1. Tianjian Zhou U of Chicago/UT Austin
Bayesian Nonparametric Models for Biomedical Data Analysis — Inference for Tumor Heterogeneity and Missing Data
Tianjian Zhou
U of Chicago/UT Austin

2. Tumor Heterogeneity A C G A C G

3. Tumor Heterogeneity A C G A C G A C G A C G

4. Tumor Heterogeneity A C G A C G A C G A C G A G A C

5. Tumor Heterogeneity A C G A C G A C G A C G A G A C A G A C

6. Tumor Heterogeneity A C G A C G A C G A C G A G A C A G A C A G A C

7. Tumor Heterogeneity A C G A C G A C G A C G A G A C A G A C A G A C A

8. Tumor Heterogeneity A C G A C G A C G A C G A G A C A G A C A G A C A

9. Tumor Heterogeneity A C G A C G A C G A C G A G A C A G A C A G A C A

10. Tumor Heterogeneity A C G A C G A C G A C G A G A C A G A C A G A C A

11. Tumor Heterogeneity A C G A C G A C G A C G A G A C A G A C A G A C A

12. Tumor Heterogeneity Subclone 1 (Normal) Subclone 2 Subclone 3 A C G A

13. Population frequencies
Tumor Heterogeneity A C G A C G A G A C A G C T
Subclone 1
Subclone 2
Subclone 3
Population frequencies
2/ / /10

14. Why Study Tumor Heterogeneity
Mutations are bad, want to eliminate mutated cells A C G A C G A G A C A G C T A C G A C G A G A C A G C T
Therapy 1: targets the mutation at the 1st locus
Therapy 2: targets the mutation at the 2nd locus

15. Why Study Tumor Heterogeneity
Mutations are bad, want to eliminate mutated cells A C G A C G A G A C A C G A C G
Therapy 1: targets the mutation at the 1st locus
Therapy 2: targets the mutation at the 2nd locus

16. Subclones/DNA sequences are unobserved/latent
Inference for Tumor Heterogeneity
????
????
????
???? ?
Subclone 1
Subclone 2
Subclone ?
Subclones/DNA sequences are unobserved/latent

17. ? Inference for Tumor Heterogeneity Data: Short DNA reads
Mixture of signals from many cells
Quantities of interest A C G ? ? ?
???? A G A C C A G A C ?
# of subclones
Phylogenetic relationship
Genotypes
Population frequencies
Two DNA strands G A C T C A G A G A

18. ? Inference for Tumor Heterogeneity Data: Short DNA reads
Mixture of signals from many cells
Quantities of interest A C G ? ? ?
???? A G A C C A G A C
Proximal
mutations ?
# of subclones
Phylogenetic relationship
Genotypes
Population frequencies
Two DNA strands G A C T C A G A G A
Proximal
mutations

19. ? Inference for Tumor Heterogeneity Data: Short DNA reads
Mixture of signals from many cells
Quantities of interest A C G ? ? ?
???? A G A C C A G A C
Mutation
Pair 1 ?
# of subclones
Phylogenetic relationship
Genotypes
Population frequencies
Two DNA strands G A C T C A G A G A
Mutation
Pair 2

20. Population frequencies
Representation of Subclones A C G A C G A G A C A G C T
Subclone 1
Subclone 2
Subclone 3
Population frequencies
2/ / /10

21. Population frequencies
Representation of Subclones
1: mutation; 0: no mutation (reference) 1 1 1 1
Subclone 1
Subclone 2
Subclone 3
Population frequencies
2/ / /10

22. Population frequencies
Representation of Subclones
Mutation pairs (MP)
1: mutation; 0: no mutation (reference) 1 1 1
MP1
Subclone 1
Subclone 2
Subclone 3
Population frequencies
2/ / /10

23. Population frequencies
Representation of Subclones
Mutation pairs (MP)
1: mutation; 0: no mutation (reference) 1 1
MP2
Subclone 1
Subclone 2
Subclone 3
Population frequencies
2/ / /10

24. Representation of Subclones1 1 1 1
Latent factor matrix Z
Factor loadings w
# of subclones C 3
Phylogenetic tree T
1 → 2 → 3
0 0
1 0
0 1
MP1
MP2
2/10
3/10
5/10
S S S3
S S S3

25. Prior Model Latent factor matrix Z|T, C T: 1 → 2 → 3 0 0 MP1 MP2 MP3
S S S3

26. Prior Model Latent factor matrix Z|T, C T: 1 → 2 → 3 0 0
# of new mutations
𝐴 2 ~ Trunc−Poi 𝜆
MP1
MP2
MP3
MP4
S S S3

27. Prior Model Latent factor matrix Z|T, C T: 1 → 2 → 3 0 0
# of new mutations
𝐴 2 ~ Trunc−Poi 𝜆
𝐴 2 =3
MP1
MP2
MP3
MP4
S S S3

28. Prior Model Latent factor matrix Z|T, C T: 1 → 2 → 3 0 0
# of new mutations
𝐴 2 ~ Trunc−Poi 𝜆
𝐴 2 =3
MP1
MP2
MP3
MP4
S S S3

29. Prior Model Latent factor matrix Z|T, C T: 1 → 2 → 3 0 0 1 0 0 1 MP1
S S S3

30. Prior Model Latent factor matrix Z|T, C T: 1 → 2 → 3 0 0 1 0 0 1
# of new mutations
𝐴 3 ~ Trunc−Poi 𝜆
MP1
MP2
MP3
MP4
S S S3

31. Prior Model Latent factor matrix Z|T, C T: 1 → 2 → 3 0 0 1 0 0 1
# of new mutations
𝐴 3 ~ Trunc−Poi 𝜆
𝐴 3 =4
MP1
MP2
MP3
MP4
S S S3

32. Prior Model Latent factor matrix Z|T, C T: 1 → 2 → 3 0 0 1 0 0 1
# of new mutations
𝐴 3 ~ Trunc−Poi 𝜆
𝐴 3 =4
MP1
MP2
MP3
MP4
S S S3

33. Prior Model Latent factor matrix Z|T, C T: 1 → 2 → 3 0 0 1 0 0 1 1 1
MP1
MP2
MP3
MP4
S S S3

34. ? Inference for Tumor Heterogeneity Data: Short DNA reads
Mixture of signals from many cells
Quantities of interest A C G ? ? ?
???? A G A C C A G A C
Mutation
Pair 1 ?
# of subclones
Phylogenetic relationship
Genotypes
Population frequencies
Two DNA strands G A C T C A G A G A
Mutation
Pair 2

35. Sampling Model A C A C A C A G A C A G C
2/10
3/10
5/10

36. Sampling Model A C A G C
5/10

37. Sampling Model A C A G
5/10

38. Sampling Model A C A G
5/10

39. Sampling Model A C A G A C A C A G A C A G C
2/10
3/10
5/10

40. Sampling Model A C A G A C A C
2/10

41. Sampling Model A C A G A C
2/10

42. Sampling Model A C A G A C
2/10

43. Sampling Model A C A G A C A C A C A G A C A G C
2/10
3/10
5/10

44. Sampling Model A C A G A C A G C
5/10

45. Sampling Model A C A G A C C
5/10

46. Sampling Model A C A G A C C
5/10

47. Sampling Model A C A G A C C A C A C A G A C A G C
2/10
3/10
5/10

48. Sampling Model A C A G A C C A G A C
3/10

49. Sampling Model A C A G A C C A G
3/10

50. Sampling Model A C A G A C C A G
3/10

51. Sampling Model A C A G A C C A G A C A C A G A C A G C
2/10
3/10
5/10

52. Sampling Model A C A G A C C A G A G A C
3/10

53. Sampling Model A C A G A C C A G A C
3/10

54. Sampling Model A C A G A C C A G A C
3/10

55. Sampling Model A C 2/10 3/10 5/10 A G A C C A G A C A C A C A G A C A

56. Sampling Model A C G 2/10 3/10 5/10 A G A C C A G A C G A C T C A G A

57. Sampling Model A C G 2/10 3/10 5/10 1 1 1 1 1 A C G A C G A G A C A G1 1 1 A C G A C G A G A C A G C T
2/10
3/10
5/10 1 1

58. ? Inference for Tumor Heterogeneity Data: Short DNA reads
Mixture of signals from many cells
Quantities of interest A C G ? ? ?
???? A G A C C A G A C
Mutation
Pair 1 ?
# of subclones
Phylogenetic relationship
Genotypes
Population frequencies
Two DNA strands G A C T C A G A G A
Mutation
Pair 2
Sampling model & Prior model → Posterior inference

59. TCGA Lung Cancer Data Malignant hyper-mutated subclone
Small population frequency

60. Concluding Remark
The use of mutation pairs strengthens inference for tumor heterogeneity

61. Inference for Missing Data
𝒚: Longitudinal outcomes after
treated by a test drug
𝑠: Dropout time
𝑠=4
Treatment
Effect
E 𝑌 6 − 𝑌 1
Outcome
𝑠=5 1 2 3 4 5 6
Time

62. Inference for Missing Data
Biased if not MCAR
Inefficient
Can’t do sensitivity analysis
Treatment
Effect
E 𝑌 6 − 𝑌 1
Outcome 1 2 3 4 5 6
Time

63. Inference for Missing Data
Treatment
Effect
E 𝑌 6 − 𝑌 1
Outcome 1 2 3 4 5 6
Time

64. Inference for Missing Data
𝒗= (Age: 66, Height: 185, Weight: 78, Gender: M)
Dropout due to lack of efficacy
𝒗= (Age: 41, Height: 170, Weight: 62, Gender: F)
Outcome
𝒗= (Age: 29, Height: 166, Weight: 54, Gender: F)
Dropout due to pregnancy
𝒗= (Age: 33, Height: 159, Weight: 49, Gender: F) 1 2 3 4 5 6
Time

65. Extrapolation Factorization
Joint model for 𝒚,𝑠 and 𝒗
𝑝 𝒚,𝑠,𝒗 =𝑝 𝒚 mis 𝒚 obs ,𝑠,𝒗 𝑝 𝒚 obs ,𝑠,𝒗
Observed data distribution: identified and can be estimated semi/non-parametrically
Extrapolation distribution: not identified without uncheckable assumptions (e.g. MAR, missing non-future dependent NFD)

66. Observed Data Distribution: Pattern Mixture Modeling
𝑝 𝒚 obs ,𝑠,𝒗 =𝑝 𝒚 obs 𝑠,𝒗 𝑝 𝑠 𝒗 𝑝 𝒗
𝑝 𝒚 obs 𝑠,𝒗 Gaussian process (GP) & autoregressive (AR) & conditional autoregressive (CAR) priors
𝑝 𝑠 𝒗 Bayesian additive regression trees (BART)
𝑝 𝒗 Bayesian bootstrap

67. Extrapolation Distribution: Identifying Restrictions
Missing at random (MAR):
𝑝 𝒚 mis 𝒚 obs ,𝑠,𝒗 is fully identified by 𝑝 𝒚 obs ,𝑠,𝒗
Non-future dependent (NFD):
𝑝 𝒚 mis 𝒚 obs ,𝑠,𝒗 is partially identified by 𝑝 𝒚 obs ,𝑠,𝒗 . Put informative priors on non-identified parameters

68. Monte Carlo Integration/G-Computation
E 𝑡 𝒚 = ∫ 𝒚 𝑡 𝒚 𝑝 𝒚 d𝒚
= ∫ 𝒚 𝑡 𝒚 ∑ 𝑠 ∫ 𝒗 𝑝 𝒚 mis 𝒚 obs ,𝑠,𝒗 𝑝 𝒚 obs 𝑠,𝒗 𝑝 𝑠 𝒗 𝑝 𝒗 ⅆ𝒗 d𝒚
𝒗= (Age: 33, Height: 166, Weight: 54, Gender: F)
Outcome 1 2 3 4 5 6
Time

69. Monte Carlo Integration/G-Computation
E 𝑡 𝒚 = ∫ 𝒚 𝑡 𝒚 𝑝 𝒚 d𝒚
= ∫ 𝒚 𝑡 𝒚 ∑ 𝑠 ∫ 𝒗 𝑝 𝒚 mis 𝒚 obs ,𝑠,𝒗 𝑝 𝒚 obs 𝑠,𝒗 𝑝 𝑠 𝒗 𝑝 𝒗 ⅆ𝒗 d𝒚
𝒗= (Age: 33, Height: 166, Weight: 54, Gender: F)
𝑠=5
Outcome 1 2 3 4 5 6
Time

70. Monte Carlo Integration/G-Computation
E 𝑡 𝒚 = ∫ 𝒚 𝑡 𝒚 𝑝 𝒚 d𝒚
= ∫ 𝒚 𝑡 𝒚 ∑ 𝑠 ∫ 𝒗 𝑝 𝒚 mis 𝒚 obs ,𝑠,𝒗 𝑝 𝒚 obs 𝑠,𝒗 𝑝 𝑠 𝒗 𝑝 𝒗 ⅆ𝒗 d𝒚
𝒗= (Age: 33, Height: 166, Weight: 54, Gender: F)
𝑠=5
Outcome 1 2 3 4 5 6
Time

71. Monte Carlo Integration/G-Computation
E 𝑡 𝒚 = ∫ 𝒚 𝑡 𝒚 𝑝 𝒚 d𝒚
= ∫ 𝒚 𝑡 𝒚 ∑ 𝑠 ∫ 𝒗 𝑝 𝒚 mis 𝒚 obs ,𝑠,𝒗 𝑝 𝒚 obs 𝑠,𝒗 𝑝 𝑠 𝒗 𝑝 𝒗 ⅆ𝒗 d𝒚
𝒗= (Age: 33, Height: 166, Weight: 54, Gender: F)
𝑠=5
Outcome 1 2 3 4 5 6
Time

72. Monte Carlo Integration/G-Computation
E 𝑡 𝒚 = ∫ 𝒚 𝑡 𝒚 𝑝 𝒚 d𝒚
= ∫ 𝒚 𝑡 𝒚 ∑ 𝑠 ∫ 𝒗 𝑝 𝒚 mis 𝒚 obs ,𝑠,𝒗 𝑝 𝒚 obs 𝑠,𝒗 𝑝 𝑠 𝒗 𝑝 𝒗 ⅆ𝒗 d𝒚
𝒗= (Age: 37, Height: 163, Weight: 51, Gender: F)
Outcome 1 2 3 4 5 6
Time

73. Monte Carlo Integration/G-Computation
E 𝑡 𝒚 = ∫ 𝒚 𝑡 𝒚 𝑝 𝒚 d𝒚
= ∫ 𝒚 𝑡 𝒚 ∑ 𝑠 ∫ 𝒗 𝑝 𝒚 mis 𝒚 obs ,𝑠,𝒗 𝑝 𝒚 obs 𝑠,𝒗 𝑝 𝑠 𝒗 𝑝 𝒗 ⅆ𝒗 d𝒚
𝒗= (Age: 37, Height: 163, Weight: 51, Gender: F)
𝑠=6
Outcome 1 2 3 4 5 6
Time

74. Monte Carlo Integration/G-Computation
E 𝑡 𝒚 = ∫ 𝒚 𝑡 𝒚 𝑝 𝒚 d𝒚
= ∫ 𝒚 𝑡 𝒚 ∑ 𝑠 ∫ 𝒗 𝑝 𝒚 mis 𝒚 obs ,𝑠,𝒗 𝑝 𝒚 obs 𝑠,𝒗 𝑝 𝑠 𝒗 𝑝 𝒗 ⅆ𝒗 d𝒚
𝒗= (Age: 37, Height: 163, Weight: 51, Gender: F)
𝑠=6
Outcome 1 2 3 4 5 6
Time

75. Monte Carlo Integration/G-Computation
E 𝑡 𝒚 = ∫ 𝒚 𝑡 𝒚 𝑝 𝒚 d𝒚
= ∫ 𝒚 𝑡 𝒚 ∑ 𝑠 ∫ 𝒗 𝑝 𝒚 mis 𝒚 obs ,𝑠,𝒗 𝑝 𝒚 obs 𝑠,𝒗 𝑝 𝑠 𝒗 𝑝 𝒗 ⅆ𝒗 d𝒚
𝒗= (Age: 66, Height: 185, Weight: 78, Gender: M)
Outcome 1 2 3 4 5 6
Time

76. Monte Carlo Integration/G-Computation
E 𝑡 𝒚 = ∫ 𝒚 𝑡 𝒚 𝑝 𝒚 d𝒚
= ∫ 𝒚 𝑡 𝒚 ∑ 𝑠 ∫ 𝒗 𝑝 𝒚 mis 𝒚 obs ,𝑠,𝒗 𝑝 𝒚 obs 𝑠,𝒗 𝑝 𝑠 𝒗 𝑝 𝒗 ⅆ𝒗 d𝒚
𝒗= (Age: 66, Height: 185, Weight: 78, Gender: M)
𝑠=3
Outcome 1 2 3 4 5 6
Time

77. Monte Carlo Integration/G-Computation
E 𝑡 𝒚 = ∫ 𝒚 𝑡 𝒚 𝑝 𝒚 d𝒚
= ∫ 𝒚 𝑡 𝒚 ∑ 𝑠 ∫ 𝒗 𝑝 𝒚 mis 𝒚 obs ,𝑠,𝒗 𝑝 𝒚 obs 𝑠,𝒗 𝑝 𝑠 𝒗 𝑝 𝒗 ⅆ𝒗 d𝒚
𝒗= (Age: 66, Height: 185, Weight: 78, Gender: M)
𝑠=3
Outcome 1 2 3 4 5 6
Time

78. Monte Carlo Integration/G-Computation
E 𝑡 𝒚 = ∫ 𝒚 𝑡 𝒚 𝑝 𝒚 d𝒚
= ∫ 𝒚 𝑡 𝒚 ∑ 𝑠 ∫ 𝒗 𝑝 𝒚 mis 𝒚 obs ,𝑠,𝒗 𝑝 𝒚 obs 𝑠,𝒗 𝑝 𝑠 𝒗 𝑝 𝒗 ⅆ𝒗 d𝒚
𝒗= (Age: 66, Height: 185, Weight: 78, Gender: M)
𝑠=3
Outcome 1 2 3 4 5 6
Time

79. Monte Carlo Integration/G-Computation
E 𝑡 𝒚 = ∫ 𝒚 𝑡 𝒚 𝑝 𝒚 d𝒚
= ∫ 𝒚 𝑡 𝒚 ∑ 𝑠 ∫ 𝒗 𝑝 𝒚 mis 𝒚 obs ,𝑠,𝒗 𝑝 𝒚 obs 𝑠,𝒗 𝑝 𝑠 𝒗 𝑝 𝒗 ⅆ𝒗 d𝒚
Outcome
𝑦 6 − 𝑦 1 1 2 3 4 5 6
Time

80. Schizophrenia Dataset
Test drug improvement over placebo
A negative value represents an improvement
Conclusion: no evidence that the test drug performs better than placebo

81. Sensitivity Analysis
Vary uncheckable assumptions and see whether conclusion differs

82. Concluding Remark
The model specifications (GP/AR/CAR/BART) nicely exploit the data structure and lead to improvement over simple parametric approaches

83. References
Zhou, T., Müller, P., Sengupta, S. and Ji, Y. (2019) PairClone: A Bayesian subclone caller based on mutation pairs. Journal of the Royal Statistical Society: Series C (Applied Statistics), 68(3),
Zhou, T., Sengupta, S., Müller, P., and Ji, Y. (2019) TreeClone: Reconstruction of tumor subclone phylogeny based on mutation pairs using next generation sequencing data. The Annals of Applied Statistics, 13(2),
Zhou, T., Daniels, M. J. and Müller, P. (2019) A Semiparametric Bayesian Approach to Dropout in Longitudinal Studies with Auxiliary Covariates. Journal of Computational and Graphical Statistics, forthcoming.

84. Thank you! Questions & comments: tjzhou@uchicago.edu
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