applied dose-response models in weed science presented by william j. price, ph.d

Applied Dose-Response Models in Weed Science

Presented by

William J. Price, Ph.D.

Statistical ProgramsCollege of Agricultural and Life Sciences

University of Idaho

Acknowledgments

• Research partially funded by USDA-ARS Hatch Project IDA01412, Idaho Agricultural Experiment Station.

• Collaborators:

• Bahman Shafii, Ph. D., Director, Statistical Programs, University of Idaho.

• Steven Seefeldt, Ph. D., USDA -ARS, University of Alaska Fairbanks.

OutlineGeneral Considerations

• Available Software

Estimation: • SAS Procedures

• PROBIT, NLIN, and NLMIXED• Response Distributions

• Normal, Poisson, Binomial• Alternative Models

• Logistic vs Exponential• Extensions to Models and Techniques

• Parameterization• Treatment Structure and Comparison• Random Effect Mixed Models

• Bayesian Methods• MCMC

Outline

Calibration: • Parametric• Non-Parametric

Conclusions

References

General Considerations

• Statistical Estimation Software

• S+

• R

• Statistica, etc.

• Sigma Plot, AXUM, etc.

• SAS

Estimation: Procedures

Probit Analysis• Data description

• Vernalization study.

• Fixed number of wheat plants• 6 to 10 wheat plants per replication and temperature.• (SAS: plants).

• Five temperatures (doses):• 0, -10, -12, -14, and -16 degrees celcius• (SAS: temp = temperature + 17).

• Number of wheat plants alive after 2 weeks recorded • (SAS: alive2wk).

Temperature

0

.25

.50

.75

1.00

-16 0

Prop

ortio

n A

live

Vernalization Data

proc probit data=freeze log optc lackfit inversecl;model alive2wk/plants = temp/distribution=logistic ;predpplot var=temp;

• Code:

• SAS Procedure: PROC PROBIT.


proc probit data=freeze log optc lackfit inversecl;model alive2wk/plants = temp/distribution=logistic;predpplot var=temp;


Probit Analysis

Probit Procedure

Model Information

Data Set WORK.FREEZEEvents Variable alive2wkTrials Variable plantsNumber of Observations 20Number of Events 122Number of Trials 195Name of Distribution LogisticLog Likelihood -83.4877251

Number of Observations Read 20Number of Observations Used 20Number of Events 122Number of Trials 195Missing Values 0

Algorithm converged.

Goodness-of-Fit Tests

Statistic Value DF Pr > ChiSq

Pearson Chi-Square 18.7054 17 0.3457L.R. Chi-Square 22.6138 17 0.1623

Response-Covariate Profile

Response Levels 2Number of Covariate Values 20

Type III Analysis of Effects

WaldEffect DF Chi-Square Pr > ChiSq

Ln(temp) 1 15.2620 <.0001

PROC PROBIT Output

PROC PROBIT Output (cont)

Analysis of Parameter Estimates

Standard 95% Confidence Chi-Parameter DF Estimate Error Limits Square Pr > ChiSq

Intercept 1 -6.9144 2.0126 -10.8590 -2.9699 11.80 0.0006Ln(temp) 1 4.5094 1.1543 2.2470 6.7717 15.26 <.0001_C_ 1 0.2258 0.0623 0.1037 0.3480

Probability temp 95% Fiducial Limits

0.01 1.67252 0.48881 2.54641 0.02 1.95482 0.66670 2.83256 . . . . . . . . . . . . 0.40 4.23519 3.00063 4.96946 0.45 4.43197 3.25697 5.16920 0.50 4.63365 3.52170 5.38443 0.55 4.84451 3.79739 5.62420 0.60 5.06959 4.08622 5.90119 . . . . . . . . . . . . 0.99 12.83734 9.46539 30.51920

PROC PROBIT Output (cont)

Temperature (C)

-16 0

0

.25

.50

.75

1.00

OPTC = 0.23

Prop

ortio

n A

live

• Probit Limitations• Proportional Data.• Maximum set to 1.0.• Limited number of response models.

• Probit Advantages• Automatic Goodness of Fit test.• Easily computed percentiles.• Ability to do treatment comparisons.• Graphic output.

PROC PROBIT (cont)

• Code:

proc nlin data=freeze ; parms I = 8 B = -4.5 C = .12; bounds B<0;

mu = C + (1-C)/(1 + exp(B*(ltemp-log(I))));model per2 = mu;output out=pred p=pred;

Nonlinear Least Squares

• SAS Procedure: PROC NLIN





The NLIN ProcedureDependent Variable per2Method: Gauss-Newton

Iterative Phase Sum of Iter I B C Squares

0 4.6000 -4.6000 0.2200 0.3767 1 4.6147 -4.6464 0.2268 0.3765 2 4.6119 -4.6366 0.2264 0.3765 3 4.6125 -4.6388 0.2265 0.3765 4 4.6123 -4.6383 0.2265 0.3765 5 4.6124 -4.6384 0.2265 0.3765

NOTE: Convergence criterion met.

Estimation Summary

Method Gauss-NewtonIterations 5R 3.628E-6PPC(B) 4.559E-6RPC(B) 0.000021Object 2.25E-10Objective 0.37647Observations Read 20Observations Used 20Observations Missing 0

PROC NLIN Output

Sum of Mean ApproxSource DF Squares Square F Value Pr > F

Model 3 9.6353 3.2118 145.03 <.0001Error 17 0.3765 0.0221Uncorrected Total 20 10.0117

ApproxParameter Estimate Std Error Approximate 95% Confidence Limits

I 4.6124 0.4511 3.6607 5.5640B -4.6384 1.7176 -8.2622 -1.0147C 0.2265 0.0723 0.0739 0.3791

Approximate Correlation Matrix I B C

I 1.0000000 -0.4991207 0.6394756B -0.4991207 1.0000000 -0.5126022C 0.6394756 -0.5126022 1.0000000

PROC NLIN Output (cont)

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0.9

1.0

Ln (temp)0 1 2 3

Prop

ortio

n A

live

PROC NLIN Output (cont)

• NLIN Limitations• Assumes normally distributed response.• Approximate tests.• Treatment comparisons not automatic.

• NLIN Advantages• Not restricted to proportional data.• Maximum may be any value.• Response models not limited.

PROC NLIN (cont)

proc nlmixed data=freeze corr; parms I = 4.5 B = -4.8 C = .228; bounds B<0, I>0;

mu = C + (1-C)/(1 + exp(B*(ltemp-log(I))));

model alive2wk ~ binomial(plants, mu);predict plants*mu out=pred1;predict mu out=pred2;

• Code:



model alive2wk ~ binomial(plants, mu);predict plants*mu out=pred1;predict mu out=pred2;

proc nlmixed data=freeze corr;parms I = 4.5 B = -4.8 C = .228;bounds B<0, I>0;


model alive2wk ~ binomial(plants, mu);predict plants*mu out=pred1;

predict mu out=pred2;









Maximum Likelihood

• SAS Procedure: PROC NLMIXED

Dependent Variable alive2wkDistribution for Dependent Variable BinomialOptimization Technique Dual Quasi-Newton

Dimensions

Observations Used 20Observations Not Used 0Total Observations 20Parameters 3

Iteration History

Iter Calls NegLogLike Diff MaxGrad Slope

1 4 28.3612297 0.034097 1.623059 -27.1026 2 5 28.2115599 0.14967 0.572223 -3.97671 3 10 28.1847135 1.433E-8 0.000016 -2.67E-8

NOTE: GCONV convergence criterion satisfied.

Fit Statistics

-2 Log Likelihood 56.4 AIC (smaller is better) 62.4 AICC (smaller is better) 63.9 BIC (smaller is better) 65.4

PROC NLMIXED Output

PROC NLMIXED Output (cont)

Parameter Estimates

StandardParm Estimate Error DF t Value Pr > |t| Alpha Lower Upper Gradient

I 4.6337 0.4151 20 11.16 <.0001 0.05 3.7677 5.4996 4.629E-6B -4.5094 1.1543 20 -3.91 0.0009 0.05 -6.9172 -2.1016 -2.62E-6C 0.2258 0.06231 20 3.62 0.0017 0.05 0.09585 0.3558 0.000016

Correlation Matrix of Parameter Estimates

Row Parameter I B C

1 I 1.0000 -0.5276 0.5848 2 B -0.5276 1.0000 -0.3896 3 C 0.5848 -0.3896 1.0000

1

2

3

4

5

6

7

8

9

10

Ln(temp)0 1 2 3

Num

ber o

f sur

vivi

n g p

lant

sPROC NLMIXED Output (cont)

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Ln(temp)0 1 2 3

Procedure Comparisons

Probit

NLinNLMixedPr

opor

tion

Aliv

e

• All three procedures can produce similar results.• Binomial or proportional data.• Maximum response of 1.0.

• PROBIT limited in models and response types.

• NLIN and NLMIXED provide nonlinear solutions.

• NLMIXED most flexible for responses and models.

Procedure Comparisons

Estimation: Response Distributions

NLMIXED : Normal Data

• Binomial - yes/no data.• Normal - continuous data.• Poisson - discrete count data.• User defined - any data.

• NLMIXED probability distributions:

• Example: Seefeldt, et al. 1995

• Wild oat resistance• Treated with fenoxaprop/2,4-D/MCPA (SAS: dose).• Dry weights at 2 weeks (SAS: adj_wt).

0.00

0.02

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0.28

Dry

Wei

ght (

g)

Biotype C

0.010 0.100 1.000 10.000Dose (kg ai/ha)


proc nlmixed data=seefeldt;parms C=.04 D=.2 B=3 I=.1 sig=.021;bounds C>0, D>0, B>0, sig>0;

if dose = 0 then mu = D;else mu = C + (D-C)/(1 + exp(B*(ldose-log(I))));model adj_wt ~ normal(mu, sig**2);predict mu out=fitted;

• Assume dry weight to be normally distributed with mean mu and variance sig2.

• Must model sig explicitly.


Iteration History

Iter Calls NegLogLike Diff MaxGrad Slope

1 14 -63.769085 17.90341 925.7938 -4745664 2 19 -90.116746 26.34766 3490.083 -1912.68 3 66 -122.58572 4.511E-8 0.040557 -1E-7

NOTE: GCONV convergence criterion satisfied.

Fit Statistics

-2 Log Likelihood -245.2AIC (smaller is better) -235.2AICC (smaller is better) -233.9BIC (smaller is better) -225.1

Parameter Estimates

StandardParm Estimate Error DF t Value Pr > |t| Lower Upper C 0.04823 0.01893 51 2.55 0.0137 0.01030 0.08616 D 0.1836 0.00794 51 23.12 <.0001 0.1677 0.1996 B 1.3283 0.4510 51 2.95 0.0047 0.4246 2.2321 I 1.1669 0.3539 51 3.30 0.0017 0.4576 1.8762 sig 0.02937 0.00280 51 10.49 <.0001 0.02376 0.03498


Dry

Wei

ght (

g)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

0.26

0.28

0.010 0.100 1.000 10.000Dose (kg ai/ha)

Biotype C


NLMIXED : Poisson data

• Data description

• Simulated injury study.

• Harmony sprayed on pea plants.• measured the number of branches/plant.• (SAS: branches).

• Ten doses:• 0 to 0.125 lbs ai/A.• (SAS: trt).

Num

ber o

f Bra

nche

s

0

10

20

30

40

50

60

70

80

90

100

0.0001 0.0010 0.0100 0.1000 1.0000Harmony Dose (lb ai/A)

Variety C


proc nlmixed data=pea; parms D=10 C=70 B=.8254 I=.01; bounds D>0, B>0;

if trt = 0 then mu = D;else mu = C + (D-C)/(1 + exp(B*(ltrt-log(I))));

model branches ~ poisson(mu);

predict mu out=pred;

• Assume the number of branches to be distributed as a Poisson variable.

• In the Poisson distribution, mean = variance = mu.


Parameter Estimates

StandardParm Estimate Error DF t Value Pr > |t| Alpha Lower Upper

C 11.6834 1.0136 38 11.53 <.0001 0.05 9.6315 13.7352 I 0.01075 0.0013 38 8.41 <.0001 0.05 0.0081 0.0133 B 1.9860 0.3246 38 6.12 <.0001 0.05 1.3289 2.6432 D 69.1586 3.3894 38 20.40 <.0001 0.05 62.2972 76.0201


0

10

20

30

40

50

60

70

80

90

100

Num

ber o

f Bra

nche

s

0.0001 0.0010 0.0100 0.1000 1.0000

Variety C

Harmony Dose (lb ai/A)


Estimation: Alternative Models

• Normal: yij = (1/2) exp((x-)2/2

• Logistic: yij = 1 / (1 + exp( -dosei - ))

• Modified Logistic: yij = C + (D-C) / (1 + exp( -Bdosei - ))) (e.g. Seefeldt et al. 1995)

• Gompertz: yij = 0 (1 - exp(exp(-(dose))))

yij = 0 exp(-(dose))• Exponential:

yij = 0 [1 - exp(-(dose))]

Common Dose-response Models

• Logistic: yij = 1 / (1 + exp( -dosei - ))

• Modified Logistic: yij = C + (D-C) / (1 + exp( -Bdosei - I))) (e.g. Seefeldt et al. 1995)

NLMIXED: Alternative Models

yij = C + (D - C) / (1 + exp(B(dosei - I)))

• Log-logistic Model

yij = (a-c) exp(-bdose) + c

• Exponential Model

Example:

Exponential Model for Pea Biomass

• A linear pattern of data on a log scale.• Implies an exponential model, e.g.

Biomass = (a-c) exp(-bdose) + c

where a is an intercept term, c is a lower limit and b is a rate parameter.

• The 50th percentile for this model is given by: I50 = ln(((a/2) - c)/(a - c))/(-b)


• Example: Pea Data

• Fit log-logistic model to biomass measurements.

proc nlmixed data=pea corr maxiter=2000; parms D=.5966 I=0.01 B=.51 C=.04 sig=.09; bounds D>0, B>0;

if trt = 0 then mu = D;else mu = C + (D-C)/(1 + exp(B*(ltrt-log(I))));

model bio ~ normal(mu, sig**2);

Log-logistic Model for Pea Biomass

Parameter Estimates


D 1.2751 4.3499 38 0.29 0.7710 0.05 -7.5308 10.0810 I 0.02792 0.4771 38 0.06 0.9536 0.05 -0.9379 0.9937 B 0.1261 0.6405 38 0.20 0.8449 0.05 -1.1704 1.4227 C -0.7546 5.9311 38 -0.13 0.8994 0.05 -12.7614 11.2522 sig 0.09240 0.01060 38 8.72 <.0001 0.05 0.07094 0.1139


Row Parameter D I B C sig

1 D 1.0000 0.6108 -0.9903 -0.9568 -0.00074 2 I 0.6108 1.0000 -0.7133 -0.8146 -0.00306 3 B -0.9903 -0.7133 1.0000 0.9873 0.001188 4 C -0.9568 -0.8146 0.9873 1.0000 0.001668 5 sig -0.00074 -0.00306 0.001188 0.001668 1.0000

Bio

mas

s (g

/pla

nt)

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0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0001 0.0010 0.0100 0.1000

Log-logistic Model for Pea Biomass



yij = C + (D - C) / (1 + exp(B(dosei - I)))

• Log-logistic Model

yij = (a-c) exp(-bdose) + c

• Exponential Model

Example:


proc nlmixed data=pea corr; parms a=.5217 b=106.5 c = .2026 sig=.09;

mu =(a-c)*exp(-b*trt) + c;

model bio ~ normal(mu, sig**2);predict mu out=pred;

estimate ’I50' log(((a/2)-c)/(a-c))/(-b);

Parameter Estimates


a 0.5260 0.03375 38 15.59 <.0001 0.05 0.4577 0.5943 b 106.50 51.3184 38 2.08 0.0448 0.05 2.6113 210.39 c 0.2026 0.03622 38 5.59 <.0001 0.05 0.1293 0.2760 sig 0.09861 0.01131 38 8.72 <.0001 0.05 0.07571 0.1215


Row Parameter a b c sig

1 a 1.0000 0.6473 0.2864 0.000060 2 b 0.6473 1.0000 0.6696 0.000091 3 c 0.2864 0.6696 1.0000 0.000062 4 sig 0.000060 0.000091 0.000062 1.0000

Additional Estimates

StandardLabel Estimate Error DF t Value Pr > |t| Alpha Lower Upper

I50 0.01576 0.006829 38 2.31 0.0265 0.05 0.001937 0.02958



0.1

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0.0001 0.0010 0.0100 0.1000

Bio

mas

s (g

/pla

nt)


Estimation: Extensions

Reparameterization

Estimation: Reparameterization

• The log-logistic model can be generalized to estimate any percentile as (Schabenberger, 1999):

yij = C + k(D - C) / (k + exp(B( dosei – I(1-Q) )))

where I(1-Q) is the dose required to reachthe Qth percentile, and k is given by :

k = Q/(1 - Q)

• Example: • Seefeldt data, biotype C.• Estimate the 90th percentile, e.g. I10

k = Q/(1 - Q) = 0.9/(1.0 - 0.9) = 9.0

Q = 0.9


Parameter Estimates


C 0.04823 0.01893 51 2.55 0.0136 0.05 0.01030 0.08616 D 0.1836 0.007942 51 23.12 <.0001 0.05 0.1677 0.1996 B 1.3284 0.4510 51 2.95 0.0047 0.05 0.4246 2.2321 I 0.2214 0.1098 51 2.02 0.0487 0.05 0.00134 0.4415 sig 0.02605 0.002484 51 10.49 <.0001 0.05 0.02107 0.03103

proc nlmixed data=seefeldt;parms C=.04 D=.2 B=3 I=.1 sig=.021;bounds C>0, D>0, B>0, sig>0;k = 9.0;if dose = 0 then mu = d;else mu = C + k*(D-C)/(k + exp(B*(ldose-log(I))));model adj_wt ~ normal(mu, sig**2);


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Dry

Wei

ght (

g)

0.010 0.100 1.000 10.000Dose (kg ai/ha)

I10 I50

Biotype C



Treatment Comparison

Estimation: Treatment Comparison

• SAS Data Step statements.• Build full model.• Estimates pooled or heterogeneous error.• Estimate and Contrast statements for treatment

comparisons.

• NLMIXED can accommodate treatment structure:

• Example: Seefeldt, et al. 1995

• Wild oat resistance to fenoxaprop (SAS: dose).• Dry weights at 2 weeks (SAS: adj_wt).• Three biotypes ; w, b, and c (SAS: biotype).

• Specify a full model with:• Independent parameters B and I for each

biotype.• Common parameter values for C, D, and sig.

Wt = C + (D - C) / (1 + exp(B(dosei - I)))

where B and I are dependent on biotype.


proc nlmixed data=seefeldt;parms C=.02757 D=.1734 Iw=.1188 Ib=.2221 Ic=1.8936

Bw=4.9642 Bb=2.9461 Bc=1.223 sig=.021;

bounds C>0, D>0, Bc>0, Bw>0, Bb>0, sig>0;

if biotype = ’w' then do; I = Iw; B=Bw;

end;else if biotype = 'c' then do;

I = Ic; B=Bc; end;else if biotype = 'b' then do;

I = Ib; B=Bb; end;

if dose = 0 then mu = D;else mu = C + (D-C)/(1 + exp(B*(ldose-log(I)))); model adj_wt ~ normal(mu, sig**2);

contrast 'Iw vs Ic' Iw-Ic;contrast 'Iw vs Ib' Iw-Ib;contrast 'Ib vs Ic' Ib-Ic;predict mu out=fitted;


Bw=4.9642 Bb=2.9461 Bc=1.223 sig=.021;





I = Ib; B=Bb; end;

if dose = 0 then mu = D;else mu = C + (D-C)/(1 + exp(B*(ldose-log(I))));model adj_wt ~ normal(mu, sig**2);



Bw=4.9642 Bb=2.9461 Bc=1.223 sig=.021;





I = Ib; B=Bb; end;

if dose = 0 then mu = D;else mu = C + (D-C)/(1 + exp(B*(ldose-log(I))));model adj_wt ~ normal(mu, sig**2);



Parameter Estimates


C 0.02757 0.003604 156 7.65 <.0001 0.05 0.02045 0.03469D 0.1734 0.004979 156 34.82 <.0001 0.05 0.1635 0.1832Iw 0.1188 0.01605 156 7.40 <.0001 0.05 0.08710 0.1505Ib 0.2221 0.01863 156 11.92 <.0001 0.05 0.1853 0.2589Ic 1.8937 0.3140 156 6.03 <.0001 0.05 1.2736 2.5139 Bw 4.9642 1.3211 156 3.76 0.0002 0.05 2.3548 7.5737 Bb 2.9461 1.1567 156 2.55 0.0118 0.05 0.6613 5.2309 Bc 1.2230 0.2016 156 6.07 <.0001 0.05 0.8248 1.6213 sig 0.02626 0.001450 156 18.11 <.0001 0.05 0.02339 0.02912

Contrasts

Num Den Label DF DF F Value Pr > F

Iw vs Ic 1 156 32.64 <.0001 Iw vs Ib 1 156 21.25 <.0001 Ib vs Ic 1 156 28.98 <.0001

Bw vs Bc 1 156 8.27 0.0046 Bw vs Bb 1 156 1.63 0.2030 Bb vs Bc 1 156 2.32 0.1300


Biotypebcw

Pred

icte

d w

eigh

t (g)

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0.010 0.100 1.000 10.000Dose (kg ai/ha)



Random Effects

Estimation: Random Effects• Random effects may also be modeled.• Locations.• Years.• Experiments/replications.• See for example Nielson, et al 2004.

• Components estimated.• Within effects - Variances.• Between effects - Covariances.

• Requires caution.• Parsimony.• Estimation problems.

• Example: Fungus Gnat Data.• Evaluate efficacy of rapeseed meal (SAS: dose).• Three experiments.• Separate runs for each block (SAS: block).

• Measured egg hatch (SAS: mort).• High variability in natural mortality from

run to run.

Estimation: Random Effects

proc nlmixed maxiter=1000 data=gnat2; parms I=.245 B=-3.58 c=.295; bounds B<0, C>0, I>0;

if dose = 0 then mu = c;else mu = c + (1-c)/(1 + exp(B*(ldose-log(I))));model mort ~ binomial(20,mu);

predict mu*20 out=pred1;run;

• Fixed effects model:


Fit Statistics

-2 Log Likelihood 1593.6 AIC (smaller is better) 1599.6 AICC (smaller is better) 1599.8 BIC (smaller is better) 1609.5

Parameter Estimates


I 0.2387 0.01117 200 21.38 <.0001 0.05 0.2167 0.2607B -1.6369 0.1708 200 -9.59 <.0001 0.05 -1.9736 -1.3002c 0.2567 0.01610 200 15.94 <.0001 0.05 0.2249 0.2884

• Fixed effects model:


Mo r

talit

y

0

10

20

Rapeseed Meal (mg)0.01 0.10

Fixed Effects Model


• Random effects model:• Let natural mortality parameter, C, be random.

proc nlmixed maxiter=1000 data=gnat2; parms I=.2582 B=-1.78 C=.2866 sigC=.17; bounds B<0, c>0, I>0;

Ce = C + e;

if dose = 0 then mu = Ce;else mu = Ce + (1-Ce)/(1 + exp(B*(ldose-log(I))));model mort ~ binomial(20,mu);

random e~normal(0, sigC**2) subject=block;


Fit Statistics -2 Log Likelihood 1182.9 AIC (smaller is better) 1190.9 AICC (smaller is better) 1191.1 BIC (smaller is better) 1192.1

Parameter Estimates StandardParm Estimate Error DF t Value Pr > |t| Alpha Lower Upper

I 0.2580 0.009979 9 25.85 <.0001 0.05 0.2354 0.2805B -1.7834 0.1668 9 -10.69 <.0001 0.05 -2.1606 -1.4062C 0.2861 0.06377 9 4.49 0.0015 0.05 0.1419 0.4304sigC 0.1981 0.04511 9 4.39 0.0017 0.05 0.09609 0.3002

• Random effects model


Mo r

talit

y

0

10

20


Random Effects Model


Mo r

talit

y

0

10

20


Random and Fixed Effects Models

Fixed EffectsRandom Effects


• In general, random effects models:• Are useful with identifiable sources of variability.• Increase overall variability.• Improve measures of fit.

• However:• They may not be parsimonious.• They can be difficult to fit.


Estimation: Bayesian Methods

• Data

• The herbicide trichlopyr was applied to lettuceseed.

• Seven herbicide rates of 0 to 1 ml/L applied.

• Three replications of 30 seeds/petri dish used.

• Germinated or not germinated recorded after 10 d.

Lettuce Mortality

• Assuming a binomial response, i.e. germinated or not.

• Logistic model:yij = M*C / (C + exp(- (dosei - )))

• SAS: PROC MCMC• Markov Chain Monte Carlo• Resampling process• Requires specification of prior distributions:

• M• •

• C set to 1.0 ( estimates LD50 ).

Estimation: Bayesian

Estimation: Bayesianproc mcmc data=steve seed=53197 ntu=1000 nmc=20000; ods select TADpanel PostSummaries; ods output PosteriorSample = ests;

parms M=1 B=2.5 gamma=0.22;Q=.5;C = (Q)/(m - Q);mu = M*C/(C + exp(B*(ldose - log(gamma + 0.03))));

prior M ~ uniform(0.8, 1.0); prior B ~ uniform(1, 5); prior gamma ~ uniform(.15, 1.50);

model yes ~ binomial(total, mu);

proc mcmc data=steve seed=53197 ntu=1000 nmc=20000; ods select TADpanel PostSummaries; ods output PosteriorSample = ests;




proc mcmc data=steve seed=53197 ntu=1000 nmc=20000; ods select TADpanel PostSummaries; ods output PosteriorSample = ests;





• Output from PROC MCMC is sparse.

• Limited textual information.

• Diagnostic plots.

• Results plots.


• PROC MCMC: Textual output

The MCMC Procedure

Posterior Summaries

StandardDeviation Percentiles

Parameter N Mean 25% 50% 75%

M 20000 0.9310 0.0153 0.9216 0.9320 0.9416B 20000 2.5413 0.3532 2.2898 2.5194 2.7652Gamma 20000 0.5132 0.0390 0.4870 0.5112 0.5375


• PROC MCMC: Diagnostic plots


• PROC MCMC: Estimated curve

Calibration: Maximum Likelihood


• Using the NLMIXED code from earlier.

• Reparameterize for the observed response (50% here).

• Estimate the percentile (gamma, ) and it’s standard error, se.

• Assuming normality:

~ N( , 2 = se2 )

^ ^

Calibration: Bayesian Parametric

Calibration: Parametric Bayesian

• Using the MCMC code from earlier.


• Estimate the percentile (gamma, ) and the associated posterior distribution.

Calibration: Parametric Bayesian

Calibration: Bayesian Non-Parametric

Calibration: Non-Parametric Bayesian

• Using the methods of Mukhopadhyay (2000) and PROC MCMC.

• A prior form of logistic assumed. Could be from previous experiment or a “best guess”.


• Estimate the percentile (gamma, ) and the associated posterior distribution.

Calibration: Non-Parametric Bayesian• An assumed prior logistic dose-response curve.

Calibration: Non-Parametric Bayesian• The estimated dose-response curve (blue).

Calibration: Procedure Comparison


Maximum Likelihood

Parametric Bayesian

Non-Parametric Bayesian


Method Estimate Lower* Upper*

MLE 0.5060 0.4264 0.6856

Parametric Bayesian 0.5132 0.4408 0.5957

Nonparametric Bayesian 0.4155 0.2875 0.6300

* Upper and Lower 95% Bounds. Confidence intervals for MLE, Credible Intervals for Bayesian methods

• Estimates and bounds for the unknown dose.

References• Kottas, A., M. D. Branco, and A. E. Gelfand. 2002. A Nonparametric

Bayesian Modeling Approach for Cytogenetic Dosimetry. Biometrics 58, 593-600.

• Mukhopadhyay, S. 2000. Bayesian Nonparametric Inference on the Dose Level with Specified Response Rate. Biometrics 56, 220-226.

• Nielson, O. K., C. Ritz, J. C. Streibig. 2004. Nonlinear mixed-model regression to analyze herbicide dose-response relationships. Weed Technonlogy, 18: 30-37.

• Price, W. J. and B. Shafii. 2005. Bayesian Analysis of Dose-response Calibration Curves. Proceedings of the Seventeenth Annual Kansas State University Conference on Applied Statistics in Agriculture [CDROM], April 25-27, 2005. Manhattan Kansas.

• Ratkowsky, D. A. 1989. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 241 pp.

• SAS Inst. Inc. 2009. SAS OnlineDoc, Version 9.2, Cary, NC.

• Schabenberger,O., B. E. Tharp, J. J. Kells, and D. Penner. 1999. Statistical tests for hormesis and effective dosages in herbicide dose response. Agron. J. 91: 713-721.

• Seefeldt, S.S., J. E. Jensen, and P. Fuerst. 1995. Log-logistic analysis of herbicide dose-response relationships. Weed Technol. 9:218-227.

• Shafii, B. and W. J. Price. 2006. Bayesian approaches to dose-response calibration models. Abstract: Proceedings of the XXIII

International Biometrics Conference [CDROM], July 16 - 21, 2006. Montreal, Quebec Canada.

References

Questions / Comments

http://www.uidaho.edu/ag/statprog

applied dose-response models in weed science presented by william j. price, ph.d

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