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Multidimensional heritability analysis of neuroanatomical shape

Jingwei Li

Brain Imaging Genetics

Genetic Variation

Behavior Cognition Neuroanatomy

Brain Imaging Genetics

Genetic Variation

Neuroanatomy

Descriptors of Brain Structures

• One-dimensional descriptors (Hibar2015; Stein2012; Sabuncu2012)

– Volume

– Surface area

– …

• Drawbacks

– Limited when capturing the anatomical variation

Same area


• Multi-dimensional shape descriptor: truncated Laplace-Beltrami Spectrum (LBS)• 𝜓: R𝑛 → R𝑛+𝑘 is the local parametrization of a submonifold 𝑀 of R𝑛+𝑘

𝑔𝑖𝑗 =< 𝜕𝑖𝜓, 𝜕𝑗𝜓 >, 𝐺 = 𝑔𝑖𝑗 𝑛×𝑛, 𝑊 = det𝐺, 𝑔𝑖𝑗 = 𝐺−1 𝑖, 𝑗

• If 𝑓 and 𝜙 are real-valued functions defined on 𝑀, then

𝛻 𝑓, 𝜙 = 𝑖,𝑗 𝑔𝑖,𝑗 𝜕𝑖𝑓 𝜕𝑗𝜙, Δ𝑓 =

1

𝑊 𝑖,𝑗 𝜕𝑖 𝑔

𝑖𝑗𝑊 𝜕𝑗𝑓

where 𝛻 𝑓, 𝜙 ≔< grad 𝑓, grad 𝜙 > and Δ𝑓 ≔ div grad 𝑓 .

• Solve Laplacian eigenvalue problem: Δ𝑓 = 𝜆𝑓Nabla operator Laplace-Beltrami operator

eigenfunction eigenvalue


• Multi-dimensional shape descriptor: truncated Laplace-Beltrami Spectrum (LBS)

Translate Laplacian eigenvalue problem: 𝚫𝒇 = 𝝀𝒇 to a variational problem:

• 𝜙Δ𝑓 𝑑𝜎 = − 𝛻 𝑓, 𝜙 𝑑𝜎

• Since 𝛻 𝑓, 𝜙 = 𝑖,𝑗 𝑔𝑖,𝑗 𝜕𝑖𝑓 𝜕𝑗𝜙 and 𝜙Δ𝑓 𝑑𝜎 = 𝜙 𝜆𝑓 𝑑𝜎 = −𝜆 𝜙𝑓𝑑𝜎

𝑖,𝑗 𝑔𝑖,𝑗 𝜕𝑖𝑓 𝜕𝑗𝜙 𝑑𝜎 = 𝜆 𝜙𝑓𝑑𝜎

variational problem

Green formula



Discretization of 𝑖,𝑗 𝑔𝑖,𝑗 𝜕𝑖𝑓 𝜕𝑗𝜙 𝑑𝜎 = 𝜆 𝜙𝑓𝑑𝜎:

• Choose 𝑛 linearly independent form functions: 𝜙1 𝑥 , 𝜙2 𝑥 ,… , 𝜙𝑛 𝑥 as basis functions (e.g. 𝑥, 𝑥2, 𝑥3, …) defined on the parameter space.

• Any eigenfunction 𝑓 can be approximately projected to the basis functions:𝑓 𝑥 ≈ 𝐹 𝑥 = 𝑈1𝜙1 𝑥 + ⋯+ 𝑈𝑛𝜙𝑛 𝑥

• To solve 𝑈 ⋅ , substitute 𝑓 and 𝜙 ⋅ into the variational problem.

• Define 𝐴 = 𝑎𝑙𝑚 𝑛×𝑛 = 𝑗,𝑘 𝜕𝑗𝐹𝑙 𝜕𝑘𝐹𝑚 𝑔𝑗𝑘𝑑𝜎𝑛×𝑛

and 𝐵 =

𝑏𝑙𝑚 𝑛×𝑛 = 𝐹𝑙𝐹𝑚𝑑𝜎 𝑛×𝑛

=> 𝐴𝑈 = 𝜆𝐵𝑈General eigenvalue problem



– Solve a Laplacian eigenvalue problem defined based on the brain region

– Obtain the first 𝑀 eigenvalues

• Properties (Reuter 2006):

– Isometric invariant

• For planar shapes and 3D-solids:isometry congruency(identical after rigid body transformation)

• For surface:isometry ≠ congruency



– Solve a Laplacian eigenvalue problem defined based on the brain region

– Obtain the first 𝑀 eigenvalues

• Properties (Reuter 2006):

– Isometric invariant

– scaling a n-dimensional manifold by the factor 𝑎 results in

scaled eigenvalues by the factor 1

𝑎2

– … In this paper, eigenvalues are scaled:

𝜆𝑖,𝑚 = 𝜆𝑖,𝑚 ⋅ 𝑉𝑖2/3

𝑖: subject; 𝑚: dimension

Heritability

• A phenotype/trait can be influenced by genetic and environmental effects.

• Heritability: how much of the variation in a phenotype/trait is due to variation in genetic factors.

Main Idea of This Paper

• Truncated LBS is more representative for a shape compared to volume.

• Use truncated LBS as descriptors for 12 brain regions to compute heritability. Compare that with volume-based heritability.

• To adapt truncated LBS into GCTA (Genome-wide Complex Trait Analysis) (Yang 2011) heritability model, propose a multi-dimensional heritability model.

GCTA heritability model

𝑦 = 𝑔 + 𝑐 + 𝑒

𝑔~𝑁 0, 𝜎𝐴2𝐾 𝑐~𝑁 0, 𝜎𝐶

2Λ 𝑒~𝑁 0, 𝜎𝐸2𝐼

Additive genetic component Common environmental component

Unique environmental component

𝑁 × 1 trait vector(𝑁: #subjects)


𝑦 = 𝑔 + 𝑐 + 𝑒



K: genetic similarity matrix

•Familial study: 𝐾 = 2 × 𝐾𝑖𝑛𝑠ℎ𝑖𝑝 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑠. E.g. parent-offspring (0.5), identical twins (1), full siblings (0.5), half siblings (0.25)

•Unrelated subjects study: genome-side single-nucleotide polymorphism (SNP) data


𝑦 = 𝑔 + 𝑐 + 𝑒



What is Single-Nucleotide Polymorphism (SNP):

• Each locus on a DNA sequence is a single nucleotide adenine (A), thymine (T), cytosine (C), or guanine (G).

• SNP: a DNA sequence variation occurring when the types of single nucleotide in the genome (or other shared sequence) differs between individuals or paired chromosomes in one subject. E.g., AAGCCTA and AAGCTTA.

• SNP can leads to alleles (variants of a given gene).

• Each SNP can have 3 genotypes: AA, Aa, aa (denoted as 0-2)


𝑦 = 𝑔 + 𝑐 + 𝑒



How to compute genetic similarity from SNP:

• 𝑋(#subjects x #SNPs).

• Standardize each column of 𝑋 (mean 0, variance 1).

• 𝐾 =𝑋𝑋𝑇

#𝑆𝑁𝑃𝑠

0 ⋯ 22 ⋯ 1⋮1 ⋯

⋮0


𝑦 = 𝑔 + 𝑐 + 𝑒



Λ: shared environment matrix between the subjects

•Familial study: e.g., twins & non-twin siblings (1)

•Unrelated subjects study: Λ vanishes


𝑦 = 𝑔 + 𝑐 + 𝑒



Identical matrix


𝑦 = 𝑔 + 𝑐 + 𝑒



ℎ2 =𝜎𝐴2

𝜎𝐴2 + 𝜎𝐶

2 + 𝜎𝐸2

ℎ2: the variance in the trait explained by the variance in additive genetic component

heritability

Multi-dimensional traits heritability model

𝑌 = 𝐺 + 𝐶 + 𝐸

𝑣𝑒𝑐 𝐺 ~𝑁 0, Σ𝐴 ⨂𝐾 , 𝑣𝑒𝑐 𝐶 ~𝑁 0, Σ𝐶 ⨂Λ , 𝑣𝑒𝑐 𝐸 ~𝑁 0, Σ𝐸 ⨂𝐼

𝑁 ×𝑀 trait matrix(𝑁: #subjects)(𝑀: #dimensions)

Σ𝐴 = 𝜎𝐴𝑟𝑠 𝑀×𝑀: 𝜎𝐴𝑟𝑠 is

the genetic covariance between 𝑟-th and 𝑠-thdimensions in traits

Σ𝐶 = 𝜎𝐶𝑟𝑠 𝑀×𝑀: 𝜎𝐶𝑟𝑠 is

the common environmental covariance between 𝑟-thand 𝑠-th dimensions in traits

Σ𝐸 = 𝜎𝐸𝑟𝑠 𝑀×𝑀: 𝜎𝐸𝑟𝑠 is

the unique environmental covariance between 𝑟-thand 𝑠-th dimensions in traits


𝑌 = 𝐺 + 𝐶 + 𝐸


⨂: Kronecker product

Σ𝐴𝑟𝑠⨂𝐾 =

𝜎𝐴11𝐾 𝜎𝐴12𝐾 ⋯ 𝜎𝐴1𝑀𝐾

𝜎𝐴21𝐾 𝜎𝐴22𝐾 ⋯ 𝜎𝐴2𝑀𝐾

⋮ ⋮ ⋮𝜎𝐴𝑀1𝐾 𝜎𝐴𝑀2𝐾 ⋯ 𝜎𝐴𝑀𝑀𝐾


𝑌 = 𝐺 + 𝐶 + 𝐸


𝑣𝑒𝑐 𝑎1, 𝑎2, ⋯ , 𝑎𝑘 =

𝑎1 𝑎2⋮ 𝑎𝑘


𝑌 = 𝐺 + 𝐶 + 𝐸


ℎ2 =tr Σ𝐴

tr Σ𝐴 + tr Σ𝐶 + tr Σ𝐸=

𝑚=1

𝑀

𝛾𝑚ℎ𝑚2

where 𝛾𝑚 =𝜎𝐴𝑚𝑚 + 𝜎𝐶𝑚𝑚 + 𝜎𝐸𝑚𝑚

𝑝=1𝑀 𝜎𝐴𝑝𝑝 + 𝜎𝐶𝑝𝑝 + 𝜎𝐸𝑝𝑝

ℎ𝑚2 =

𝜎𝐴𝑚𝑚

𝜎𝐴𝑚𝑚 + 𝜎𝐶𝑚𝑚 + 𝜎𝐸𝑚𝑚

The multi-dimensional trait heritability is a weighted average of the heritability of each dimension.

heritability


• Properties

– Invariant to rotations of data

𝑌𝑇 = 𝐺𝑇 + 𝐶𝑇 + 𝐸𝑇

ℎ𝑇2 = ℎ2

𝑌 = 𝐺 + 𝐶 + 𝐸 (1)

(2)

heritability from model (1)heritability from model (2)

𝑇𝑇𝑇 = 𝑇𝑇𝑇 = 𝐼

Consider covariates

• Sometimes, we want to study the effects after controlling some nuisance variables by regressing them out.

• E.g., age, gender, handness

Consider covariates

𝑌 = 𝑋𝐵 + 𝐺 + 𝐶 + 𝐸


𝑌 = 𝑈𝑇𝑌 = 𝑈𝑇𝐺 + 𝑈𝑇𝐶 + 𝑈𝑇𝐸 = 𝐺 + 𝐶 + 𝐸

𝑣𝑒𝑐 𝐺 ~𝑁 0, Σ𝐴 ⨂ 𝑈𝑇𝐾𝑈 , 𝑣𝑒𝑐 𝐶 ~𝑁 0, Σ𝐶 ⨂ 𝑈𝑇Λ𝑈 ,

𝑣𝑒𝑐 𝐸 ~𝑁 0, Σ𝐸 ⨂𝐼

𝑈𝑇𝑋 = 0𝑈𝑇𝑈 = 𝐼

𝑈𝑈𝑇 = 𝐼 − 𝑋 𝑋𝑇𝑋 −1𝑋𝑇

𝑈: 𝑁 × 𝑁 − 𝑞

Covariates (𝑁 × 𝑞)

Analysis

• Datasets:

– Genomics Superstruct Project (GSP; N = 1320) – unrelated subjects

– Human Connectome Project (HCP; N = 590)• 72 monozygotic twin pairs

• 69 dizygotic twin pairs

• 253 full siblings of twins

• 55 singletons

• 12 brain structures

• Traits

– Volume

– Truncated LBS

Volume heritability (GSP data)

• Before multiple comparisons correction: 3/12 brain structures are significant• After multiple comparisons correction: none is significant• Most structures: parametric & nonparametric p values are similar => standard errors

estimates are accurate

Volume heritability (GSP data)

Test-retest reliability:• Lin’s concordance correlation coefficient

𝜌𝑐 =2𝜌𝜎𝑥𝜎𝑦

𝜎𝑥2 + 𝜎𝑦

2 + 𝜇𝑥 − 𝜇𝑦2

variance mean

correlation coefficient

𝑥, 𝑦: use repeated runs on separate days of the same set of subjects

Truncated LBS heritability (GSP data)

• Before multiple comparisons correction: 7/12 brain structures are significant• After multiple comparisons correction: 5/12 brain structures are significant• Most structures: parametric & nonparametric p values are similar => standard errors

estimates are accurate• Smaller standard error than volume-based heritability


Test-retest reliability:• Averaged Lin’s concordance correlation coefficient

across 𝑀 dimensions

𝜌𝑐 =2𝜌𝜎𝑥𝜎𝑦

𝜎𝑥2 + 𝜎𝑦

2 + 𝜇𝑥 − 𝜇𝑦2

variance mean

correlation coefficient

𝑥, 𝑦: use repeated runs on separate days of the same set of subjects


Truncated LBS heritability (HCP data)

• Only significant brain structures results are shown

• Consistently higher than GSP dataset– Possible reason: in unrelated subjects only the variation of some

common SNPs are captured.

Structure 𝒉𝟐 Standard Error

Accumbens area 0.309 0.162

Caudate 0.583 0.124

Cerebellum 0.653 0.120

Corpus Callosum 0.558 0.136

Hippocampus 0.363 0.190

Third Ventricle 0.536 0.134

Putamen 0.483 0.212

Visualizing principal mode of shape variation

• PCA is a kind of rotation of data. The first PC of LBS explains a large percentage of shape variation.

• Heritability model: (1) invariant to rotation; (2) heritability of multi-dimensional trait = weighted average of each dimension’s heritability

• The heritability of truncated LBS is the weighted average of the first M PCs’ heritability.

Visualizing principal mode of shape variationProcedures (for one brain structure)1. Register each subject’s mask (1 – in structure, 0 – out of structure)

to a common used template.

2. Create a population average of structure surface for plotting– A weighted average of all subjects’ registered mask image

– Weight: Gaussian kernel • center: average of first PC

• distance: subject-specific corresponding first PC <-> center

• Width: resulting 500 shapes have non-0 weights

– The isosurface with 0.5 in the averaged map

3. Use the same Gaussian kernel, generate averaged maps by including the shapes around +2 standard deviation of the first PC (-2 s.d. as well)

4. Plot the difference between the two maps in step 3 on the surface generated in step 2.

Visualizing principal mode of shape variation

Red: shapes around +2 s.d. are larger than -2 s.d.

Blue: shapes around -2 s.d. are larger than +2 s.d.

Strengths

• Use truncated LBS instead of volume as features

– Capture more shape variation

– Isometry invariance

– Does not require any registration or mapping (Reuter 2006 & 2009)

• Generalize the concept of heritability into multi-dimensional phenotypes

– Other applications (multi-tests of one behavior; disease study)

Strengths

• Variability of heritability estimation

– Multi-dimensional trait heritability model < original GCTA model (unrelated subject dataset)

– Heritability estimates are more accurate, more significant

• Propose a visualization method for shape variation

– Interpretation: shape variation along the first PC axis of the shape descriptor

Weakness

• Optimal number of eigenvalue may not be 50

– Only 30, 50, 70 are tested

– Error bars for difference number of eigenvalues are not shown

– Other number except 50 (used in paper) could lead to higher heritability and smaller error bars

Weakness

• Optimal number of eigenvalue can be different for different brain structures

– Amygdala: heritability is similar for 30, 50, 70 eigenvalues (even decrease)

– 3rd-ventricle: heritability increases from 0.4 to 0.6

Weakness

• Links between proposed visualization method and LBS heritability are not clear.

• Only volume-based GCTA heritability is compared to the new method and new model.

– More comparisons with the literature (e.g., Gilmore 2010; Baare 2001)

Backup: invariant to rotations of data

𝑐𝑜𝑣 𝑣𝑒𝑐 𝐺𝑇

= 𝑐𝑜𝑣 𝑇𝑇⨂𝐼 𝑣𝑒𝑐 𝐺

= 𝑇𝑇⨂𝐼 𝑣𝑒𝑐 𝐺 𝑇⨂𝐼

= 𝑇𝑇⨂𝐼 Σ𝐴⨂𝐾 𝑇⨂𝐼

= 𝑇𝑇Σ𝐴𝑇 ⨂𝐾

Similarly, 𝑐𝑜𝑣 𝑣𝑒𝑐 𝐶𝑇 = 𝑇𝑇Σ𝐶𝑇 ⨂Λ, 𝑐𝑜𝑣 𝑣𝑒𝑐 𝐸𝑇 = 𝑇𝑇Σ𝐸𝑇 ⨂𝐼

ℎ𝑇2 =

𝑡𝑟 𝑇𝑇Σ𝐴𝑇

𝑡𝑟 𝑇𝑇Σ𝐴𝑇 + 𝑡𝑟 𝑇𝑇Σ𝐶𝑇 + 𝑡𝑟 𝑇𝑇Σ𝐸𝑇

=𝑡𝑟 Σ𝐴 𝑇𝑇𝑇

𝑡𝑟 Σ𝐴 𝑇𝑇𝑇 + 𝑡𝑟 Σ𝐶 𝑇𝑇𝑇 + 𝑡𝑟 Σ𝐸(𝑇𝑇𝑇)

=𝑡𝑟 Σ𝐴

𝑡𝑟 Σ𝐴 + 𝑡𝑟 Σ𝐶 + 𝑡𝑟 Σ𝐸= ℎ2

Theorem: 𝑣𝑒𝑐 𝐴𝑋𝐵 = 𝐵𝑇⨂𝐴 𝑣𝑒𝑐 𝑋Here 𝐴 = 𝐼, 𝑋 = 𝐺, 𝐵 = 𝑇

• 𝐴⨂𝐵 𝑇 = 𝐴𝑇⨂𝐵𝑇

• 𝑐𝑜𝑣 𝐴𝑋 = 𝐴𝑐𝑜𝑣 𝑋 𝐴𝑇

𝐴⨂𝐵 𝐶⨂𝐷 = 𝐴𝐶⨂𝐵𝐷

• 𝑡𝑟 𝐴𝐵𝐶 = 𝑡𝑟 𝐵𝐶𝐴 = 𝑡𝑟 𝐶𝐴𝐵• Associative property of matrix

multiplication

Backup: multi-dimensional trait heritability is a weighted average of heritability of each dimension

ℎ2 =𝑡𝑟 Σ𝐴

𝑡𝑟 Σ𝐴 + Σ𝐶 + Σ𝐸

= 𝑚=1𝑀 𝜎𝐴𝑚𝑚

𝑝=1𝑀 𝜎𝐴𝑝𝑝 + 𝑝=1

𝑀 𝜎𝐶𝑝𝑝 + 𝑝=1𝑀 𝜎𝐸𝑝𝑝

=

𝑚=1

𝑀𝜎𝐴𝑚𝑚 + 𝜎𝐶𝑚𝑚 + 𝜎𝐸𝑚𝑚

𝑝=1𝑀 𝜎𝐴𝑝𝑝 + 𝜎𝐶𝑝𝑝 + 𝜎𝐸𝑝𝑝

⋅𝜎𝐴𝑚𝑚

𝜎𝐴𝑚𝑚 + 𝜎𝐶𝑚𝑚 + 𝜎𝐸𝑚𝑚

=

𝑚=1

𝑀

𝛾𝑚ℎ𝑚2

Backup: moment-matching estimator for unrelated subjects (no shared environmental component)𝑐𝑜𝑣 𝑦𝑟 , 𝑦𝑠 = 𝜎𝐴𝑟𝑠𝐾 + 𝜎𝐸𝑟𝑠𝐼 ⟹ 𝑦𝑟𝑦𝑠

𝑇 = 𝜎𝐴𝑟𝑠𝐾 + 𝜎𝐸𝑟𝑠𝐼

To estimate 𝜎𝐴𝑟𝑠, 𝜎𝐸𝑟𝑠, use a regression model:

𝑣𝑒𝑐 𝑦𝑟𝑦𝑠𝑇 = 𝜎𝐴𝑟𝑠𝑣𝑒𝑐 𝐾 + 𝜎𝐸𝑟𝑠𝑣𝑒𝑐 𝐼

⟹ 𝑦𝑠⨂𝑦𝑟 = 𝜎𝐴𝑟𝑠𝑣𝑒𝑐 𝐾 + 𝜎𝐸𝑟𝑠𝑣𝑒𝑐 𝐼

⟹ 𝑣𝑒𝑐 𝐾 𝑇 𝑦𝑠⨂𝑦𝑟 = 𝜎𝐴𝑟𝑠𝑣𝑒𝑐 𝐾 𝑇𝑣𝑒𝑐 𝐾 + 𝜎𝐸𝑟𝑠𝑣𝑒𝑐 𝐾 𝑇𝑣𝑒𝑐 𝐼

𝑣𝑒𝑐 𝐼 𝑇 𝑦𝑠⨂𝑦𝑟 = 𝜎𝐴𝑟𝑠𝑣𝑒𝑐 𝐼 𝑇𝑣𝑒𝑐 𝐾 + 𝜎𝐸𝑟𝑠𝑣𝑒𝑐 𝐼 𝑇𝑣𝑒𝑐 𝐼

⟹ 𝑦𝑠⨂𝑦𝑟

𝑇𝑣𝑒𝑐 𝐾 = 𝜎𝐴𝑟𝑠𝑣𝑒𝑐 𝐾 𝑇𝑣𝑒𝑐 𝐾 + 𝜎𝐸𝑟𝑠𝑣𝑒𝑐 𝐼 𝑇𝑣𝑒𝑐 𝐾

𝑦𝑠⨂𝑦𝑟𝑇𝑣𝑒𝑐 𝐼 = 𝜎𝐴𝑟𝑠𝑣𝑒𝑐 𝐾 𝑇𝑣𝑒𝑐 𝐼 + 𝜎𝐸𝑟𝑠𝑣𝑒𝑐 𝐼 𝑇𝑣𝑒𝑐 𝐼

⟹ 𝑦𝑠𝑇⨂𝑦𝑟

𝑇 𝑣𝑒𝑐 𝐾 = 𝜎𝐴𝑟𝑠𝑣𝑒𝑐 𝐾 𝑇𝑣𝑒𝑐 𝐾 + 𝜎𝐸𝑟𝑠𝑣𝑒𝑐 𝐼 𝑇𝑣𝑒𝑐 𝐾

𝑦𝑠𝑇⨂𝑦𝑟

𝑇 𝑣𝑒𝑐 𝐼 = 𝜎𝐴𝑟𝑠𝑣𝑒𝑐 𝐾 𝑇𝑣𝑒𝑐 𝐼 + 𝜎𝐸𝑟𝑠𝑣𝑒𝑐 𝐼 𝑇𝑣𝑒𝑐 𝐼

⟹ 𝑦𝑟𝑇𝐾𝑦𝑠 = 𝜎𝐴𝑟𝑠𝑡𝑟 𝐾

2 + 𝜎𝐸𝑟𝑠𝑡𝑟 𝐾

𝑦𝑟𝑇𝑦𝑠 = 𝜎𝐴𝑟𝑠𝑡𝑟 𝐾 + 𝜎𝐸𝑟𝑠𝑡𝑟 𝐼

Backup: moment-matching estimator for unrelated subjects (no shared environmental component)

⟹𝜎𝐴𝑟𝑠𝜎𝐸𝑟𝑠

=𝑡𝑟 𝐾2 𝑡𝑟 𝐾

𝑡𝑟 𝐾 𝑡𝑟 𝐼

−1𝑦𝑟𝑇𝐾𝑦𝑠𝑦𝑟𝑇𝑦𝑠

⟹

𝜎𝐴𝑟𝑠 =𝑦𝑟𝑇 𝑁𝐾 − 𝑡𝑟 𝐾 𝐼 𝑦𝑠𝑁𝑡𝑟 𝐾2 − 𝑡𝑟2 𝐾

≔𝑦𝑟𝑇 𝐾 − 𝜏𝐼 𝑦𝑠

𝜈𝐾

𝜎𝐸𝑟𝑠 =𝑦𝑟𝑇 𝑡𝑟 𝐾2 𝐼 − 𝑡𝑟 𝐾 𝐾 𝑦𝑠𝑁𝑡𝑟 𝐾2 − 𝑡𝑟2[𝐾]

=𝑦𝑟𝑇 𝜅𝐼 − 𝜏𝐾 𝑦𝑠

𝜈𝐾

where 𝜏 = 𝑡𝑟 𝐾𝑁, 𝜅 = 𝑡𝑟 𝐾2

𝑁, 𝜈𝐾 = 𝑡𝑟 𝐾2 − 𝑡𝑟2 𝐾𝑁 = 𝑁 𝜅 − 𝜏

⟹ Σ𝐴 =𝑌𝑇 𝐾 − 𝜏𝐼 𝑌

𝜈𝐾, Σ𝐸 =

𝑌𝑇 𝜅𝐼 − 𝜏𝐾 𝑌

𝜈𝐾

Backup: sampling variance of the point estimator

𝑄𝐴 ≔𝐾 − 𝜏𝐼

𝜈𝐾, 𝑄𝐸 ≔

𝜅𝐼 − 𝜏𝐾

𝜈𝐾

𝑡𝐴 ≔ 𝑡𝑟 Σ𝐴 = 𝑡𝑟 𝑌𝑇𝑄𝐴𝑌 , 𝑡𝐸 = 𝑡𝑟 Σ𝐸 = 𝑡𝑟 𝑌𝑇𝑄𝐸𝑌 , 𝑡 =𝑡𝐴𝑡𝐸

The heritability is a function of 𝑡: 𝑓 𝑡 =𝑡𝐴

𝑡𝐴+𝑡𝐸

𝑣𝑎𝑟 ℎ𝑆𝑁𝑃2 = 𝑣𝑎𝑟 𝑓 𝑡 ≈

𝜕𝑓 𝑡

𝜕𝑡𝑐𝑜𝑣 𝑡

𝜕𝑓 𝑡

𝜕𝑡𝑇

where 𝜕𝑓 𝑡

𝜕𝑡=

𝜕𝑓 𝑡

𝜕𝑡,𝜕𝑓 𝑡

𝜕𝑡=

𝑡𝐸

𝑡𝐴+𝑡𝐸2 ,

−𝑡𝐴

𝑡𝐴+𝑡𝐸2

Define 𝑉𝑟𝑠 = 𝑐𝑜𝑣 𝑦𝑟 , 𝑦𝑠 = 𝜎𝐴𝑟𝑠𝐾 + 𝜎𝐸𝑟𝑠𝐼


𝑐𝑜𝑣 𝑡𝑟 𝑌𝑇𝑄𝛼𝑌 , 𝑡𝑟 𝑌𝑇𝑄𝛽𝑌

= 𝑟,𝑠=1

𝑀

𝑐𝑜𝑣 𝑦𝑟𝑇𝑄𝛼𝑦𝑟 , 𝑦𝑠

𝑇𝑄𝛽𝑦𝑠

= 2 𝑟,𝑠=1

𝑀

𝑡𝑟 𝑄𝛼𝑉𝑟𝑠𝑄𝛽𝑉𝑟𝑠

⟹ 𝑐𝑜𝑣 𝑡 = 2 𝑟,𝑠=1

𝑀 𝑡𝑟 𝑄𝐴𝑉𝑟𝑠𝑄𝐴𝑉𝑟𝑠 𝑡𝑟 𝑄𝐴𝑉𝑟𝑠𝑄𝐸𝑉𝑟𝑠𝑡𝑟 𝑄𝐸𝑉𝑟𝑠𝑄𝐴𝑉𝑟𝑠 𝑡𝑟 𝑄𝐸𝑉𝑟𝑠𝑄𝐸𝑉𝑟𝑠

≈ 2 𝑟,𝑠=1

𝑀

𝜎𝐴𝑟𝑠 + 𝜎𝐸𝑟𝑠2 𝑡𝑟 𝑄𝐴

2 𝑡𝑟 𝑄𝐴𝑄𝐸𝑡𝑟 𝑄𝐸𝑄𝐴 𝑡𝑟 𝑄𝐸

2

=2𝑡𝑟 Σ𝐴 + Σ𝐸

2

𝜈𝐾

1 −𝜏−𝜏 𝜅

≈2𝑡𝑟 Σ𝐴 + Σ𝐸

2

𝜈𝐾

1 −1−1 1

Quadratic form of statistics:𝑐𝑜𝑣 𝜖𝑇Λ1𝜖, 𝜖

𝑇Λ2𝜖 = 2𝑡𝑟 Λ1ΣΛ2Σ + 4𝜇𝑇Λ1ΣΛ2𝜇Here 𝜇 = 0

𝑉𝑟𝑠 = 𝜎𝐴𝑟𝑠𝐾 + 𝜎𝐸𝑟𝑠𝐼

≈ 𝜎𝐴𝑟𝑠𝐼 + 𝜎𝐸𝑟𝑠𝐼

𝐾 ≈ 𝐼⟹ 𝜏 ≈ 1, 𝜅 ≈ 1


𝑡𝑟 𝑄𝐴2 = 𝑡𝑟

𝐾 − 𝜏𝐼 2

𝜈𝐾= 𝑡𝑟

𝐾 −𝑡𝑟 𝐾𝑁

𝐼2

𝑡𝑟 𝐾2 −𝑡𝑟2 𝐾𝑁

2 = 𝑡𝑟𝐾2 − 2

𝑡𝑟 𝐾𝑁 𝐾𝐼 +

𝑡𝑟2 𝐾𝑁2 𝐼


2

=𝑡𝑟 𝐾2 − 2

𝑡𝑟2 𝐾𝑁 +

𝑡𝑟2 𝐾𝑁


2 =1

𝜈𝐾

𝑡𝑟 𝑄𝐴𝑄𝐸 = 𝑡𝑟𝐾 − 𝜏𝐼 𝜅𝐼 − 𝜏𝐾

𝜈𝐾2 =

𝑡𝑟 𝜅𝐾𝐼 − 𝜏𝐾2 − 𝜏𝐾𝐼2 + 𝜏2𝐼𝐾

𝜈𝐾2

=

𝑡𝑟 𝐾2

𝑁 𝑡𝑟 𝐾 −𝑡𝑟 𝐾𝑁 𝑡𝑟 𝐾2 −


𝑡𝑟3 𝐾𝑁2

𝜈𝐾2

=

𝑡𝑟 𝐾𝑁

𝑡𝑟 𝐾2 − 𝑡𝑟 𝐾2 − 𝑡𝑟 𝐾 +𝑡𝑟2 𝐾𝑁


2 = −𝜏

𝜈𝐾


𝑡𝑟 𝑄𝐸2 =

𝑡𝑟 𝜅𝐼 − 𝜏𝐾 2

𝜈𝐾2 =

𝑡𝑟 𝜅2𝐼 − 2𝜅𝜏𝐾 + 𝜏2𝐾2

𝜈𝐾2

=𝜅

𝑡𝑟 𝐾2

𝑁 𝑁 − 2𝑡𝑟2 𝐾𝑁 +

𝑡𝑟2 𝐾𝑁2

𝑁𝑡𝑟 𝐾2 𝑡𝑟 𝐾2

𝜈𝐾2

=𝜅 𝑡𝑟 𝐾2 − 2


𝑡𝑟2 𝐾𝑁

𝜈𝐾 𝑡𝑟 𝐾2 −𝑡𝑟2 𝐾𝑁

=𝜅

𝜈𝐾

𝑣𝑎𝑟 ℎ𝑆𝑁𝑃2 = 𝑣𝑎𝑟 𝑓 𝑡 ≈

𝜕𝑓 𝑡

𝜕𝑡𝑐𝑜𝑣 𝑡

𝜕𝑓 𝑡

𝜕𝑡𝑇

≈2𝑡𝑟 Σ𝐴 + Σ𝐸

2

𝜈𝐾 𝑡𝐴 + 𝑡𝐸 4𝑡𝐸 , −𝑡𝐴

1 −1−1 1

𝑡𝐸−𝑡𝐴


2

𝜈𝐾 𝑡𝐴 + 𝑡𝐸 4𝑡𝐴 + 𝑡𝐸

2


2

𝜈𝐾 𝑡𝑟 Σ𝐴 + 𝑡𝑟 Σ𝐸2=

2

𝜈𝐾⋅𝑡𝑟 Σ𝐴 + Σ𝐸

2

𝑡𝑟 Σ𝐴 + Σ𝐸2=

2

𝜈𝐾⋅𝑡𝑟 Σ𝑃

2

𝑡𝑟 Σ𝑃2


For univariate trait, 𝑡𝑟 Σ𝑃2 = 𝑡𝑟2 Σ𝑃 , ⟹ 𝑣𝑎𝑟 ℎ𝑆𝑁𝑃

2 =2

𝜈𝐾

For multi-dimensional trait,

𝑡𝑟 Σ𝑃2

𝑡𝑟2 Σ𝑃=

𝑖=1𝑀 𝜆𝑖

2

𝑖=1𝑀 𝜆𝑖

2 ≤ 1 ⟹ 𝑣𝑎𝑟 ℎ𝑆𝑁𝑃2 ≤

2

𝜈𝐾