Accelerated Permutation Inference for Pedigree Heritability Inference with
Family-Based Neuroimaging data
Habib Ganjgahi1, Anderson Winkler2, David C. Glahn3, John Blangero4, Peter Kochunov5, and Thomas Nichols1
1University
of Warwick
2
University of Oxford
3
Yale University
4
5
University of Texas Rio Grande Valley
University of Maryland
Results
• WLS and ML heritability estimators are compareable for large samples.
• The parametric inference is conservative.
• False positive rates with P2 permutation almost exact, P1 conservative..
PP plot for Uncorrected P-values
0.2
0.6
0.15
0.4
0.1
0.2
0.05
0
0
=
nS
8
13
6
62
=
nS
8
85
=
nS
=1
7
49
8
13
bias h2: ML vs WLS
0.02
0.03
0
0.025
-0.02
0.02
-0.04
0.015
-0.06
0.01
-0.08
0.005
-0.1
=
nS
6
62
nS
8
85
1
S=
n
26
n
6
S=
8
S=
58
=
n
n
nS
1
S=
58
6
S=
8
S=
n
n
100
80
80
40
4
=1
TL,ML TL,WLS TW,ML TW,WLS
Figure 1b: h2 Test FP (top left) and Power
FWE Rates
Observed -log(P-values)
2
0.04
0.03
1.5
0.02
• Var(ε ) = Σ =
8
2.5
2
1.5
1
0.5
1
1.5
2
2.5
3
3.5
TS
4
2
1.5
1
0.5
0
0
0
2.5
TW,WLS
0.5
1
1.5
2
2.5
3
Theoretical -log(P-values)
3.5
0
0.5
1
1.5
2
2.5
3
3.5
Theoretical -log(P-values)
Test Statistics
Figure 2a.
Figure 2b.
Figure 2c.
• Voxel-wise FWE p-values (2a), Cluster-wise FWE rates (2b) and cluster-wise FWE p-values (2c) for different
cluster forming thresholds. Rates are nominal except for the higher cluster forming thresholds of the Wald
test (Fig 2b,c).
(2)
σA2Dg
• MLE estimation of
ML invariance property.
6
3
0.5
0.01
Theoretical -log(P-values)
• S: kinship matrix eigenvalue decomposition Φ = SDgS 0 where, N×N matrix S and diagonal
Dg are eigenvector and eigenvalue matrices respectively.
+ σE2 I: transformed data variance, a diagonal matrix.
β, σA2 and σE2 are same in the original and transformed
4
PP plot for T W,WLS Max Cluster Size
3.5
P=0.05
P=0.01
P=0.005
P=0.001
3
0.05
2.5
PP plot for T S Max Cluster Size
3.5
Observed -log(P-values)
P=0.05
P=0.01
P=0.005
P=0.001
0.06
W,WLS
3
0
⇒ Y ∗ = X ∗β + ε ∗,
2
Figure 1c: Uncorrected P-value
Performance
0
• Y ∗ and X ∗: transformed trait and covariates respectively.
∗
0
Theoretical -log(P-values)
Observed -log(P-values)
T
0
Eigen-simplified polygenic model: Applying an orthogonal transformation based on the
eigenvectors of the kinship matrix to the Eq (1) gives
∗
0
TL,ML TL,WLS TW,ML TW,WLS
Test Statisics, h2 =0.6
0.07
0.5
• Φ: N × N kinship matrix; 2Φij is the genetic relatedness of subjects i and j.
S 0Y = S 0X β + S 0g + S 0ε
1
TS
1
• σA2 and σE2 : additive genetic and environmental effects, respectively.
3
Cluster-wise Inference
PP plot for Maximum Statistic
3.5
(1)
4
2
Test Statisics, h2 =0.4
nS
5
40
0
TS
6
Image-wise Simulations
Methods
Var(ε) = Σ = σA2(2Φ) + σE2 I
60
20
TS
Y = Xβ + ε
TL,ML TL,WLS TW,ML TW,WLS
Test Statisics, h2 =0.2
100
97
Figure 1a: h2 Estimation Accuracy
4
Polygenic model:
40
Test Statisics, h2 =0
60
7
Parametric TW,WLS
TS
0
26
Parametric TS
0
=1
8
Permutation T W,WLS
60
TL,ML TL,WLS TW,ML TW,WLS
20
38
Permutation T S
P1
P2
Parametric
20
nS
ML: H2=0
ML: H2=0.2
ML: H2=0.4
ML: H2=0.6
ML: H2=0.8
WLS: H2=0
WLS: H2=0.2
WLS: H2=0.4
WLS: H2=0.6
WLS: H2=0.8
97
4
=1
2
7
49
0
38
3
TS
MSE h2: ML vs WLS
0.035
80
0
=
0.04
4
1
nS
nS
100
Rejection Rates
0.8
5
Observed -log(P-values)
sd h2: ML vs WLS
Rejection Rates
Mean h2: ML vs WLS
Rejection Rates
The prerequisite of any genetic analysis is establishing the heritability of your trait of
interest. In neuroimaging genetics, the large number of voxel-wise measurements present
a challenge both in terms of computational intensity, and the need to control false positive
risk over multiple tests. There is a gap in existing tools, as standard neuroimaging software
cannot estimate heritability, and yet standard quantitative genetics tools cannot provide
essential neuroimaging inferences, like FWE corrected voxel- or cluster-wise P-values.
Moreover, available heritability tools rely on P-values that can be inaccurate due to use
of asymptotic inference methods. Hence, there is a need for alternative, computationally
efficient inference procedures that make fewer assumptions. Permutation methods can
provide exact control of false positive rates and FWE corrected voxel and cluster-wise
inferences. Blangero (2013) introduced a method to accelerate maximum likelihood heritability estimation. However, this advance doesn’t eliminate iterative optimization, possible
convergence problems, nor the use of asymptotic P-values. In the present work (Ganjgahi
2015), we expanded upon Blangero (2013) to derive approximate, non-iterative estimates
and test statistics based on the first iteration of Newton’s method, develop corresponding
Wald and Score tests, and find P-values from a permutation test.
Rejection Rates
Introduction
Real Data Analysis
• ML and WLS heritability estimators have same distribution (Fig 3b)
• WLS estimator showing a slight but consistent trend towards underestimation relative to ML (Fig 3a).
• Score Test is slightly less sensitive than the other tests (Fig 3c).
model due to
Histogram of non-zero h 2 (89.1% of h 2 > 0)
4
Parameter Estimation
3
2
• A non-iterative heritability estimator can be created based on 1-step optimization of the
likelihood, equivalent to a WLS regression of squared residuals on the kinship matrix
eigenvalues.
1
• Amemiya (1977) showed that such estimator is asymptotically normal and consistent.
n
o
−1 0 ∗
∗0 ∗ −1 ∗0 ∗
0
β̂OLS = X X
X Y , θ̂OLS = max 0, U U
U fOLS , ε̂OLS = Y ∗ − X ∗β̂OLS;
1
β̂WLS = X
∗0
(Σ̂∗OLS)−1X ∗
−1
∗0
X (Σ̂∗OLS)−1Y ∗,
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.9
1
ML
Histogram of non-zero h 2 (89.1% of h 2 > 0)
4
3
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
WLS
Figure 3a
Figure 3b
Figure 3c
−1
−1
0 ∗2 −1 ∗
θ̂WLS = max 0, U 0(Σ̂∗2
)
U
U
(Σ̂OLS) fOLS ,
OLS
• β and θ = (σA2, σE2) are the transformed model fixed and random parameters.
∗
: squared OLS residuals.
• fOLS
• U = [1, λg]: N ×2 auxiliary model design matrix, where 1 is a vector of ones and λg = {λgi}
is a vector of kinship matrix eigenvalues.
2
2
2
2
2
• Σ̂∗OLS is formed by θ̂OLS = (σA,OLS
, σE,OLS
). ĥ2WLS = σ̂A,WLS
/ (σ̂A,WLS
+ σ̂E,WLS
).
Test Statistics:
• The Wald test
ã Fully converge ML estimator.
ã One-step WLS estimator (corresponds to a generalized sums of squares):
−1
1 2
0
∗−1
∗−1 0 ∗−1
0 ∗−1
∗−1
TW,WLS = (σ̂A,WLS)2(Σ̂∗−1
λ
)
I
−
Σ̂
1
(
Σ̂
1)
(
Σ̂
1)
1
Σ̂
Σ̂
OLS g
OLS
OLS
OLS
OLS
OLSλg.
2
Figure 3e
Figure 3f
• Real data results, voxel-wise heritability estimates for ML
(top) and WLS (bottom) (Fig 3d).
• Likelihood Ratio Test (LRT):
• Voxel-wise 5% FWE significant heritability, for LRT (top),
WLS Wald (middle) and the score (bottom) tests (Fig 3e).
ã TL,ML: based on fully converged estimator.
ã TL,WLS: based on WLS estimator.
• Cluster-wise 5% FWE inference results for uncorrected
P=0.01% cluster forming threshold, LRT (top row), WLS
Wald (middle) and the score test (bottom) (Fig 3f).
• The Score test: Takes the form of an auxiliary model explained sums of square.
!2 2
0
1 σ̂A,OLS
11
0
TS =
λg I −
λg ,
2
2 σ̂OLS
N
Figure 3d
Conclusion
Hypothesis Testing
• Parametic Test: Asymptotic 50 : 50 mixture χ with 1 and 0 degrees of freedom.
In this work we presented fast and powerful method to estimate heritability that enables standard spatial inference for genetic analysis tools optimized for imaging research, such as the SOLAR/SOLAReclipse
(Kochunov2013).
2
• Permutation Test: two variants
ã P1: Permute the kinship structure, refitting the model with Φ∗ = PΦP 0.
∗
ã P2: Ỹ ∗ = X ∗β̂OLS + Pε̂OLS
.
where P is a permutation matrix, β̂OLS is the fixed effect parameter estimation and
the residulas under the null hypothesis.
References
∗
ε̂OLS
is
T. Amemiya. A note on a heteroscedastic model. Journal of Econometrics, 6(3):365-370, 1977.
J. Blangero et al. A kernel of truth: statistical advances in polygenic variance component models for complex human pedigrees., volume 81. Academic Press, 2013.
H Ganjgahi et al, (2015). Fast and powerful heritability inference for family-based neuroimaging studies. NeuroImage, In press. P. Kochunov, T.E, Nichols (2013). SOLAR-Eclipse computational tools for imaging genetic and
mega-genetic analysis. 19th Annual Meeting of the Organization for Human Brain Mapping.
Electronic copy of this poster: http://warwick.ac.uk/tenichols/ohbm