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Reid D. Landes Trey Spencer Ian A. Zelaya

A Bayesian random coefficient nonlinear regression for a split-plot experiment for detecting differences in the half-life of a compound. Reid D. Landes Trey Spencer Ian A. Zelaya. Outline. Background and objective Description of experiment Classical analysis plan Bayesian analysis plan

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Reid D. Landes Trey Spencer Ian A. Zelaya

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  1. A Bayesian random coefficient nonlinear regression for a split-plot experiment for detecting differences in the half-life of a compound Reid D. Landes Trey Spencer Ian A. Zelaya

  2. Outline • Background and objective • Description of experiment • Classical analysis plan • Bayesian analysis plan • Hierarchical model (HM) • Results • Discussion

  3. Background • Shikimic acid (SA) is often studied in plant sciences to monitor effects of glyphosate (an active ingredient in herbicides) • SA starts to break down after it is separated from plant tissue treated with glyphosate • The SA chromophore has a half-life (G) • SA is often quantified with spectrophotometric methods • Optical density ( y ) is recorded

  4. Background • G is thought to (at least be) dependent upon temperature (Tmp) • Typically 2 Tmps used: 4 & 23 degrees C • Maybe G is dependent upon the solution used in preparing the samples (Prep) • Typically 2 types: A and B

  5. Background & Objective • Lab tech can prepare and process… • many samples if G is long • not as many samples if G is short • What Prep*Tmp combination prolongs G?

  6. Experiment Design • Treatment structure: 3 factors @ 2 levels • Prep: A & B • Tmp: 4 & 23 degrees Celsius • Amt: 30 & 60 units of SA • Laid out in a split plot design

  7. Experiment DesignRun i, i =1,2 Whole Plot E.U. (Prep*Tmp) Split Plot E.U. (Prep*Tmp*Amt) Sampling Unit (tube)

  8. Comment • If we could directly observe half-life from a tube (say g), then use ANOVA

  9. Observable data y

  10. Model for tube data • yhijkl – optical density from Prep*Tmp h in Runi with Amt j from Tube kat time l • xl– time (0, 1,…, 14, 28, 50) hours • ahijk – lower asymptote ( y at infinite time ) • bhijk – range ( y0 – a ) • ghijk – half life • ehijkl– additive mean-zero error

  11. Classical Analysis Plan Since time values were consistent for each tube, and there was no missing data… • Estimate g for each tube via nonlinear regression • Fit estimated g’s with linear mixed model having ANOVA described above • Make inference on Prep*Tmp treatment effects, Ghwith h=1,2,3,4

  12. Bayesian Analysis Plan • Model ydata with a random coefficients nonlinear regression (a HM) • Assign priors to variance parameters and whole plot treatment effects • Use MCMC simulation in Bayes HM to obtain empirical posterior distributions • Make inference on Gh from [Gh| y] with h=1,2,3,4

  13. Sampling Unit g ~ Dsn(G, tg) g

  14. Split Plot E.U. G G ~ Dsn(g, tG)

  15. Whole Plot E.U. g g ~ Dsn(G, tg)

  16. Bayes Hierarchical Model Overview

  17. Bayes Hierarchical Model yhijkl xl te ahijk bhijk ghijk ta tb tg Ghij Ahij Bhij

  18. Bayes Hierarchical Model Ahij Ghij Bhij Amtj tB tG ghi tA ahi bhi Ah Bh Gh ta tb tg

  19. BHM Levels 1, 2, & 3

  20. BHM Levels 4 & 5

  21. MCMC in BHM • MCMC simulation via WinBUGS 1.4 • Ran 3 chains with initial values chosen from the data • Burn-in = 6000 • Thinning rate = every 50th • 6000 iterates to approximate posteriors • Assessed posterior convergence with Brooks-Gelman-Rubin plots

  22. 5 4 Bayesian 3 Classical 2 1 0 -1 -2 -3 -4 Prep Main Effect Tmp Main Effect Prep*Tmp Interaction 95% CI’s forPrep & Tmp Effects Hours

  23. 9 8 Bayesian 7 Classical 6 5 4 3 2 1 0 A4 A23 B4 B23 95% CI’s for G HALF L IFE

  24. 10 8 Bayesian 6 Classical 4 2 0 -2 -4 -6 A4- A4- A4- A23- A23- B4- A23 B4 B23 B4 B23 B23 All Pairwise Comparisons Hours

  25. Posterior Probabilities of Ranks Most Probable Ranking Sequence: P({B23, A23, B4, A4} | y) = 0.917

  26. Addressing the Objective • Use Prep A at 4 degrees (A4) • Given the data, the probability that A4 is the best is 0.950 • From Posterior Probabilities of Rank, posterior probabilities for TMP=4 or PREP=A are easily calculated • Pr(TMP=4 > TMP=23 | y ) = [(.047+.950)+(.932+.049)]/2 = 0.989

  27. Discussion • Classical approach makes sense thanks to • Consistently spaced time points for each tube • No missing y data • Classical approach is easy to implement • Bayesian approach is flexible • Inconsistent time spacing and missing data are no problem

  28. Bayesian vs. Classical • Bayesian approach • has one model for all the data • produces tighter interval estimates • Allows more meaningful probability statements that address the objective • Classical ANOVA approach is widely accepted, esp. by non-statistician reviewers.

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