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March  2020, 2(1): 35-54. doi: 10.3934/fods.2020003

Bayesian inference for latent chain graphs

Deng Lu 1, , Maria De Iorio 2, , Ajay Jasra 3, and  Gary L. Rosner 4,

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

Published  March 2020

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.

Keywords: Chain graphs, Bayesian inference, sequential Monte Carlo.
Mathematics Subject Classification: 62F15.
Citation: Deng Lu, Maria De Iorio, Ajay Jasra, Gary L. Rosner. Bayesian inference for latent chain graphs. Foundations of Data Science, 2020, 2 (1) : 35-54. doi: 10.3934/fods.2020003
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

K. Bollen, Structural Equation Models with Latent Variables, Wiley: New York, 1989. doi: 10.1002/9781118619179.  Google Scholar

[8]

C. M. Carvalho and M. West, Dynamic matrix-variate graphical modelso, Bayesian Anal., 2 (2007), 69-97.  doi: 10.1214/07-BA204.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. L. Lauritzen and N. Wermuth, Mixed Interaction Models, Institut for Elektroniske Systemer, Aalborg Universitetscenter, 1984. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

C. McCarter and S. Kim, On sparse Gaussian chain graph models, Advances in Neural Information Processing Systems (NIPS), 2 (2014), 3212-3220.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

H. Wang, Scaling It Up: Stochastic search structure learning in graphical models, Bayes. Anal, 10 (2015), 351-377.  doi: 10.1214/14-BA916.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

K. Bollen, Structural Equation Models with Latent Variables, Wiley: New York, 1989. doi: 10.1002/9781118619179.  Google Scholar

[8]

C. M. Carvalho and M. West, Dynamic matrix-variate graphical modelso, Bayesian Anal., 2 (2007), 69-97.  doi: 10.1214/07-BA204.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. L. Lauritzen and N. Wermuth, Mixed Interaction Models, Institut for Elektroniske Systemer, Aalborg Universitetscenter, 1984. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

C. McCarter and S. Kim, On sparse Gaussian chain graph models, Advances in Neural Information Processing Systems (NIPS), 2 (2014), 3212-3220.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

H. Wang, Scaling It Up: Stochastic search structure learning in graphical models, Bayes. Anal, 10 (2015), 351-377.  doi: 10.1214/14-BA916.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Simulation results for the independent case: (a) ESS in each SMC step; (b) plot of $ \Omega[1,1] $ across on particles; (c) acceptance rates in each SMC step; (d) distribution of the log(target) (i.e., log of $ \pi(B,\Omega,(a_{ij})_{i<j}\mid y_{1:m},\alpha) $) at the end of the algorithm
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12]">Figure 2.  Chain Graph Estimate presented in [12]
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Figure 3.  Empirical Graph
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Figure 4.  posterior estimated chain graph using a Dirichlet prior with $ \alpha = (0.39 , 0.25 , 0.36 , 0.05) $
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Figure 5.  posterior estimated chain graph using a Dirichlet prior with $ \alpha = (1 , 1 , 1 , 1) $
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Figure 6.  posterior estimated chain graph using a Dirichlet prior with $ \alpha = (1 , 3 , 3 , 3) $
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Figure 7.  Chain graph with highest posterior probability
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Table 1.  Posterior probability $ \mathbb{P} (a_{ij} = 0 | y_{1:m},\alpha), 1\leq i < j \leq p $
Nodes $ 2 $ $ 3 $ $ 4 $ $ 5 $ $ 6 $ $ 7 $ $ 8 $ $ 9 $ $ 10 $
$ \; \; 1 $ 0.882 0.892 0.920 0.932 0.920 0.870 0.874 0.908 0.934
$ \; \; 2 $ 0.902 0.906 0.828 0.898 0.934 0.784 0.804 0.906
$ \; \; 3 $ 0.914 0.890 0.880 0.890 0.908 0.900 0.890
$ \; \; 4 $ 0.900 0.866 0.882 0.770 0.918 0.900
$ \; \; 5 $ 0.788 0.952 0.912 0.830 0.918
$ \; \; 6 $ 0.806 0.932 0.906 0.908
$ \; \; 7 $ 0.904 0.738 0.916
$ \; \; 8 $ 0.918 0.740
$ \; \; 9 $ 0.952
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Nodes $ 2 $ $ 3 $ $ 4 $ $ 5 $ $ 6 $ $ 7 $ $ 8 $ $ 9 $ $ 10 $
$ \; \; 1 $ 0.882 0.892 0.920 0.932 0.920 0.870 0.874 0.908 0.934
$ \; \; 2 $ 0.902 0.906 0.828 0.898 0.934 0.784 0.804 0.906
$ \; \; 3 $ 0.914 0.890 0.880 0.890 0.908 0.900 0.890
$ \; \; 4 $ 0.900 0.866 0.882 0.770 0.918 0.900
$ \; \; 5 $ 0.788 0.952 0.912 0.830 0.918
$ \; \; 6 $ 0.806 0.932 0.906 0.908
$ \; \; 7 $ 0.904 0.738 0.916
$ \; \; 8 $ 0.918 0.740
$ \; \; 9 $ 0.952
Table 2.  The adjacency matrix corresponding to chain graph in Figure 2
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
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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
Table 3.  The adjacency matrix corresponding to the chain graph in Figure 3
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
Table Options

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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
Table 4.  Summaries of different chain graphs using package SEM
Base chain graph Chain graph selected by SIN
Edge p-value Edge p-value Edge p-value
strat — spend 1.630e-14 pacc $ \rightarrow $ salar 1.137e-06 strat — spend 1.630e-14
strat $ \rightarrow $ salar 1.382e-06 pacc $ \rightarrow $ rejr 1.470e-03 strat — salar 1.082e-05
spend — salar 2.629e-11 strat $ \rightarrow $ apgra 8.237e-02 strat $ \rightarrow $ top10 1.935e-09
strat $ \rightarrow $ top10 4.743e-07 spend $ \rightarrow $ apgra 6.067e-02 spend — salar 7.156e-13
spend $ \rightarrow $ top10 2.822e-28 salar $ \rightarrow $ apgra 4.794e-03 spend $ \rightarrow $ top10 5.979e-34
top10 — salar 8.931e-03 top10 $ \rightarrow $ apgra 4.253e-01 spend $ \rightarrow $ tstsc 3.995e-12
strat $ \rightarrow $ tstsc 3.140e-03 tstsc $ \rightarrow $ apgra 1.096e-10 spend $ \rightarrow $ rejr 2.909e-03
spend $ \rightarrow $ tstsc 6.634e-01 pacc $ \rightarrow $ apgra 2.711e-03 salar $ \rightarrow $ tstsc 2.350e-05
salar $ \rightarrow $ tstsc 2.008e-04 salar $ \rightarrow $ rejr 1.323e-03
top10 $ \rightarrow $ tstsc 6.831e-19 salar $ \rightarrow $ pacc 1.827e-14
strat $ \rightarrow $ rejr 1.954e-01 salar $ \rightarrow $ apgra 1.570e-02
spend $ \rightarrow $ rejr 3.621e-03 top10 — tstsc 1.256e-09
salar $ \rightarrow $ rejr 2.575e-04 top10 — pacc 5.020e-01
top10 — rejr 1.816e-04 tstsc — rejr 8.297e-03
tstsc — rejr 1.003e-02 tstsc $ \rightarrow $ apgra 8.352e-19
strat $ \rightarrow $ pacc 2.585e-02 rejr — pacc 5.617e-03
spend $ \rightarrow $ pacc 4.109e-07 pacc $ \rightarrow $ apgra 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
($ \alpha=(0.39 , 0.25 , 0.36 , 0.05) $) ($ \alpha=(1,1,1,1) $) ($ \alpha=(1,3,3,3) $)
Edge p-value Edge p-value Edge p-value
strat — spend 1.630e-14 spend $ \rightarrow $ strat 2.150e-52 spend $ \rightarrow $ strat 1.536e-42
strat $ \rightarrow $ salar 1.727e-06 strat $ \rightarrow $ salar 9.127e-07 salar $ \rightarrow $ strat 4.952e-06
strat $ \rightarrow $ top10 3.597e-09 spend $ \rightarrow $ salar 2.484e-24 strat $ \rightarrow $ top10 1.636e-07
spend $ \rightarrow $ salar 4.956e-33 top10 $ \rightarrow $ spend 2.068e-25 tstsc $ \rightarrow $ strat 8.237e-01
spend $ \rightarrow $ top10 1.086e-33 salar $ \rightarrow $ pacc 7.304e-10 salar $ \rightarrow $ spend 9.659e-11
spend $ \rightarrow $ tstsc 4.059e-12 apgra $ \rightarrow $ salar 3.751e-11 spend $ \rightarrow $ top10 3.881e-10
salar $ \rightarrow $ tstsc 5.524e-05 top10 — tstsc 2.163e-14 tstsc $ \rightarrow $ spend 2.886e-08
salar $ \rightarrow $ pacc 1.012e-18 top10 $ \rightarrow $ rejr 2.504e-03 tstsc $ \rightarrow $ salar 3.654e-27
salar $ \rightarrow $ apgra 3.201e-05 tstsc $ \rightarrow $ rejr 2.847e-04 salar $ \rightarrow $ rejr 8.844e-02
top10 — tstsc 2.295e-10 tstsc $ \rightarrow $ apgra 1.428e-43 salar $ \rightarrow $ pacc 1.062e-08
top10 $ \rightarrow $ rejr 1.951e-03 rejr $ \rightarrow $ pacc 1.480e-04 salar $ \rightarrow $ apgra 1.036e-04
tstsc $ \rightarrow $ rejr 2.348e-04 apgra $ \rightarrow $ pacc 3.587e-03 tstsc $ \rightarrow $ top10 1.033e-20
tstsc $ \rightarrow $ apgra 1.367e-18 top10 $ \rightarrow $ rejr 9.160e-03
rejr $ \rightarrow $ pacc 1.032e-03 tstsc $ \rightarrow $ rejr 5.443e-03
pacc — apgra 5.315e-03 tstsc $ \rightarrow $ apgra 2.644e-17
rejr $ \rightarrow $ pacc 3.048e-04
apgra $ \rightarrow $ pacc 5.759e-03
AIC BIC AIC BIC AIC BIC
61.372 -50.522 102.93 -18.169 58.401 -47.356
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Base chain graph Chain graph selected by SIN
Edge p-value Edge p-value Edge p-value
strat — spend 1.630e-14 pacc $ \rightarrow $ salar 1.137e-06 strat — spend 1.630e-14
strat $ \rightarrow $ salar 1.382e-06 pacc $ \rightarrow $ rejr 1.470e-03 strat — salar 1.082e-05
spend — salar 2.629e-11 strat $ \rightarrow $ apgra 8.237e-02 strat $ \rightarrow $ top10 1.935e-09
strat $ \rightarrow $ top10 4.743e-07 spend $ \rightarrow $ apgra 6.067e-02 spend — salar 7.156e-13
spend $ \rightarrow $ top10 2.822e-28 salar $ \rightarrow $ apgra 4.794e-03 spend $ \rightarrow $ top10 5.979e-34
top10 — salar 8.931e-03 top10 $ \rightarrow $ apgra 4.253e-01 spend $ \rightarrow $ tstsc 3.995e-12
strat $ \rightarrow $ tstsc 3.140e-03 tstsc $ \rightarrow $ apgra 1.096e-10 spend $ \rightarrow $ rejr 2.909e-03
spend $ \rightarrow $ tstsc 6.634e-01 pacc $ \rightarrow $ apgra 2.711e-03 salar $ \rightarrow $ tstsc 2.350e-05
salar $ \rightarrow $ tstsc 2.008e-04 salar $ \rightarrow $ rejr 1.323e-03
top10 $ \rightarrow $ tstsc 6.831e-19 salar $ \rightarrow $ pacc 1.827e-14
strat $ \rightarrow $ rejr 1.954e-01 salar $ \rightarrow $ apgra 1.570e-02
spend $ \rightarrow $ rejr 3.621e-03 top10 — tstsc 1.256e-09
salar $ \rightarrow $ rejr 2.575e-04 top10 — pacc 5.020e-01
top10 — rejr 1.816e-04 tstsc — rejr 8.297e-03
tstsc — rejr 1.003e-02 tstsc $ \rightarrow $ apgra 8.352e-19
strat $ \rightarrow $ pacc 2.585e-02 rejr — pacc 5.617e-03
spend $ \rightarrow $ pacc 4.109e-07 pacc $ \rightarrow $ apgra 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
($ \alpha=(0.39 , 0.25 , 0.36 , 0.05) $) ($ \alpha=(1,1,1,1) $) ($ \alpha=(1,3,3,3) $)
Edge p-value Edge p-value Edge p-value
strat — spend 1.630e-14 spend $ \rightarrow $ strat 2.150e-52 spend $ \rightarrow $ strat 1.536e-42
strat $ \rightarrow $ salar 1.727e-06 strat $ \rightarrow $ salar 9.127e-07 salar $ \rightarrow $ strat 4.952e-06
strat $ \rightarrow $ top10 3.597e-09 spend $ \rightarrow $ salar 2.484e-24 strat $ \rightarrow $ top10 1.636e-07
spend $ \rightarrow $ salar 4.956e-33 top10 $ \rightarrow $ spend 2.068e-25 tstsc $ \rightarrow $ strat 8.237e-01
spend $ \rightarrow $ top10 1.086e-33 salar $ \rightarrow $ pacc 7.304e-10 salar $ \rightarrow $ spend 9.659e-11
spend $ \rightarrow $ tstsc 4.059e-12 apgra $ \rightarrow $ salar 3.751e-11 spend $ \rightarrow $ top10 3.881e-10
salar $ \rightarrow $ tstsc 5.524e-05 top10 — tstsc 2.163e-14 tstsc $ \rightarrow $ spend 2.886e-08
salar $ \rightarrow $ pacc 1.012e-18 top10 $ \rightarrow $ rejr 2.504e-03 tstsc $ \rightarrow $ salar 3.654e-27
salar $ \rightarrow $ apgra 3.201e-05 tstsc $ \rightarrow $ rejr 2.847e-04 salar $ \rightarrow $ rejr 8.844e-02
top10 — tstsc 2.295e-10 tstsc $ \rightarrow $ apgra 1.428e-43 salar $ \rightarrow $ pacc 1.062e-08
top10 $ \rightarrow $ rejr 1.951e-03 rejr $ \rightarrow $ pacc 1.480e-04 salar $ \rightarrow $ apgra 1.036e-04
tstsc $ \rightarrow $ rejr 2.348e-04 apgra $ \rightarrow $ pacc 3.587e-03 tstsc $ \rightarrow $ top10 1.033e-20
tstsc $ \rightarrow $ apgra 1.367e-18 top10 $ \rightarrow $ rejr 9.160e-03
rejr $ \rightarrow $ pacc 1.032e-03 tstsc $ \rightarrow $ rejr 5.443e-03
pacc — apgra 5.315e-03 tstsc $ \rightarrow $ apgra 2.644e-17
rejr $ \rightarrow $ pacc 3.048e-04
apgra $ \rightarrow $ pacc 5.759e-03
AIC BIC AIC BIC AIC BIC
61.372 -50.522 102.93 -18.169 58.401 -47.356
Table 5.  Tissues and cell types examined in the PK studies
Compound Compartment Notation
TFV Blood plasma TFV$ _{plasma} $
TFV Rectal biopsy tissue TFV$ _{tissue} $
TFV Rectal fluid TFV$ _{rectal} $
TFVdp Rectal biopsy tissue TFVdp$ _{tissue} $
TFVdp Total mononuclear cells in rectal tissue Total$ _{\text{MMC}} $
TFVdp CD4$ ^+ $ lymphocytes from MMC CD4$ ^+_{\text{MMC}} $
TFVdp CD4$ ^- $ lymphocytes from MMC CD4$ ^-_{\text{MMC}} $
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Compound Compartment Notation
TFV Blood plasma TFV$ _{plasma} $
TFV Rectal biopsy tissue TFV$ _{tissue} $
TFV Rectal fluid TFV$ _{rectal} $
TFVdp Rectal biopsy tissue TFVdp$ _{tissue} $
TFVdp Total mononuclear cells in rectal tissue Total$ _{\text{MMC}} $
TFVdp CD4$ ^+ $ lymphocytes from MMC CD4$ ^+_{\text{MMC}} $
TFVdp CD4$ ^- $ lymphocytes from MMC CD4$ ^-_{\text{MMC}} $
Table 6.  Summaries of two different chain graphs using package SEM
Chain graph selected by algorithm Chain graph selected by algorithm
($ \alpha=(1,1,1,1) $) ($ \alpha=(1,3,3,3) $)
Edge p-value Edge p-value
CD4$ ^-_{\text{MMC}} $ $ \rightarrow $ Total$ _{\text{MMC}} $ 9.881e-19 CD4$ ^+_{\text{MMC}} $ $ \rightarrow $ TFVdp$ _{tissue} $ 5.687e-01
CD4$ ^-_{\text{MMC}} $ $ \rightarrow $ CD4$ ^+_{\text{MMC}} $ 4.091e-81 CD4$ ^+_{\text{MMC}} $ $ \rightarrow $ CD4$ ^-_{\text{MMC}} $ 2.941e-46
CD4$ ^-_{\text{MMC}} $ $ \rightarrow $ TFV$ _{rectal} $ 1.496e-01 CD4$ ^+_{\text{MMC}} $ $ \rightarrow $ TFV$ _{plasma} $ 1.210e-02
Total$ _{\text{MMC}} $ — TFVdp$ _{tissue} $ 1.589e-02 CD4$ ^+_{\text{MMC}} $ $ \rightarrow $ Total$ _{\text{MMC}} $ 7.028e-10
TFVdp$ _{tissue} $ — CD4$ ^+_{\text{MMC}} $ 1.812e-02 CD4$ ^+_{\text{MMC}} $ $ \rightarrow $ TFV$ _{rectal} $ 1.874e-03
CD4$ ^+_{\text{MMC}} $ — TFV$ _{rectal} $ 8.583e-03 TFVdp$ _{tissue} $ — CD4$ ^-_{\text{MMC}} $ 8.477e-02
Total$ _{\text{MMC}} $ $ \rightarrow $ TFV$ _{plasma} $ 1.815e-03 TFVdp$ _{tissue} $ $ \rightarrow $ TFV$ _{rectal} $ 2.991e-01
CD4$ ^+_{\text{MMC}} $ $ \rightarrow $ TFV$ _{tissue} $ 7.043e-01 TFVdp$ _{tissue} $ $ \rightarrow $ TFV$ _{plasma} $ 2.584e-01
TFVdp$ _{tissue} $ $ \rightarrow $ Total$ _{\text{MMC}} $ 1.162e-13
CD4$ ^-_{\text{MMC}} $ $ \rightarrow $ Total$ _{\text{MMC}} $ 2.352e-22
TFV$ _{plasma} $ $ \rightarrow $ TFV$ _{tissue} $ 4.259e-01
TFV$ _{plasma} $ — Total$ _{\text{MMC}} $ 5.719e-02
TFV$ _{tissue} $ $ \rightarrow $ TFV$ _{rectal} $ 7.550e-01
Total$ _{\text{MMC}} $ $ \rightarrow $ TFV$ _{rectal} $ 2.337e-02
AIC BIC AIC BIC
Inf Inf 46.875 -11.909
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Chain graph selected by algorithm Chain graph selected by algorithm
($ \alpha=(1,1,1,1) $) ($ \alpha=(1,3,3,3) $)
Edge p-value Edge p-value
CD4$ ^-_{\text{MMC}} $ $ \rightarrow $ Total$ _{\text{MMC}} $ 9.881e-19 CD4$ ^+_{\text{MMC}} $ $ \rightarrow $ TFVdp$ _{tissue} $ 5.687e-01
CD4$ ^-_{\text{MMC}} $ $ \rightarrow $ CD4$ ^+_{\text{MMC}} $ 4.091e-81 CD4$ ^+_{\text{MMC}} $ $ \rightarrow $ CD4$ ^-_{\text{MMC}} $ 2.941e-46
CD4$ ^-_{\text{MMC}} $ $ \rightarrow $ TFV$ _{rectal} $ 1.496e-01 CD4$ ^+_{\text{MMC}} $ $ \rightarrow $ TFV$ _{plasma} $ 1.210e-02
Total$ _{\text{MMC}} $ — TFVdp$ _{tissue} $ 1.589e-02 CD4$ ^+_{\text{MMC}} $ $ \rightarrow $ Total$ _{\text{MMC}} $ 7.028e-10
TFVdp$ _{tissue} $ — CD4$ ^+_{\text{MMC}} $ 1.812e-02 CD4$ ^+_{\text{MMC}} $ $ \rightarrow $ TFV$ _{rectal} $ 1.874e-03
CD4$ ^+_{\text{MMC}} $ — TFV$ _{rectal} $ 8.583e-03 TFVdp$ _{tissue} $ — CD4$ ^-_{\text{MMC}} $ 8.477e-02
Total$ _{\text{MMC}} $ $ \rightarrow $ TFV$ _{plasma} $ 1.815e-03 TFVdp$ _{tissue} $ $ \rightarrow $ TFV$ _{rectal} $ 2.991e-01
CD4$ ^+_{\text{MMC}} $ $ \rightarrow $ TFV$ _{tissue} $ 7.043e-01 TFVdp$ _{tissue} $ $ \rightarrow $ TFV$ _{plasma} $ 2.584e-01
TFVdp$ _{tissue} $ $ \rightarrow $ Total$ _{\text{MMC}} $ 1.162e-13
CD4$ ^-_{\text{MMC}} $ $ \rightarrow $ Total$ _{\text{MMC}} $ 2.352e-22
TFV$ _{plasma} $ $ \rightarrow $ TFV$ _{tissue} $ 4.259e-01
TFV$ _{plasma} $ — Total$ _{\text{MMC}} $ 5.719e-02
TFV$ _{tissue} $ $ \rightarrow $ TFV$ _{rectal} $ 7.550e-01
Total$ _{\text{MMC}} $ $ \rightarrow $ TFV$ _{rectal} $ 2.337e-02
AIC BIC AIC BIC
Inf Inf 46.875 -11.909
Table 7.  Summaries of modification indices for the model corresponding to the chain graph obtained under prior with $ \alpha = (1,3,3,3) $
5 largest modification indices, A matrix 5 largest modification indices, P matrix
(regression coefficients) (variances/covariances)
TFV$ _{rectal} $ $ \rightarrow $ Total$ _{\text{MMC}} $ 3.281 TFV$ _{rectal} $ — Total$ _{\text{MMC}} $ 3.394
TFV$ _{rectal} $ $ \rightarrow $ TFV$ _{plasma} $ 2.219 TFV$ _{rectal} $ — TFV$ _{plasma} $ 2.881
CD4$ ^-_{\text{MMC}} $ $ \rightarrow $ TFV$ _{rectal} $ 0.709 TFV$ _{rectal} $ — CD4$ ^-_{\text{MMC}} $ 0.709
TFV$ _{rectal} $ $ \rightarrow $ TFVdp$ _{tissue} $ 0.654 TFV$ _{rectal} $ — TFVdp$ _{tissue} $ 0.709
TFV$ _{rectal} $ $ \rightarrow $ CD4$ ^-_{\text{MMC}} $ 0.555 TFV$ _{tissue} $ — TFVdp$ _{tissue} $ 0.389
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5 largest modification indices, A matrix 5 largest modification indices, P matrix
(regression coefficients) (variances/covariances)
TFV$ _{rectal} $ $ \rightarrow $ Total$ _{\text{MMC}} $ 3.281 TFV$ _{rectal} $ — Total$ _{\text{MMC}} $ 3.394
TFV$ _{rectal} $ $ \rightarrow $ TFV$ _{plasma} $ 2.219 TFV$ _{rectal} $ — TFV$ _{plasma} $ 2.881
CD4$ ^-_{\text{MMC}} $ $ \rightarrow $ TFV$ _{rectal} $ 0.709 TFV$ _{rectal} $ — CD4$ ^-_{\text{MMC}} $ 0.709
TFV$ _{rectal} $ $ \rightarrow $ TFVdp$ _{tissue} $ 0.654 TFV$ _{rectal} $ — TFVdp$ _{tissue} $ 0.709
TFV$ _{rectal} $ $ \rightarrow $ CD4$ ^-_{\text{MMC}} $ 0.555 TFV$ _{tissue} $ — TFVdp$ _{tissue} $ 0.389
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