Background Information for Linear Structural Equation Models

SUPPLEMENTARY INFORMATION II

BPA Performance on DAG conversion

Structural equation models (SEMs) are statistical causal models, which are associated with a directed graph (cyclic or acylic ) representing the causal structure of the model. The vertices of the graph are the variables, and there is a directed edge from Xi to Xj if the coefficient of Xi in the structural equation for Xj is non-zero. In order to test the performance of BPA on conversion of graph with cycles into DAGs, we have generated discrete datasets of size 1000 from 20 synthetic directed graphs containing cyclic paths following a SEM model. We have used TETRAD version IV software (http://www.phil.cmu.edu/projects/tetrad/tetrad4.html).

Properties of synthetic cyclic graphs and their corresponding acylic collapsed versions are given below:

Properties of Synthetic Cyclic Graphs

Name / size / order / density / max degree / average degree
Cyclic Graph 1 / 16 / 18 / 0.15 / 4 / 2.25
Cyclic Graph 2 / 17 / 20 / 0.147059 / 4 / 2.352941
Cyclic Graph 3 / 20 / 23 / 0.121053 / 4 / 2.3
Cyclic Graph 4 / 25 / 28 / 0.093333 / 4 / 2.24
Cyclic Graph 5 / 15 / 16 / 0.152381 / 4 / 2.133333
Cyclic Graph 6 / 17 / 20 / 0.147059 / 5 / 2.352941
Cyclic Graph 7 / 30 / 37 / 0.085057 / 4 / 2.466667
Cyclic Graph 8 / 26 / 30 / 0.092308 / 4 / 2.307692
Cyclic Graph 9 / 27 / 31 / 0.088319 / 4 / 2.296296
Cyclic Graph 10 / 18 / 22 / 0.143791 / 4 / 2.444444
Cyclic Graph 11 / 19 / 24 / 0.140351 / 4 / 2.526316
Cyclic Graph 12 / 28 / 32 / 0.084656 / 4 / 2.285714
Cyclic Graph 13 / 25 / 34 / 0.113333 / 5 / 2.72
Cyclic Graph 14 / 30 / 33 / 0.075862 / 4 / 2.2
Cyclic Graph 15 / 17 / 20 / 0.147059 / 4 / 2.352941
Cyclic Graph 16 / 24 / 29 / 0.105072 / 4 / 2.416667
Cyclic Graph 17 / 20 / 27 / 0.142105 / 5 / 2.7
Cyclic Graph 18 / 18 / 22 / 0.143791 / 4 / 2.444444
Cyclic Graph 19 / 29 / 36 / 0.08867 / 5 / 2.482759
Cyclic Graph 20 / 17 / 21 / 0.154412 / 4 / 2.470588

Properties of DAGs (after conversion)

Name / size / order / density / max degree / average degree
DAG 1 / 16 / 22 / 0.183333 / 6 / 2.75
DAG 2 / 17 / 36 / 0.264706 / 7 / 4.235294
DAG 3 / 20 / 58 / 0.305263 / 12 / 5.8
DAG 4 / 25 / 36 / 0.12 / 7 / 2.88
DAG 5 / 15 / 23 / 0.219048 / 6 / 3.066667
DAG 6 / 17 / 45 / 0.330882 / 10 / 5.294118
DAG 7 / 30 / 89 / 0.204598 / 13 / 5.933333
DAG 8 / 26 / 36 / 0.110769 / 6 / 2.769231
DAG 9 / 27 / 66 / 0.188034 / 12 / 4.888889
DAG 10 / 18 / 135 / 0.882353 / 17 / 15
DAG 11 / 19 / 110 / 0.643275 / 15 / 11.57895
DAG 12 / 28 / 81 / 0.214286 / 14 / 5.785714
DAG 13 / 25 / 137 / 0.456667 / 18 / 10.96
DAG 14 / 30 / 95 / 0.218391 / 13 / 6.333333
DAG 15 / 17 / 35 / 0.257353 / 9 / 4.117647
DAG 16 / 24 / 69 / 0.25 / 10 / 5.75
DAG 17 / 20 / 77 / 0.405263 / 13 / 7.7
DAG 18 / 18 / 135 / 0.882353 / 17 / 15
DAG 19 / 29 / 103 / 0.253695 / 18 / 7.103448
DAG 20 / 17 / 63 / 0.463235 / 11 / 7.411765

BPA is run for these synthetic cyclic graphs (i.e. during BPA procedure cyclic graphs are converted to DAGs and BDe score is calculated) using data that follows underlying SEM with 1000 bootstraps. All synthetic networks have yielded lowest attainable p-value < 0.001.

BPA Results of Synthetic Graphs and Data

Name / BDe Score / P-Value
DAG 1 / -5.27E+03 / < 0.001
DAG 2 / -4.19E+03 / < 0.001
DAG 3 / -4.95E+03 / < 0.001
DAG 4 / -7.18E+03 / < 0.001
DAG 5 / -2.29E+03 / < 0.001
DAG 6 / -5.91E+03 / < 0.001
DAG 7 / -9.59E+03 / < 0.001
DAG 8 / -7.45E+03 / < 0.001
DAG 9 / -6.25E+03 / < 0.001
DAG 10 / -7.36E+03 / < 0.001
DAG 11 / -6.97E+03 / < 0.001
DAG 12 / -8.09E+03 / < 0.001
DAG 13 / -1.17E+04 / < 0.001
DAG 14 / -1.01E+04 / < 0.001
DAG 15 / -4.64E+03 / < 0.001
DAG 16 / -1.11E+04 / < 0.001
DAG 17 / -7.31E+03 / < 0.001
DAG 18 / -7.65E+03 / < 0.001
DAG 19 / -9.26E+03 / < 0.001
DAG 20 / -4.50E+03 / < 0.001

Model Parameters

Graph and configuration (edge coefficients) of SEM for all 20 networks are given below:

Model 1:

X10 / X12 / -1.1718 / X6 / X5 / -0.7466
X11 / X13 / -1.3494 / X7 / X15 / -0.638
X12 / X11 / -0.9805 / X7 / X6 / -1.4866
X15 / X16 / 0.5303 / X8 / X4 / -0.6136
X16 / X7 / 1.4882 / X8 / X9 / 1.2116
X2 / X14 / -1.466
X2 / X4 / 0.7461
X3 / X1 / 1.0446
X3 / X8 / 1.1618
X4 / X3 / 0.5777
X4 / X7 / 1.3977
X5 / X13 / -1.2141
X6 / X10 / -1.0197

Model2:

X10 / X16 / -0.6955 / X7 / X15 / -0.6622
X11 / X12 / -0.5678 / X7 / X6 / -0.5627
X12 / X10 / -0.519 / X8 / X17 / 1.2722
X13 / X11 / -1.3508 / X8 / X4 / 1.0952
X14 / X5 / -0.7367 / X8 / X9 / 1.4937
X15 / X16 / 1.1749
X16 / X7 / -0.6018
X2 / X14 / 1.498
X3 / X1 / 1.1624
X3 / X8 / -0.5643
X4 / X2 / 1.1988
X4 / X7 / 1.0989
X5 / X13 / 0.9351
X5 / X6 / 1.1223
X6 / X10 / -1.359

Model 3:

X1 / X2 / -1.1455 / X4 / X10 / 1.218
X10 / X6 / -1.2653 / X4 / X7 / 1.3944
X10 / X16 / 1.3587 / X5 / X13 / -1.0187
X11 / X12 / -0.7167 / X6 / X12 / 0.577
X13 / X11 / 1.0654 / X7 / X15 / 0.8018
X14 / X4 / 0.5486 / X7 / X9 / -0.7216
X14 / X5 / -1.4341 / X8 / X3 / -0.8724
X15 / X16 / -1.3761 / X9 / X8 / -1.3299
X15 / X17 / -1.0681
X16 / X20 / -0.7457
X18 / X10 / -1.2341
X19 / X2 / 0.5338
X2 / X14 / 0.9695
X20 / X18 / 0.5915
X3 / X19 / -1.1406

Model 4:

X1 / X2 / -0.8758 / X17 / X23 / 1.0016 / X3 / X7 / -0.722
X1 / X3 / -1.0887 / X19 / X20 / -1.027 / X4 / X6 / -0.791
X10 / X17 / -1.1148 / X2 / X4 / -1.2894 / X6 / X11 / 0.614
X11 / X21 / -0.6417 / X20 / X18 / -1.1133 / X7 / X16 / 0.6298
X11 / X13 / -1.0236 / X21 / X12 / -0.6466 / X7 / X6 / -1.4537
X12 / X11 / 0.7281 / X21 / X25 / 0.8271 / X8 / X9 / 0.9245
X13 / X5 / 0.5333 / X22 / X10 / 1.1587 / X9 / X7 / 1.2396
X14 / X5 / -0.9998 / X22 / X12 / 1.0546 / X9 / X15 / 0.8036
X15 / X16 / 1.289 / X24 / X18 / 0.6677
X16 / X24 / -1.0633 / X25 / X22 / -1.4364

Model 5:

X10 / X12 / -0.6432
X11 / X6 / -1.1792
X12 / X11 / -0.7858
X2 / X14 / -0.577
X2 / X4 / 1.28
X3 / X1 / 1.4902
X3 / X8 / -1.0506
X4 / X3 / -1.074
X4 / X7 / -0.9171
X5 / X13 / -0.8619
X6 / X10 / 0.6658
X6 / X5 / 0.6495
X7 / X15 / -1.0161
X7 / X6 / 1.1632
X8 / X4 / 0.9387
X8 / X9 / -1.2285

Model 6:

X10 / X12 / -1.169 / X6 / X17 / -0.6736
X11 / X13 / 0.5704 / X6 / X10 / 1.0329
X12 / X11 / 1.4675 / X6 / X5 / -0.8738
X15 / X16 / 0.7437 / X7 / X15 / -1.4077
X16 / X7 / 1.4869 / X7 / X6 / 0.8677
X17 / X4 / 1.4594 / X8 / X4 / -1.4363
X2 / X14 / 0.9464 / X8 / X9 / 0.8858
X2 / X4 / -1.0311
X3 / X1 / -1.394
X3 / X8 / -1.064
X4 / X3 / -0.626
X4 / X7 / -0.7892
X5 / X13 / -1.4897

Model 7:

Model 8:

X10 / X12 / -1.2943 / X2 / X14 / 1.0593 / X5 / X13 / -1.0216
X11 / X13 / -1.2915 / X2 / X4 / 1.4540 / X6 / X10 / 1.2852
X11 / X17 / -1.3780 / X20 / X21 / 1.2718 / X6 / X5 / 0.7203
X12 / X11 / -0.7954 / X22 / X21 / 0.6029 / X7 / X15 / -1.3577
X12 / X18 / 0.5883 / X25 / X23 / 0.9875 / X7 / X6 / 1.2014
X15 / X16 / 0.9325 / X26 / X23 / -1.2377 / X8 / X22 / 1.4806
X16 / X7 / 1.1566 / X26 / X24 / -1.2541 / X8 / X4 / -1.0832
X17 / X24 / -0.9469 / X3 / X1 / 1.2806 / X8 / X9 / 0.9469
X18 / X25 / 0.6231 / X3 / X8 / 0.9973
X19 / X16 / -1.3185 / X4 / X3 / -0.6473
X19 / X20 / 1.4193 / X4 / X7 / 0.5359

Model 9:

X1 / X2 / 1.0633 / X22 / X8 / -0.9347
X10 / X24 / -0.9480 / X24 / X25 / 1.4125
X10 / X15 / -1.3198 / X25 / X11 / 1.1395
X10 / X23 / 1.3168 / X27 / X11 / -1.4310
X11 / X16 / 1.2539 / X3 / X18 / 0.6200
X12 / X6 / -0.5829 / X4 / X8 / 1.3506
X13 / X4 / 0.6812 / X4 / X9 / 1.3010
X14 / X13 / 1.2269 / X5 / X6 / -0.7387
X16 / X10 / 0.7487 / X6 / X19 / -1.3513
X17 / X20 / 0.5736 / X7 / X19 / -1.1549
X19 / X17 / 0.7149 / X7 / X26 / 0.5997
X2 / X18 / 0.6525 / X8 / X16 / -0.7479
X2 / X5 / 1.1253 / X8 / X6 / -1.3485
X20 / X12 / 0.5800 / X9 / X14 / 0.7262
X20 / X27 / -1.1650 / X9 / X22 / -1.1455
X21 / X4 / -0.5551

Model 10:

X1 / X16 / 1.3629 / X3 / X6 / 0.7173
X10 / X9 / -1.4951 / X4 / X1 / -0.8031
X11 / X6 / -1.4308 / X5 / X3 / -0.9583
X11 / X18 / -0.7692 / X6 / X8 / 0.9614
X12 / X11 / 1.1996 / X7 / X3 / 0.8717
X12 / X7 / -1.2991 / X8 / X10 / 0.9772
X13 / X2 / -0.6794 / X9 / X5 / 1.2429
X14 / X12 / -0.7293
X15 / X4 / -0.5183
X15 / X2 / 0.9462
X16 / X5 / 1.4299
X17 / X13 / 1.1971
X17 / X1 / 1.4014
X2 / X12 / -0.7236
X3 / X15 / 0.7663

Model 11:

X1 / X17 / -0.5640 / X2 / X16 / -0.7353
X1 / X11 / -1.2634 / X3 / X7 / 1.1123
X10 / X5 / -1.4539 / X4 / X15 / 0.6065
X10 / X13 / -0.6365 / X5 / X12 / 0.8969
X11 / X17 / -0.8058 / X6 / X3 / 0.8332
X12 / X16 / 0.6780 / X7 / X14 / -0.9985
X13 / X4 / -0.5211 / X7 / X11 / -0.7332
X14 / X10 / 0.5966 / X7 / X6 / 0.5644
X16 / X3 / 1.4062 / X8 / X15 / -0.5933
X16 / X8 / -1.3496 / X9 / X18 / -0.8384
X17 / X2 / -1.3333
X18 / X2 / 1.0999
X19 / X9 / -0.9152
X2 / X19 / -1.2501

Model 12:

X10 / X11 / 0.5348 / X21 / X1 / 1.0394
X11 / X12 / 1.2009 / X21 / X4 / 0.8313
X12 / X25 / -0.9173 / X22 / X14 / -0.6558
X12 / X20 / -1.2101 / X23 / X15 / 0.8044
X13 / X4 / -0.9038 / X23 / X24 / -1.3538
X14 / X13 / 1.4268 / X24 / X10 / 0.5313
X14 / X9 / -1.0585 / X25 / X28 / -0.7897
X15 / X22 / 0.7003 / X25 / X27 / -1.1561
X16 / X15 / 1.3337 / X27 / X12 / 1.1292
X17 / X19 / 1.1189 / X28 / X24 / -0.5337
X17 / X26 / -0.5383 / X3 / X1 / -0.9293
X18 / X3 / 0.9042 / X4 / X5 / -1.1338
X19 / X20 / -0.7964 / X6 / X16 / -0.9981
X2 / X19 / -1.2118 / X7 / X18 / 0.6197
X2 / X6 / 0.5560 / X8 / X6 / -0.8629
X20 / X17 / -0.9073 / X9 / X8 / 0.5255

Model 13:

X1 / X3 / 0.7136 / X16 / X25 / 0.9996 / X3 / X4 / -0.5334
X1 / X2 / -0.5244 / X17 / X13 / 1.0469 / X4 / X7 / -1.2307
X10 / X16 / -0.5951 / X17 / X24 / -0.7521 / X4 / X9 / 1.3023
X10 / X18 / 0.9133 / X18 / X23 / 0.7774 / X5 / X4 / -0.5202
X10 / X12 / -1.3389 / X2 / X5 / 0.9794 / X5 / X6 / -1.1503
X11 / X6 / 0.7951 / X20 / X16 / 1.1196 / X6 / X10 / -0.5515
X12 / X11 / -0.5908 / X21 / X20 / -1.4195 / X7 / X5 / 0.6150
X12 / X24 / -1.3451 / X22 / X21 / -1.3127 / X8 / X22 / -1.1198
X13 / X14 / -0.7129 / X22 / X15 / 0.9149 / X9 / X7 / 1.1926
X15 / X9 / -0.5940 / X23 / X24 / 1.3917 / X9 / X8 / -1.1581
X15 / X16 / 1.4170 / X24 / X18 / -0.5997
X16 / X7 / 0.7256 / X25 / X19 / -0.8231

Model 14:

X1 / X11 / 0.9384 / X15 / X8 / -0.5550 / X27 / X16 / -1.4576
X10 / X9 / -1.0105 / X16 / X7 / 0.7197 / X27 / X11 / -0.9284
X10 / X26 / 0.6168 / X17 / X24 / -1.1545 / X28 / X7 / 1.0656
X11 / X12 / 0.8705 / X2 / X10 / 1.3049 / X29 / X17 / -0.8555
X12 / X23 / -1.0333 / X2 / X3 / -0.7884 / X29 / X5 / 0.6542
X13 / X30 / -1.1956 / X20 / X15 / -0.7236 / X30 / X6 / 1.4201
X13 / X23 / -1.4057 / X20 / X21 / 1.3575 / X4 / X27 / 0.6673
X14 / X28 / 1.0978 / X21 / X22 / -0.6532 / X4 / X3 / 0.5314
X14 / X20 / 1.4981 / X23 / X14 / -1.4291 / X6 / X27 / -1.1318
X15 / X19 / -0.7040 / X24 / X18 / -0.8464 / X7 / X13 / 1.2079
X15 / X18 / -1.4290 / X26 / X25 / -0.5346 / X9 / X4 / 0.7807

Model 15: