Rodriguez_NRMC_Presentation

Ameliorating Statistical Methodologies as Genomic Data Burgeon:Refined Proportional Odds Model with Application to New Dravet Dataset

Ivan Rodriguez∗,⊥, †Joseph C. Watkins, Ph.D.∗,⊥,§

∗The University of Arizona⊥Department of Mathematics†UROC-PREP/STAR Program

§Graduate Interdisciplinary Program in Statistics, Chair

October 1, 2016

Focus

Ivan Rodriguez Ameliorating Statistical Methodologies October 1, 2016

Focus


Research Overview

Challenge: making sense of this abundant data.Motivation: ≈150,000 newborns diagnosed with genetic diseaseannually (Nussbaum, McInnes, & Willard, 2007).Objectives:

Match data and diagnosis by improving existing technique.Apply model to new and exclusive dataset.


Research Overview




Research Overview




Research Overview




Research Overview




Research Overview




Methods: Ordinal Categorical Data Analysis

Analysis of data with non-arbitrary categorical ordering.Relevant example: disease severity scale.Complications:

Assigning numeric values to categories.Nonequidistance between categories.

Naïve solution: dichotomize ordinal outcome.
































Methods: Proportional Odds Model, General

Better method: the proportional odds model (McCullagh, 1980).Extends binary logistic regression (Cox, 1958).Celebrated method for ordinal data analysis (Bender & Grouven,1998).Applications: surveys, quality assurance, radiology, clinical research(McCullagh, 1999).

logit[P(Yi ≤ j | Xi)

]= θj − βTXi , j ∈ (1, . . . , J − 1),

logit(π) = log(

π

1− π

).





]= θj − βTXi , j ∈ (1, . . . , J − 1),

logit(π) = log(

π

1− π

).





]= θj − βTXi , j ∈ (1, . . . , J − 1),

logit(π) = log(

π

1− π

).





]= θj − βTXi , j ∈ (1, . . . , J − 1),

logit(π) = log(

π

1− π

).





]= θj − βTXi , j ∈ (1, . . . , J − 1),

logit(π) = log(

π

1− π

).





]= θj − βTXi , j ∈ (1, . . . , J − 1),

logit(π) = log(

π

1− π

).


Methods: Proportional Odds Model, Limitations

Great on paper, but not in practice.Proportional odds assumption often violated (Long & Freese, 2006).

A standard workaround: modify the model.Refine the latent variable.Fine-tune the null hypothesis.






















Methods: Latent Variable

Variables that are inferred, not directly observed.The focus is to make better inferences.

Y ∗ = βT + ε,

P(Y ≤ j | X ) = 1exp(βTX − θj) + 1.




Y ∗ = βT + ε,

P(Y ≤ j | X ) = 1exp(βTX − θj) + 1.




Y ∗ = βT + ε,

P(Y ≤ j | X ) = 1exp(βTX − θj) + 1.




Y ∗ = βT + ε,

P(Y ≤ j | X ) = 1exp(βTX − θj) + 1.


Methods: Hypothesis Testing

Null versus alternative hypotheses: H0 against HA.Traditionally, H0 is the status quo.

H0 : β1 = · · · = βq = 0.

β = τξ, τ ∈ F,Si = ξTxi .

H0 : τ = 0,

P(Y ≤ j | X

)= 1

exp(Sτ − θj) + 1.




H0 : β1 = · · · = βq = 0.


H0 : τ = 0,

P(Y ≤ j | X

)= 1





H0 : β1 = · · · = βq = 0.


H0 : τ = 0,

P(Y ≤ j | X

)= 1





H0 : β1 = · · · = βq = 0.


H0 : τ = 0,

P(Y ≤ j | X

)= 1





H0 : β1 = · · · = βq = 0.


H0 : τ = 0,

P(Y ≤ j | X

)= 1





H0 : β1 = · · · = βq = 0.


H0 : τ = 0,

P(Y ≤ j | X

)= 1



Methods: Score Function

Allows for quantification of performance of model.

u(θ1, . . . , θJ−1, τ) = −J∑

j=1

nj∑i=1

Sij[1− ψ

(θj − τSij

)− ψ

(θj−1 − τSij

)],

ψ(t) = 11 + exp(−t) .




u(θ1, . . . , θJ−1, τ) = −J∑

j=1

nj∑i=1

Sij[1− ψ

(θj − τSij

)− ψ

(θj−1 − τSij

)],

ψ(t) = 11 + exp(−t) .




u(θ1, . . . , θJ−1, τ) = −J∑

j=1

nj∑i=1

Sij[1− ψ

(θj − τSij

)− ψ

(θj−1 − τSij

)],

ψ(t) = 11 + exp(−t) .


Methods: Simulations

Criteria: type I error frequency and power.Algorithm:1. Generate genotype data.2. Obtain error terms.3. Fix latent variables.4. Produce ordinal categorical responses.5. Estimate θj under modified H0.6. Plug θ̂j into score function.7. Receive p-values.





























Methods: Application

Response: Dravet syndrome patient severity.Predictor: 12 stress-related single nucleotide polymorphisms.Sample size: 22 relatively isolated individuals.Categories: 2, mild and severe.Other data: sex, status, IQ, allele count.

















Results: The Proposed Model Is Successful

Type I error and power comparable to:Sequence kernel association test (Wu et al., 2011).Optimized sequence kernel association test (Lee et al., 2012).

In terms of power, outperforms:Variable threshold test (Price et al., 2010).Cohort allelic sums test (Morgenthaler & Thilly, 2007).Cumulative minor-allele test (Zawistowski et al., 2010).






























Results: Stress and Dravet Are Intricately Correlated

Rare phenotypes prevalent for young severe patients.Several genes protect or exacerbate Dravet.

Likely varies on case-by-case basis.

Stress-Dravet link contingent on sample heterogeneity.






















Discussion

Preliminary evaluation of model and dataset analysis.Severe modifying genes significantly determine quality-of-life.Identification of modifying genes is paramount.

Provides impetus for new medication and treatment.

Personalized care will rise with genomic information.


Discussion





Discussion





Discussion





Discussion





Discussion





In Conclusion

Focus:Improving the proportional odds model.Unknown link between stress and Dravet.

Takeaways:The proposed model is formidable.A new stress-Dravet link has been established.


In Conclusion




In Conclusion




In Conclusion




In Conclusion




In Conclusion




In Conclusion




Acknowledgments

Joseph C. Watkins, Ph.D.Miao Zhang, M.S.Michael Hammer, Ph.D., and the Hammer Lab.Andrew Huerta, Ph.D. and Reneé Reynolds, M.A.Andrew Carnie, Ph.D.

This research was supported in part by the Western Alliance to ExpandStudent Opportunities (WAESO) Louis Stokes Alliance for MinorityParticipation (LSAMP) National Science Foundation (NSF) CooperativeAgreement No. HRD-1101728.


Acknowledgments

Joseph C. Watkins, Ph.D.

Miao Zhang, M.S.Michael Hammer, Ph.D., and the Hammer Lab.Andrew Huerta, Ph.D. and Reneé Reynolds, M.A.Andrew Carnie, Ph.D.



Acknowledgments

Joseph C. Watkins, Ph.D.Miao Zhang, M.S.

Michael Hammer, Ph.D., and the Hammer Lab.Andrew Huerta, Ph.D. and Reneé Reynolds, M.A.Andrew Carnie, Ph.D.



Acknowledgments

Joseph C. Watkins, Ph.D.Miao Zhang, M.S.Michael Hammer, Ph.D., and the Hammer Lab.

Andrew Huerta, Ph.D. and Reneé Reynolds, M.A.Andrew Carnie, Ph.D.



Acknowledgments

Joseph C. Watkins, Ph.D.Miao Zhang, M.S.Michael Hammer, Ph.D., and the Hammer Lab.Andrew Huerta, Ph.D. and Reneé Reynolds, M.A.

Andrew Carnie, Ph.D.



Acknowledgments




Acknowledgments




References

Bender, R., & Grouven, U. (1998). Using binary logistic regression models for ordinal data with non-proportionalodds. Journal of Clinical Epidemiology, 51(10), 809–816. doi:10.1016/S0895-4356(98)00066-3

Cox, D. R. (1958). The regression analysis of binary sequences (with discussion). Journal of the Royal StatisticalSociety, Series B, 20, 215–242.

Lee, S., Emond, M. J., Bamshad, M. J., Barnes, K. C., Rieder, M. J., Nickerson, D. A., NHLBI GO Exome SequencingProject, . . . , Lin, X. (2012). Optimal unified approach for rare-variant association testing with application tosmall-sample case-control whole-exome sequencing studies. The American Journal of Human Genetics, 91(2),224–237. doi:10.1016/j.ajhg.2012.06.007

Long, J. S., & Freese, J. (2006). Regression models for categorical dependent variables using Stata. College Station,TX: Stata Press.

McCullagh, P. (1980). Regression models for ordinal data. Journal of the Royal Statistical Society, 42(2), 109–142.McCullagh, P. (1999). The proportional odds model. In P. Armitage, Encyclopedia of Biostatistics Vol. 5 (3560–

3563). Hoboken, NJ: John Wiley & Sons.Morgenthaler, S., & Thilly, W. G. (2007). A strategy to discover genes that carry multi-allelic or mono-allelic risk

for common diseases: A cohort allelic sums test (CAST). Mutation Research, 615(1–2), 28–56. doi:10.1016/j.mrfmmm.2006.09.003

Nussbaum, R. L., McInnes, R. R., & Willard, H. F. (2007). Thompson & Thompson genetics in medicine (6th ed.).Philadelphia, PA: W. B. Saunders. doi:10.1016/S0015-0282(02)03084-4

Price, A. L., Kryukov, G. V., de Bakker, P. I., Purcell, S. M., Staples, J., Wei, L. J., & Sunyaev, S. R. (2010).Pooled association tests for rare variants in exon-resequencing studies. The American Journal of Human Genetics,86(6), 832–838. doi:10.1016/j.ajhg.2010.04.005

Wu, M. C., Lee, S., Cai, T., Li, Y., Boehnke, M., & Lin, X. (2011). Rare-variant association testing for sequencingdata with the sequence kernel association test. The American Journal of Human Genetics, 89(1), 82–93. doi:10.1016/j.ajhg.2011.05.029

Zawistowski, M., Gopalakrishnan, S., Ding, J., Li, Y., Grimm, S., & Zöllner, S. (2010). Extending rare-varianttesting strategies: Analysis of noncoding sequence and imputed genotypes. The American Journal of HumanGenetics, 87(5), 604–617. doi:10.1016/j.ajhg.2010.10.012


Questions?

Ivan Rodriguez: 〈[email protected]〉 .


Rodriguez_NRMC_Presentation

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