1. Advancing Pharmacogenomics Analysis of Drug Response in Early-phase Clinical Trials
Hong Zhang, Judong Shen & Devan V. Mehrotra
Early Development Statistics, BARDS, Merck
07/29/2019

2. Pharmacogenomics (PGx) towards precision medicine
Genetic markers as stratifying factors
Adams et al. Clinical Pharmacogenomics
Applications in Nephrology. CJASN 2018
Schork, Personalized medicine: Time for one-person trials, Nature 2015.

3. Costs/benefits of PGx
Based on a prospective, open-label, randomized controlled trial (n=110).
Elliot et al. Clinical impact of pharmacogenetic profiling with a clinical decision support tool in polypharmacy home health patients: A prospective pilot randomized controlled trial . PLOS ONE 2017

4. Clinical PGx strategy in Merck
Phase I
Phase II
Phase III
Stratified Trial
Enriched for responders
YES
Inform
Discovery
Genetic Marker/
Positive Result
Candidate Genes
Drug metabolism,
drug targets
Primary Discovery
GWAS + WES
Continued Discovery
GWAS, targeted
genotyping, +/- WES NO
GENETICS
Genetic variation explains PK variability
Genetic variation Explains variable safety and efficacy
Validation: Phase II GWAS “hit” predicts for response in Phase 3
IMMUNO-ONCOLOGY
Biomarker Validation, Mechanism of Action, Novel Biomarkers, New Targets
Modified from Urban et al., 2014

5. PGx in early-phase clinical development
Objective: identify genetic markers that impact drug responses
Drug exposure: Pharmacokinetics (PK) parameters (i.e., Cmax, AUC, etc.)
Candidate (ADME) gene study used in phase 1 study.
Drug safety and efficacy
Candidate gene study and Genome Wide Association Study (GWAS) used in phase 2 or later phase.
Approaches: test the association between genetic variants and drug responses
Treatment arm only vs. both arms;
Single variant vs. multiple variants;
Single trait vs. multiple traits.
Statistical challenges
Inflated type I error and limited power for discovery due to small sample size;
Bias adjustment of the effect size for PGx confirmatory study;
Lack of actionable findings.

6. PGx single variant regression model
Consider a generalized linear model in the PGx scenario
𝑔 𝐸 𝑌 = 𝛽 0 + 𝛽 𝑇 𝑇+ 𝛽 𝑋 𝑋+ 𝛽 𝐺 𝐺+ 𝛽 𝐺𝑇 𝐺𝑇,
where 𝑌 is the response, 𝑇 is the treatment indicator, 𝑋 are control covariates, 𝐺 is the genotype, 𝐺𝑇 is the genotype-by-treatment interaction. We want to test the joint effect:
𝐻 0 : 𝛽 𝐺 = 𝛽 𝐺𝑇 =0 v.s. 𝐻 1 : 𝛽 𝐺 ≠0 𝑜𝑟 𝛽 𝐺𝑇 ≠0,
Notation
T is binary, placebo T=0, treatment T=1.
G or single nucleotide polymorphism (SNP) is coded as 0, 1, or 2. ……

7. Currently existing single variant joint test methods
Likelihood ratio test: 2 𝑙 0 − 𝑙 1 ∼ 𝜒 2 2 , where 𝑙 𝑖 is the maximized log likelihood under 𝐻 𝑖 , 𝑖=0,1.
F test (for continuous trait): (𝑛−𝑝)(𝑅𝑆 𝑆 0 −𝑅𝑆 𝑆 1 ) 2𝑅𝑆 𝑆 1 ∼𝐹(2, 𝑛−𝑝) where 𝑅𝑆 𝑆 𝑖 is the residual sum of squares under 𝐻 𝑖 , 𝑖=0,1, 𝑛 is the sample size and 𝑝 is the number of variables.
Firth’s penalized likelihood ratio test (for binary trait): 2 𝑙 0 − 𝑙 1 ∼ 𝜒 2 2 , where 𝑙 is the maximized penalized likelihood with penalty function 1 2 ln⁡det⁡(𝐼), where 𝐼 is the Fisher’s information matrix.
Firth’s method corrects the bias of maximum likelihood estimation (except for linear regression) to improve inference on small samples.

8. Motivation for developing alterative methods of LRT and Firth
MK phase 2 trial, continuous trait, N=118
MK phase 3 trial, binary trait, N=704

9. 𝑔 𝐸 𝑌 = 𝛽 0 + 𝛽 𝑇 𝑇+ 𝛽 𝑋 𝑋+ 𝛽 𝐺 𝐺+ 𝛽 𝐺𝑇 𝐺𝑇.
Weighted joint test
Recall a generalized linear model
𝑔 𝐸 𝑌 = 𝛽 0 + 𝛽 𝑇 𝑇+ 𝛽 𝑋 𝑋+ 𝛽 𝐺 𝐺+ 𝛽 𝐺𝑇 𝐺𝑇.
Define composite variable 𝑍 𝑤 =𝑤𝐺+(1−|𝑤|)𝐺𝑇, −1≤𝑤≤1.
The marginal score statistic:
𝑆 𝑤 =𝑍 𝑤 ′ 𝑢,
where the residuals 𝑢=𝑌− 𝜇 , 𝜇 is the estimator of 𝐸 𝑌 under 𝐻 0 .
We can also conduct a 1df LRT based on 𝑍 𝑤 :
2 𝑙 0 − 𝑙 1 ∼ 𝜒 1 2 ,
where 𝑙 0 is the maximized likelihood under 𝐻 0 , and 𝑙 1 is the maximized likelihood under
𝐻 1 :𝑔 𝐸 𝑌 = 𝛽 0 + 𝛽 𝑇 𝑇+ 𝛽 𝑋 𝑋+𝛽 𝑍 𝑤
* Dey R, Schmidt EM, Abecasis GR, Lee S. A fast and accurate algorithm to test for binary phenotypes and its application to PheWAS. Am. J. Hum. Genet. 2017;2:014.

10. Weighted joint test: another perspective
𝑤 2
Recall a generalized linear model
𝑔 𝐸 𝑌
= 𝛽 0 + 𝛽 𝑇 𝑇+ 𝛽 𝑋 𝑋+ 𝛽 𝐺 𝐺+ 𝛽 𝐺𝑇 𝐺𝑇
= 𝛽 0 + 𝛽 𝑇 𝑇+ 𝛽 𝑋 𝑋+ (|𝛽 𝐺 |+| 𝛽 𝐺𝑇 |)( 𝛽 𝐺 |𝛽 𝐺 |+ |𝛽 𝐺𝑇 | 𝐺+ 𝛽 𝐺𝑇 |𝛽 𝐺 |+| 𝛽 𝐺𝑇 | 𝐺𝑇)
= 𝛽 0 + 𝛽 𝑇 𝑇+ 𝛽 𝑋 𝑋+𝛽 𝑤 1 𝐺+ 𝑤 2 𝐺𝑇 ,
where 𝛽= |𝛽 𝐺 |+ |𝛽 𝐺𝑇 | and 𝑤 1 , 𝑤 2 ∈ −1, 1 , 𝑤 𝑤 2 ≡1.
In light of this transformation, the marginal score test is the score test of a single variable 𝑍 𝑤 = 𝑤 1 𝐺+ 𝑤 2 𝐺𝑇 that combines the main term and the interaction term according to some weight.
The optimal weights are of course 𝑤 1 = 𝛽 𝐺 |𝛽 𝐺 |+ |𝛽 𝐺𝑇 | and 𝑤 2 = 𝛽 𝐺𝑇 |𝛽 𝐺 |+| 𝛽 𝐺𝑇 | , which we never know – therefore we developed an adaptive way to approximate the optimal weight.
𝑤 1

11. 𝑃 𝑚𝑖𝑛𝑃>𝑝 =𝑃( 𝑆 𝑤 < 𝐹 𝑤 −1 𝑝 , 𝑤= 𝑤 1 ,…, 𝑤 𝑑 ).
New method 1: Adaptive Weighted jOint Test (AWOT)
To choose the optimal weight, we propose an adaptive approach by grid search. For example,
Equal-step weights: 𝑤 1 =−1, −0.9, …0, 0.1,…,1.
Weights focus on interaction effect: 𝑤 1 =−1, −0.5,−0.1, −0.05, −0.01, 0, 0.01,0.05, 0.1, 0.5, 1.
Let 𝑝 𝑤 be the p-value of 𝑆 𝑤 = (𝑤 1 𝐺+ 𝑤 2 𝐺𝑇)′𝑢, the adaptive score test statistic is
𝑚𝑖𝑛𝑃= min 𝑤 𝑝 𝑤 .
The p-value of minP can be calculated by noting that
𝑃 𝑚𝑖𝑛𝑃>𝑝 =𝑃( 𝑆 𝑤 < 𝐹 𝑤 −1 𝑝 , 𝑤= 𝑤 1 ,…, 𝑤 𝑑 ).
𝑆 𝑤 , 𝑤= 𝑤 1 ,…, 𝑤 𝑑 , follows a multivariate normal distribution with mean zero and covariance matrix:
Σ 𝑖𝑗 = 𝑍 𝑤 𝑖 ′ 𝐶𝑜𝑣(𝑢) 𝑍 𝑤 𝑗 ′ .

12. New method 2: Omnibus Test - Cauchy Weighted jOint Test (CWOT)
minP approach is difficult to extend to 1df-LRT since the joint distribution of score and LRT are unknown.
To include LRT to cover more signal patterns, we used the Cauchy p-value combination method*
𝑇= 1 2𝑑 𝑖=1 𝑑 𝑗=1 2 tan 𝜋 0.5− 𝑝 𝑖𝑗 ,
where 𝑝 𝑖𝑗 , 𝑖=1,…,𝑑, 𝑗=1,2 are p-values of 1df-LRT (𝑗=1) or 1df score test (𝑗=2) based on weight 𝑤 𝑖 . Then
𝑃 𝑇>𝑡 ≈𝑃 𝐶>𝑡 = 1 2 − tan −1 𝑡 𝜋 ,
for large 𝑡, where 𝐶 is standard Cauchy random variable.
* Yaowu Liu & Jun Xie (2019) Cauchy Combination Test: A Powerful Test With Analytic p-Value Calculation Under Arbitrary Dependency Structures, Journal of the American Statistical Association, DOI: /

13. AWOT real data analysis results for continuous trait (CT)
MK phase 2 trial, continuous trait, N=118, LRT
MK phase 2 trial, continuous trait, N=118, AWOT

14. Simulation results: type I error, joint test for continuous trait
MAF n
𝜶=𝟓×𝟏 𝟎 −𝟖
𝜶=𝟏×𝟏 𝟎 −𝟓
CWOT_Score
AWOT
LRT FT
0.05
200
1.07E-08
1.87E-08
1.44E-07
4.80E-08
5.64E-06
8.66E-06
1.86E-05
9.90E-06
0.15
1.60E-08
4.00E-08
1.72E-07
6.00E-08
5.87E-06
8.83E-06
1.89E-05
1.01E-05
0.25
8.00E-09
2.50E-08
1.91E-07
4.10E-08
5.99E-06
8.84E-06
1.88E-05
0.03
500
2.01E-08
8.97E-08
5.79E-08
8.34E-06
8.54E-06
1.31E-05
3.20E-08
3.40E-08
8.60E-08
4.60E-08
8.42E-06
8.63E-06
3.10E-08
3.30E-08
8.11E-08
5.01E-08
8.31E-06
8.37E-06
3.00E-08
7.90E-08
4.50E-08
0.01
1000
4.40E-08
4.70E-08
6.90E-08
5.70E-08
9.22E-06
9.07E-06
1.14E-05
9.99E-06
4.90E-08
7.30E-08
5.50E-08
9.08E-06
8.92E-06
1.13E-05
9.89E-06
3.70E-08
3.80E-08
6.10E-08
1.12E-05
9.86E-06
5.00E-08
7.10E-08
6.20E-08
9.32E-06
9.09E-06
9.94E-06
3.50E-08
6.40E-08
9.31E-06
9.11E-06
1.00E-05
Type I error ≥ 1.3α is marked in red, 𝛽 𝑇 =0.5, number of simulations = 1E9
CWOT_Score: CWOT by using only score test p-values from each weight
FT: F test
LRT tends to generate inflated type I error while AWOT has much better type I error control (slightly conservative).
F test controls type I error the best.

15. Simulation results: power, joint test for continuous trait, 𝜶=𝟓×𝟏 𝟎 −𝟖

16. CWOT real data analysis results for binary trait (BT)
MK phase 3 trial, binary trait, N=704

17. Simulation results: type I error, joint test for binary trait, 𝜶=𝟓×𝟏 𝟎 −𝟖
MAF n
CWOT
LRT
0.05
200
3.40E-08
2.90E-08
3.30E-08
3.80E-08
4.64E-08
0.15
5.30E-08
1.02E-07
4.60E-08
6.90E-08
2.93E-08
6.59E-08
0.25
5.78E-08
7.68E-08
5.43E-08
9.57E-08
2.59E-08
7.97E-08
0.03
500
5.31E-08
6.50E-08
4.81E-08
3.94E-08
2.66E-08
1.98E-08
5.56E-08
7.35E-08
4.00E-08
3.70E-08
3.23E-08
3.35E-08
5.10E-08
5.70E-08
4.10E-08
7.50E-08
2.73E-08
4.48E-08
5.80E-08
4.40E-08
6.70E-08
2.98E-08
6.52E-08
0.01
1000
2.00E-08
2.10E-08
5.00E-08
3.41E-08
5.60E-08
7.90E-08
5.20E-08
3.49E-08
2.48E-08
6.10E-08
6.00E-08
3.11E-08
2.45E-08
4.80E-08
4.36E-08
8.49E-08
3.89E-08
6.75E-08
Type I error ≥ 6.5E-8 (2SE) is marked in red, number of simulations = 1E9
LRT tends to generate inflated type I error while CWOT controls
type I error well (slightly conservative).

18. Simulation results: type I error, joint test for binary trait, 𝜶=𝟏×𝟏 𝟎 −𝟓
MAF n
AWOT
CWOT
LRT
Firth
0.05
200
9.60E-06
1.21E-05
1.33E-05
1.40E-06
9.20E-06
8.80E-06
8.00E-06
5.20E-06
6.60E-06
6.10E-06
6.30E-06
5.00E-06
0.15
9.80E-06
1.40E-05
1.59E-05
6.80E-06
8.60E-06
6.20E-06
3.80E-06
6.97E-06
1.06E-05
5.60E-06
0.25
9.00E-06
1.09E-05
1.32E-05
1.12E-05
1.61E-05
7.20E-06
2.20E-06
1.02E-05
1.56E-05
4.60E-06
0.03
500
1.10E-05
1.26E-05
1.43E-05
1.18E-05
8.50E-06
6.90E-06
5.35E-06
4.24E-06
7.40E-06
1.22E-05
1.19E-05
1.04E-05
5.40E-06
6.00E-06
1.14E-05
1.00E-05
7.80E-06
1.05E-05
1.17E-05
8.20E-06
1.16E-05
9.90E-06
1.30E-05
4.20E-06
8.59E-06
1.52E-05
0.01
1000
8.90E-06
1.01E-05
1.80E-06
8.40E-06
3.40E-06
4.80E-06
1.28E-05
6.40E-06
9.40E-06
8.38E-06
6.16E-06
7.00E-06
1.29E-05
1.08E-05
1.44E-05
7.82E-06
8.73E-06
9.35E-06
1.38E-05
9.30E-06
1.20E-05
9.50E-06
1.11E-05
7.41E-06
9.34E-06
Type I error ≥ 1.3E-05 (2SE) is marked in red, number of simulations = 5E6
Compared to CWOT, LRT tends to generate inflated type I error while CWOT controls type I error well.
Firth method has the most conservative type I error rate.

19. Simulation results: power, joint test for binary trait, 𝜶=𝟓×𝟏 𝟎 −𝟖
In the most interesting PGx scenarios, that is the main effect is weak while the interaction effect is large and in the same direction with the treatment effect, the proposed CWOT method has larger power than 2-df LRT and Firth test.
MAF = 15%, 𝛽 𝑇 = 0.5, 𝛽 0 = -1.75, sample size is adjusted for proper power.
G=0: SNP-; G=1: SNP+

20. Conclusions and challenges
Joint test for association studies
LRT tends to generate inflated type I error while A/CWOT controls type I error well.
Firth method has the most conservative type I error rate.
The proposed A/CWOT has higher power than 2df- LRT in the most interesting PGx signal patterns.
Challenges of PGx in early-phase clinical trials
Winner’s curse adjustment is needed for designing confirmatory study.
PGx signals may be too weak in early-phase studies.
Prediction/stratification using (many) genetic markers: polygenic score?

21. Thank you!

22. Appendix: Score test with SPA
Let 𝜇 be the estimator of 𝐸 𝑌 under the null model. Define the residuals
𝑢=𝑌− 𝜇 .
The marginal score statistic
𝑆= 𝐺 ′ 𝑢.
The null distribution of 𝑆 𝑤 can be approximated by normal distribution.
The accuracy can be further improved by saddle point approximation (SPA*) for binary trait, because the cumulant-generating function 𝐾(𝑡) of 𝑆 can be explicitly written out. Then
SPA calculates the p-value of 𝑆 by
P 𝑆>s = 𝐹 (𝑠)≈1−Φ 𝑥+ 1 𝑥 log 𝑦 𝑥 ,
where 𝑥=𝑠𝑖𝑔𝑛 𝑡 2( 𝑡 𝑠−𝐾( 𝑡 )) , 𝑦= 𝑡 𝐾 ′′ ( 𝑡 ) , 𝑡 is the solution of 𝐾 ′ 𝑡 =𝑠.
* Dey R, Schmidt EM, Abecasis GR, Lee S. A fast and accurate algorithm to test for binary phenotypes and its application to PheWAS. Am. J. Hum. Genet. 2017;2:014.