Bayesian approach to discriminant problems for count data with application to multilocus short tandem repeat dataset

Koji Tsukuda; Shuhei Mano; Toshimichi Yamamoto

doi:10.1515/sagmb-2018-0044

Article

Bayesian approach to discriminant problems for count data with application to multilocus short tandem repeat dataset

Koji Tsukuda , Shuhei Mano and Toshimichi Yamamoto

Published/Copyright: May 4, 2020

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From the journal Statistical Applications in Genetics and Molecular Biology Volume 19 Issue 2

Abstract

Short Tandem Repeats (STRs) are a type of DNA polymorphism. This study considers discriminant analysis to determine the population of test individuals using an STR database containing the lengths of STRs observed at more than one locus. The discriminant method based on the Bayes factor is discussed and an improved method is proposed. The main issues are to develop a method that is relatively robust to sample size imbalance, identify a procedure to select loci, and treat the parameter in the prior distribution. A previous study achieved a classification accuracy of 0.748 for the g-mean (geometric mean of classification accuracies for two populations) and 0.867 for the AUC (area under the receiver operating characteristic curve). We improve the maximum values for the g-mean to 0.830 and the AUC to 0.935. Computer simulations indicate that the previous method is susceptible to sample size imbalance, whereas the proposed method is more robust while achieving almost identical classification accuracy. Furthermore, the results confirm that threshold adjustment is an effective countermeasure to sample size imbalance.

Keywords: Bayes factor; Dirichlet–multinomial model; DNA polymorphism; locus selection; sample size imbalance

Corresponding author: Koji Tsukuda, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, 153-8902, Tokyo, Japan; and Faculty of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka-shi, 819-0395, Fukuoka, Japan, E-mail: tsukuda@math.kyushu-u.ac.jp

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: 16H02791

Award Identifier / Grant number: 17H04148

Award Identifier / Grant number: 18K13454

Acknowledgments

The authors would like to express their gratitude to Professor Naruya Saitou who provided part of the data used in this study, and to the referees who provided a lot of insightful comments. This work was partly supported by Japan Society for the Promotion of Science KAKENHI Grant Number 16H02791, 17H04148 and 18K13454. This study was partly carried out when the first author was a member of Biostatistics center, Kurume University.

Appendix A

Selecting prior distribution

We examine the following prior distributions:

Bayes–Laplace uniform prior: γ=1;
Jeffreys prior(Jeffreys 1946): γ=1/2;
Perks prior(Perks 1947): γ=1/d;
Minimax prior(Trybuła 1958): γ=n/d;

where d denotes the number of classes and n denotes the sample size. With respect to the Perks and minimax priors, we consider two methods to acquire the label of classes as is mentioned in Subsection 3.2. In the data analysis, Perks (C) and minimax (C) denote methods in which the class labels are acquired as the union of labels of each population; Perks (NC) and minimax (NC) denote methods in which the class labels are acquired by each population.

Figure 2 shows the AUC using the above prior distributions. Perks (C) is optimal with respect to PM1, and Jeffreys prior is optimal with respect to the other methods. In all other cases, the AUCs are not improved over the case in which the DP is estimated by maximizing the marginal likelihood. Overall, in our analysis, the results indicate that these prior distributions do not improve the accuracy.

Appendix B

Simulation results

In this section, we show the tables that represent the results of the simulation presented in Section 7.

Table 10:

Simulation results (L~=0). Numbers are the average true classification accuracies for population A (PopA) and population B (PopB) and average g-means (GM). Threshold value of Bayes factor is denoted by t.

parameters(r,γ,n2)	method	t=1			t=tp			t=tg
parameters(r,γ,n2)	method	PopA	PopB	GM	PopA	PopB	GM	PopA	PopB	GM
r=1,γ=5	Y-C	0.514	0.521	0.504
n2=100	P1-C	0.492	0.529	0.495
r=1,γ=5	Y-C	0.885	0.106	0.245	0.799	0.202	0.359	0.511	0.487	0.468
n2=25	P1-C	0.656	0.358	0.459	0.480	0.520	0.475	0.503	0.520	0.482
r=1,γ=1	Y-C	0.508	0.499	0.488
n2=100	P1-C	0.497	0.489	0.478
r=1,γ=1	Y-C	0.860	0.144	0.310	0.759	0.247	0.403	0.501	0.485	0.469
n2=25	P1-C	0.653	0.354	0.458	0.461	0.515	0.468	0.504	0.489	0.475
r=1,γ=1/2	Y-C	0.481	0.510	0.477
n2=100	P1-C	0.494	0.510	0.485
r=1,γ=1/2	Y-C	0.834	0.195	0.350	0.706	0.309	0.447	0.531	0.489	0.479
n2=25	P1-C	0.659	0.349	0.456	0.480	0.538	0.489	0.508	0.519	0.492
r=0,γ=5	Y-C	0.655	0.629	0.631
n2=100	P1-C	0.653	0.627	0.628
r=0,γ=5	Y-C	0.923	0.193	0.372	0.861	0.289	0.470	0.613	0.594	0.579
n2=25	P1-C	0.736	0.456	0.561	0.572	0.635	0.588	0.591	0.611	0.579
r=0,γ=1	Y-C	0.892	0.896	0.891
n2=100	P1-C	0.892	0.887	0.887
r=0,γ=1	Y-C	0.973	0.512	0.693	0.954	0.614	0.757	0.839	0.799	0.811
n2=25	P1-C	0.897	0.732	0.804	0.827	0.822	0.819	0.846	0.806	0.820
r=0,γ=1/2	Y-C	0.970	0.969	0.969
n2=100	P1-C	0.972	0.969	0.970
r=0,γ=1/2	Y-C	0.988	0.762	0.864	0.981	0.825	0.897	0.939	0.908	0.921
n2=25	P1-C	0.966	0.866	0.912	0.929	0.920	0.922	0.926	0.915	0.918
r=1/2,γ=5	Y-C	0.549	0.556	0.540
n2=100	P1-C	0.558	0.541	0.540
r=1/2,γ=5	Y-C	0.894	0.109	0.248	0.824	0.209	0.385	0.549	0.520	0.505
n2=25	P1-C	0.680	0.360	0.478	0.527	0.547	0.514	0.527	0.532	0.501
r=1/2,γ=1	Y-C	0.699	0.648	0.664
n2=100	P1-C	0.689	0.628	0.648
r=1/2,γ=1	Y-C	0.918	0.201	0.400	0.851	0.316	0.502	0.630	0.599	0.591
n2=25	P1-C	0.740	0.443	0.556	0.598	0.610	0.590	0.601	0.610	0.588
r=1/2,γ=1/2	Y-C	0.779	0.751	0.758
n2=100	P1-C	0.787	0.760	0.767
r=1/2,γ=1/2	Y-C	0.920	0.346	0.535	0.871	0.473	0.623	0.719	0.685	0.687
n2=25	P1-C	0.807	0.597	0.679	0.667	0.726	0.682	0.709	0.699	0.691

Table 11:

Simulation results (L~=7,17). γ is always 1. Optimization to determine tgdoes not converge in the search range for several cases.

parameters (r,n2,L~)	method	t=1			t=tp			t=tg
parameters (r,n2,L~)	method	PopA	PopB	GM	PopA	PopB	GM	PopA	PopB	GM
r=1,n2=100	Y-C	0.551	0.490	0.505
L~=7	P1-C	0.543	0.492	0.505
r=1,n2=25	Y-C	0.978	0.037	0.101	0.960	0.058	0.147	0.871	0.167	0.324
L~=7	P1-C	0.792	0.258	0.420	0.711	0.327	0.460	0.562	0.482	0.486
r=0,n2=100	Y-C	0.809	0.827	0.814
L~=7	P1-C	0.813	0.817	0.811
r=0,n2=25	Y-C	0.993	0.113	0.277	0.985	0.149	0.338	0.942	0.332	0.534
L~=7	P1-C	0.900	0.469	0.635	0.855	0.577	0.693	0.755	0.705	0.718
r=1,n2=100	Y-C	0.491	0.529	0.498
L~=17	P1-C	0.490	0.518	0.491
r=1,n2=25	Y-C	0.995	0.003	0.009	0.994	0.003	0.009	0.992	0.003	0.009
L~=17	P1-C	0.855	0.151	0.293	0.803	0.206	0.357	0.630	0.390	0.471
r=0,n2=100	Y-C	0.761	0.761	0.754
L~=17	P1-C	0.754	0.766	0.753
r=0,n2=25	Y-C	0.998	0.013	0.041	0.997	0.017	0.052	0.997	0.021	0.063
L~=17	P1-C	0.930	0.293	0.488	0.901	0.359	0.546	0.780	0.569	0.653

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Received: 2018-08-03

Accepted: 2020-03-23

Published Online: 2020-05-04

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Articles in the same Issue

https://doi.org/10.1515/sagmb-2018-0044

Keywords for this article

Bayes factor; Dirichlet–multinomial model; DNA polymorphism; locus selection; sample size imbalance