Quantifying and Mitigating the Effect of Preferential Sampling on Phylodynamic Inference

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1 Quantifying and Mitigating the Effect of Preferential Sampling on Phylodynamic Inference Michael D. Karcher Department of Statistics University of Washington, Seattle April 2015 joint work (and slide construction) with Vladimir Minin (UW Stat) Michael D. Karcher (UW, Statistics) April

2 Background Relative genetic diversity New York H3N2 HA NA H1N1 HA NA Year Michael D. Karcher (UW, Statistics) April

3 The Coalescent # Michael D. Karcher (UW, Statistics) April

4 The Coalescent # Michael D. Karcher (UW, Statistics) April

5 The Coalescent Time t 1 t 2 t 3 t 4 u 2 u 3 u 4 # Michael D. Karcher (UW, Statistics) April

6 The Coalescent Time t 1 t 2 u 2 Pr(u k > t t k+1 ) = e (k 2) t+t k+1 t k+1 1 Ne(u) du u 3 t 3 t 4 u 4 N(t) = N Constant population size N(t) = Ne 100t Exponential growth Michael D. Karcher (UW, Statistics) April

7 Coalescent Notation Michael D. Karcher (UW, Statistics) April

8 Coalescent Notation Sampling times: s 1 s 2 s 3 s 4 s 5 Number sampled: n 1 =2 n 2 =1 n 3 =2 n 4 =1 n 5 =1 Michael D. Karcher (UW, Statistics) April

9 Coalescent Notation Sampling times: s 1 s 2 s 3 s 4 s 5 Number sampled: n 1 =2 n 2 =1 n 3 =2 n 4 =1 n 5 =1 Coalescent times: t 1 t 2 t 3 t 4 t 5 t 6 t 7 Michael D. Karcher (UW, Statistics) April

10 Coalescent Notation Intervals: I 0,2 I 0,3 I 1,3 I 0,4 I 0,5 I 0,6 I 1,6 I 0,7 I 1,7 I 2,7 Sampling times: Number sampled: s 1 n 1 =2 s 2 s 3 s 4 s 5 n 2 =1 n 3 =2 n 4 =1 n 5 =1 Coalescent times: t 1 t 2 t 3 t 4 t 5 t 6 t 7 Michael D. Karcher (UW, Statistics) April

11 Coalescent as a Point Process where The coalescent is a non-homogeneous CTMC and can be viewed as a Markov point process on [0, ). Assume genealogy G is known. The likelihood of observing the coalescent times: Pr(t 1,..., t n 1 ) = Pr(t k 1 t k ) = C 0,k N e (t k 1 ) exp and C i,k = ( n i,k ) 2. [ n 1 k=1 I 0,k Pr(t k 1 t k ), C 0,k dt m N e (t) i=1 I i,k ] C i,k dt, N e (t) Michael D. Karcher (UW, Statistics) April

12 Discretization Continuous formulation is intractable Construct fine, regular grid x = {x j } B j=1 with grid width w Let γ j = N e (x j ) Construct piecewise constant approximation N γ(t) = B i=1 γ i1 t [xi w/2,x i +w/2) Substitute N γ(t) for N e (t) and Pr(γ τ) for Pr(N e (t) θ) Effective population size Time N e (t) N γ (t) Michael D. Karcher (UW, Statistics) April

13 Motivation Effective Population Size Truth Estimate Credible Region Sampling events Coalescent events Time Michael D. Karcher (UW, Statistics) April

14 Posteriors P (γ, τ G) P (G γ) P (γ τ) P (τ) P (γ, τ, β G, s) P (G s, γ) P (s γ, β) P (γ τ) P (τ) P (β) γ - effective population size trajectory P (G s, γ) - coalescent likelihood P (s γ, β) - sampling likelihood G - genealogy with branch lengths τ, β - hyper-parameters s - sampling schedule Michael D. Karcher (UW, Statistics) April

15 Posteriors P (γ, τ G) P (G γ) P (γ τ) P (τ) P (γ, τ, β G, s) P (G s, γ) P (s γ, β) P (γ τ) P (τ) P (β) γ - effective population size trajectory P (G s, γ) - coalescent likelihood P (s γ, β) - sampling likelihood G - genealogy with branch lengths τ, β - hyper-parameters s - sampling schedule Michael D. Karcher (UW, Statistics) April

16 Modeling Sampling Times Model sampling times as an inhomogeneous Poisson process Intensity proportional to a power (β 1 ) of N γ(t) Model includes uniform sampling, when β 1 = 0 s γ, β PoisProc [0,s0 ](β 0 Nγ(t) β 1 ) n log P (s γ, β) = n log β 0 + β 1 log Nγ(t)(s i ) i=1 sn s 0 β 0 N γ(t) β 1 dr Michael D. Karcher (UW, Statistics) April

17 Bayesian nonparametric phylodynamic reconstruction Bayesian nonparametric phylodynamic reconstruction (): Phylodynamic method implemented using INLA (Integrated Nested Laplace Approximation) as in [Palacios and Minin, ] P (γ, τ G) P (G γ) P (γ τ) P (τ) Bayesian nonparametric phylodynamic reconstruction with preferential sampling (-PS): As above, but accounting for preferential sampling P (γ, τ, β G, s) P (G s, γ) P (s γ, β) P (γ τ) P (τ) P (β) log P (s γ, β) = n log β 0 + n i=1 β 1 log Nγ(t)(s i ) s n s 0 β 0 Nγ(t) β 1dr Michael D. Karcher (UW, Statistics) April

18 Example comparison output Effective Population Size Time PS Truth Estimate Credible Region Sampling events Coalescent events Time Michael D. Karcher (UW, Statistics) April

19 Simulation Study Periodic population trajectory 512 simulated genealogies 500 individuals sampled per genealogy per sampling schedule Uniform schedule distributes sampling events uniformly across sampling window Proportional schedule samples with intensity proportional to N e (t) on sampling window Michael D. Karcher (UW, Statistics) April

20 Measures of performance Local measures (at grid points) MRE: Mean relative error O T T MRW: Mean relative width of 95% credible interval Global measures (integrated over interval) MRD: Mean relative deviation O T T MRW: Mean relative width of 95% credible interval Michael D. Karcher (UW, Statistics) April

21 Mean local measures Uniform (48, 6) Proportional (48, 6) Uniform (6, 0) Proportional (6, 0) Effective Population Size Mean Relative Error PS Mean Relative Width Sampling events Coalescent events Time Time Time Time Michael D. Karcher (UW, Statistics) April

22 Mean global measures Uniform (48, 6) Proportional (48, 6) Uniform (6, 0) Proportional (6, 0) -PS -PS -PS -PS MRD MRW Mean Relative Deviation Uniform t = (48, 6) Proportional Uniform t = (6, 0) Proportional Mean Relative Width PS Uniform Proportional Uniform Proportional Michael D. Karcher (UW, Statistics) April

23 Case Studies New York influenza data The original study examines H3N2 and H1N1 influenza dynamics in New York and New Zealand. We examine H3N2 in New York. Regional influenza data The original study examined H3N2 influenza seasonal migration between world regions. We examine the regions separately. # PANGEA-HIV data Simulated sequence dataset modeling an HIV epidemic in Africa, in order to test different phylodynamic methods. Michael D. Karcher (UW, Statistics) April

24 New York influenza data Applying and -PS to Rambaut et al [] s New York influenza data, we see clearer seasonality and narrower credible regions. MRW β 1 95% -PS credible interval (4.09, 6.56) Ne(t) Sampling events PS Mean Relative Width PS Coalescent events Jan 1993 Jan 1994 Jan 1995 Jan 1996 Jan 1997 Jan 1998 Jan 1999 Jan Jan 2001 Jan Jan 2003 Jan Jan 2005 Jan 1993 Jan 1994 Jan 1995 Jan 1996 Jan 1997 Jan 1998 Jan 1999 Jan Jan 2001 Jan Jan 2003 Jan Jan 2005 Jan 1993 Jan 1994 Jan 1995 Jan 1996 Jan 1997 Jan 1998 Jan 1999 Jan Jan 2001 Jan Jan 2003 Jan Jan 2005 Michael D. Karcher (UW, Statistics) April

25 Regional influenza data (North China) Applying and -PS to Zinder et al. [2014] s regional influenza data for Northern China, we do not see clear preferential sampling and only see minor gains in precision. MRW β 1 95% -PS credible interval (0.43, 1.37) PS Mean Relative Width PS Ne(t) Sampling events Coalescent events Michael D. Karcher (UW, Statistics) April

26 PANGEA-HIV data Applying and -PS to PANGEA-HIV data, we see preferential sampling and narrower credible regions close to the end of the simulation. MRW β 1 95% -PS credible interval (0.50, 1.08) PS Mean Relative Width Ne(t) 1e 01 1e+01 1e+03 1e+05 Sampling events Coalescent events 1e 01 1e+01 1e+03 1e PS Michael D. Karcher (UW, Statistics) April

27 Results summary Unrecognized preferential sampling can introduce systematic estimation bias In the presence of preferential sampling, modeling sampling times gives better accuracy and precision. In the absence of preferential sampling, modeling sampling times performs no worse than the equivalent conditional method. Modeling sampling times performs better close to the most recent time, which is useful for designing epidemiological interventions. Michael D. Karcher (UW, Statistics) April

28 Future plans Extend sampling event model to include covariates Implement sampling event model in BEAST Infer population trajectories and genealogies from sequences Construct posterior predictive check for BEAST implementation Michael D. Karcher (UW, Statistics) April

29 References M. S. Gill, et al. Improving Bayesian population dynamics inference: a coalescent-based model for multiple loci. Molecular Biology and Evolution, 30(3): , J. A. Palacios and V. N. Minin. Integrated nested Laplace approximation for Bayesian nonparametric phylodynamics. Proceedings of the Twenty-Eighth International Conference on Uncertainty in Artificial Intelligence, ,. J. A. Palacios and V. N. Minin. Gaussian Process-Based Bayesian Nonparametric Inference of Population Size Trajectories from Gene Genealogies. Biometrics, 69, 8 18, A. Rambaut, et. al. The genomic and epidemiological dynamics of human influenze A virus. Nature 453, ,. D. Zinder, T. Bedford, and E. B. Baskerville. Seasonality in the migration and establishment of H3N2 Influenza lineages with epidemic growth and decline. arxiv preprint, pages 1 18, URL Michael D. Karcher (UW, Statistics) April

30 Measures of performance Mean relative error MRE i (t) = ˆN e,i (t) N e (t) N e (t) Mean relative deviation MRD i = 1 b ˆN e,i (t) N e (t) b a a N e (t) Michael D. Karcher (UW, Statistics) April

31 Measures of performance Mean relative width of the 95% credible interval MRW i (t) = ˆN e,i,0.975 (t) ˆN e,i,0.025 (t) N e (t) MRW i = 1 b b a a ˆN e,i,0.975 (t) ˆN e,i,0.025 (t) dt N e (t) Michael D. Karcher (UW, Statistics) April

32 Empirical measures of performance Empirical mean relative width of the 95% credible interval EMRW i (t) = ˆN e,i,0.975 (t) ˆN e,i,0.025 (t) ˆN e (t) EMRW i = 1 b b a a ˆN e,i,0.975 (t) ˆN e,i,0.025 (t) dt ˆN e (t) Michael D. Karcher (UW, Statistics) April

33 Negative control simulations (piecewise constant) Effective Population Size Mean Relative Error Mean Relative Width Time Time Time Time Michael D. Karcher (UW, Statistics) April

34 Negative control simulations (Gaussian process) Effective Population Size Mean Relative Error Mean Relative Width Time Time Time Time Michael D. Karcher (UW, Statistics) April

35 Results summary EMRW β 1 95% -PS credible interval New York influenza (4.04, 6.41) PANGEA (0.50, 1.08) Regional influenza USA-Canada (0.41, 1.27) South America (-0.49, 0.95) Europe (0.18, 1.53) India (0.17, 1.29) Japan-Korea (0.35, 1.26) North China (0.43, 1.36) South China (-0.12, 0.74) Southeast Asia (-0.15, 0.68) Oceania (0.20, 1.25) Michael D. Karcher (UW, Statistics) April

36 Regional influenza data (USA & Canada) Applying and -PS to Zinder et al. [2014] s Regional influenza data for USA and Canada, we do not see clear preferential sampling and only see minor gains in precision. PS Mean Relative Width Ne(t) 5e 02 5e 01 5e+00 5e+01 Sampling events Coalescent events 5e 02 5e 01 5e+00 5e PS Michael D. Karcher (UW, Statistics) April

37 Regional influenza data (South America) Applying and -PS to Zinder et al. [2014] s Regional influenza data for South America, we do not see clear preferential sampling and in fact see minor losses in precision. PS Mean Relative Width Ne(t) Sampling events PS Coalescent events Michael D. Karcher (UW, Statistics) April

38 Regional influenza data (Europe) Applying and -PS to Zinder et al. [2014] s Regional influenza data for Europe, we do not see clear preferential sampling and only see minor gains in precision. PS Mean Relative Width PS Ne(t) Sampling events Coalescent events Michael D. Karcher (UW, Statistics) April

39 Regional influenza data (South China) Applying and -PS to Zinder et al. [2014] s Regional influenza data for Southern China, we do not see clear preferential sampling and only see minor gains in precision. PS Mean Relative Width Ne(t) Sampling events Coalescent events PS Michael D. Karcher (UW, Statistics) April

40 Regional influenza data (Southeast Asia) Applying and -PS to Zinder et al. [2014] s Regional influenza data for Southeast Asia, we do not see clear preferential sampling and do not see a change in precision. PS Mean Relative Width Ne(t) Sampling events Coalescent events PS Michael D. Karcher (UW, Statistics) April

41 Regional influenza data (India) Applying and -PS to Zinder et al. [2014] s Regional influenza data for India, we do not see clear preferential sampling, but we do see some gains in precision. PS Mean Relative Width Ne(t) Sampling events Coalescent events PS Michael D. Karcher (UW, Statistics) April

42 Regional influenza data (Japan & Korea) Applying and -PS to Zinder et al. [2014] s Regional influenza data for Japan and Korea, we do not see clear preferential sampling and only see minor gains in precision. PS Mean Relative Width Ne(t) Sampling events Coalescent events PS Michael D. Karcher (UW, Statistics) April

43 Regional influenza data (Oceania) Applying and -PS to Zinder et al. [2014] s Regional influenza data for Oceania, we do not see clear preferential sampling and only see minor gains in precision. PS Mean Relative Width Ne(t) Sampling events Coalescent events PS Michael D. Karcher (UW, Statistics) April

44 Sequence data Population trajectory accggaaacgcgcgaaatttacacggggg accggaaacgcgcgaaatttacacggggg accggaaacgcgcgaaatttacacggggg Sequence Data accggaaacgcgcgaaatttacacggggg accggaaacgcgcgaaatttacacggggg Genealogy Pop. N(t) Dynamics Time P (G, Q, N e (t), θ D) P (D G, Q) P (Q) P (G N e (t)) P (N e (t) θ) P (θ) G - genealogy with branch lengths N e (t) - effective population size trajectory P (G N e (t)) - coalescent prior Q - substitution matrix D - sequence data θ - hyper-parameters Michael D. Karcher (UW, Statistics) April

Seasonality in the migration and establishment of H3N2 Influenza lineages with epidemic growth and decline

Zinder et al. BMC Evolutionary Biology (2014) 14:3 DOI 10.1186/s12862-014-0272-2 RESEARCH ARTICLE Open Access Seasonality in the migration and establishment of H3N2 Influenza lineages with epidemic growth