| |
True covariance structure
|
---|
| |
IND
|
AR1
|
UNS
|
---|
| |
LMM working covariance
| | |
LMM working covariance
| | |
LMM working covariance
| | |
---|
( m, n )
|
β
3
|
IND
|
AR1
|
UNS
|
SMA
|
UMA
|
IND
|
AR1
|
UNS
|
SMA
|
UMA
|
IND
|
AR1
|
UNS
|
SMA
|
UMA
|
---|
(5,5)
|
0
|
0.057
|
0.065
|
0.222
|
0.043
|
0.043
|
0.192
|
0.079
|
0.270
|
0.047
|
0.055
|
0.044
|
0.074
|
0.251
|
0.042
|
0.068
|
|
0.040
|
0.074
|
0.087
|
0.260
|
0.065
|
0.052
|
0.240
|
0.126
|
0.276
|
0.076
|
0.061
|
0.095
|
0.160
|
0.358
|
0.099
|
0.086
|
|
0.080
|
0.209
|
0.232
|
0.399
|
0.167
|
0.095
|
0.424
|
0.242
|
0.391
|
0.141
|
0.071
|
0.278
|
0.361
|
0.589
|
0.247
|
0.114
|
(5,10)
|
0
|
0.050
|
0.062
|
0.099
|
0.049
|
0.059
|
0.185
|
0.064
|
0.100
|
0.047
|
0.050
|
0.040
|
0.060
|
0.100
|
0.044
|
0.042
|
|
0.040
|
0.128
|
0.135
|
0.196
|
0.111
|
0.090
|
0.381
|
0.136
|
0.202
|
0.099
|
0.076
|
0.155
|
0.202
|
0.316
|
0.161
|
0.117
|
|
0.080
|
0.351
|
0.362
|
0.412
|
0.329
|
0.166
|
0.776
|
0.405
|
0.478
|
0.317
|
0.166
|
0.545
|
0.621
|
0.766
|
0.569
|
0.442
|
(5,30)
|
0
|
0.049
|
0.054
|
0.062
|
0.050
|
0.056
|
0.188
|
0.055
|
0.057
|
0.055
|
0.048
|
0.045
|
0.064
|
0.056
|
0.056
|
0.062
|
|
0.040
|
0.295
|
0.305
|
0.316
|
0.285
|
0.171
|
0.501
|
0.297
|
0.302
|
0.263
|
0.156
|
0.465
|
0.532
|
0.637
|
0.488
|
0.401
|
|
0.080
|
0.807
|
0.810
|
0.823
|
0.779
|
0.522
|
0.973
|
0.828
|
0.840
|
0.786
|
0.580
|
0.968
|
0.980
|
0.995
|
0.971
|
0.976
|
(5,50)
|
0
|
0.038
|
0.039
|
0.045
|
0.041
|
0.039
|
0.176
|
0.059
|
0.058
|
0.054
|
0.056
|
0.039
|
0.062
|
0.049
|
0.048
|
0.040
|
|
0.040
|
0.421
|
0.425
|
0.429
|
0.414
|
0.235
|
0.685
|
0.445
|
0.446
|
0.393
|
0.239
|
0.685
|
0.745
|
0.780
|
0.717
|
0.674
|
|
0.080
|
0.952
|
0.952
|
0.954
|
0.948
|
0.824
|
1.000
|
0.954
|
0.957
|
0.946
|
0.838
|
0.998
|
0.998
|
1.000
|
0.998
|
0.998
|
(10,5)
|
0
|
0.059
|
0.065
|
--*
|
0.052
|
--†
|
0.300
|
0.072
|
--*
|
0.039
|
--†
|
0.050
|
0.059
|
--*
|
0.042
|
--†
|
|
0.020
|
0.129
|
0.138
|
--*
|
0.100
|
--†
|
0.412
|
0.123
|
--*
|
0.071
|
--†
|
0.172
|
0.189
|
--*
|
0.159
|
--†
|
|
0.040
|
0.362
|
0.371
|
--*
|
0.280
|
--†
|
0.540
|
0.248
|
--*
|
0.146
|
--†
|
0.545
|
0.545
|
--*
|
0.418
|
--†
|
(10,10)
|
0
|
0.050
|
0.053
|
0.275
|
0.055
|
0.045
|
0.309
|
0.045
|
0.241
|
0.040
|
0.039
|
0.044
|
0.044
|
0.281
|
0.049
|
0.044
|
|
0.020
|
0.177
|
0.177
|
0.400
|
0.184
|
0.078
|
0.453
|
0.162
|
0.375
|
0.129
|
0.066
|
0.284
|
0.289
|
0.493
|
0.275
|
0.113
|
|
0.040
|
0.323
|
0.326
|
0.641
|
0.576
|
0.178
|
0.707
|
0.391
|
0.616
|
0.299
|
0.097
|
0.828
|
0.825
|
0.840
|
0.772
|
0.254
|
(10,30)
|
0
|
0.050
|
0.051
|
0.097
|
0.047
|
0.048
|
0.278
|
0.051
|
0.088
|
0.060
|
0.057
|
0.059
|
0.059
|
0.095
|
0.053
|
0.049
|
|
0.020
|
0.502
|
0.502
|
0.587
|
0.494
|
0.191
|
0.596
|
0.297
|
0.344
|
0.259
|
0.112
|
0.660
|
0.660
|
0.774
|
0.673
|
0.320
|
|
0.040
|
0.977
|
0.972
|
1.000
|
0.977
|
0.725
|
0.980
|
0.802
|
0.866
|
0.747
|
0.370
|
1.000
|
1.000
|
1.000
|
0.997
|
0.943
|
(10,50)
|
0
|
0.052
|
0.052
|
0.075
|
0.049
|
0.047
|
0.283
|
0.058
|
0.072
|
0.049
|
0.065
|
0.050
|
0.052
|
0.078
|
0.048
|
0.049
|
|
0.020
|
0.698
|
0.696
|
0.708
|
0.686
|
0.351
|
0.762
|
0.460
|
0.471
|
0.399
|
0.180
|
0.876
|
0.877
|
0.909
|
0.887
|
0.600
|
|
0.040
|
0.998
|
0.998
|
1.000
|
0.998
|
0.949
|
1.000
|
0.966
|
0.970
|
0.938
|
0.686
|
1.000
|
1.000
|
1.000
|
1.000
|
1.000
|
(20,5)
|
0
|
0.049
|
0.055
|
--*
|
0.044
|
--†
|
0.340
|
0.052
|
--*
|
0.048
|
--†
|
0.050
|
0.054
|
--*
|
0.054
|
--†
|
|
0.010
|
0.208
|
0.217
|
--*
|
0.157
|
--†
|
0.436
|
0.119
|
--*
|
0.083
|
--†
|
0.210
|
0.218
|
--*
|
0.211
|
--†
|
|
0.020
|
0.614
|
0.619
|
--*
|
0.489
|
--†
|
0.647
|
0.288
|
--*
|
0.200
|
--†
|
0.641
|
0.639
|
--*
|
0.655
|
--†
|
(20,10)
|
0
|
0.050
|
0.057
|
--*
|
0.051
|
--†
|
0.329
|
0.053
|
--*
|
0.043
|
--†
|
0.049
|
0.053
|
--*
|
0.054
|
--†
|
|
0.010
|
0.361
|
0.361
|
--*
|
0.316
|
--†
|
0.788
|
0.154
|
--*
|
0.122
|
--†
|
0.446
|
0.448
|
--*
|
0.445
|
--†
|
|
0.020
|
0.911
|
0.908
|
--*
|
0.863
|
--†
|
0.803
|
0.460
|
--*
|
0.382
|
--†
|
0.972
|
0.971
|
--*
|
0.955
|
--†
|
(20,30)
|
0
|
0.057
|
0.055
|
0.178
|
0.063
|
0.065
|
0.350
|
0.048
|
0.169
|
0.044
|
0.041
|
0.052
|
0.052
|
0.181
|
0.048
|
0.050
|
|
0.010
|
0.808
|
0.808
|
0.928
|
0.795
|
0.258
|
0.736
|
0.384
|
0.444
|
0.340
|
0.101
|
0.914
|
0.914
|
1.000
|
0.927
|
0.399
|
|
0.020
|
1.000
|
1.000
|
1.000
|
1.000
|
0.871
|
1.000
|
0.914
|
1.000
|
0.855
|
0.293
|
1.000
|
1.000
|
1.000
|
1.000
|
0.987
|
(20,50)
|
0
|
0.049
|
0.046
|
0.095
|
0.046
|
0.049
|
0.346
|
0.042
|
0.089
|
0.042
|
0.051
|
0.049
|
0.050
|
0.104
|
0.049
|
0.052
|
|
0.010
|
0.935
|
0.932
|
0.986
|
0.926
|
0.442
|
0.833
|
0.525
|
0.550
|
0.482
|
0.154
|
0.990
|
0.992
|
1.000
|
0.992
|
0.715
|
|
0.020
|
1.000
|
1.000
|
1.000
|
1.000
|
0.997
|
1.000
|
0.986
|
1.000
|
0.969
|
0.597
|
1.000
|
1.000
|
1.000
|
1.000
|
1.000
|
- *: Since m was large relative to the sample size (N = 2n), modeling with large number of parameters was discarded.
- †: Since N-k <m-1, the UMA could not be conducted