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Semi-supervised classification on graphs using explicit diffusion dynamics
Bayesian inference for latent chain graphs
1. | Department of Statistics & Applied Probability, National University of Singapore, Singapore, 117546, SG |
2. | Yale-NUS, Singapore, 138527, SG. & , Department of Statistical Science, University College London, UK |
3. | Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal, 23955, KSA |
4. | Oncology Biostatistics and Bioinformatics, Sidney Kimmel Comprehensive Cancer Center, Johns Hopkins University, Baltimore, MD 21205, USA |
In this article we consider Bayesian inference for partially observed Andersson-Madigan-Perlman (AMP) Gaussian chain graph (CG) models. Such models are of particular interest in applications such as biological networks and financial time series. The model itself features a variety of constraints which make both prior modeling and computational inference challenging. We develop a framework for the aforementioned challenges, using a sequential Monte Carlo (SMC) method for statistical inference. Our approach is illustrated on both simulated data as well as real case studies from university graduation rates and a pharmacokinetics study.
References:
[1] |
K. Q. Abdool, K. S. S. Abdool and J. A. Frohlich,
Effectiveness and safety of tenofovir gel, an antiretroviral microbicide, for the prevention of HIV infection in women, Science, 329 (2010), 1168-1174.
doi: 10.1126/science.1193748. |
[2] |
S. A. Andersson, D. Madigan and M. D. Perlman,
Alternative Markov properties for chain graphs, Scand. J. Statist., 28 (2001), 33-85.
doi: 10.1111/1467-9469.00224. |
[3] |
P. A. Anton, R. D. Cranston, A. Kashuba, C. W. Hendrix, N. N. Bumpus, N. R. Harman, J. Elliott, L. Janocko, E. Khanukhova, R. Dennis, W. G. Cumberland, C. Ju, A. C. Dieguez, C. Mauck and I. McGowan,
RMP-02/MTN-006: A phase rectal safety, acceptability, pharmacokinetic, and pharmacodynamic study of tenofovir 1% gel compared with oral tenofovir disoproxil fumarate, AIDS Res Hum Retroviruses, 28 (2012), 1412-1421.
doi: 10.1089/aid.2012.0262. |
[4] |
J. M. Baeten, D. Donnell and P. Ndase,
Antiretroviral prophylaxis for HIV prevention in heterosexual men and women, N Engl J Med, 367 (2012), 399-410.
doi: 10.1056/NEJMoa1108524. |
[5] |
A. Beskos, A. Jasra, N. Kantas and A. Thiery,
On the convergence of adaptive sequential Monte Carlo, Ann. Appl. Probab., 26 (2016), 1111-1146.
doi: 10.1214/15-AAP1113. |
[6] |
B. C. Boerebach, K. M. Lombarts, C. Keijzer, M. J. Heineman and O. A. Arah, The teacher, the physician and the person: How faculty's teaching performance influences their role modeling, PLoS One, 7 (2012), e32089.
doi: 10.1371/journal.pone.0032089. |
[7] |
K. Bollen, Structural Equation Models with Latent Variables, Wiley: New York, 1989.
doi: 10.1002/9781118619179. |
[8] |
C. M. Carvalho and M. West,
Dynamic matrix-variate graphical modelso, Bayesian Anal., 2 (2007), 69-97.
doi: 10.1214/07-BA204. |
[9] |
H. Chun, X. Zhang and H. Zhao,
Gene regulation network inference with joint sparse Gaussian graphical models, J. Comp. Graph. Statist., 24 (2015), 954-974.
doi: 10.1080/10618600.2014.956876. |
[10] |
P. Del Moral, A. Doucet and A. Jasra,
Sequential Monte Carlo samplers, J. Roy. Statist. Soc. Ser. B, 68 (2006), 411-436.
doi: 10.1111/j.1467-9868.2006.00553.x. |
[11] |
A. Dobra, C. Hans, B. Jones, J. R. Nevins, G. Yao and M. West,
Sparse graphical models for exploring gene expression data, J. Mult. Anal., 90 (2004), 196-212.
doi: 10.1016/j.jmva.2004.02.009. |
[12] |
M. Drton and M. Eichler,
Maximum Likelihood Estimation in Gaussian Chain Graph Models under the Alternative Markov Property, Scand. J. Statist., 33 (2006), 247-257.
doi: 10.1111/j.1467-9469.2006.00482.x. |
[13] |
M. Drton and M. D. Perlman,
A SINful approach to Gaussian graphical model selection, Journal of Statistical Planning and Inference, 138 (2008), 1179-1200.
doi: 10.1016/j.jspi.2007.05.035. |
[14] |
M. J. Druzdel and C. Glymour, Causal inferences from databases: Why universities lose students, in Computation, Causation, and Discovery (eds C. Glymour and G. F. Cooper), AAAI Press, Menlo Park, CA., (1999), 521–539. |
[15] |
A. Jasra, D. A. Stephens, A. Doucet and T. Tsagaris,
Inference for Lévy driven stochastic volatility models via adaptive sequential Monte Carlo, Scand. J. Statist., 38 (2011), 1-22.
doi: 10.1111/j.1467-9469.2010.00723.x. |
[16] |
G. Kanayama, H. G. Pope and J. I. Hudson,
Associations of anabolic-androgenic steroid use with other behavioral disorders: an analysis using directed acyclic graphs, Psychol Med, 48 (2018), 2601-2608.
doi: 10.1017/S0033291718000508. |
[17] |
S. L. Lauritzen and T. S. Richardson,
Chain graph models and their causal interpretations, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64 (2002), 321-348.
doi: 10.1111/1467-9868.00340. |
[18] |
S. L. Lauritzen and D. J. Spiegelhalter,
Local computations with probabilities on graphical structures and their applications to expert systems (with discussion), J. R. Statist. Soc. B, 50 (1988), 157-224.
doi: 10.1111/j.2517-6161.1988.tb01721.x. |
[19] |
S. L. Lauritzen and N. Wermuth, Mixed Interaction Models, Institut for Elektroniske Systemer, Aalborg Universitetscenter, 1984. |
[20] |
S. L. Lauritzen and N. Wermuth,
Graphical models for association between variables, some of which are qualitative and some quantitative, Ann. Statist, 17 (1989), 31-57.
doi: 10.1214/aos/1176347003. |
[21] |
A. Lenkoski and A. Dobra,
Computational aspects related to inference in Gaussian graphical models with the G-Wishart prior, Journal of Computational and Graphical Statistics, 20 (2011), 140-157.
doi: 10.1198/jcgs.2010.08181. |
[22] |
M. Levitz, M. D. Perlman and D. Madigan,
Separation and completeness properties for AMP chain graph Markov models, Annals of statistics, 29 (2001), 1751-1784.
doi: 10.1214/aos/1015345961. |
[23] |
C. McCarter and S. Kim,
On sparse Gaussian chain graph models, Advances in Neural Information Processing Systems (NIPS), 2 (2014), 3212-3220.
|
[24] |
J. Pearl, A constraint propagation approach to probabilistic reasoning, in Uncertainty in Artificial Intelligence (eds. L. M. Kanal and J. Lemmer), North-Holland, Amsterdam, (1986), 357–370. |
[25] |
J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, The Morgan Kaufmann Series in Representation and Reasoning. Morgan Kaufmann, San Mateo, CA, 1988. |
[26] |
J. M. Pena, Learning marginal AMP chain graphs under faithfulness, in European Workshop on Probabilistic Graphical Models (eds. Linda C. van der Gaag and Ad J. Feelders), Springer, (2014), 382–395. |
[27] |
N. Richardson-Harman, C. W. Hendrix, N. N. Bumpus, C. Mauck, R. D. Cranston, K. Yang, J. Elliott, K. Tanner and I. McGowan, Correlation between compartmental tenofovir concentrations and an ex vivo rectal biopsy model of tissue infectibility in the RMP-02/MTN-006 phase 1 study, PLoS One, 9 (2014), e111507.
doi: 10.1371/journal.pone.0111507. |
[28] |
R. Silva, A MCMC approach for learning the structure of gaussian acyclic directed mixed graphs, in Statistical Models for Data Analysis (eds. P. Giudici, S. Ingrassia and M. Vichi), Springer: New York, (2013), 343–351.
doi: 10.1007/978-3-319-00032-9_39. |
[29] |
R. Silva and Z. Ghahramani,
The Hidden Life of Latent Variables: Bayesian learning with mixed graph models, J. Mach. Learn. Res., 10 (2009), 1187-1238.
|
[30] |
D. Sonntag and J. M. Pena,
On expressiveness of the chain graph interpretations, International Journal of Approximate Reasoning, 68 (2016), 91-107.
doi: 10.1016/j.ijar.2015.07.009. |
[31] |
L. Tan, A. Jasra, M. De Iorio and T. Ebbels,
Bayesian Inference for multiple Gaussian graphical models, Ann. Appl. Stat., 11 (2017), 2222-2251.
doi: 10.1214/17-AOAS1076. |
[32] |
H. Wang,
Scaling It Up: Stochastic search structure learning in graphical models, Bayes. Anal, 10 (2015), 351-377.
doi: 10.1214/14-BA916. |
[33] |
H. Wang, C. Reesony and C. M. Carvalho,
Dynamic financial index models: Modeling conditional dependencies via graphs, Bayesian Anal., 6 (2011), 639-663.
doi: 10.1214/11-BA624. |
[34] |
N. Wermuth,
Linear recursive equations, covariance selection and path analysis, J. Am. Statist. Assoc, 75 (1980), 963-972.
doi: 10.1080/01621459.1980.10477580. |
[35] |
N. Wermuth and and S. L. Lauritzen,
On substantive research hypotheses, conditional independence graphs and graphical chain models (with discussion), J. Roy. Statist. Soc. Ser. B, 52 (1990), 21-72.
doi: 10.1111/j.2517-6161.1990.tb01771.x. |
[36] |
K. H. Yang, H. Hendrix, N. Bumpus and J. Elliott, et. al, A multi-compartment single and multiple dose pharmacokinetic comparison of rectally applied tenofovir 1% gel and oral tenofovir disoproxil fumarate, PLOS One, 9 (2014), e106196.
doi: 10.1371/journal.pone.0106196. |
[37] |
Y. Zhou, A. M. Johansen and J. A. Aston,
Towards Automatic Model Comparison: An Adaptive Sequence Monte Carlo Approach, J. Comp. Graph. Statist., 25 (2016), 701-726.
doi: 10.1080/10618600.2015.1060885. |
show all references
References:
[1] |
K. Q. Abdool, K. S. S. Abdool and J. A. Frohlich,
Effectiveness and safety of tenofovir gel, an antiretroviral microbicide, for the prevention of HIV infection in women, Science, 329 (2010), 1168-1174.
doi: 10.1126/science.1193748. |
[2] |
S. A. Andersson, D. Madigan and M. D. Perlman,
Alternative Markov properties for chain graphs, Scand. J. Statist., 28 (2001), 33-85.
doi: 10.1111/1467-9469.00224. |
[3] |
P. A. Anton, R. D. Cranston, A. Kashuba, C. W. Hendrix, N. N. Bumpus, N. R. Harman, J. Elliott, L. Janocko, E. Khanukhova, R. Dennis, W. G. Cumberland, C. Ju, A. C. Dieguez, C. Mauck and I. McGowan,
RMP-02/MTN-006: A phase rectal safety, acceptability, pharmacokinetic, and pharmacodynamic study of tenofovir 1% gel compared with oral tenofovir disoproxil fumarate, AIDS Res Hum Retroviruses, 28 (2012), 1412-1421.
doi: 10.1089/aid.2012.0262. |
[4] |
J. M. Baeten, D. Donnell and P. Ndase,
Antiretroviral prophylaxis for HIV prevention in heterosexual men and women, N Engl J Med, 367 (2012), 399-410.
doi: 10.1056/NEJMoa1108524. |
[5] |
A. Beskos, A. Jasra, N. Kantas and A. Thiery,
On the convergence of adaptive sequential Monte Carlo, Ann. Appl. Probab., 26 (2016), 1111-1146.
doi: 10.1214/15-AAP1113. |
[6] |
B. C. Boerebach, K. M. Lombarts, C. Keijzer, M. J. Heineman and O. A. Arah, The teacher, the physician and the person: How faculty's teaching performance influences their role modeling, PLoS One, 7 (2012), e32089.
doi: 10.1371/journal.pone.0032089. |
[7] |
K. Bollen, Structural Equation Models with Latent Variables, Wiley: New York, 1989.
doi: 10.1002/9781118619179. |
[8] |
C. M. Carvalho and M. West,
Dynamic matrix-variate graphical modelso, Bayesian Anal., 2 (2007), 69-97.
doi: 10.1214/07-BA204. |
[9] |
H. Chun, X. Zhang and H. Zhao,
Gene regulation network inference with joint sparse Gaussian graphical models, J. Comp. Graph. Statist., 24 (2015), 954-974.
doi: 10.1080/10618600.2014.956876. |
[10] |
P. Del Moral, A. Doucet and A. Jasra,
Sequential Monte Carlo samplers, J. Roy. Statist. Soc. Ser. B, 68 (2006), 411-436.
doi: 10.1111/j.1467-9868.2006.00553.x. |
[11] |
A. Dobra, C. Hans, B. Jones, J. R. Nevins, G. Yao and M. West,
Sparse graphical models for exploring gene expression data, J. Mult. Anal., 90 (2004), 196-212.
doi: 10.1016/j.jmva.2004.02.009. |
[12] |
M. Drton and M. Eichler,
Maximum Likelihood Estimation in Gaussian Chain Graph Models under the Alternative Markov Property, Scand. J. Statist., 33 (2006), 247-257.
doi: 10.1111/j.1467-9469.2006.00482.x. |
[13] |
M. Drton and M. D. Perlman,
A SINful approach to Gaussian graphical model selection, Journal of Statistical Planning and Inference, 138 (2008), 1179-1200.
doi: 10.1016/j.jspi.2007.05.035. |
[14] |
M. J. Druzdel and C. Glymour, Causal inferences from databases: Why universities lose students, in Computation, Causation, and Discovery (eds C. Glymour and G. F. Cooper), AAAI Press, Menlo Park, CA., (1999), 521–539. |
[15] |
A. Jasra, D. A. Stephens, A. Doucet and T. Tsagaris,
Inference for Lévy driven stochastic volatility models via adaptive sequential Monte Carlo, Scand. J. Statist., 38 (2011), 1-22.
doi: 10.1111/j.1467-9469.2010.00723.x. |
[16] |
G. Kanayama, H. G. Pope and J. I. Hudson,
Associations of anabolic-androgenic steroid use with other behavioral disorders: an analysis using directed acyclic graphs, Psychol Med, 48 (2018), 2601-2608.
doi: 10.1017/S0033291718000508. |
[17] |
S. L. Lauritzen and T. S. Richardson,
Chain graph models and their causal interpretations, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64 (2002), 321-348.
doi: 10.1111/1467-9868.00340. |
[18] |
S. L. Lauritzen and D. J. Spiegelhalter,
Local computations with probabilities on graphical structures and their applications to expert systems (with discussion), J. R. Statist. Soc. B, 50 (1988), 157-224.
doi: 10.1111/j.2517-6161.1988.tb01721.x. |
[19] |
S. L. Lauritzen and N. Wermuth, Mixed Interaction Models, Institut for Elektroniske Systemer, Aalborg Universitetscenter, 1984. |
[20] |
S. L. Lauritzen and N. Wermuth,
Graphical models for association between variables, some of which are qualitative and some quantitative, Ann. Statist, 17 (1989), 31-57.
doi: 10.1214/aos/1176347003. |
[21] |
A. Lenkoski and A. Dobra,
Computational aspects related to inference in Gaussian graphical models with the G-Wishart prior, Journal of Computational and Graphical Statistics, 20 (2011), 140-157.
doi: 10.1198/jcgs.2010.08181. |
[22] |
M. Levitz, M. D. Perlman and D. Madigan,
Separation and completeness properties for AMP chain graph Markov models, Annals of statistics, 29 (2001), 1751-1784.
doi: 10.1214/aos/1015345961. |
[23] |
C. McCarter and S. Kim,
On sparse Gaussian chain graph models, Advances in Neural Information Processing Systems (NIPS), 2 (2014), 3212-3220.
|
[24] |
J. Pearl, A constraint propagation approach to probabilistic reasoning, in Uncertainty in Artificial Intelligence (eds. L. M. Kanal and J. Lemmer), North-Holland, Amsterdam, (1986), 357–370. |
[25] |
J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, The Morgan Kaufmann Series in Representation and Reasoning. Morgan Kaufmann, San Mateo, CA, 1988. |
[26] |
J. M. Pena, Learning marginal AMP chain graphs under faithfulness, in European Workshop on Probabilistic Graphical Models (eds. Linda C. van der Gaag and Ad J. Feelders), Springer, (2014), 382–395. |
[27] |
N. Richardson-Harman, C. W. Hendrix, N. N. Bumpus, C. Mauck, R. D. Cranston, K. Yang, J. Elliott, K. Tanner and I. McGowan, Correlation between compartmental tenofovir concentrations and an ex vivo rectal biopsy model of tissue infectibility in the RMP-02/MTN-006 phase 1 study, PLoS One, 9 (2014), e111507.
doi: 10.1371/journal.pone.0111507. |
[28] |
R. Silva, A MCMC approach for learning the structure of gaussian acyclic directed mixed graphs, in Statistical Models for Data Analysis (eds. P. Giudici, S. Ingrassia and M. Vichi), Springer: New York, (2013), 343–351.
doi: 10.1007/978-3-319-00032-9_39. |
[29] |
R. Silva and Z. Ghahramani,
The Hidden Life of Latent Variables: Bayesian learning with mixed graph models, J. Mach. Learn. Res., 10 (2009), 1187-1238.
|
[30] |
D. Sonntag and J. M. Pena,
On expressiveness of the chain graph interpretations, International Journal of Approximate Reasoning, 68 (2016), 91-107.
doi: 10.1016/j.ijar.2015.07.009. |
[31] |
L. Tan, A. Jasra, M. De Iorio and T. Ebbels,
Bayesian Inference for multiple Gaussian graphical models, Ann. Appl. Stat., 11 (2017), 2222-2251.
doi: 10.1214/17-AOAS1076. |
[32] |
H. Wang,
Scaling It Up: Stochastic search structure learning in graphical models, Bayes. Anal, 10 (2015), 351-377.
doi: 10.1214/14-BA916. |
[33] |
H. Wang, C. Reesony and C. M. Carvalho,
Dynamic financial index models: Modeling conditional dependencies via graphs, Bayesian Anal., 6 (2011), 639-663.
doi: 10.1214/11-BA624. |
[34] |
N. Wermuth,
Linear recursive equations, covariance selection and path analysis, J. Am. Statist. Assoc, 75 (1980), 963-972.
doi: 10.1080/01621459.1980.10477580. |
[35] |
N. Wermuth and and S. L. Lauritzen,
On substantive research hypotheses, conditional independence graphs and graphical chain models (with discussion), J. Roy. Statist. Soc. Ser. B, 52 (1990), 21-72.
doi: 10.1111/j.2517-6161.1990.tb01771.x. |
[36] |
K. H. Yang, H. Hendrix, N. Bumpus and J. Elliott, et. al, A multi-compartment single and multiple dose pharmacokinetic comparison of rectally applied tenofovir 1% gel and oral tenofovir disoproxil fumarate, PLOS One, 9 (2014), e106196.
doi: 10.1371/journal.pone.0106196. |
[37] |
Y. Zhou, A. M. Johansen and J. A. Aston,
Towards Automatic Model Comparison: An Adaptive Sequence Monte Carlo Approach, J. Comp. Graph. Statist., 25 (2016), 701-726.
doi: 10.1080/10618600.2015.1060885. |






Nodes | |||||||||
0.882 | 0.892 | 0.920 | 0.932 | 0.920 | 0.870 | 0.874 | 0.908 | 0.934 | |
0.902 | 0.906 | 0.828 | 0.898 | 0.934 | 0.784 | 0.804 | 0.906 | ||
0.914 | 0.890 | 0.880 | 0.890 | 0.908 | 0.900 | 0.890 | |||
0.900 | 0.866 | 0.882 | 0.770 | 0.918 | 0.900 | ||||
0.788 | 0.952 | 0.912 | 0.830 | 0.918 | |||||
0.806 | 0.932 | 0.906 | 0.908 | ||||||
0.904 | 0.738 | 0.916 | |||||||
0.918 | 0.740 | ||||||||
0.952 |
Nodes | |||||||||
0.882 | 0.892 | 0.920 | 0.932 | 0.920 | 0.870 | 0.874 | 0.908 | 0.934 | |
0.902 | 0.906 | 0.828 | 0.898 | 0.934 | 0.784 | 0.804 | 0.906 | ||
0.914 | 0.890 | 0.880 | 0.890 | 0.908 | 0.900 | 0.890 | |||
0.900 | 0.866 | 0.882 | 0.770 | 0.918 | 0.900 | ||||
0.788 | 0.952 | 0.912 | 0.830 | 0.918 | |||||
0.806 | 0.932 | 0.906 | 0.908 | ||||||
0.904 | 0.738 | 0.916 | |||||||
0.918 | 0.740 | ||||||||
0.952 |
strat | spend | salar | top10 | tstsc | rejr | pacc | apgra | |
strat | 0 | 1 | 1 | 2 | 0 | 0 | 0 | 0 |
spend | 1 | 0 | 1 | 2 | 2 | 2 | 0 | 0 |
salar | 1 | 1 | 0 | 0 | 2 | 2 | 2 | 2 |
top10 | 3 | 3 | 0 | 0 | 1 | 0 | 1 | 0 |
tstsc | 0 | 3 | 3 | 1 | 0 | 1 | 0 | 2 |
rejr | 0 | 3 | 3 | 0 | 1 | 0 | 1 | 0 |
pacc | 0 | 0 | 3 | 1 | 0 | 1 | 0 | 2 |
apgra | 0 | 0 | 3 | 0 | 3 | 0 | 3 | 0 |
strat | spend | salar | top10 | tstsc | rejr | pacc | apgra | |
strat | 0 | 1 | 1 | 2 | 0 | 0 | 0 | 0 |
spend | 1 | 0 | 1 | 2 | 2 | 2 | 0 | 0 |
salar | 1 | 1 | 0 | 0 | 2 | 2 | 2 | 2 |
top10 | 3 | 3 | 0 | 0 | 1 | 0 | 1 | 0 |
tstsc | 0 | 3 | 3 | 1 | 0 | 1 | 0 | 2 |
rejr | 0 | 3 | 3 | 0 | 1 | 0 | 1 | 0 |
pacc | 0 | 0 | 3 | 1 | 0 | 1 | 0 | 2 |
apgra | 0 | 0 | 3 | 0 | 3 | 0 | 3 | 0 |
strat | spend | salar | top10 | tstsc | rejr | pacc | apgra | |
strat | 0 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
spend | 1 | 0 | 1 | 2 | 2 | 2 | 2 | 2 |
salar | 3 | 1 | 0 | 1 | 2 | 2 | 3 | 2 |
top10 | 3 | 3 | 1 | 0 | 2 | 1 | 0 | 2 |
tstsc | 3 | 3 | 3 | 3 | 0 | 1 | 0 | 2 |
rejr | 3 | 3 | 3 | 1 | 1 | 0 | 3 | 0 |
pacc | 3 | 3 | 2 | 0 | 0 | 2 | 0 | 2 |
apgra | 3 | 3 | 3 | 3 | 3 | 0 | 3 | 0 |
strat | spend | salar | top10 | tstsc | rejr | pacc | apgra | |
strat | 0 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
spend | 1 | 0 | 1 | 2 | 2 | 2 | 2 | 2 |
salar | 3 | 1 | 0 | 1 | 2 | 2 | 3 | 2 |
top10 | 3 | 3 | 1 | 0 | 2 | 1 | 0 | 2 |
tstsc | 3 | 3 | 3 | 3 | 0 | 1 | 0 | 2 |
rejr | 3 | 3 | 3 | 1 | 1 | 0 | 3 | 0 |
pacc | 3 | 3 | 2 | 0 | 0 | 2 | 0 | 2 |
apgra | 3 | 3 | 3 | 3 | 3 | 0 | 3 | 0 |
Base chain graph | Chain graph selected by SIN | ||||
Edge | p-value | Edge | p-value | Edge | p-value |
strat — spend | 1.630e-14 | pacc |
1.137e-06 | strat — spend | 1.630e-14 |
strat |
1.382e-06 | pacc |
1.470e-03 | strat — salar | 1.082e-05 |
spend — salar | 2.629e-11 | strat |
8.237e-02 | strat |
1.935e-09 |
strat |
4.743e-07 | spend |
6.067e-02 | spend — salar | 7.156e-13 |
spend |
2.822e-28 | salar |
4.794e-03 | spend |
5.979e-34 |
top10 — salar | 8.931e-03 | top10 |
4.253e-01 | spend |
3.995e-12 |
strat |
3.140e-03 | tstsc |
1.096e-10 | spend |
2.909e-03 |
spend |
6.634e-01 | pacc |
2.711e-03 | salar |
2.350e-05 |
salar |
2.008e-04 | salar |
1.323e-03 | ||
top10 |
6.831e-19 | salar |
1.827e-14 | ||
strat |
1.954e-01 | salar |
1.570e-02 | ||
spend |
3.621e-03 | top10 — tstsc | 1.256e-09 | ||
salar |
2.575e-04 | top10 — pacc | 5.020e-01 | ||
top10 — rejr | 1.816e-04 | tstsc — rejr | 8.297e-03 | ||
tstsc — rejr | 1.003e-02 | tstsc |
8.352e-19 | ||
strat |
2.585e-02 | rejr — pacc | 5.617e-03 | ||
spend |
4.109e-07 | pacc |
5.481e-03 | ||
AIC | BIC | AIC | BIC | ||
67.887 | -13.319 | 80.838 | -24.919 | ||
Chain graph selected by algorithm | Chain graph selected by algorithm | Chain graph selected by algorithm | |||
( |
( |
( |
|||
Edge | p-value | Edge | p-value | Edge | p-value |
strat — spend | 1.630e-14 | spend |
2.150e-52 | spend |
1.536e-42 |
strat |
1.727e-06 | strat |
9.127e-07 | salar |
4.952e-06 |
strat |
3.597e-09 | spend |
2.484e-24 | strat |
1.636e-07 |
spend |
4.956e-33 | top10 |
2.068e-25 | tstsc |
8.237e-01 |
spend |
1.086e-33 | salar |
7.304e-10 | salar |
9.659e-11 |
spend |
4.059e-12 | apgra |
3.751e-11 | spend |
3.881e-10 |
salar |
5.524e-05 | top10 — tstsc | 2.163e-14 | tstsc |
2.886e-08 |
salar |
1.012e-18 | top10 |
2.504e-03 | tstsc |
3.654e-27 |
salar |
3.201e-05 | tstsc |
2.847e-04 | salar |
8.844e-02 |
top10 — tstsc | 2.295e-10 | tstsc |
1.428e-43 | salar |
1.062e-08 |
top10 |
1.951e-03 | rejr |
1.480e-04 | salar |
1.036e-04 |
tstsc |
2.348e-04 | apgra |
3.587e-03 | tstsc |
1.033e-20 |
tstsc |
1.367e-18 | top10 |
9.160e-03 | ||
rejr |
1.032e-03 | tstsc |
5.443e-03 | ||
pacc — apgra | 5.315e-03 | tstsc |
2.644e-17 | ||
rejr |
3.048e-04 | ||||
apgra |
5.759e-03 | ||||
AIC | BIC | AIC | BIC | AIC | BIC |
61.372 | -50.522 | 102.93 | -18.169 | 58.401 | -47.356 |
Base chain graph | Chain graph selected by SIN | ||||
Edge | p-value | Edge | p-value | Edge | p-value |
strat — spend | 1.630e-14 | pacc |
1.137e-06 | strat — spend | 1.630e-14 |
strat |
1.382e-06 | pacc |
1.470e-03 | strat — salar | 1.082e-05 |
spend — salar | 2.629e-11 | strat |
8.237e-02 | strat |
1.935e-09 |
strat |
4.743e-07 | spend |
6.067e-02 | spend — salar | 7.156e-13 |
spend |
2.822e-28 | salar |
4.794e-03 | spend |
5.979e-34 |
top10 — salar | 8.931e-03 | top10 |
4.253e-01 | spend |
3.995e-12 |
strat |
3.140e-03 | tstsc |
1.096e-10 | spend |
2.909e-03 |
spend |
6.634e-01 | pacc |
2.711e-03 | salar |
2.350e-05 |
salar |
2.008e-04 | salar |
1.323e-03 | ||
top10 |
6.831e-19 | salar |
1.827e-14 | ||
strat |
1.954e-01 | salar |
1.570e-02 | ||
spend |
3.621e-03 | top10 — tstsc | 1.256e-09 | ||
salar |
2.575e-04 | top10 — pacc | 5.020e-01 | ||
top10 — rejr | 1.816e-04 | tstsc — rejr | 8.297e-03 | ||
tstsc — rejr | 1.003e-02 | tstsc |
8.352e-19 | ||
strat |
2.585e-02 | rejr — pacc | 5.617e-03 | ||
spend |
4.109e-07 | pacc |
5.481e-03 | ||
AIC | BIC | AIC | BIC | ||
67.887 | -13.319 | 80.838 | -24.919 | ||
Chain graph selected by algorithm | Chain graph selected by algorithm | Chain graph selected by algorithm | |||
( |
( |
( |
|||
Edge | p-value | Edge | p-value | Edge | p-value |
strat — spend | 1.630e-14 | spend |
2.150e-52 | spend |
1.536e-42 |
strat |
1.727e-06 | strat |
9.127e-07 | salar |
4.952e-06 |
strat |
3.597e-09 | spend |
2.484e-24 | strat |
1.636e-07 |
spend |
4.956e-33 | top10 |
2.068e-25 | tstsc |
8.237e-01 |
spend |
1.086e-33 | salar |
7.304e-10 | salar |
9.659e-11 |
spend |
4.059e-12 | apgra |
3.751e-11 | spend |
3.881e-10 |
salar |
5.524e-05 | top10 — tstsc | 2.163e-14 | tstsc |
2.886e-08 |
salar |
1.012e-18 | top10 |
2.504e-03 | tstsc |
3.654e-27 |
salar |
3.201e-05 | tstsc |
2.847e-04 | salar |
8.844e-02 |
top10 — tstsc | 2.295e-10 | tstsc |
1.428e-43 | salar |
1.062e-08 |
top10 |
1.951e-03 | rejr |
1.480e-04 | salar |
1.036e-04 |
tstsc |
2.348e-04 | apgra |
3.587e-03 | tstsc |
1.033e-20 |
tstsc |
1.367e-18 | top10 |
9.160e-03 | ||
rejr |
1.032e-03 | tstsc |
5.443e-03 | ||
pacc — apgra | 5.315e-03 | tstsc |
2.644e-17 | ||
rejr |
3.048e-04 | ||||
apgra |
5.759e-03 | ||||
AIC | BIC | AIC | BIC | AIC | BIC |
61.372 | -50.522 | 102.93 | -18.169 | 58.401 | -47.356 |
Compound | Compartment | Notation |
TFV | Blood plasma | TFV |
TFV | Rectal biopsy tissue | TFV |
TFV | Rectal fluid | TFV |
TFVdp | Rectal biopsy tissue | TFVdp |
TFVdp | Total mononuclear cells in rectal tissue | Total |
TFVdp | CD4 |
CD4 |
TFVdp | CD4 |
CD4 |
Compound | Compartment | Notation |
TFV | Blood plasma | TFV |
TFV | Rectal biopsy tissue | TFV |
TFV | Rectal fluid | TFV |
TFVdp | Rectal biopsy tissue | TFVdp |
TFVdp | Total mononuclear cells in rectal tissue | Total |
TFVdp | CD4 |
CD4 |
TFVdp | CD4 |
CD4 |
Chain graph selected by algorithm | Chain graph selected by algorithm | ||
( |
( |
||
Edge | p-value | Edge | p-value |
CD4 |
9.881e-19 | CD4 |
5.687e-01 |
CD4 |
4.091e-81 | CD4 |
2.941e-46 |
CD4 |
1.496e-01 | CD4 |
1.210e-02 |
Total |
1.589e-02 | CD4 |
7.028e-10 |
TFVdp |
1.812e-02 | CD4 |
1.874e-03 |
CD4 |
8.583e-03 | TFVdp |
8.477e-02 |
Total |
1.815e-03 | TFVdp |
2.991e-01 |
CD4 |
7.043e-01 | TFVdp |
2.584e-01 |
TFVdp |
1.162e-13 | ||
CD4 |
2.352e-22 | ||
TFV |
4.259e-01 | ||
TFV |
5.719e-02 | ||
TFV |
7.550e-01 | ||
Total |
2.337e-02 | ||
AIC | BIC | AIC | BIC |
Inf | Inf | 46.875 | -11.909 |
Chain graph selected by algorithm | Chain graph selected by algorithm | ||
( |
( |
||
Edge | p-value | Edge | p-value |
CD4 |
9.881e-19 | CD4 |
5.687e-01 |
CD4 |
4.091e-81 | CD4 |
2.941e-46 |
CD4 |
1.496e-01 | CD4 |
1.210e-02 |
Total |
1.589e-02 | CD4 |
7.028e-10 |
TFVdp |
1.812e-02 | CD4 |
1.874e-03 |
CD4 |
8.583e-03 | TFVdp |
8.477e-02 |
Total |
1.815e-03 | TFVdp |
2.991e-01 |
CD4 |
7.043e-01 | TFVdp |
2.584e-01 |
TFVdp |
1.162e-13 | ||
CD4 |
2.352e-22 | ||
TFV |
4.259e-01 | ||
TFV |
5.719e-02 | ||
TFV |
7.550e-01 | ||
Total |
2.337e-02 | ||
AIC | BIC | AIC | BIC |
Inf | Inf | 46.875 | -11.909 |
5 largest modification indices, A matrix | 5 largest modification indices, P matrix | ||
(regression coefficients) | (variances/covariances) | ||
TFV |
3.281 | TFV |
3.394 |
TFV |
2.219 | TFV |
2.881 |
CD4 |
0.709 | TFV |
0.709 |
TFV |
0.654 | TFV |
0.709 |
TFV |
0.555 | TFV |
0.389 |
5 largest modification indices, A matrix | 5 largest modification indices, P matrix | ||
(regression coefficients) | (variances/covariances) | ||
TFV |
3.281 | TFV |
3.394 |
TFV |
2.219 | TFV |
2.881 |
CD4 |
0.709 | TFV |
0.709 |
TFV |
0.654 | TFV |
0.709 |
TFV |
0.555 | TFV |
0.389 |
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