A Constrained Instrumental Variable Approach, and its Application to Mendelian Randomization with Pleiotropy

Lai J iang (Lady Davis Ins titute & McGill), Karim Oualkacha (UQAM),

Celia Greenwood (Lady Davis Ins titute & McGill)

Montreal Caus al Inference Works hop, J uly 2016

Outline

1. The problem

2. Solutions and es timators

3. Simulations – s imple example

4. Simulations - an example borrowed from reality

5. Finally …

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The Problem

Instrumental Variables

We are interes ted in the causal effect of X (phenotype) on Y (dis ease)

U: unobserved variable that may confound the as socia tion between X and Y

G: a variable (genotype) we want to use as an ins trumental variable

DAG: U

G X Y

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Core Conditions for Instrumental Variable analysis

U

G X Y 1) G independent of U 2) G not marginally independent of X 3) G and Y are independent conditional on X and U

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Potential violations of assumptions in MR

Linkage Disequilibrium (Condition 3 or Condition 1)

Genetic Heterogeneity (Weak IV)

Population Stra tifica tion (Condition 3)

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G2 U

G1 X Y

G1

G2

G3

X

U

Y

G1

P

X Y

U

Pleiotropy A SNP (G) is associated with another intermediate phenotype (Z, other than X) which also has an effect on the disease Y.

Z

G X Y

U

G X Y

Z U (a) (b)

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• Condition (3) – Y and G independent given X and U – is violated in (a). • Condition (1) – G and U independent – is violated in (b).

(Original) Problem

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Estimators

Causal effect estimation: IV assumptions satisfied

One SNP

G X Y

U

δ β

ρ η

In the presence of pleiotropy

Z

G X

U

δ

ω

β

ξ η

ρ

Y

In the presence of pleiotropy

Y

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Z

G X

U

Potential solutions when there is pleiotropy

1. Naïve Method (TSLS us ing a ll SNPs ) {Biased} [Baum 2003] 2. Adjus t for G for Z, us e Gres as IV {Unbiased} 3. Limited Information Maximum Likelihood. (LIML) {Biased} [Burges s 2013] 4. Stepwise Selection Methods -- s elect subset of G {Incons is tent} [Bai 2008] 5. Inverse Probability Weights [Cole 2008]. 6. Cons tra ined Ins trumental Variables

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1. Naïve method in practice (#1, #2)

1. Select SNPs s trongly as socia ted with phenotype of interes t X

2. Remove SNPs s trongly as socia ted with pleiotropic phenotypes Z

3. Remove weak SNPs to avoid overfitting & bias

4. If the resulting ins trumental variable G s till has a s trong pleiotropic effect,

cons ider regres s ing G|Z and use Gres ins tead of G (ie. Method #2)

5. TSLS model fitting us ing G (or Gres) -> X -> Y

3. LIML

Limited Information Maximum Likelihood uses ins truments to rectify bias in a regres s ion Y ~ X, a ris ing when X is correla ted with res iduals

• i.e. Pleiotropy leads to correla tion between X and Z

LIML takes into account the covariances of the errors

More s table to weak ins trumental variable bias

• Z is not used

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4. Stepwise Selection Methods.

Step by s tep methods (such as (Bai, 2008)):

G: Construct a new IV based on a subset S

Select new features into S With respect to X | G

Delete features in S with respect to Z | G

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Implementation details a re crucial here

1. Specific criteria (s ignificance thres holds ) for s election and deletion?

2. How to s ummarize the information in the final s et S? PCA? PLS? Etc…

3. Stop s trategy?

4. Starting S (previous publis hed s ources )?

5. Backward? Forward? Bidirectional?

5. Inverse Probability Weight Adjusted Regression (IPWAR)

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Constrained Instrumental Variable Method

Constrained Instrumental Variable

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Y

Z

G X

U

Constrained Instrumental Variable

-

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Constrained Instrumental Variable

• When 𝑝 < 𝑛 there is an exact solution

• In fact there are 𝑝 − 𝑘 orthogonal solutions

• 𝐺𝑟𝑟𝑟 = 𝐼𝑛 − 𝑃𝑍 𝐺, the projection of 𝐺 to subspace orthogonal to columns of 𝑍

• We search for a direction in Span(𝐺), 𝐺∗, mos t informative for 𝑋, orthogonal to 𝑍

• 𝐺∗ belongs to Span(𝐺𝑟𝑟𝑟)

• 𝑆𝑝𝑆𝑛 𝐺𝑟𝑟𝑟 = 𝑆𝑝𝑆𝑛 𝐺∗ + 𝑆𝑝𝑆𝑛(𝑐𝑐𝑐𝑝𝑐𝑐𝑐𝑐𝑛𝑐 𝑐𝑜 𝐺∗𝑖𝑛 𝐺𝑟𝑟𝑟 𝑠𝑠𝑠𝑠𝑝𝑆𝑐𝑐)

• Complement is not informative for X

• Hence, CIV has identical performance to s tra ightforward 𝐺𝑟𝑟𝑟

Constrained Instrumental Variable

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What about when 𝑝 > 𝑛?

Penalized Constrained Instrumental Variable

L0 penalty

LASSO penalty

Non-convex Penalties

…...

Penalization

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Which penalty?

• LASSO (L1) penalty does not a lways yield a sparse solution automatically for this problem.

• Neither do non-convex penalties (e.g. SCAD)

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L0 penalty

• L0 penalty: 𝑠 0 = ∑ 𝑠𝑗0

𝑗 < 𝜆 directly enforces spars ity

• However, greedy s earch for optimal 𝑠, 𝜆 is 𝑛𝑝-hard in computational complexity

• Alternatively, cons ider smoothed approximate L0 penalties as 𝜎 → 0:

𝑜𝜎 𝑥 = 𝑐𝑥𝑝 −𝑥2

2𝜎2 → �1, 𝑥 = 00, 𝑥 ≠ 0

So 𝑠 0 ≈ 𝑝 − ∑ 𝑜𝜎(𝑠𝑗)𝑗

Implementation

Two methods :

1. If 𝐩 < 𝐧, linear a lgebra solution

2. If 𝐩 > 𝐧, we propose a numerical solution for penalized CIV Start from an initial gues s 𝑠, and an initial penalty 𝜎 = max |𝑠| Iteratively decreas e 𝜎 → 𝜎𝑚𝑚𝑛 a long a chos en s equence Cros s validation to choos e 𝜆 for each 𝜎 bas ed on MSE of (𝑌|𝐺,𝑠)

3. Boots trap for s tandard errors

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Simulations

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• Naive TSLS

• Adjusting each G for Z, then TSLS

• LIML

• Stepwise Selection Methods (forward, backward)

• Inverse Probability Weight Adjusted Regression (IPWAR)

• CCA/sparseCCA

• CIV (Constrained Instrumental Variable)

• Smoothed CIV

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Methods compared

Bias as a function of 𝜉 • N=200 • 20 SNPs • 5 SNPs as s ociated with X. • Among thes e 5: • 2 SNPs als o as s ociated with Z • 𝛽: Effect 𝑋 → 𝑌

Conditional TSLS/ G res iduals CCA, CCA_LASSO LIML Backward/ Forward IPWAR CIV/Smoothed CIV

Bias and standard error 𝝃=1 𝝃=5

Estimate Mean square error

Estimate Mean square error

G residual 1.09 0.01 1.08 0.01

IPWAR 2.00 1.21 15.30 2.20

CIV 1.09 0.01 1.09 0.02

CIV smooth 1.01 0.01 1.05 0.01

SNPs selected over simulations

Method SNP1 SNP2 SNP3 SNP4 SNP5 Others

CIV 0.76 0.76 1.00 1.0 1.0 ~0.2

Smoothed CIV

0.00 0.00 0.29 0.62 0.29 0.0

Note: Applied hard threshold: 0.1*max(estimate)

SNPs selected CIV Smoothed CIV

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Contrived Example

LDL-HDL-CHD

HDL/LDL Simulation

Bias Bootstrap

variance Lower_CI Upper_CI

Naive -0.090 0.025 -0.145 -0.034

G residuals 0.004 0.027 -0.050 0.059

LIML -0.112 0.027 -0.169 -0.066

IPWAR 0.066 0.025 0.011 0.121

Backward -0.008 0.042 -0.094 0.079

Forward 0.006 0.055 -0.097 0.109

CCA_LASSO -0.090 0.030 -0.147 -0.033

CCA -0.090 0.028 -0.147 -0.032

CIV 0.002 0.031 -0.052 0.057

TRUE -0.0002 0.031 -0.067 0.066

HDL/LDL Simulation

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Finally…

Comments

• The smoothing CIV selects SNPs as socia ted with X and not Z in a non-greedy s earch for the optimal subset

• G_res , CCA etc s elect SNPs differently and will keep overlapping SNPs

• For 𝑝 > 𝑛 or 𝑝 large • CIV will work • G_res iduals will not work or will become uns table

• 𝑝 > 𝑛 is a very unlikely Mendelian randomization s cenario

Questions

• Other applications of the smoothed L0 penalty? • We create one ins trument

• What about creating s everal ins truments through the s ame kinds of projections (p-k)?

• Links to • Targeted learning • Multiple invalid ins truments (averages of weak ins truments )

(New) problem

• Computationally efficient variable s election subject to linear cons tra ints

Thank you! Acknowledgements: CIHR Ludmer Centre for Neuroinformatics and Mental Health Brent Richards & his group

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