chains               2
  Random seed for the first chain                                0
  Starting value information                           UNPERTURBED
  Treatment of categorical mediator                         LATENT
  Algorithm used for Markov chain Monte Carlo           GIBBS(PX1)
  Convergence criterion                                  0.500D-01
  Maximum number of iterations                               50000
  K-th iteration used for thinning                               1

Input data file(s)
  ex9.31.dat
Input data format  FREE


SUMMARY OF DATA

     Number of clusters                        100

       Size (s)    Cluster ID with Size s

         50        1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
                   22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
                   40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57
                   58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
                   76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93
                   94 95 96 97 98 99 100



COVARIANCE COVERAGE OF DATA

Minimum covariance coverage value   0.100

     Number of missing data patterns             2


     PROPORTION OF DATA PRESENT


           Covariance Coverage
              Y             X             W             XM
              ________      ________      ________      ________
 Y              1.000
 X              1.000         1.000
 W              1.000         1.000         1.000
 XM             1.000         1.000         1.000         1.000



UNIVARIATE SAMPLE STATISTICS


     UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS

         Variable/         Mean/     Skewness/   Minimum/ % with                Percentiles
        Sample Size      Variance    Kurtosis    Maximum  Min/Max      20%/60%    40%/80%    Median

     Y                     0.013       0.287     -10.377    0.02%      -1.551     -0.526     -0.098
            5000.000       4.351       1.600      10.792    0.02%       0.390      1.530
     X                     0.000       0.017      -3.681    0.02%      -0.966     -0.305      0.006
            5000.000       1.241      -0.140       4.237    0.02%       0.284      0.952
     W                    -0.055       0.514      -2.098    1.00%      -0.760     -0.416     -0.285
             100.000       0.744       0.072       2.415    1.00%       0.022      0.682
     XM                    0.136      -0.054      -2.490    1.00%      -0.741     -0.031      0.173
             100.000       0.963       0.139       2.544    1.00%       0.418      0.829


THE MODEL ESTIMATION TERMINATED NORMALLY

     USE THE FBITERATIONS OPTION TO INCREASE THE NUMBER OF ITERATIONS BY A FACTOR
     OF AT LEAST TWO TO CHECK CONVERGENCE AND THAT THE PSR VALUE DOES NOT INCREASE.



MODEL FIT INFORMATION

Number of Free Parameters                              22

Information Criteria

          Deviance (DIC)                        14729.136
          Estimated Number of Parameters (pD)     318.156



MODEL RESULTS

                                Posterior  One-Tailed         95% C.I.
                    Estimate       S.D.      P-Value   Lower 2.5%  Upper 2.5%  Significance

Within Level

Between Level

 SY         ON
    W                  0.120       0.016      0.000       0.090       0.151      *
    XM                 0.054       0.014      0.000       0.027       0.082      *

 SX         ON
    W                  0.289       0.101      0.004       0.090       0.483      *
    XM                 0.343       0.089      0.001       0.174       0.519      *

 LOGV       ON
    W                  0.325       0.046      0.000       0.234       0.412      *
    XM                 0.053       0.041      0.086      -0.024       0.137

 Y          ON
    W                  0.493       0.092      0.000       0.305       0.666      *
    XM                 0.392       0.082      0.000       0.236       0.556      *

 Y        WITH
    SY                -0.002       0.011      0.401      -0.026       0.020
    SX                -0.067       0.074      0.157      -0.221       0.070
    LOGV              -0.006       0.034      0.431      -0.074       0.062

 SY       WITH
    SX                -0.009       0.012      0.210      -0.034       0.014
    LOGV               0.002       0.006      0.352      -0.009       0.014

 SX       WITH
    LOGV               0.000       0.035      0.495      -0.068       0.067

 Intercepts
    Y                 -0.012       0.080      0.444      -0.172       0.145
    SY                 0.190       0.014      0.000       0.163       0.217      *
    SX                 0.772       0.086      0.000       0.605       0.939      *
    LOGV               0.053       0.039      0.080      -0.022       0.130

 Residual Variances
    Y                  0.592       0.099      0.000       0.433       0.823      *
    SY                 0.008       0.003      0.000       0.004       0.015      *
    SX                 0.697       0.114      0.000       0.520       0.957      *
    LOGV               0.104       0.023      0.000       0.066       0.155      *


TECHNICAL 1 OUTPUT


     PARAMETER SPECIFICATION FOR WITHIN


           NU
              Y             X             Y&1
              ________      ________      ________
                    0             0             0


           LAMBDA
              Y             X             Y&1
              ________      ________      ________
 Y                  0             0             0
 X                  0             0             0
 Y&1                0             0             0


           THETA
              Y             X             Y&1
              ________      ________      ________
 Y                  0
 X                  0             0
 Y&1                0             0             0


           ALPHA
              Y             X             Y&1
              ________      ________      ________
                    0             0             0


           BETA
              Y             X             Y&1
              ________      ________      ________
 Y                  0             0             0
 X                  0             0             0
 Y&1                0             0             0


           PSI
              Y             X             Y&1
              ________      ________      ________
 Y                  0
 X                  0             0
 Y&1                0             0             0


     PARAMETER SPECIFICATION FOR BETWEEN


           NU
              Y             W             XM
              ________      ________      ________
                    0             0             0


           LAMBDA
              SY            SX            LOGV          Y             W
              ________      ________      ________      ________      ________
 Y                  0             0             0             0             0
 W                  0             0             0             0             0
 XM                 0             0             0             0             0


           LAMBDA
              XM
              ________
 Y                  0
 W                  0
 XM                 0


           THETA
              Y             W             XM
              ________      ________      ________
 Y                  0
 W                  0             0
 XM                 0             0             0


           ALPHA
              SY            SX            LOGV          Y             W
              ________      ________      ________      ________      ________
                    1             2             3             4             0


           ALPHA
              XM
              ________
                    0


           BETA
              SY            SX            LOGV          Y             W
              ________      ________      ________      ________      ________
 SY                 0             0             0             0             5
 SX                 0             0             0             0             7
 LOGV               0             0             0             0             9
 Y                  0             0             0             0            11
 W                  0             0             0             0             0
 XM                 0             0             0             0             0


           BETA
              XM
              ________
 SY                 6
 SX                 8
 LOGV              10
 Y                 12
 W                  0
 XM                 0


           PSI
              SY            SX            LOGV          Y             W
              ________      ________      ________      ________      ________
 SY                13
 SX                14            15
 LOGV              16            17            18
 Y                 19            20            21            22
 W                  0             0             0             0             0
 XM                 0             0             0             0             0


           PSI
              XM
              ________
 XM                 0


     STARTING VALUES FOR WITHIN


           NU
              Y             X             Y&1
              ________      ________      ________
                0.000         0.000         0.000


           LAMBDA
              Y             X             Y&1
              ________      ________      ________
 Y              1.000         0.000         0.000
 X              0.000         1.000         0.000
 Y&1            0.000         0.000         1.000


           THETA
              Y             X             Y&1
              ________      ________      ________
 Y              0.000
 X              0.000         0.000
 Y&1            0.000         0.000         0.000


           ALPHA
              Y             X             Y&1
              ________      ________      ________
                0.000         0.000         0.000


           BETA
              Y             X             Y&1
              ________      ________      ________
 Y              0.000         0.000         0.000
 X              0.000         0.000         0.000
 Y&1            0.000         0.000         0.000


           PSI
              Y             X             Y&1
              ________      ________      ________
 Y              0.000
 X              0.000         0.620
 Y&1            0.000         0.000         2.175


     STARTING VALUES FOR BETWEEN


           NU
              Y             W             XM
              ________      ________      ________
                0.000         0.000         0.000


           LAMBDA
              SY            SX            LOGV          Y             W
              ________      ________      ________      ________      ________
 Y              0.000         0.000         0.000         1.000         0.000
 W              0.000         0.000         0.000         0.000         1.000
 XM             0.000         0.000         0.000         0.000         0.000


           LAMBDA
              XM
              ________
 Y              0.000
 W              0.000
 XM             1.000


           THETA
              Y             W             XM
              ________      ________      ________
 Y              0.000
 W              0.000         0.000
 XM             0.000         0.000         0.000


           ALPHA
              SY            SX            LOGV          Y             W
              ________      ________      ________      ________      ________
                0.000         0.000         0.000         0.013         0.000


           ALPHA
              XM
              ________
                0.000


           BETA
              SY            SX            LOGV          Y             W
              ________      ________      ________      ________      ________
 SY             0.000         0.000         0.000         0.000         0.000
 SX             0.000         0.000         0.000         0.000         0.000
 LOGV           0.000         0.000         0.000         0.000         0.000
 Y              0.000         0.000         0.000         0.000         0.000
 W              0.000         0.000         0.000         0.000         0.000
 XM             0.000         0.000         0.000         0.000         0.000


           BETA
              XM
              ________
 SY             0.000
 SX             0.000
 LOGV           0.000
 Y              0.000
 W              0.000
 XM             0.000


           PSI
              SY            SX            LOGV          Y             W
              ________      ________      ________      ________      ________
 SY             1.000
 SX             0.000         1.000
 LOGV           0.000         0.000         1.000
 Y              0.000         0.000         0.000         2.175
 W              0.000         0.000         0.000         0.000         0.372
 XM             0.000         0.000         0.000         0.000         0.000


           PSI
              XM
              ________
 XM             0.481



     PRIORS FOR ALL PARAMETERS            PRIOR MEAN      PRIOR VARIANCE     PRIOR STD. DEV.

     Parameter 1~N(0.000,infinity)           0.0000            infinity            infinity
     Parameter 2~N(0.000,infinity)           0.0000            infinity            infinity
     Parameter 3~N(0.000,infinity)           0.0000            infinity            infinity
     Parameter 4~N(0.000,infinity)           0.0000            infinity            infinity
     Parameter 5~N(0.000,infinity)           0.0000            infinity            infinity
     Parameter 6~N(0.000,infinity)           0.0000            infinity            infinity
     Parameter 7~N(0.000,infinity)           0.0000            infinity            infinity
     Parameter 8~N(0.000,infinity)           0.0000            infinity            infinity
     Parameter 9~N(0.000,infinity)           0.0000            infinity            infinity
     Parameter 10~N(0.000,infinity)          0.0000            infinity            infinity
     Parameter 11~N(0.000,infinity)          0.0000            infinity            infinity
     Parameter 12~N(0.000,infinity)          0.0000            infinity            infinity
     Parameter 13~IW(0.000,-5)             infinity            infinity            infinity
     Parameter 14~IW(0.000,-5)             infinity            infinity            infinity
     Parameter 15~IW(0.000,-5)             infinity            infinity            infinity
     Parameter 16~IW(0.000,-5)             infinity            infinity            infinity
     Parameter 17~IW(0.000,-5)             infinity            infinity            infinity
     Parameter 18~IW(0.000,-5)             infinity            infinity            infinity
     Parameter 19~IW(0.000,-5)             infinity            infinity            infinity
     Parameter 20~IW(0.000,-5)             infinity            infinity            infinity
     Parameter 21~IW(0.000,-5)             infinity            infinity            infinity
     Parameter 22~IW(0.000,-5)             infinity            infinity            infinity


TECHNICAL 8 OUTPUT



     Kolmogorov-Smirnov comparing posterior distributions across chains 1 and 2 using 100 draws.





     Parameter   KS Statistic P-value
     Parameter 11    0.1400    0.2606
     Parameter 22    0.1100    0.5560
     Parameter 12    0.1000    0.6766
     Parameter 2    0.1000    0.6766
     Parameter 10    0.0700    0.9610
     Parameter 8    0.0600    0.9921
     Parameter 9    0.0600    0.9921
     Parameter 15    0.0500    0.9995
     Parameter 20    0.0500    0.9995
     Parameter 4    0.0400    1.0000
     Parameter 7    0.0300    1.0000
     Parameter 17    0.0200    1.0000
     Parameter 3    0.0200    1.0000
     Parameter 21    0.0200    1.0000
     Parameter 14    0.0100    1.0000
     Parameter 18    0.0100    1.0000
     Parameter 1    0.0100    1.0000
     Parameter 5    0.0100    1.0000
     Parameter 6    0.0100    1.0000
     Parameter 13    0.0000    1.0000
     Parameter 16    0.0000    1.0000
     Parameter 19    0.0000    1.0000



     Simulated prior distributions

     Parameter       Prior Mean  Prior Variance  Prior Std. Dev.


     Parameter 1 Improper Prior
     Parameter 2 Improper Prior
     Parameter 3 Improper Prior
     Parameter 4 Improper Prior
     Parameter 5 Improper Prior
     Parameter 6 Improper Prior
     Parameter 7 Improper Prior
     Parameter 8 Improper Prior
     Parameter 9 Improper Prior
     Parameter 10 Improper Prior
     Parameter 11 Improper Prior
     Parameter 12 Improper Prior
     Parameter 13 Improper Prior
     Parameter 14 Improper Prior
     Parameter 15 Improper Prior
     Parameter 16 Improper Prior
     Parameter 17 Improper Prior
     Parameter 18 Improper Prior
     Parameter 19 Improper Prior
     Parameter 20 Improper Prior
     Parameter 21 Improper Prior
     Parameter 22 Improper Prior


   TECHNICAL 8 OUTPUT FOR BAYES ESTIMATION

     CHAIN    BSEED
     1        0
     2        285380

                     POTENTIAL       PARAMETER WITH
     ITERATION    SCALE REDUCTION      HIGHEST PSR
     100              1.221               1
     200              1.121               5
     300              1.028               10
     400              1.018               21
     500              1.028               17
     600              1.014               21
     700              1.008               21
     800              1.010               16
     900              1.017               21
     1000             1.024               21
     1100             1.012               16
     1200             1.010               21
     1300             1.011               21
     1400             1.006               21
     1500             1.003               17
     1600             1.003               9
     1700             1.009               9
     1800             1.008               9
     1900             1.007               10
     2000             1.004               9


PLOT INFORMATION

The following plots are available:

  Histograms (sample values)
  Scatterplots (sample values)
  Between-level histograms (sample values, sample means/variances)
  Between-level scatterplots (sample values, sample means/variances)
  Time series plots (sample values, ACF, PACF)
  Bayesian posterior parameter distributions
  Bayesian posterior parameter trace plots
  Bayesian autocorrelation plots

     Beginning Time:  05:06:54
        Ending Time:  05:07:19
       Elapsed Time:  00:00:25



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