QTL mapping in mice - Biostatistics and Medical Informaticskbroman/talks/qtl_nz.pdf ·...
Transcript of QTL mapping in mice - Biostatistics and Medical Informaticskbroman/talks/qtl_nz.pdf ·...
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QTL mapping in mice
Karl W Broman
Department of BiostatisticsJohns Hopkins University
Baltimore, Maryland, USA
www.biostat.jhsph.edu/˜kbroman
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Outline
� Experiments, data, and goals
� Models
� ANOVA at marker loci
� Interval mapping
� LOD scores, LOD thresholds
� Mapping multiple QTLs
� Simulations
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Backcross experiment
P1
A A
P2
B B
F1
A B
BCBC BC BC
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Intercross experiment
P1
A A
P2
B B
F1
A B
F2F2 F2 F2
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Trait distributions
30
40
50
60
70
A B F1 BC
Strain
Phe
noty
pe
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Data and Goals
Phenotypes: ��� = trait value for mouse
�Genotypes: � �� = 1/0 if mouse
�
is BB/AB at marker
�
(for a backcross)Genetic map: Locations of markers
Goals:
� Identify the (or at least one) genomic regions(QTLs) that contribute to variation in the trait.
� Form confidence intervals for QTL locations.
� Estimate QTL effects.
Note: QTL = “quantitative trait locus”
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Why?
Mice: Find gene
� Drug targets, biochemical basis
Agronomy: Selection for improvement
Flies: Genetic architecture
� Evolution
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80
60
40
20
0
Chromosome
Loca
tion
(cM
)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 X
Genetic map
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20 40 60 80 100 120
20
40
60
80
100
120
Markers
Indi
vidu
als
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 X
Genotype data
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Statistical structure
QTL
Markers Phenotype
Covariates
The missing data problem:
Markers QTL
The model selection problem:
QTL, covariates � phenotype
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Models: Recombination
We assume no crossover interference.
� � Points of exchange (crossovers) are accordingto a Poisson process.
� � The
��� �
(marker genotypes) form a Markovchain
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Example
A B − B A A
B A A − B A
A A A A A B
?
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Models: Genotype Phenotype
Let � = phenotype� = whole genome genotype
Imagine a small number of QTLs with genotypes ������ � � � ��� .(
� �
distinct genotypes)
E
� � � � � � ��� "!# # # ! �%$ var� � � � � � & '� !# # # ! �$
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Models: Genotype Phenotype
Homoscedasticity (constant variance): & '� ( & '
Normally distributed residual variation: � � �) * � � � � & ' �
.
Additivity: � � !# # # ! �$ � � + ���, � -� �� ( �� � .
or
/
)
Epistasis: Any deviations from additivity.
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The simplest method: ANOVA
� Split mice into groupsaccording to genotypeat a marker.
� Do a t-test / ANOVA.
� Repeat for each marker.
40
50
60
70
80
Phe
noty
pe
BB AB
Genotype at D1M30
BB AB
Genotype at D2M99
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ANOVA at marker loci
Advantages
� Simple.
� Easily incorporatecovariates.
� Easily extended to morecomplex models.
� Doesn’t require a geneticmap.
Disadvantages
� Must exclude individualswith missing genotype data.
� Imperfect information aboutQTL location.
� Suffers in low density scans.� Only considers one QTL at a
time.
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Interval mapping (IM)
Lander & Botstein (1989)
� Take account of missing genotype data
� Interpolate between markers
� Maximum likelihood under a mixture model
0 20 40 60
0
2
4
6
8
Map position (cM)
LOD
sco
re
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Interval mapping (IM)
Lander & Botstein (1989)
� Assume a single QTL model.
� Each position in the genome, one at a time, is posited as theputative QTL.
� Let 0 � 1/0 if the (unobserved) QTL genotype is BB/AB.Assume �) * � ��1 � & �
� Given genotypes at linked markers, �) mixture of normal dist’nswith mixing proportion
243 � 0 � . �marker data
�
:
QTL genotype576 578 BB ABBB BB
9: ; <>= ? 9: ; <>@ ?A 9: ; < ? <= <>@ A 9 : ; < ?
BB AB9: ; <= ? <>@ A < <= 9: ; <>@ ?A <
AB BB <= 9: ; <>@ ?A < 9: ; <= ? <>@ A <
AB AB <= <@ A 9: ; < ? 9 : ; <= ? 9: ; <@ ?A 9: ; < ?
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The normal mixtures
576 578B
7 cM 13 cM
C Two markers separated by 20 cM,with the QTL closer to the leftmarker.
C The figure at right show the dis-tributions of the phenotype condi-tional on the genotypes at the twomarkers.
C The dashed curves correspond tothe components of the mixtures.
20 40 60 80 100
Phenotype
BB/BB
BB/AB
AB/BB
AB/AB
µ0
µ0
µ0
µ0
µ1
µ1
µ1
µ1
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Interval mapping (continued)
Let D � � 243 � 0�� � . �
marker data
�
�� � 0�� ) * � �1E � & ' �
243 � ��� �
marker data� ��F � � ��� & � � D � G � ��� H � �� & � + � . � D � � G � �� H �F � & �
where
G � � H �� & � � density of normal distribution
Log likelihood:
I � ��F � � ��� & � � � JLKM 243 � ��� �
marker data� ��F � � ��� & �
Maximum likelihood estimates (MLEs) of � F , � � , &:EM algorithm.
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LOD scores
The LOD score is a measure of the strength of evidence for thepresence of a QTL at a particular location.
LOD
� 0 � � JLKM � F likelihood ratio comparing the hypothesis of aQTL at position 0 versus that of no QTL
� JKM . /243 � � �QTL at 0� N ��F 1 � N � � 1 � N &1 �
243 � � �no QTL� N �� N & �
N �F 1 � N � � 1 � N &1 are the MLEs, assuming a single QTL at position 0.
No QTL model: The phenotypes are independent and identicallydistributed (iid)
* � �� & ' �
.
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An example LOD curve
0 20 40 60 80 100
0
2
4
6
8
10
12
Chromosome position (cM)
LOD
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0 500 1000 1500 2000 2500
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Map position (cM)
lod
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19LOD curves
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Interval mapping
Advantages
� Takes proper account ofmissing data.
� Allows examination ofpositions between markers.
� Gives improved estimatesof QTL effects.
� Provides pretty graphs.
Disadvantages
� Increased computationtime.
� Requires specializedsoftware.
� Difficult to generalize.� Only considers one QTL at
a time.
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LOD thresholds
Large LOD scores indicate evidence for the presence of a QTL.
Q: How large is large?
We consider the distribution of the LOD score under the nullhypothesis of no QTL.
Key point: We must make some adjustment for our examination ofmultiple putative QTL locations.
We seek the distribution of the maximum LOD score, genome-wide. The 95th %ile of this distribution serves as a genome-wideLOD threshold.
Estimating the threshold: simulations, analytical calculations, per-mutation (randomization) tests.
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Null distribution of the LOD score
C Null distribution derived bycomputer simulation of backcrosswith genome of typical size.
C Solid curve: distribution of LODscore at any one point.
C Dashed curve: distribution ofmaximum LOD score,genome-wide.
0 1 2 3 4
LOD score
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Permutation tests
mice
markers
genotypedata
phenotypes
O LOD
9QP ?(a set of curves)
O 5SRT UVXW LOD
9 P ?
C Permute/shuffle the phenotypes; keep the genotype data intact.
C Calculate LOD
Y 9QP ? ; Z 5 YR T UVXW LODY 9 P ?
C We wish to compare the observed5
to the distribution of
5 Y
.
C [�\ 9 5 Y] 5 ?
is a genome-wide P-value.
C The 95th %ile of
5 Y
is a genome-wide LOD threshold.
C We can’t look at all ^ _ possible permutations, but a random set of 1000 is feasi-ble and provides reasonable estimates of P-values and thresholds.
C Value: conditions on observed phenotypes, marker density, and pattern of miss-ing data; doesn’t rely on normality assumptions or asymptotics.
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Permutation distribution
maximum LOD score
0 1 2 3 4 5 6 7
95th percentile
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Multiple QTL methods
Why consider multiple QTLs at once?
� Reduce residual variation.
� Separate linked QTLs.
� Investigate interactions between QTLs (epistasis).
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Epistasis in a backcross
Additive QTLs
Interacting QTLs
0
20
40
60
80
100
Ave
. phe
noty
pe
A HQTL 1
A
H
QTL 2
0
20
40
60
80
100
Ave
. phe
noty
pe
A HQTL 1
A
H
QTL 2
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Epistasis in an intercross
Additive QTLs
Interacting QTLs
0
20
40
60
80
100
Ave
. phe
noty
pe
A H BQTL 1
AH
BQTL 2
0
20
40
60
80
100
Ave
. phe
noty
pe
A H BQTL 1
A
H
BQTL 2
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Abstractions / simplifications
� Complete marker data
� QTLs are at the marker loci
� QTLs act additively
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The problem
n backcross mice; M markers
�� � = genotype (1/0) of mouse � at marker �
�� = phenotype (trait value) of mouse �
�� � � +5
� R :-� ��� +�` � Which
-� a � /?
b Model selection in regression
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How is this problem different?
� Relationship among the x’s
� Find a good model vs. minimize prediction error
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Model selection
� Select class of models– Additive models
– Add’ve plus pairwise interactions
– Regression trees
� Compare models
– BIC c 9Qd ? R e�f g RSS
9Qd ?ih j d j 9k e�f g ^^ ?
– Sequential permutation tests
– Estimate of prediction error
� Search model space– Forward selection (FS)
– Backward elimination (BE)
– FS followed by BE
– MCMC� Assess performance
– Maximize no. QTLs found;control false positive rate
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Why BIC l?
� For a fixed no. markers, letting m n, BIC o is consistent.
� There exists a prior (on models + coefficients) for whichBIC o is the –log posterior.
� BIC o is essentially equivalent to use of a threshold on theconditional LOD score
� It performs well.
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Choice of l
Smaller
l
: include more loci; higher false positive rate
Larger
l
: include fewer loci; lower false positive rate
Let L = 95% genome-wide LOD threshold(compare single-QTL models to the null model)
Choose
l
= 2 L / JLKM :p mWith this choice of
l, in the absence of QTLs, we’ll
include at least one extraneous locus, 5% of thetime.
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Simulations
� Backcross with n=250
� No crossover interference
� 9 chr, each 100 cM
� Markers at 10 cM spacing;complete genotype data
� 7 QTLs
– One pair in coupling– One pair in repulsion– Three unlinked QTLs
� Heritability = 50%
� 2000 simulation replicates
1
2
3
4
5
6
7
8
9
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Methods
� ANOVA at marker loci
� Composite interval mapping (CIM)
� Forward selection with permutation tests
� Forward selection with BICk
� Backward elimination with BICk
� FS followed by BE with BICk
� MCMC with BICk
� A selected marker is deemed correct if it is within10 cM of a QTL (i.e., correct or adjacent)
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0 1 2 3 4 5 6 7
Correct
Ave no. chosen
ANOVA
35
79
11
CIM
fs, perm
fs
be
fs/be
mcmc
BIC
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0.0
0.5
1.0
1.5
2.0
QTLs linked in couplingA
ve n
o. c
hose
n
AN
OV
A 3 5 7 9 11
CIM fs, p
erm fs be
fs/b
e
mcm
c
BIC
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0.0
0.5
1.0
1.5
2.0
QTLs linked in repulsionA
ve n
o. c
hose
n
AN
OV
A 3 5 7 9 11
CIM fs, p
erm fs be
fs/b
e
mcm
c
BIC
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0.0
0.5
1.0
1.5
2.0
2.5
3.0
Other QTLsA
ve n
o. c
hose
n
AN
OV
A 3 5 7 9 11
CIM fs, p
erm fs be
fs/b
e
mcm
c
BIC
![Page 44: QTL mapping in mice - Biostatistics and Medical Informaticskbroman/talks/qtl_nz.pdf · 2003-01-14 · Interval mapping (IM) Lander & Botstein (1989) Assume a single QTL model. Each](https://reader036.fdocuments.net/reader036/viewer/2022062920/5f0243fb7e708231d4036776/html5/thumbnails/44.jpg)
0.0
0.1
0.2
0.3
0.4
Extraneous linkedA
ve n
o. c
hose
n
AN
OV
A 3 5 7 9 11
CIM fs, p
erm fs be
fs/b
e
mcm
c
BIC
![Page 45: QTL mapping in mice - Biostatistics and Medical Informaticskbroman/talks/qtl_nz.pdf · 2003-01-14 · Interval mapping (IM) Lander & Botstein (1989) Assume a single QTL model. Each](https://reader036.fdocuments.net/reader036/viewer/2022062920/5f0243fb7e708231d4036776/html5/thumbnails/45.jpg)
0.00
0.01
0.02
0.03
0.04
0.05
0.06
Extraneous unlinkedA
ve n
o. c
hose
n
AN
OV
A 3 5 7 9 11
CIM fs, p
erm fs be
fs/b
e
mcm
c
BIC
![Page 46: QTL mapping in mice - Biostatistics and Medical Informaticskbroman/talks/qtl_nz.pdf · 2003-01-14 · Interval mapping (IM) Lander & Botstein (1989) Assume a single QTL model. Each](https://reader036.fdocuments.net/reader036/viewer/2022062920/5f0243fb7e708231d4036776/html5/thumbnails/46.jpg)
Summary
� QTL mapping is a model selection problem.
� Key issue: the comparison of models.
� Large-scale simulations are important.
� More refined procedures do not necessarily giveimproved results.
� BICk with forward selection followed by backwardelimination works quite well (in the case of additiveQTLs).
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Acknowledgements
Terry Speed, University of California, Berkeley, and WEHI
Gary Churchill, The Jackson Laboratory
Saunak Sen, University of California, San Francisco
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References
C Broman KW (2001) Review of statistical methods for QTL mapping in experi-mental crosses. Lab Animal 30(7):44–52
Review for non-statisticians
C Broman KW, Speed TP (1999) A review of methods for identifying QTLs inexperimental crosses. In: Seillier-Moiseiwitch F (ed) Statistics in MolecularBiology. IMS Lecture Notes—Monograph Series. Vol 33, pp. 114–142
Older, more statistical review.
C Lander ES, Botstein D (1989) Mapping Mendelian factors underlying quantita-tive traits using RFLP linkage maps. Genetics 121:185–199
The seminal paper.
C Churchill GA, Doerge RW (1994) Empirical threshold values for quantitative traitmapping. Genetics 138:963–971
LOD thresholds by permutation tests.
C Miller AJ (2002) Subset selection in regression, 2nd edition. Chapman & Hall,New York.
A reasonably good book on model selection in regression.
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C Strickberger MW (1985) Genetics, 3rd edition. Macmillan, New York, chapter11.
An old but excellent general genetics textbook with a very interesting discussionof epistasis.
C Broman KW, Speed TP (2002) A model selection approach for the identificationof quantitative trait loci in experimental crosses (with discussion). J Roy StatSoc B 64:641–656, 737–775
Contains the simulation study described above.