```                  Eyesight example: test symmetry                  1
19:28 Wednesday, November 29, 2017

The FREQ Procedure

Table of r by l

r         l

Frequency|       1|       2|       3|       4|  Total
---------+--------+--------+--------+--------+
1 |    492 |    179 |     82 |     36 |    789
---------+--------+--------+--------+--------+
2 |    205 |   1772 |    362 |    117 |   2456
---------+--------+--------+--------+--------+
3 |     78 |    432 |   1512 |    234 |   2256
---------+--------+--------+--------+--------+
4 |     66 |    124 |    266 |   1520 |   1976
---------+--------+--------+--------+--------+
Total         841     2507     2222     1907     7477

Statistics for Table of r by l

Test of Symmetry
------------------------
Statistic (S)    19.1066
DF                     6
Pr > S            0.0040

Kappa Statistics

Statistic          Value       ASE     95% Confidence Limits
------------------------------------------------------------
Simple Kappa      0.5954    0.0073       0.5811       0.6097
Weighted Kappa    0.6524    0.0071       0.6385       0.6662

Sample Size = 7477
Eyesight example: test symmetry                  2
19:28 Wednesday, November 29, 2017

The FREQ Procedure

Table of w97 by w05

w97       w05

Frequency|
Percent  |
Row Pct  |
Col Pct  |C       |L       |LD      |RU      |  Total
---------+--------+--------+--------+--------+
C        |     12 |      0 |      0 |      0 |     12
|  16.22 |   0.00 |   0.00 |   0.00 |  16.22
| 100.00 |   0.00 |   0.00 |   0.00 |
|  57.14 |   0.00 |   0.00 |   0.00 |
---------+--------+--------+--------+--------+
L        |      9 |     44 |      2 |      1 |     56
|  12.16 |  59.46 |   2.70 |   1.35 |  75.68
|  16.07 |  78.57 |   3.57 |   1.79 |
|  42.86 | 100.00 |  25.00 | 100.00 |
---------+--------+--------+--------+--------+
LD       |      0 |      0 |      6 |      0 |      6
|   0.00 |   0.00 |   8.11 |   0.00 |   8.11
|   0.00 |   0.00 | 100.00 |   0.00 |
|   0.00 |   0.00 |  75.00 |   0.00 |
---------+--------+--------+--------+--------+
Total          21       44        8        1       74
28.38    59.46    10.81     1.35   100.00
Eyesight example: test symmetry                  3
19:28 Wednesday, November 29, 2017

The FREQ Procedure

Table of w97 by w05c

w97       w05c

Frequency|
Percent  |
Row Pct  |
Col Pct  |C       |L       |LD      |  Total
---------+--------+--------+--------+
C        |     12 |      0 |      0 |     12
|  16.22 |   0.00 |   0.00 |  16.22
| 100.00 |   0.00 |   0.00 |
|  57.14 |   0.00 |   0.00 |
---------+--------+--------+--------+
L        |      9 |     45 |      2 |     56
|  12.16 |  60.81 |   2.70 |  75.68
|  16.07 |  80.36 |   3.57 |
|  42.86 | 100.00 |  25.00 |
---------+--------+--------+--------+
LD       |      0 |      0 |      6 |      6
|   0.00 |   0.00 |   8.11 |   8.11
|   0.00 |   0.00 | 100.00 |
|   0.00 |   0.00 |  75.00 |
---------+--------+--------+--------+
Total          21       45        8       74
28.38    60.81    10.81   100.00

Statistics for Table of w97 by w05c

Test of Symmetry
------------------------
Statistic (S)    11.0000
DF                     3
Pr > S            0.0117

Kappa Statistics

Statistic          Value       ASE     95% Confidence Limits
------------------------------------------------------------
Simple Kappa      0.6935    0.0829       0.5311       0.8560
Weighted Kappa    0.7172    0.0789       0.5625       0.8719

Sample Size = 74
Eyesight example: fit symmetry via genmod             4
19:28 Wednesday, November 29, 2017

The GENMOD Procedure

Model Information

Data Set              WORK.NEWEYES
Distribution               Poisson
Dependent Variable           count

Number of Observations Used          16

Class Level Information

Class      Levels    Values

md             10    -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

Criteria For Assessing Goodness Of Fit

Criterion                     DF           Value        Value/DF

Deviance                       6         19.2492          3.2082
Scaled Deviance                6         19.2492          3.2082
Pearson Chi-Square             6         19.1065          3.1844
Scaled Pearson X2              6         19.1065          3.1844
Log Likelihood                        42820.6478
Full Log Likelihood                     -68.3141
AIC (smaller is better)                 156.6282
AICC (smaller is better)                200.6282
BIC (smaller is better)                 164.3541

Algorithm converged.

Analysis Of Maximum Likelihood Parameter Estimates

Standard   Wald 95% Confidence
Parameter         DF   Estimate      Error          Limits

Intercept          0     0.0000     0.0000     0.0000     0.0000
md          -10    1     5.5215     0.0447     5.4338     5.6091
md          -9     1     4.7916     0.0644     4.6654     4.9179
md          -8     1     5.9839     0.0355     5.9144     6.0535
md          -7     1     3.9318     0.0990     3.7378     4.1259
md          -6     1     4.3820     0.0791     4.2271     4.5370
md          -5     1     5.2575     0.0510     5.1575     5.3575
md          -4     1     7.3265     0.0256     7.2762     7.3767
md          -3     1     7.3212     0.0257     7.2708     7.3716
md          -2     1     7.4799     0.0238     7.4333     7.5264
md          -1     1     6.1985     0.0451     6.1101     6.2868
Scale              0     1.0000     0.0000     1.0000     1.0000

Analysis Of Maximum Likelihood
Parameter Estimates

Wald
Parameter         Chi-Square    Pr > ChiSq

Intercept                .           .
md          -10      15243.3        <.0001
md          -9       5533.34        <.0001
md          -8       28431.1        <.0001
md          -7       1576.84        <.0001
md          -6       3072.35        <.0001
md          -5       10614.2        <.0001
md          -4       81589.2        <.0001
md          -3       81042.9        <.0001
md          -2       99140.5        <.0001
md          -1       18903.2        <.0001
Scale

NOTE: The scale parameter was held fixed.

Lagrange Multiplier Statistics

Parameter     Chi-Square    Pr > ChiSq

Intercept          .             .
Eyesight example: fit symmetry via genmod             5
19:28 Wednesday, November 29, 2017

Obs    r    l    count     md    outpred

1    4    4     1520     -4    1520.00
2    4    3      266    -10     250.00
3    4    2      124     -9     120.50
4    4    1       66     -7      51.00
5    3    4      234    -10     250.00
6    3    3     1512     -3    1512.00
7    3    2      432     -8     397.00
8    3    1       78     -6      80.00
9    2    4      117     -9     120.50
10    2    3      362     -8     397.00
11    2    2     1772     -2    1772.00
12    2    1      205     -5     192.00
13    1    4       36     -7      51.00
14    1    3       82     -6      80.00
15    1    2      179     -5     192.00
16    1    1      492     -1     492.00
Eyesight example: fit sat. model to test symmetry         6
19:28 Wednesday, November 29, 2017

The GENMOD Procedure

Model Information

Data Set              WORK.NEWEYES
Distribution               Poisson
Dependent Variable           count

Number of Observations Used          16

Class Level Information

Class      Levels    Values

r               4    1 2 3 4
l               4    1 2 3 4

Parameter Information

Parameter       Effect       r    l

Prm1            Intercept
Prm2            r*l          1    1
Prm3            r*l          1    2
Prm4            r*l          1    3
Prm5            r*l          1    4
Prm6            r*l          2    1
Prm7            r*l          2    2
Prm8            r*l          2    3
Prm9            r*l          2    4
Prm10           r*l          3    1
Prm11           r*l          3    2
Prm12           r*l          3    3
Prm13           r*l          3    4
Prm14           r*l          4    1
Prm15           r*l          4    2
Prm16           r*l          4    3
Prm17           r*l          4    4

Criteria For Assessing Goodness Of Fit

Criterion                     DF           Value        Value/DF

Deviance                       0          0.0000           .
Scaled Deviance                0          0.0000           .
Pearson Chi-Square             .          0.0000           .
Scaled Pearson X2              .          0.0000           .
Log Likelihood                        42830.2723
Full Log Likelihood                     -58.6895
AIC (smaller is better)                 149.3790
AICC (smaller is better)                   .
BIC (smaller is better)                 161.7405

Algorithm converged.

Analysis Of Maximum Likelihood Parameter Estimates

Standard   Wald 95% Confidence
Parameter           DF   Estimate      Error          Limits

Intercept            0     0.0000     0.0000     0.0000     0.0000
r*l         1   1    1     6.1985     0.0451     6.1101     6.2868
r*l         1   2    1     5.1874     0.0747     5.0409     5.3339
r*l         1   3    1     4.4067     0.1104     4.1903     4.6232
r*l         1   4    1     3.5835     0.1667     3.2569     3.9102
r*l         2   1    1     5.3230     0.0698     5.1861     5.4599
r*l         2   2    1     7.4799     0.0238     7.4333     7.5264
r*l         2   3    1     5.8916     0.0526     5.7886     5.9947
r*l         2   4    1     4.7622     0.0925     4.5810     4.9434
r*l         3   1    1     4.3567     0.1132     4.1348     4.5786
r*l         3   2    1     6.0684     0.0481     5.9741     6.1627
r*l         3   3    1     7.3212     0.0257     7.2708     7.3716
r*l         3   4    1     5.4553     0.0654     5.3272     5.5834
r*l         4   1    1     4.1897     0.1231     3.9484     4.4309
r*l         4   2    1     4.8203     0.0898     4.6443     4.9963
r*l         4   3    1     5.5835     0.0613     5.4633     5.7037
r*l         4   4    1     7.3265     0.0256     7.2762     7.3767
Scale                0     1.0000     0.0000     1.0000     1.0000

Analysis Of Maximum Likelihood
Parameter Estimates

Wald
Parameter           Chi-Square    Pr > ChiSq

Intercept                  .           .
r*l         1   1      18903.2        <.0001
r*l         1   2      4816.71        <.0001
r*l         1   3      1592.37        <.0001
r*l         1   4       462.30        <.0001
r*l         2   1      5808.56        <.0001
r*l         2   2      99140.5        <.0001
r*l         2   3      12565.6        <.0001
r*l         2   4      2653.36        <.0001
r*l         3   1      1480.51        <.0001
r*l         3   2      15908.7        <.0001
r*l         3   3      81042.9        <.0001
r*l         3   4      6963.96        <.0001
r*l         4   1      1158.51        <.0001
r*l         4   2      2881.15        <.0001
r*l         4   3      8292.66        <.0001
r*l         4   4      81589.2        <.0001
Scale

NOTE: The scale parameter was held fixed.

Lagrange Multiplier Statistics

Parameter     Chi-Square    Pr > ChiSq

Intercept          .             .

Contrast Results

Chi-
Contrast        DF     Square    Pr > ChiSq    Type

r0               6      19.25        0.0038    LR
Eyesight example: fit sat. model to test quasisym.        7
19:28 Wednesday, November 29, 2017

The GENMOD Procedure

Model Information

Data Set              WORK.NEWEYES
Distribution               Poisson
Dependent Variable           count

Number of Observations Used          16

Class Level Information

Class      Levels    Values

r               4    1 2 3 4
l               4    1 2 3 4

Parameter Information

Parameter       Effect       r    l

Prm1            Intercept
Prm2            r*l          1    1
Prm3            r*l          1    2
Prm4            r*l          1    3
Prm5            r*l          1    4
Prm6            r*l          2    1
Prm7            r*l          2    2
Prm8            r*l          2    3
Prm9            r*l          2    4
Prm10           r*l          3    1
Prm11           r*l          3    2
Prm12           r*l          3    3
Prm13           r*l          3    4
Prm14           r*l          4    1
Prm15           r*l          4    2
Prm16           r*l          4    3
Prm17           r*l          4    4

Criteria For Assessing Goodness Of Fit

Criterion                     DF           Value        Value/DF

Deviance                       0          0.0000           .
Scaled Deviance                0          0.0000           .
Pearson Chi-Square             .          0.0000           .
Scaled Pearson X2              .          0.0000           .
Log Likelihood                        42830.2723
Full Log Likelihood                     -58.6895
AIC (smaller is better)                 149.3790
AICC (smaller is better)                   .
BIC (smaller is better)                 161.7405

Algorithm converged.

Analysis Of Maximum Likelihood Parameter Estimates

Standard   Wald 95% Confidence
Parameter           DF   Estimate      Error          Limits

Intercept            0     0.0000     0.0000     0.0000     0.0000
r*l         1   1    1     6.1985     0.0451     6.1101     6.2868
r*l         1   2    1     5.1874     0.0747     5.0409     5.3339
r*l         1   3    1     4.4067     0.1104     4.1903     4.6232
r*l         1   4    1     3.5835     0.1667     3.2569     3.9102
r*l         2   1    1     5.3230     0.0698     5.1861     5.4599
r*l         2   2    1     7.4799     0.0238     7.4333     7.5264
r*l         2   3    1     5.8916     0.0526     5.7886     5.9947
r*l         2   4    1     4.7622     0.0925     4.5810     4.9434
r*l         3   1    1     4.3567     0.1132     4.1348     4.5786
r*l         3   2    1     6.0684     0.0481     5.9741     6.1627
r*l         3   3    1     7.3212     0.0257     7.2708     7.3716
r*l         3   4    1     5.4553     0.0654     5.3272     5.5834
r*l         4   1    1     4.1897     0.1231     3.9484     4.4309
r*l         4   2    1     4.8203     0.0898     4.6443     4.9963
r*l         4   3    1     5.5835     0.0613     5.4633     5.7037
r*l         4   4    1     7.3265     0.0256     7.2762     7.3767
Scale                0     1.0000     0.0000     1.0000     1.0000

Analysis Of Maximum Likelihood
Parameter Estimates

Wald
Parameter           Chi-Square    Pr > ChiSq

Intercept                  .           .
r*l         1   1      18903.2        <.0001
r*l         1   2      4816.71        <.0001
r*l         1   3      1592.37        <.0001
r*l         1   4       462.30        <.0001
r*l         2   1      5808.56        <.0001
r*l         2   2      99140.5        <.0001
r*l         2   3      12565.6        <.0001
r*l         2   4      2653.36        <.0001
r*l         3   1      1480.51        <.0001
r*l         3   2      15908.7        <.0001
r*l         3   3      81042.9        <.0001
r*l         3   4      6963.96        <.0001
r*l         4   1      1158.51        <.0001
r*l         4   2      2881.15        <.0001
r*l         4   3      8292.66        <.0001
r*l         4   4      81589.2        <.0001
Scale

NOTE: The scale parameter was held fixed.

Lagrange Multiplier Statistics

Parameter     Chi-Square    Pr > ChiSq

Intercept          .             .

Contrast Results

Chi-
Contrast        DF     Square    Pr > ChiSq    Type

r0               3       7.27        0.0638    LR
Quasisymmetry via CATMOD                     8
19:28 Wednesday, November 29, 2017

The CATMOD Procedure

Data Summary

Response           r*l         Response Levels    16
Weight Variable    count       Populations         1
Data Set           NEWEYES     Total Frequency  7477
Frequency Missing  0           Observations       16

Population Profiles

Sample    Sample Size
---------------------
1            7477

Response Profiles

Response    r    l
------------------
1       1    1
2       1    2
3       1    3
4       1    4
5       2    1
6       2    2
7       2    3
8       2    4
9       3    1
10       3    2
11       3    3
12       3    4
13       4    1
14       4    2
15       4    3
16       4    4

Analysis of Variance

Source       DF   Chi-Square    Pr > ChiSq
------------------------------------------
Residual      3         7.22        0.0651
Marginal symmetry via CATMOD                   9
19:28 Wednesday, November 29, 2017

The CATMOD Procedure

Data Summary

Response           r*l         Response Levels    16
Weight Variable    count       Populations         1
Data Set           NEWEYES     Total Frequency  7477
Frequency Missing  0           Observations       16

Population Profiles

Sample    Sample Size
---------------------
1            7477

Response Profiles

Response    r    l
------------------
1       1    1
2       1    2
3       1    3
4       1    4
5       2    1
6       2    2
7       2    3
8       2    4
9       3    1
10       3    2
11       3    3
12       3    4
13       4    1
14       4    2
15       4    3
16       4    4

Analysis of Variance

Source       DF   Chi-Square    Pr > ChiSq
------------------------------------------
Residual      3        11.98        0.0075
Quasiindependence via GENMOD                  10
19:28 Wednesday, November 29, 2017

The GENMOD Procedure

Model Information

Data Set              WORK.NEWEYES
Distribution               Poisson
Dependent Variable           count

Number of Observations Used          16

Class Level Information

Class      Levels    Values

l               4    1 2 3 4
r               4    1 2 3 4
d               5    1 2 3 4 5

Criteria For Assessing Goodness Of Fit

Criterion                     DF           Value        Value/DF

Deviance                       5        199.1062         39.8212
Scaled Deviance                5        199.1062         39.8212
Pearson Chi-Square             5        198.0094         39.6019
Scaled Pearson X2              5        198.0094         39.6019
Log Likelihood                        42730.7193
Full Log Likelihood                    -158.2426
AIC (smaller is better)                 338.4852
AICC (smaller is better)                404.4852
BIC (smaller is better)                 346.9837

Algorithm converged.

Analysis Of Maximum Likelihood Parameter Estimates

Standard       Wald 95%             Wald
Parameter     DF  Estimate     Error   Confidence Limits  Chi-Square

Intercept      1    4.3884    0.0745    4.2423    4.5345     3465.63
r          1   1   -0.4570    0.0755   -0.6050   -0.3090       36.62
r          2   1    0.6872    0.0647    0.5604    0.8140      112.83
r          3   1    0.7818    0.0641    0.6561    0.9074      148.78
r          4   0    0.0000    0.0000    0.0000    0.0000         .
l          1   1   -0.1769    0.0748   -0.3235   -0.0303        5.59
l          2   1    0.8716    0.0669    0.7405    1.0027      169.77
l          3   1    0.8900    0.0679    0.7569    1.0230      171.87
l          4   0    0.0000    0.0000    0.0000    0.0000         .
d          1   1    2.4439    0.0946    2.2586    2.6293      668.01
d          2   1    1.5326    0.0700    1.3954    1.6698      479.57
d          3   1    1.2610    0.0704    1.1230    1.3990      320.69
d          4   1    2.9381    0.0788    2.7835    3.0926     1388.97
d          5   0    0.0000    0.0000    0.0000    0.0000         .
Scale          0    1.0000    0.0000    1.0000    1.0000

Analysis Of Maximum
Likelihood Parameter
Estimates

Parameter     Pr > ChiSq

Intercept         <.0001
r          1      <.0001
r          2      <.0001
r          3      <.0001
r          4       .
l          1      0.0181
l          2      <.0001
l          3      <.0001
l          4       .
d          1      <.0001
d          2      <.0001
d          3      <.0001
d          4      <.0001
d          5       .
Scale

NOTE: The scale parameter was held fixed.

LR Statistics For Type 3 Analysis

Chi-
Source           DF     Square    Pr > ChiSq

r                 3     464.77        <.0001
l                 3     453.16        <.0001
d                 4    6472.41        <.0001
Quasiindependence via GENMOD                  11
19:28 Wednesday, November 29, 2017

The FREQ Procedure

Statistics for Table of r by l

Statistic                              Value       ASE
------------------------------------------------------
Gamma                                 0.7982    0.0080
Kendall's Tau-b                       0.6583    0.0080
Stuart's Tau-c                        0.6327    0.0079

Somers' D C|R                         0.6590    0.0080
Somers' D R|C                         0.6576    0.0080

Pearson Correlation                   0.7027    0.0084
Spearman Correlation                  0.7064    0.0082
Polychoric Correlation                0.7797    0.0060

Lambda Asymmetric C|R                 0.5612    0.0087
Lambda Asymmetric R|C                 0.5656    0.0085
Lambda Symmetric                      0.5634    0.0084

Uncertainty Coefficient C|R           0.3377    0.0075
Uncertainty Coefficient R|C           0.3390    0.0075
Uncertainty Coefficient Symmetric     0.3383    0.0075

Sample Size = 7477
```