Some challenges in the analysis of microbiome data. Shyamal Peddada Biostatistics and Computational Biology Branch NIEHS, NIH

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1 Some challenes in the analysis of micobiome data Shyamal Peddada Biostatistics and Computational Bioloy Banch NIEHS, NIH

2 Diet, Exposue, Stess, etc. Micobiome Genes Disease

3 Outline Some scientific questions of inteest Motivatin example Noweian infant ut micobiome study The data Thee impotant featues of the data Compaison of two o moe ecosystems Statistical issues & Methods Illustation of the methods Estimation of pecision and coelation matices Concludin emaks & open eseach poblems 3

4 Why Study Micobiome? Humans have 0 times moe mico-oanisms than human cells. (~ 00 tillion to ~ 0 tillion). Numeous health outcomes ae possibly influenced by (o linked to) micobiome: IBD/IBS Obesity Neuoloical diseases Cance Obesity Asthma Autism Moto skills Etc. 4

5 Some Geneal Questions of Scientific Inteest How does the extenal envionment affects the intenal envionment? o Diet Micobiome Micobial composition and health: healthy vs. sick individuals o Testin o Classification o Suvival analysis Netwoks and associations amon taxa Genetics x micobiome x nutition x infections x social x psycholoical factos x envionnemental chemicals Functional micobiomics 5

6 Noweian Micofloa (NoMIC) Study: Infant ut [D. Meete Eesbo, Noweian Institute of Public Health] Lonitudinal study (53 infants) Mode of delivey (C-section/vainal) Tem (full/patial tem) Exposue to antibiotics (ae, type) Fecal samples obtained at aes: Day 4, 0, 30, 0, y, ys, 7 ys 6

7 Some Specific Questions Association between vaious health outcomes and ut micobiome composition Tempoal chanes in infant ut micobiome composition Effect of vaious factos on ut micobiome composition: C-section Exposue to antibiotics Tem (full o patial) Gut micobiome composition and shot chain fatty acids Diet and matenal ut micobiome [Micobiome, 06] 7

8 Micobial Composition Vaies Spatially 8 Lozupone et al. (03)

9 Some Pominent Micobes by Location 9 Lozupone et al. (03)

10 Data Ecosystem (e.. ut): Sequence the specimen A andom specimen Read counts of 6S RNA fo each Opeational Taxonomic Unit (OTU) 0

11 OTUs Summaized at Diffeent Levels of Phyloeny OTU Genus level Phyla level OTU_ OTU_ Genus OTU_3 OTU_4 Phylum OTU_5 Genus OTU_6 OTU_7 Genus Phylum p OTU_m Teminoloy used in this talk: Taxa

12 OTU Abundance Table OTU Subect Subect Subect n OTU_ OTU_ OTU_3 OTU_4 OTU_m O O O 3 O 4 O O O3 O 4 Om Om O n O n O 3 n O 4n Omn

13 Abundance Vs. Relative Abundance Abundance of 5 taxa: Ecosystem Relative abundance of 5 taxa: Ecosystem Unobsevable Abundance of 5 taxa: Specimen Relative abundance of 5 taxa: Specimen 3 Obsevable

14 A Sinle Taxon Can Chane all Relative Abundances Abundance of 5 taxa: Ecosystem I Relative abundance of 5 taxa: Ecosystem I Abundance of 5 taxa: Ecosystem II Relative abundance of 5 taxa: Ecosystem II 4

15 Not Sufficient to Compae Relative Abundances Reseache may be inteested in identifyin taxa whose abundance chaned between the ecosystems 5

16 Thee impotant featues of the data 6

17 Featue : Unequal libay sizes Total numbe of eads can vay dastically acoss specimens fom the same ecosystem Taxa Sample Sample Sample 3 Taxon Taxon Taxon Taxon Column sum

18 Raefaction 8

19 Basic idea Step : Identify the smallest libay that is consideed to be defective. Let the size of this libay be L Step : Discad all samples with libay sizes less than L Step 3: Subsample (without eplacement) the emainin samples such that each sample has a libay size L 9

20 A toy example Oiinal data: Suppose L = 500 Raefied data: OTU Subect Subect Subect 3 OTU_ OTU_ OTU_ OTU_ Libay size OTU Subect Subect OTU_ OTU_ OTU_ OTU_ Libay size Dop 0

21 Compaison of nomalization methods

22 Compaison of vaious nomalization methods Raw data metaenomeseq DESeq Raefied 0K lib

23 Featue : Relative Abundances in a Simplex Non-neative Sum to Hence they ae points inside a simplex (compositional data) Taxon\Baby Baby A Baby B Baby C Bifidobacteim Bifidobacteium Blautia C A Steptococcus Othes Steptococcus B Blautia Standad statistical methods (e.. ANOVA, Kuskal-Wallis) may not be applicable diectly 3

24 Some Common Stateies fo Compain (Relative) Abundance Methods based on Diichlet-multinomial distibution RNA-Seq liteatue ANOVA 4

25 Diichlet-Multinomial Distibution Then a easonable pobability model conditional on libay size and the elative abundance: O O O. ( O m i O, O,,..., O ( O., ~ Multi ( O., ) i m )',,..., m ) To model exta-multinomial vaiability, often usin Diichlet distibution: is modeled iid ~ Diichlet( ) 5

26 A Consequence Unde the D-M model paiwise coelations of (elative) abundances amon all pais of taxa ae neative! This is not because of the Bioloy but it is an atifact of the sum constaint on the andom vaiables! Bioloically not easonable! Mosiman (Biometika, 96) 6

27 Featue 3: Zeo Counts As many as 80% of the enties in a OTU table may consist of zeos! 7

28 Zeo Counts: Some Common Stateies. Add a small positive constant, called pseudo count k O i O k i Pawlowsky-Glahn and Buccianti (0), Xia et al. (0), Mandal et al. (05), etc.. Impute usin some pobability model Pawlowsky-Glahn and Buccianti (0) 3. Zeo inflated models Relative Abundance: Zeo inflated beta eession (Chen, Li, 06) Abundance: Zeo inflated Gaussian model (Paulson, 03) 8

29 Types of Zeos We defined/identified 3 types of zeos: Stuctual zeos: Taxa absent because of the expeimental condition. E.. some taxa pesent in a ain foest may not be pesent in a deset Outlies: Thee is some latent vaiable in the data that poduces exteme obsevations. E.. unknown to the investiato a ecent smoke is mixed in the sample of non-smokes Samplin zeos: Caused by samplin depth o libay size. Low abundance taxa ae not epesented Kaul et al., in eview 9

30 Identification of Zeos: Stuctual zeos th taxon in the th oup is declaed to be a stuctual zeo in the th oup if pˆ.96 pˆ ( n pˆ ) 0 pˆ popotion of zeos in the sample 30

31 Identification of Zeos: Outlie zeos Y i ( ) : th th Lo-atio of taxon in the oup, elative to a efeence taxon, fo the th i obsevation Model ( i,,..., n ) : Y iid i ~ N (, ) ( ) N(, ) Usin Peddada and Hwan (00) classify obsevations into at most clustes 3

32 Outlies: Two examples Potential outlie zeos 3

33 Identification of Zeos: Samplin Zeos If the zeos ae neithe unifom zeos no outlie zeos then they ae classified as samplin zeos Caused by samplin depth Samplin zeos ae imputed by 33

34 Testin poblem Compae two o moe oups 34

35 Analysis of Composition of Micobiomes (ANCOM) Mandal et al. (05) 35

36 A Sinle Taxon Can Chane all Relative Abundances Abundance of 5 taxa: Ecosystem I Relative abundance of 5 taxa: Ecosystem I Abundance of 5 taxa: Ecosystem II Relative abundance of 5 taxa: Ecosystem II 36

37 Fom Simplex to Euclidean space Let i, i,,..., m, denote the elative abundance of taxa Pefom the followin tansfomation Y i ln i m ln O O i m The tansfomed data belon to the Euclidean space Note: The above tansfomation is applied within each specimen Aitchison (980) 37

38 Lemma Fo i,..., m, let i i ) ln( )) d i Assumption: Amon d, d,..., d m at least ae zeo [i.e. abundance of at least taxa does not chane] Lemma: Suppose fo a taxon ) ln( )) ) ln( )) Relative abundance fo all Then )) )) Abundance 38

39 An illustation 39

40 Relative Abundance Data Can Be Used to Infe About Abundance: Illustation of the Lemma Abundance Table Taxon Ecosystem Ecosystem Taxon Taxon 4 4 Taxon3 0 0 Taxon Taxon Sum Relative Abundance Table Lo Relative Abundance Ratios Taxon Ecosystem Ecosystem Lo(Taxon/Taxon) Lo(Taxon/Taxon3) Lo(Taxon/Taxon4) Lo(Taxon/Taxon5) Taxon Ecosystem Ecosystem Taxon Taxon.04.0 Taxon Taxon4.0.5 W #{Distinct lo - atios} Taxon

41 Relative Abundance Data Can Be Used to Infe About Abundance: Illustation of the Lemma Abundance Table Taxon Ecosystem Ecosystem Taxon Taxon 4 4 Taxon3 0 0 Taxon Taxon Sum Relative Abundance Table Taxon Ecosystem Ecosystem Taxon Taxon.04.0 Lo Relative Abundance Ratios Taxon Ecosystem Ecosystem Lo(Taxon/Taxon) Lo(Taxon/Taxon3) Lo(Taxon/Taxon4) Lo(Taxon/Taxon5) W #{Distinct lo - atios} W #{Distinct lo - atios} Taxon Taxon4.0.5 Taxon

42 Relative Abundance Data Can Be Used to Infe About Abundance: Illustation of the Lemma Abundance Table Taxon Ecosystem Ecosystem Taxon Taxon 4 4 Taxon3 0 0 Taxon Taxon Sum Relative Abundance Table Lo Relative Abundance Ratios Taxon Ecosystem Ecosystem Lo(Taxon3/Taxon) Lo(Taxon3/Taxon) Lo(Taxon3/Taxon4) Lo(Taxon3/Taxon5) Taxon Ecosystem Ecosystem Taxon Taxon.04.0 Taxon Taxon4.0.5 Taxon W #{Distinct lo - atios} W #{Distinct lo - atios} W3 #{Distinct lo - atios} 4

43 Relative Abundance Data Can Be Used to Infe About Abundance: Illustation of the Lemma Abundance Table Taxon Ecosystem Ecosystem Taxon Taxon 4 4 Taxon3 0 0 Taxon Taxon Sum Relative Abundance Table Lo Relative Abundance Ratios Taxon Ecosystem Ecosystem Lo(Taxon4/Taxon) Lo(Taxon4/Taxon).6 3. Lo(Taxon4/Taxon3) Lo(Taxon4/Taxon5) Taxon Ecosystem Ecosystem Taxon Taxon.04.0 Taxon Taxon4.0.5 Taxon W #{Distinct lo - atios} W #{Distinct lo - atios} W3 #{Distinct lo - atios} W #{Distinct lo - atios}

44 Relative Abundance Data Can Be Used to Infe About Abundance: Illustation of the Lemma Abundance Table Taxon Ecosystem Ecosystem Taxon Taxon 4 4 Taxon3 0 0 Taxon Taxon Sum Relative Abundance Table Lo Relative Abundance Ratios Taxon Ecosystem Ecosystem Lo(Taxon5/Taxon) Lo(Taxon5/Taxon) Lo(Taxon5/Taxon3).87.4 Lo(Taxon5/Taxon4) Taxon Ecosystem Ecosystem Taxon Taxon.04.0 Taxon Taxon4.0.5 Taxon W #{Distinct lo - atios} W #{Distinct lo - atios} W3 #{Distinct lo - atios} W #{Distinct lo - atios} 4 W5 #{Distinct lo - atios}

45 Testin poblem th,..., m, Fo the taxon, we test the followin hypotheses, elative to each efeence taxon : H 0 : ) ln( )) ) ln( )) H a : ) ln( )) ) ln( )) Use: Standad paametic (e.. t-test) o non-paametic test (e.. Mann-Whitney) If thee ae covaiates o epeated measuements, then use standad methods fom linea eession 45

46 Implementation of ANCOM Step. Identify the type of zeos and deal with it suitably Step. Fo the th i taxon, test fo equality of abundance between the two ecosystems elative to each emainin m taxa m Step 3. Since tests ae bein pefomed, apply multiple testin coection Step 4. W i : Numbe of null hypotheses eected in Step 3 Step 5: Repeat the above steps fo all taxa Step 6: Usin the empiical distibution of declae the sinificance of a taxon W 46

47 A lae numbe of simulation studies wee conducted Results of one confiuation of a paticula study 47

48 Simulation Study Based on a Real Data Set in Capoaso et al., PNAS 0 Baseline data: Data on 000 taxa fom the pape Goup (contol oup): A andom sample with eplacement is dawn fom baseline data Goup (teatment oup): A andom sample with eplacement is dawn fom baseline data. Fo non-null data: Randomly spiked 5, 0, 5 o 0% taxa Amount of spikin 5 to 0% (i.e. incease in abundance) Sample sizes: 5, 0, 00 pe oup Numbe of simulations: 800 FDR nominal level: 5%. 48

49 ANCOM Contols FDR Bette Than Othe Methods Consideed T-test on abundance -test on elative abundance Metaen.Seq N = 0 Effect Size = 0% Pecent taxa spiked = 0% EdeR DESeq ANCOM Weiss et al., False Discovey Rate 49

50 ANCOM Competes Well in Tems of Powe T-test on abundance T-test on elative abundance Metaen.Seq N = 0 Effect Size = 0% Pecent taxa spiked = 0% EdeR DESeq ANCOM Weiss et al., Powe 50

51 Moe than ecosystems 5

52 Global test Suppose thee ae ecosystems (o expeimental oups) to be compaed. A wide ane of analyses can be pefomed A. Classical lobal test G H 0, : G ) ln( )) ) ln( )) H a, : G ) ln( )) ) ln( )) Not a vey useful test because eection of the null only implies thee exists at least one ecosystem that is sinificantly diffeent 5

53 Diectional tests B. Diectional tests: Often eseaches ae inteested in knowin if the (elative) abundance inceased o deceased between two ecosystems fo all pais of ecosystems 53 )) ln( ) (ln( )) ln( ) (ln( )) ln( ) (ln( )) ln( ) (ln( : )) ln( ) (ln( )) ln( ) (ln( :,,,,, 0, a E E E E H E E H Total numbe of hypotheses to be tested = ) ( G m G

54 Diectional tests B. Diectional tests: BH pocedue fo the above multiple testin poblem will be too consevative Instead one can use mdfdr contollin pocedue of Guo et al. (00). It contols the oveall FDR (unde the same assumptions as BH pocedue while bein substantially moe poweful than BH Step: Fo each taxon, pefom the followin two-sided test, usin t-test H H 0,, a,,,, : : ) ln( ) ln( )) )) ) ln( ) ln( )) )) Let p,, denote the coespondin p-value 54

55 Diectional tests Step: Let ~ p G min, p Step 3: Apply BH pocedue on the adusted p-values at a pe-specified level of sinificant,, ~ p,,,..., m R Step 4: Suppose null hypotheses ae eected out of total hypotheses in Step 3 m Step 5: Fo evey taxon p R m,, G T ( ) 0, declaed sinificant in Step 4 with, if then declae that, ) ln( )) ( ) ) ln( )) 55

56 Tests aainst a specific ecosystem C. Diectional tests aainst a pespecified ecosystem (e.. Contol oup): Often eseaches ae inteested in knowin if the (elative) abundance inceased o deceased in an ecosystem elative a pespecified ecosystem. H 0,,, contol : ) ln( )) contol ) ln( contol )) H a,,, contol : ) ) ln( )) ln( )) contol contol ) ln( ) ln( contol )) contol )) ( G) Total numbe of hypotheses to be tested = ( m )( G ) 56

57 Tests aainst a specific ecosystem C. Diectional tests: Instead of mdfdr contollin pocedue of Guo et al. (00) one can use a enealization of Dunnett s type test of Gandhi et al. (06) which is moe poweful than Guo et al. (00) 57

58 Tests fo pattens D. Test fo tends o pattens: Hih dose ABx Low dose ABx Medium dose ABx C-Section bon And No Abx exposue No ABx Vainally bon And No Abx exposue C-Section bon And Abx exposue Kaul et al. (07) Vainally bon And Abx exposue 58

59 Tests fo pattens D. Test fo tends o pattens: Hih dose ABx Medium dose ABx Low dose ABx No ABx H 0, : ) ln( )) ) ln( ))... G ) ln( G )) H a, : ) ln( )) ) ln( ))... G ) ln( G )) 59

60 Tests fo pattens D. Test fo tends o pattens: C-Section bon And No Abx exposue Vainally bon And No Abx exposue C-Section bon And Abx exposue Vainally bon And Abx exposue H 0, : ) ln( )) ) ln( )) 3 3 ) ln( )) 4 4 ) ln( )) H a, : ) ln( )) ) ln( )) ) ln( )) 3 3 ) ln( )) 4 4 ) ln( )) 4 4 ) ln( )) 60

61 Tests fo pattens D. Test fo tends o pattens: Moe eneally one can test union of all pattens of inteest usin the ode esticted infeence based methods of Peddada et al. (003), Fanan et al. (04), Jelsema and Peddada (06) H 0, : ) ln( )) ) ln( ))... G ) ln( G )) H a, : Union of all pattens of inteest Kaul et al. (07) 6

62 Note All tests descibed so fa ae applicable in a linea model settin, with covaiates pesent as well as fo epeated measuement o lonitudinal data Softwae packae is available fo unconstained tests at NIEHS. We ae expandin the softwae to deal with diectional and tend tests 6

63 Classification of Samples Usin Micobiome n m paadim 63

64 Classification of subects into two oups x ( x, x,..., x Suppose m is a vecto of taxa fo a andomly chosen subect fom a population )' Question: How to classify the subect into healthy and disease oups? Related question is how to estimate associations between vaious hih dimensional data. Fo example, coelation between ene expession and micobiome 64

65 A Vaiety of Stateies Possible Linea Disciminant Analysis (LDA) Suppot Vecto Machines [Theoy is not developed yet] Etc. 65

66 Loistic-Nomal Distibution Y i ( ) : th th Lo-atio of taxon in the oup, elative to a efeence taxon, fo the th i obsevation Moe pecisely: Y i( ) ln x x i i,,,...,m i,,...,n i : Y ~ MVN(, ) 66

67 LDA To classify a new obsevation vecto y R m into one of two populations:,, Classical LDA would classify into if Howeve y y' ( ) ( ' ' ) 67

68 Challenes In the pesent settin Zeos in the data! n m 68

69 Estimation of Mean Vecto and Pecision Matix 69 Within each oup apply zeo identification aloithm Samplin zeo impute 0 by Stuctual o outlie zeo teat as missin Let, ) ( ˆ ) ( m l Y l n l n i il l ) ˆ )( ˆ ( ), ( ˆ ), (, m im l m l n i il m l Y Y m l n ), ( ), ( ˆ ), ( ˆ ), ( ˆ,,, m l n m l n m l n m l n m l m l m l

70 Estimation of Pecision Matix Let ˆ 0 ( ˆ, s,,,..., m, s,,..., m ) Estimate the pecision matix as follows: min subect to ˆ 0 R I ( m) ( m), Cone of positive semi - definite matices 70

71 Estimation of Pecision Matix Theoem: Choose ˆ O with pobability of at least lo( m ) k then n lo( m ) n c exp( c lo( m )) Kaul et al. (06) 7

72 Some open poblems Inteation of micobiome, expession, metabolomic data etc. Estimation of associations o netwoks amon taxa Hih dimensional eession analysis fo micobiome data Simplex on simplex eession: To undestand associations between the shot chain fatty acids and micobiome Genetics x micobiome x nutition x infections x social x psycholoical factos x envionnemental chemicals Asymptotic theoy fo classification usin SVM 7

73 Acknowledements Abhishek Kaul, Post-doctoal Reseach Fellow, NIEHS Siddhatha Mandal, Public Health Foundation of India Meete Eesbo, Noweian Institute of Public Health Sophie Weiss, Univesity of Coloado, Coloado Rob Kniht, UC San Dieo, Califonia 73

74 Thank you! 74

75 75