Quantile Regression

Short Informal Mathematics (SIM) Talk in the Maths & Stats Department at the University of Limerick. Work completed as part of my BSc in Mathematical Sciences.

Transcript

1. Quantile Regression Reference range of Thyroid function test in pregnancy Kevin Brosnan University of Limerick 13th October, 2015
2. Table of Contents Table of Contents Intro to QR Thyroid Disease Existing Methods Proposed Solution Conclusions Kevin Brosnan (UL) Quantile Regression 13th October, 2015 2 / 16
3. Intro to QR Table of Contents Intro to QR Thyroid Disease Existing Methods Proposed Solution Conclusions Kevin Brosnan (UL) Quantile Regression 13th October, 2015 3 / 16
4. Intro to QR Interested in understanding the entire distribution of the data 5.5 6.0 6.5 7.0 7.5 6.0 6.5 7.0 7.5 8.0 8.5 Household Income Food Expenditure Kevin Brosnan (UL) Quantile Regression 13th October, 2015 4 / 16
5. Intro to QR What does this actually tell us? 5.5 6.0 6.5 7.0 7.5 6.0 6.5 7.0 7.5 8.0 8.5 Household Income Food Expenditure Kevin Brosnan (UL) Quantile Regression 13th October, 2015 4 / 16
6. Intro to QR Quantile Regression helps to complete the picture 5.5 6.0 6.5 7.0 7.5 6.0 6.5 7.0 7.5 8.0 8.5 Household Income Food Expenditure Kevin Brosnan (UL) Quantile Regression 13th October, 2015 4 / 16
7. Thyroid Disease Table of Contents Intro to QR Thyroid Disease Existing Methods Proposed Solution Conclusions Kevin Brosnan (UL) Quantile Regression 13th October, 2015 5 / 16
8. Thyroid Disease Thyroid Disease Affects 2% of Pregnancies Non-pregnant ranges are not valid Most Common Endocrine Condition Kevin Brosnan (UL) Quantile Regression 13th October, 2015 6 / 16
9. Thyroid Disease Thyroid Disease Thyroid Data 311 Observations Individuals from each Trimester 4 Thyroid Hormone Variables Measured Affects 2% of Pregnancies Non-pregnant ranges are not valid Most Common Endocrine Condition Kevin Brosnan (UL) Quantile Regression 13th October, 2015 6 / 16
10. Existing Methods Table of Contents Intro to QR Thyroid Disease Existing Methods Proposed Solution Conclusions Kevin Brosnan (UL) Quantile Regression 13th October, 2015 7 / 16
11. Existing Methods Theory For each given quantile of interest, the quantile coefficients are estimated by the objective function ˆ β = argmin β∈ p ρτ (yi − x β) where ρτ is a check function Estimating the quantiles independently of one another results in Crossing Quantiles - shown in the next slide Kevin Brosnan (UL) Quantile Regression 13th October, 2015 8 / 16
12. Existing Methods Application Crossing Quantiles - this shouldn’t be happening! 10 15 20 25 30 35 40 8 10 12 14 16 18 Gestation T4 Quantile Regression: T4 ~ Gestation + BrokenStick(25) Kevin Brosnan (UL) Quantile Regression 13th October, 2015 9 / 16
13. Existing Methods Application Crossing Quantiles - this shouldn’t be happening! 10 15 20 25 30 35 40 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Gestation T3 Quantile Regression: T3 ~ Gestation + Gestation2 Kevin Brosnan (UL) Quantile Regression 13th October, 2015 9 / 16
14. Existing Methods Application Crossing Quantiles - this shouldn’t be happening! 10 15 20 25 30 35 40 −1 0 1 2 3 4 5 Gestation TSH Quantile Regression: TSH0.5 ~ Gestation Kevin Brosnan (UL) Quantile Regression 13th October, 2015 9 / 16
15. Proposed Solution Table of Contents Intro to QR Thyroid Disease Existing Methods Proposed Solution Conclusions Kevin Brosnan (UL) Quantile Regression 13th October, 2015 10 / 16
16. Proposed Solution Theory The objective function defined in my proposed solution is exactly the same as was previous but with additional constraints as follows: ˆ β = argmin β∈ p ρτ (yi − x β) subject to      −ˆ β(τi ) ≥ −ˆ β(τi+1) + , if 0 < τ < 0.5, no constraints, if τ = 0.5, ˆ β(τi ) ≥ ˆ β(τi−1) + , if 0.5 < τ < 1 This removes the issue of Crossing Quantiles Kevin Brosnan (UL) Quantile Regression 13th October, 2015 11 / 16
17. Proposed Solution Theory Kevin Brosnan (UL) Quantile Regression 13th October, 2015 12 / 16
18. Proposed Solution Application Existing R Implementation 10 15 20 25 30 35 40 8 10 12 14 16 18 Gestation T4 Quantile Regression: T4 ~ Gestation + BrokenStick(25) Kevin Brosnan (UL) Quantile Regression 13th October, 2015 13 / 16
19. Proposed Solution Application Non-Crossing Quantiles - Proposed Solution 10 15 20 25 30 35 40 8 10 12 14 16 18 Gestation T4 Quantile Regression: T4 ~ Gestation + BrokenStick(25) Kevin Brosnan (UL) Quantile Regression 13th October, 2015 13 / 16
20. Proposed Solution Application Existing R Implementation 10 15 20 25 30 35 40 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Gestation T3 Quantile Regression: T3 ~ Gestation + Gestation2 Kevin Brosnan (UL) Quantile Regression 13th October, 2015 13 / 16
21. Proposed Solution Application Non-Crossing Quantiles - Proposed Solution 10 15 20 25 30 35 40 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Gestation T3 Quantile Regression: T3 ~ Gestation + Gestation2 Kevin Brosnan (UL) Quantile Regression 13th October, 2015 13 / 16
22. Proposed Solution Application Existing R Implementation 10 15 20 25 30 35 40 −1 0 1 2 3 4 5 Gestation TSH Quantile Regression: TSH0.5 ~ Gestation Kevin Brosnan (UL) Quantile Regression 13th October, 2015 13 / 16
23. Proposed Solution Application Non-Crossing Quantiles - Proposed Solution 10 15 20 25 30 35 40 −1 0 1 2 3 4 5 Gestation TSH Quantile Regression: TSH0.5 ~ Gestation Kevin Brosnan (UL) Quantile Regression 13th October, 2015 13 / 16
24. Conclusions Table of Contents Intro to QR Thyroid Disease Existing Methods Proposed Solution Conclusions Kevin Brosnan (UL) Quantile Regression 13th October, 2015 14 / 16
25. Conclusions Advantages Removes issue of crossing quantiles The quantile profiles are consistent throughout the data Computational effort is similar to that of existing methods Easy to implement Kevin Brosnan (UL) Quantile Regression 13th October, 2015 15 / 16
26. Conclusions Advantages Removes issue of crossing quantiles The quantile profiles are consistent throughout the data Computational effort is similar to that of existing methods Easy to implement Disadvantages Have not tested on simulated data Have not tested on large data sets Kevin Brosnan (UL) Quantile Regression 13th October, 2015 15 / 16
27. Q&A Thanks for listening! Kevin Brosnan (UL) Quantile Regression 13th October, 2015 16 / 16