Practical Application of Dose-response Functions in Weed Science William J. Price

transcript

Practical Application of Dose-response Functions in Weed Science

William J. Price

Statistical ProgramsCollege of Agricultural and Life Sciences

University of Idaho, Moscow, Idaho

• Statistical Estimation Software

• S+

• Statistica, etc.

• Sigma Plot, AXUM, etc.

• SAS

• 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)

Probit Maximum Likelihood• 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

Vernalization Data

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

• Code:

Probit Maximum Likelihood

• SAS Procedure: PROC PROBIT.

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

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.

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

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

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)

OPTC = 0.23

• 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

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

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)

Ln (temp)0 1 2 3

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:

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;

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

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

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

Dimensions

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

Iteration History

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

Fit Statistics

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

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

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

Ln(temp)0 1 2 3

tsPROC NLMIXED Output (cont)

Ln(temp)0 1 2 3

Procedure Comparisons

Probit

NLMixedPro

• 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

NLMIXED Example: 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).

)NLMIXED Example: Normal Data

Biotype C

0.010 0.100 1.000 10.000

Dose (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;

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

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

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

Iteration History

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

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

0.010 0.100 1.000 10.000

Dose (kg ai/ha)

Biotype C

NLMIXED: Extension of Log-logistic Model

• 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

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

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

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);

0.010 0.100 1.000 10.000

Dose (kg ai/ha)

I10I50

Biotype C

NLMIXED Example: Treatment Structure

• 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);

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

I = Ib; B=Bb; end;

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

I = Ib; B=Bb; end;

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

I = Ib; B=Bb; end;

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

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

Biotype

0.010 0.100 1.000 10.000

Dose (kg ai/ha)

NLMIXED Example: 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).

0.0001 0.0010 0.0100 0.1000 1.0000

Harmony 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;

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

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

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

0.0001 0.0010 0.0100 0.1000 1.0000

Variety C

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)

NLMIXED: Alternative Models

• 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;

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

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

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

0.0001 0.0010 0.0100 0.1000

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

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

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);

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

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

0.0001 0.0010 0.0100 0.1000

Nonlinear Mixed Models

• 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.

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;

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

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

Fit Statistics

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:

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;

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

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

Random Effects Model

Random and Fixed Effects Models

Fixed Effects

Random 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.

References

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

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

• SAS Inst. Inc. 2004. SAS OnlineDoc, Version 9, 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.

Questions / Commentshttp://www.uidaho.edu/ag/statprog

Practical Application of Dose-response Functions in Weed Science William J. Price

Documents