1. Zhi Yang, MS Department of Preventive Medicine, USC Jul 29, 2018
Statistical Approach for Investigating Change in Mutational Processes During Cancer Growth and Development
Zhi Yang, MS
Department of Preventive Medicine, USC
Jul 29, 2018
Hello everyone
My name is Zhi, I am a third year Phd student in the biostatistics program.
In project four, we also do hierarchical modeling but in tumors by using somatic mutations. More specifically, we will describe the somatic mutation with mutational signature, which is a concept I will introduce later in the talk. Therefore, we use hierarchical modeling of mutational signatures in tumors to capture the change during the tumor growth.

2. A Unifying Model to Test Difference?
HiLDA = “Hierarchical Latent Dirichlet Allocation”
Uncertainty in Proportions
Somatic mutations
pmsignature
Estimated
Proportions, 𝒒
Are 𝒒 different in two groups?
Regress 𝒒 on 𝑮 (0=branch, 1=trunk)
HiLDA
If people would like to infer the difference in signature proportions, they can take the point estimates by using any current methods, for example, R package pmsignature by assuming independence. Then, take the fractions to regress on the indicator variable group, 1 as trunk 2

3. Hierarchical Latent Dirichlet Allocation
𝒑 𝒊 𝟎
𝒑 𝒊 𝟏
Hyperprior
𝒒 𝑖 1
𝑍 𝑖,𝑗 1
𝑋 𝑖,𝑗 1
𝒇 𝑘
𝑘=1…𝐾
𝑗=1… 𝑛 𝑖 1
𝑖=1…𝑁
𝑗=1… 𝑛 𝑖 0
𝑋 𝑖,𝑗 0
𝑍 𝑖,𝑗 0
𝒒 𝑖 0
Signature
Latent
signature
assignment
Observed
Mutation
Proportions
𝜹 𝑘
Hyperprior
Branch
Trunk
Adding animation for hyperprior 3

4. Methods: HiLDA Branch - Trunk 2nd sig 3rd sig Coefficient -0.786 0.984
Group 𝒈
Tumor 𝒊
log(fractions) of Signature 𝒌
1st sig
2nd sig
3rd sig
Trunk 1
𝜶 𝟐 + 𝜸 𝟏,𝟐 𝟎
𝜶 𝟑 + 𝜸 𝟏,𝟑 𝟎 2
𝜶 𝟐 + 𝜸 𝟐,𝟐 𝟎
𝜶 𝟑 + 𝜸 𝟐,𝟑 𝟎 … 16
𝜶 𝟐 + 𝜸 𝟏𝟔,𝟐 𝟎
𝜶 𝟑 + 𝜸 𝟏𝟔,𝟑 𝟎
Branch
𝜶 𝟐 + 𝜷 𝟐 + 𝜸 𝟏,𝟐 𝟏
𝜶 𝟑 + 𝜷 𝟑 + 𝜸 𝟏,𝟑 𝟏
𝜶 𝟐 + 𝜷 𝟐 + 𝜸 𝟐,𝟐 𝟏
𝜶 𝟑 + 𝜷 𝟑 + 𝜸 𝟐,𝟑 𝟏
𝜶 𝟐 + 𝜷 𝟐 + 𝜸 𝟏𝟔,𝟐 𝟏
𝜶 𝟑 + 𝜷 𝟑 + 𝜸 𝟏𝟔,𝟑 𝟏
𝑞 𝑖,𝑘 𝑔 = 𝜙 𝑖,𝑘 𝑔 𝑘 𝜙 𝑖,𝑘 𝑔 ; 𝑙𝑜𝑔 𝑞 𝑖,𝑘 𝑔 𝑞 𝑖, 1 𝑔 = 𝜶 𝒌 + 𝜷 𝒌 + 𝜸 𝒊,𝒌 𝒈
𝜶 𝒌 : Baseline difference between 1st and 𝑘 𝑡ℎ signature
𝜷 𝒌 : Difference between two groups in 𝑘 𝑡ℎ signature
𝜸 𝒋𝒌 𝒈 : Variation for 𝑘 𝑡ℎ signature of 𝑖 𝑡ℎ tumor in 𝑔 𝑡ℎ group 4
Group 𝒈
Tumor 𝒊
log(fractions) of Signature 𝒌
1st sig
2nd sig
3rd sig
Trunk 1
𝜶 𝟐 + 𝜸 𝟏,𝟐 𝟎
𝜶 𝟑 + 𝜸 𝟏,𝟑 𝟎 2
𝜶 𝟐 + 𝜸 𝟐,𝟐 𝟎
𝜶 𝟑 + 𝜸 𝟐,𝟑 𝟎 … 16
𝜶 𝟐 + 𝜸 𝟏𝟔,𝟐 𝟎
𝜶 𝟑 + 𝜸 𝟏𝟔,𝟑 𝟎
Branch
𝜶 𝟐 + 𝜷 𝟐 + 𝜸 𝟏,𝟐 𝟏
𝜶 𝟑 + 𝜷 𝟑 + 𝜸 𝟏,𝟑 𝟏
𝜶 𝟐 + 𝜷 𝟐 + 𝜸 𝟐,𝟐 𝟏
𝜶 𝟑 + 𝜷 𝟑 + 𝜸 𝟐,𝟑 𝟏
𝜶 𝟐 + 𝜷 𝟐 + 𝜸 𝟏𝟔,𝟐 𝟏
𝜶 𝟑 + 𝜷 𝟑 + 𝜸 𝟏𝟔,𝟑 𝟏
Group 𝒈
Tumor 𝒊
log(fractions) of Signature 𝒌
1st sig
2nd sig
3rd sig
Trunk 1
𝜶 𝟐 + 𝜸 𝟏,𝟐 𝟎
𝜶 𝟑 + 𝜸 𝟏,𝟑 𝟎 2
𝜶 𝟐 + 𝜸 𝟐,𝟐 𝟎
𝜶 𝟑 + 𝜸 𝟐,𝟑 𝟎 … 16
𝜶 𝟐 + 𝜸 𝟏𝟔,𝟐 𝟎
𝜶 𝟑 + 𝜸 𝟏𝟔,𝟑 𝟎
Branch
𝜶 𝟐 + 𝜷 𝟐 + 𝜸 𝟏,𝟐 𝟏
𝜶 𝟑 + 𝜷 𝟑 + 𝜸 𝟏,𝟑 𝟏
𝜶 𝟐 + 𝜷 𝟐 + 𝜸 𝟐,𝟐 𝟏
𝜶 𝟑 + 𝜷 𝟑 + 𝜸 𝟐,𝟑 𝟏
𝜶 𝟐 + 𝜷 𝟐 + 𝜸 𝟏𝟔,𝟐 𝟏
𝜶 𝟑 + 𝜷 𝟑 + 𝜸 𝟏𝟔,𝟑 𝟏
Branch - Trunk
2nd sig
3rd sig
Coefficient
-0.786
0.984 SE
0.152
0.587
P value
<0.001
0.094

5. Results: Two-step Method v.s. HiLDA
Branch-Trunk
2nd Sig
3rd Sig
Coefficient
-0.786
0.984
-0.795
3.417 SE
0.152
0.587
0.179
1.424
P value
<0.001
0.094
0.016
The new signatures (3rd signature) tend to appear significantly more often in the branch mutations (𝑝=0.016) by using the new model (HiLDA) after considering uncertainty. 5
Branch - Trunk
2nd sig
3rd sig
Coefficient
-0.786
0.984 SE
0.152
0.587
P value
<0.001
0.094