Download Advanced Twin Workshop 2001

The Causes of Variation
Lindon Eaves and Tim York
Boulder, CO
March 2001
One Issue (Among Many!)
• Identifying genes that cause complex
diseases and genes that contribute to
variation in quantitative traits
Quantitative Trait Locus (QTL)
Any gene whose contribution to
variation in a quantitative trait is
large enough to stand out against
the background noise of other
genetic and environmental factors
Quantitative Trait
A continuously variable trait (in
which variation may be caused by
multiple genetic and/or
environmental factors); any
categorical trait in which
differences between categories
may be mapped onto variation in
a continuous trait
Common diseases
•
•
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•
•
•
Estimated life time risk c.60%
Substantial genetic component
“Non-Mendelian” inheritance
Non-genetic risk factors
Multiple interacting pathways
Most genes still not mapped
Examples
•
•
•
•
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Ischaemic heart disease (30-50%, F-M)
Breast cancer (12%, F)
Colorectal cancer (5%)
Recurrent major depression (10%)
ADHD (5%)
Non-insulin dependent diabetes (5%)
Essential hypertension (10-25%)
Even for “simple” diseases:
Number of alleles is large
(Wright et al, 1999)
• Ischaemic heart disease (LDR) >190
• Breast cancer (BRAC1) >300
• Colorectal cancer (MLN1) >140
Definitions
• Locus: One of c. 30-40,000 genes
• Allele: One of several variants of a specific
gene
• Gene: a sequence of DNA that codes for a
specific function
• Base pair: chemical “letter” of the genome (a
gene has many 1000’s of base pairs)
• Genome: all the genes considered together
Finding QTLs
• Linkage
• Association
Linkage
Finds QTLs by correlating
phenotypic similarity with genetic
similarity (“IBD”) in specific parts
of genome
Linkage
• Doesn’t depend on “guessing gene”
• Works over broad regions (good for
getting in right ball-park) and whole
genome (“genome scan”)
• Only detects large effects (>10%)
• Requires large samples (10,000’s?)
• Can’t guarantee close to gene
Association
• Looks for correlation between specific
alleles and phenotype (trait value,
disease risk)
Association
• More sensitive to small effects
• Need to “guess” gene/alleles
(“candidate gene”) or be close enough
for linkage disequilibrium with nearby
loci
• May get spurious association
(“stratification”) – need to have genetic
controls to be convinced
“Reality”:
For complex disorders and
quantitative traits
Large number of alleles at large
number of genes
Defining the Haystack
• 3x109 base pairs
• Markers every 6-10kb for association in
populations with no recent bottleneck history
• 1 SNPs per 721 b.p. (Wang et al., 1998)
• c.14 SNPs per 10kb = 1000s
haplotypes/alleles
• O (104 -105) genes
Problems
• Large number of loci and alleles/haplotypes
• Possible interactions between genes
• Possible interactions between genes and
environment
• Relatively low frequencies of individual risk
factors
• Functional form of genotype-phenotype
relations not known
• Sorting out signal from noise – minimizing
errors within budget
• Scaling of phenotype (continuous,
discontinuous)
• Spurious association (stratification)
Prepare for the worst
Need statistical approaches that
can screen enormous numbers of
loci and alleles to identify reliably
those that have impact on risk to
disease
System Chosen for Study
•
•
•
•
•
100 loci
20 loci affect outcome, 80 “nuisance” genes
257 alleles/locus
Allele frequencies c.20-0.1%
Disease genes each explain 2.5% variance in
risk (c. 2-fold risk increase)
• 40% rarest alleles increase risk
• 50% variance non-genetic
It’s a Mess!
• Don’t know which genes – might have
clues
• Don’t know which alleles – unordered
categories
• >250100 locus/allele combinations
• More predictor combinations than
people (“curse of dimensionality”)
• Reality worse
Problems
• Informatics: large volume of data
• Computational: large number of
combinations
• Statistical: large number of chance
associations
• Genetic-epidemiological: secondary
associations
How are we going
to figure it out?
Data Mining
(Steinberg and Cartel)
• Attempt to discover possibly very complex
structure in huge databases (large number of
records and large number of variables)
• Problems include classification, regression,
clustering, association (market analysis)
• Need tools to partially or fully automate the
discovery process
• Large databases support search for rare but
important patterns and interactions (epistasis,
GxE)
Some Approaches to DM
•
•
•
•
Logistic regression
Neural networks
“CART” (Breiman et al. 1984)
“MARS” (Friedman, 1991)
“MARS”
•
•
•
•
Multivariate
Adaptive
Regression
Splines
Key references
Friedman, J.H. (1991) Multivariate Adaptive
Regression Splines (with discussion), Annals
of Statistics, 19: 1-141.
Steinberg, D., Bernstein, B., Colla, P., Martin,
K., Friedman, J.H. (1999) MARS User Guide.
San Diego, CA: Salford Systems
The MARS Advantage
• Allows large number of predictors
(loci/alleles/environments) to be screened
• Non-parametric
• Continuous and discontinuous outcomes
• Systematic search for detailed interactions
• Testing and cross-validation
• Continuous and categorical predictors
• Decides best form of relationship
Example Regression Spline:
Impact of Non-Retail Business on Median Boston House Prices
Curve 1: Maximum = 19.08890
Median
20
House
Price
Model for spline:
15
b1 = max(0, INDUS - 8.140)
b2 = max(0, 8.140 - INDUS )
Y = 20.968 - 0.268 b1 + 1.802 b2
10
5
“Knot”
0
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28
INDUS
Industrial Business
Fitting functions with Splines
• Piece-wise linear regression.
– simplest form. allow regression to bend.
• “Knots” define where the function changes behavior.
• Local fit vs. Global fit.
actual data
spline with 3 knots
One predictor example
True knots at 20 and 45 (left)
Best single knot at about 35 (right)
Y
Y
10 20 30 40 50 60
X
10 20 30 40 50 60
X
10 20 30 40 50 60
10 20 30 40 50 60
10 20 30 40 50 60
10 20 30 40 50 60
Re-express variables as basis
functions
• Done to generalize the search for knots. Difficult to
illustrate splines with > one dimension.
• Core building block of MARS model
– max (0, X – c);
– example: BF1 = max(0, ENV – 5);
BF2 = max(0, ENV – 8);
0 for ENV <= 5;
1 for 5 <= ENV <= 8;
1 + 2 for ENV > 8;
• Weighted sum of basis functions used to approximate
the global function.
– ie
y = constant + 1 * BF1 + 2 * BF2 + error;
“Adaptive” Spline
• “Optimal” placement of knots
• “Optimal” selection of predictors and
interactions
Adaptive splines
• Problem:
– What is the optimal location of knots?
– How many knots do you need?
– Best to test all variable / knot locations, but
computationally burdensome.
• MARS solution:
– Develop an overfit model with too many knots.
– Remove all knots that contribute little to model
quality.
– The final model should have approximately correct
knot locations.
“Optimal”
Explains “salient” features of data
Ignores irrelevant features
Stands up to replication
- Several ways to operationalize
mathematically
MARS 2-step model building
• Step 1. Growing phase:
– begins with only a constant in the model.
– serially adds basis functions to a user defined limit. tests
each for improvement when added to the model.
– addition of basis functions until an overly large model is
found. (theoretically the true model is captured).
• Step 2. Pruning phase:
– delete basis function that contributes least to model fit.
– refit the model and delete next term, repeat.
– the most parsimonious model is selected.
• GCV criterion to select optimal model (Craven 1979).
• MARS option uses 10 fold cross-validation to estimate DF.
Cross-validation
• Protects against over fitting data.
• Develops a model on subset of data. Tests fit
on remaining set.
• Systematically assesses how many DF to
charge each variable entered into model.
– Adding a basis function will always lower MSE.
– This reduction is penalized by DF charged.
• Only backwards deletion step is penalized.
Genetic Example:
Regression spline for multi-allelic locus
Probability of disease = 0.037 + 0.114 b1.
Where:
b1 = 1 if ( LOCUS1 = 30 OR LOCUS1 = 37 OR LOCUS1 = 39 OR LOCUS1 = 43
OR LOCUS1 = 44 OR LOCUS1 = 46 OR LOCUS1 = 66 OR LOCUS1 = 73
OR LOCUS1 = 76 OR LOCUS1 = 78 OR LOCUS1 = 79 OR LOCUS1 = 80
OR LOCUS1 = 83 OR LOCUS1 = 87 OR LOCUS1 = 90 OR LOCUS1 = 95
OR LOCUS1 = 103 OR LOCUS1 = 106 OR LOCUS1 = 111 OR LOCUS1 = 113
OR LOCUS1 = 114 OR LOCUS1 = 116 OR LOCUS1 = 118 OR LOCUS1 = 128
OR LOCUS1 = 129 OR LOCUS1 = 133 OR LOCUS1 = 134 OR LOCUS1 = 139
OR LOCUS1 = 146 OR LOCUS1 = 147 OR LOCUS1 = 148 OR LOCUS1 = 170
OR LOCUS1 = 177 OR LOCUS1 = 179 OR LOCUS1 = 182 OR LOCUS1 = 183
OR LOCUS1 = 185 OR LOCUS1 = 192 OR LOCUS1 = 202 OR LOCUS1 = 208
OR LOCUS1 = 209 OR LOCUS1 = 214 OR LOCUS1 = 215 OR LOCUS1 = 218
OR LOCUS1 = 219 OR LOCUS1 = 222 OR LOCUS1 = 223 OR LOCUS1 = 226
OR LOCUS1 = 229 OR LOCUS1 = 230 OR LOCUS1 = 231 OR LOCUS1 = 232
OR LOCUS1 = 235 OR LOCUS1 = 236 OR LOCUS1 = 237 OR LOCUS1 = 240
OR LOCUS1 = 241 OR LOCUS1 = 242 OR LOCUS1 = 244 OR LOCUS1 = 253
OR LOCUS1 = 254),
b1 = 0 otherwise
What happens when nothing is going on? Including only “nuisance” loci (21-80).
N=10,000.
Validation
Loci Identified
None
23 25 30 32 35-37 40 47 50 54 55 57 64 68 72 74 76
87 89 91 92 94 96 97
10-fold cross-validation
25
Loci Identified as contributing to variation in outcome
Sample Size
1000
2000
5000
10000
Validation
Loci Identified
None
2 5-8 10-12 14-18 20 24 40 43 45 56 59 70 77 94
Split-sample
7 10 14
10-fold
14
None
2 3 5 6 8-18 20 38 45 47 69 72 80 88 95 100
Split-sample
12 14 20
10-fold
14
None
2-20 29 32 43 55 56 74
Split-sample
10 15 16 20
10–fold
2-19
None
1-20 25 26 94
Split-sample
1-20 25 94
10-fold
1-20
Correct (+) and Incorrect (-) Assignment of Alleles to High- and Low-Risk Groups by
MARS Model (N=10,000)
Low Risk
(N=30)
Locus
High Risk
(N=227)
Low Risk
(N=30)
+
-
+
-
1
29
1
146
81
2
29
1
145
3
29
1
4
30
5
Locus
High Risk
(N=227)
+
-
+
-
11
29
1
155
72
82
12
29
1
147
80
152
75
13
30
0
155
72
0
138
89
14
30
0
149
78
30
0
142
85
15
29
1
170
57
6
28
2
139
88
16
30
0
150
77
7
28
2
143
84
17
28
2
151
76
8
29
1
148
79
18
28
2
147
80
9
27
3
154
73
19
29
1
140
87
10
29
1
157
70
20
29
1
146
81
So Far:
Does quite well for largish random
samples and continuous
outcomes.
-What about disease
(dichotomous) outcomes?
-What about selected (extreme)
samples?
Generating Dichotomous Outcomes from Continuous Measure
Threshold
Prevalence
21
9.1%
22
4.9%
24
1.0%
Loci Identified by fitting MARS model to dichotomous outcomes (N=10,000)
Prevalence
No validation
10-fold cross validation
9.1%
1 2 5 6 8 9 11-17 19
4.9%
1 2 4 5 6 910 13-15 17-20
8
1.0%
1 2 5 8 9 10-17 19 56
2
16
Loci cross-validated by MARS model for extremes from sample of 10,000
screened individuals
Proportion Selected
Upper % Lower %
9.2
4.9
11.2
6.3
Total N
2024
1116
Outcome
Loci Cross-Validated
Continuous
1-3 5-10 66 88 75
Dichotomous
2 3 5-29 69
Continuous
1-3 6-10 12-15 18 20 68
Dichotomous
1-4 6-8 10-15 17 19 48
So?
• Can detect signal due to relatively large
numbers of relatively rare unordered alleles
of relatively small effect at relatively many loci
amid the noise of still more loci and
environmental effects
• “MARS” may provide elements for analyzing
such data in this and similar contexts (?microarrays, SNPs, expression arrays?)
• Works with continuous data on random
samples and dichotomous outcomes on
selected samples
GAW12 – Simulated data
• Provided for two populations:
– large general pop.
– pop. isolate – founded 20 generations ago by 100 ind.
– limited migration b/w.
• Common disease:
– prevalence of 25%. increases with age
– middle age disease, some early onset
– more common in females than males
• General population
–
–
–
–
7 genes simulated
13 to 20 kb
12 to 40 diallelic sites at start of simulation
passed through 120 to 200K of random mating:
• mutation, intragenic recombination, gene conversion – allowed
at diff. rates for diff. genes
• each gene contains a 500bp recombination hotspot – 15 to 65%
of intragenic recombinations
• 8 to 13 mutational hotspots per gene (6 – 300 x’s )
– 25% of genes isolated for 35 to 85K
generations.
GENE1
GENE5
Length (kb)
20
17
Start # of SNP
40
20
150K
165K
.01
.002
4x10-8
6x10-9
Gene conv.
.01
.002
Mean length
conv.
1000
1600
Start of rec.
hotspot / % in
10349 / 50%
4197 / 65%
# mutat. hotspot
13
8
Incr mut rate
200
20
Random Mating
Rec. rate
Mutation rate
• Isolate population
– loosely modeled after pop. history of Old Order Amish
in Lancaster Co., PA
– Founders: 200 chr.’s sampled from general pop.
– 20,000 chr.’s sampled from general pop. to create an
“outside pop”
– Isolate: children <12, mean 4 ; Outside: children <12, 1
– migration allowed b/w pop.s at each generation
• rate: migrants = 5% of current isolate size
– evolution progressed for 20 generations with
recombination (no mutations, no intragenic rec.)
– founders were then sampled to create the isolate pop.
• 23 extended pedigrees with 1,497 individuals from
each population. (1,000 living)
• Pedigrees include the proband, spouse, and all
first, second, and third degree relatives of each.
• Living individuals are provided:
–
–
–
–
–
–
affected status, fid, mid, sex
age at last exam
age of onset if affected
5 quantitative risk factors
2 environmental risk factors (binary and quantitative)
marker genotype for 1 cM whole genome screen. 2,855 total
markers with an average of 9.1 alleles
– sequence data for 7 candidate genes – 1,176 sequence variants
• 50 replicates provided for each pop.
Sequence data
• Isolate and General population
• Intron and Exon sequence from 7 candidate genes.
• Kept only those individuals with sequence data.
Each set contain 7,000 individuals. 64 mb MARS
limit.
• 5 sets of 7 randomly selected replicates (used 35
of 50 replicates provided)
• 5 associated quantitative risk factors.
• Covariates included: E1, E2, Age, Sex, Age of
onset.
• Affected status binary.
• Exon sequence coded for each individual as
having 0, 1, or 2 ancestral variants.
• If intron variant present (whether 1 or 2 copies)
given a value of 1. Coded in binary form as
haplotypes of length four.
Aff Status
E1
Q1
MG1
CG6
Age of onset
MG6
Liability
Q2
Q3
Q4
CG1
Q5
MG5
MG2
MG3
E2
MG4
Age
CG2
True Model
Isolate pop.
AFF
E1, Q1-Q5, MG6
[557]
Q1
E1, MG1 [5782]
E1, Q1-Q5, MG6 E1, Q1-Q5, MG6
[(435 547 548 557)
[(27 57 76 110)(435
5244 5268 6912 7281] 547 548 557)]
MG1 [5007]
MG1 [5782]
Q2
E1, MG1 [5782]
E1, MG1 [5007]
E1, MG1 [5782]
Q3
E1, E2
E1, E2
E1, E2
Q4
E1, AGE
E1, AGE
E1, AGE
Q5
E1, MG5 [multi-allelic] E1, MG5 [1289
3745 8657 8817]
E1, MG5 [1289
3745 8657 8817]
ONSET
MG6 [557]
none
MG6 [15625]
General pop.
Conclusions
• MARS works well to capture functional form of
disease etiology in simulated data with
dichotomous outcome.
• In most cases was within 1 Kb of functional
variant.
• Generated a predictive model that was replicable
in at least 4 of 5 data sets.
• Highly interpretable output in the form of basis
functions and Importance values.
• MARS may have problems with highly correlated
variables.
• Pattern-recognition tools can be useful to narrow
down search for genes.
Comparison of MARS and ANN
MARS
ANN
Both are non-parametric estimation schemes, allow for a high number of input
predictors, allow for interactions, & non-linear mappings.
Maximum allowable basis functions
and degree of interactions.
Type of network architecture needs to
be specified.
Models are developed fast.
Models are trained more slowly
(DeVeaux et al. 1993).
Backwards elimination stage to
remove unnecessary basis functions.
Problem of overfitting the data esp.
with small data sets.
Easily interpretable basis functions.
Local interpretation of the function.
Black box-weights have little meaning.
Diff. to interpret predictor contribution
Penalizes model complexity. Tries to
dev. a low order, interpretable model.
Non-linear transformations and high
connectivity allows for  complexity.
But the Haystack is Very
Large
• Reality worse than simulations
• More alleles at more loci
• Phenotypes more complex
(multivariate)
• More irrelevant loci (?1000’s)
• Interactions with environment and
between loci
• Spurious associations
It Needs Collaboration
Clinical
Statistical
Molecular
Epidemiological
Physiological
Developmental
Informational
Evolutionary