Practical Application of Dose-response
Functions in Weed Science
William J. Price
Statistical Programs
College of Agricultural and Life Sciences
University of Idaho, Moscow, Idaho
• Statistical Estimation Software
• SAS
• S+
• R
• Statistica, etc.
• Sigma Plot, AXUM, etc.
Common Dose-response Models
• Normal:
yij = (1/2) exp((x-)2/2
• Logistic:
yij = 1 / (1 + exp( -b1( dosei - b0 ))
• Modified Logistic:
yij = C + (D-C) / (1 + exp( -B(dosei - I)))
I)))
(e.g. Seefeldt et al. 1995)
• Gompertz:
• Exponential:
yij = b0 (1 - exp(exp(-b1(dose))))
yij = b0 exp(-b1(dose))
yij = b0 [1 - exp(-b1(dose))]
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).
Vernalization Data
1.00
Proportion Alive
.75
.50
.25
0
-16
0
Temperature
Probit Maximum Likelihood
• SAS Procedure: PROC PROBIT.
• Code:
proc
lackfit
inversecl;
proc probit
probitdata=freeze
data=freeze
data=freezelog
log
logoptc
optc
optc
lackfit
lackfit
inversecl;
inversecl;
model alive2wk/plants
= temp/distribution=logistic
;
model
alive2wk/plants
= temp/distribution=logistic
;
predpplot
var=temp;
predpplot var=temp;
PROC PROBIT Output
Probit Procedure
Goodness-of-Fit Tests
Model Information
Statistic
Data Set
Events Variable
Trials Variable
Number of Observations
Number of Events
Number of Trials
Name of Distribution
Log Likelihood
WORK.FREEZE
alive2wk
plants
20
122
195
Logistic
-83.4877251
Number of Observations Read
Number of Observations Used
Number of Events
Number of Trials
Missing Values
Pearson Chi-Square
L.R.
Chi-Square
DF
Pr > ChiSq
18.7054
22.6138
17
17
0.3457
0.1623
Response-Covariate Profile
Response Levels
Number of Covariate Values
20
20
122
195
0
2
20
Type III Analysis of Effects
Effect
Algorithm converged.
Value
Ln(temp)
DF
Wald
Chi-Square
Pr > ChiSq
1
15.2620
<.0001
PROC PROBIT Output (cont)
Analysis of Parameter Estimates
Standard
Error
Parameter DF Estimate
Intercept
Ln(temp)
_C_
1
1
1
-6.9144
4.5094
0.2258
95% Confidence
Limits
2.0126 -10.8590
1.1543
2.2470
0.0623
0.1037
Probability
temp
0.01
0.02
.
.
.
0.40
0.45
0.50
0.55
0.60
.
.
.
0.99
1.67252
1.95482
.
.
.
4.23519
4.43197
4.63365
4.84451
5.06959
.
.
.
12.83734
-2.9699
6.7717
0.3480
ChiSquare Pr > ChiSq
11.80
15.26
95% Fiducial Limits
0.48881
0.66670
.
.
.
3.00063
3.25697
3.52170
3.79739
4.08622
.
.
.
9.46539
2.54641
2.83256
.
.
.
4.96946
5.16920
5.38443
5.62420
5.90119
.
.
.
30.51920
0.0006
<.0001
PROC PROBIT Output (cont)
1.00
Proportion Alive
.75
.50
.25
OPTC = 0.23
0
-16
0
Temperature (C)
PROC PROBIT (cont)
• Probit Advantages
•
•
•
•
Automatic Goodness of Fit test.
Easily computed percentiles.
Ability to do treatment comparisons.
Graphic output.
• Probit Limitations
• Proportional Data.
• Maximum set to 1.0.
• Limited number of response models.
Nonlinear Least Squares
• SAS Procedure: PROC NLIN
• Code:
proc
proc nlin
nlindata=freeze
data=freeze; ;
parms
= -4.5
C =C.12;
parmsI I= =8 8B B
= -4.5
= .12;
bounds
boundsB<0;
B<0;
mu == CC ++(1-C)/(1
(1-C)/(1
+ exp(B*(ltemp-log(I))));
mu
+ exp(B*(ltemp-log(I))));
model per2
per2= =mu;
mu;
model
output
output out=pred
out=predp=pred;
p=pred;
PROC NLIN Output
The NLIN Procedure
Dependent Variable per2
Method: Gauss-Newton
Iterative Phase
Iter
I
B
C
Sum of
Squares
0
1
2
3
4
5
4.6000
4.6147
4.6119
4.6125
4.6123
4.6124
-4.6000
-4.6464
-4.6366
-4.6388
-4.6383
-4.6384
0.2200
0.2268
0.2264
0.2265
0.2265
0.2265
0.3767
0.3765
0.3765
0.3765
0.3765
0.3765
NOTE: Convergence criterion met.
Estimation Summary
Method
Iterations
R
PPC(B)
RPC(B)
Object
Objective
Observations Read
Observations Used
Observations Missing
Gauss-Newton
5
3.628E-6
4.559E-6
0.000021
2.25E-10
0.37647
20
20
0
PROC NLIN Output (cont)
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Uncorrected Total
3
17
20
9.6353
0.3765
10.0117
3.2118
0.0221
Parameter
I
B
C
Estimate
Approx
Std Error
4.6124
-4.6384
0.2265
0.4511
1.7176
0.0723
I
I
B
C
1.0000000
-0.4991207
0.6394756
Approx
Pr > F
145.03
<.0001
Approximate 95% Confidence Limits
3.6607
-8.2622
0.0739
Approximate Correlation Matrix
B
C
-0.4991207
1.0000000
-0.5126022
F Value
0.6394756
-0.5126022
1.0000000
5.5640
-1.0147
0.3791
PROC NLIN Output (cont)
1.0
Proportion Alive
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
Ln (temp)
2
3
PROC NLIN (cont)
• NLIN Advantages
• Not restricted to proportional data.
• Maximum may be any value.
• Response models not limited.
• NLIN Limitations
• Assumes normally distributed response.
• Approximate tests.
• Treatment comparisons not automatic.
Maximum Likelihood
• SAS Procedure: PROC NLMIXED
• Code:
proc
proc nlmixed
nlmixeddata=freeze
data=freeze
data=freezecorr;
corr;
corr;
parms
I = 4.5
= -4.8
= .228;
parms
I = B4.5
B = C-4.8
C = .228;
bounds
B<0, B<0,
I>0; I>0;
bounds
mu
+
mu
(1-C)/(1
+ exp(B*(ltemp-log(I))));
mu == C
CC +
++(1-C)/(1
(1-C)/(1
+ exp(B*(ltemp-log(I))));
exp(B*(ltemp-log(I))));
model
binomial(plants,
mu);
model
alive2wk
~ ~
binomial(plants,
mu);mu);
model alive2wk
alive2wk~
binomial(plants,
predict
plants*mu
out=pred1;
predict plants*mu
out=pred1;
predict
predict
muout=pred2;
out=pred2;
predict mu
mu
out=pred2;
PROC NLMIXED Output
Dependent Variable
Distribution for Dependent Variable
Optimization Technique
alive2wk
Binomial
Dual Quasi-Newton
Dimensions
Observations Used
Observations Not Used
Total Observations
Parameters
20
0
20
3
Iteration History
Iter
Calls
NegLogLike
Diff
MaxGrad
Slope
1
2
3
4
5
10
28.3612297
28.2115599
28.1847135
0.034097
0.14967
1.433E-8
1.623059
0.572223
0.000016
-27.1026
-3.97671
-2.67E-8
NOTE: GCONV convergence criterion satisfied.
Fit Statistics
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
56.4
62.4
63.9
65.4
PROC NLMIXED Output (cont)
Parameter Estimates
Parm
Estimate
Standard
Error
I
B
C
4.6337
-4.5094
0.2258
0.4151
1.1543
0.06231
DF
t Value
Pr > |t|
20
20
20
11.16
-3.91
3.62
<.0001
0.0009
0.0017
Alpha
0.05
0.05
0.05
Correlation Matrix of Parameter Estimates
Row
1
2
3
Parameter
I
B
C
I
B
C
1.0000
-0.5276
0.5848
-0.5276
1.0000
-0.3896
0.5848
-0.3896
1.0000
Lower
3.7677
-6.9172
0.09585
Upper
5.4996
-2.1016
0.3558
Gradient
4.629E-6
-2.62E-6
0.000016
PROC NLMIXED Output (cont)
10
9
8
7
6
5
4
3
2
1
0
1
2
Ln(temp)
3
Procedure Comparisons
1.0
0.9
Proportion Alive
0.8
0.7
0.6
Probit
0.5
NLin
NLMixed
0.4
0.3
0.2
0.1
0
1
2
Ln(temp)
3
Procedure Comparisons
• 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.
NLMIXED Example: Normal Data
• NLMIXED probability distributions:
•
•
•
•
Binomial - yes/no data.
Normal - continuous data.
Poisson - discrete count data.
User defined - any data.
• 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
0.28
Biotype C
0.26
0.24
Dry Weight (g)
0.22
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0.010
0.100
Dose (kg ai/ha)
1.000
10.000
NLMIXED Example: Normal Data
• Assume dry weight to be normally distributed with
mean mu and variance sig2.
• Must model sig explicitly.
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;
NLMIXED Example: Normal Data
Iteration History
Iter
Calls
NegLogLike
Diff
MaxGrad
Slope
1
2
3
14
19
66
-63.769085
-90.116746
-122.58572
17.90341
26.34766
4.511E-8
925.7938
3490.083
0.040557
-4745664
-1912.68
-1E-7
NOTE: GCONV convergence criterion satisfied.
Fit Statistics
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
-245.2
-235.2
-233.9
-225.1
Parameter Estimates
Parm
C
D
B
I
sig
Estimate
0.04823
0.1836
1.3283
1.1669
0.02937
Standard
Error
0.01893
0.00794
0.4510
0.3539
0.00280
DF
51
51
51
51
51
t Value
2.55
23.12
2.95
3.30
10.49
Pr > |t|
0.0137
<.0001
0.0047
0.0017
<.0001
Lower
0.01030
0.1677
0.4246
0.4576
0.02376
Upper
0.08616
0.1996
2.2321
1.8762
0.03498
NLMIXED Example: Normal Data
0.28
0.26
Biotype C
0.24
Dry Weight (g)
0.22
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0.010
0.100
Dose (kg ai/ha)
1.000
10.000
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 reach
the Qth percentile, and k is given by :
k = Q/(1 - Q)
NLMIXED: Extension of Log-logistic Model
• Example:
• Seefeldt data, biotype C.
• Estimate the 90th percentile, e.g. I10
Q = 0.9
k = Q/(1 - Q) = 0.9/(1.0 - 0.9) = 9.0
NLMIXED: Extension of Log-logistic Model
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);
Parameter Estimates
Parm
Estimate
C
D
B
0.04823
0.1836
1.3284
sig
0.02605
I
0.2214
Standard
Error
DF
t Value
Pr > |t|
Alpha
Lower
Upper
0.01893
0.007942
0.4510
0.1098
0.002484
51
51
51
51
51
2.55
23.12
2.95
2.02
10.49
0.0136
<.0001
0.0047
0.0487
<.0001
0.05
0.05
0.05
0.05
0.05
0.01030
0.1677
0.4246
0.00134
0.02107
0.08616
0.1996
2.2321
0.4415
0.03103
NLMIXED: Extension of Log-logistic Model
0.28
0.26
Biotype C
0.24
Dry Weight (g)
0.22
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
I50
I10
0.02
0.00
0.010
0.100
Dose (kg ai/ha)
1.000
10.000
NLMIXED Example: Treatment Structure
• NLMIXED can accommodate treatment structure:
•
•
•
•
SAS Data Step statements.
Build full model.
Estimates pooled or heterogeneous error.
Estimate and Contrast statements for treatment
comparisons.
• 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).
NLMIXED Example: Treatment Structure
• 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.
NLMIXED Example: Treatment Structure
proc nlmixed data=seefeldt;
parms
D=.1734
Iw=.1188
Ib=.2221
Ic=1.8936
parms C=.02757
C=.02757
D=.1734
Iw=.1188
Ib=.2221
Ic=1.8936
Bw=4.9642
Bb=2.9461
Bc=1.223
sig=.021;
Bw=4.9642 Bb=2.9461 Bc=1.223 sig=.021;
bounds
bounds C>0,
C>0, D>0,
D>0, Bc>0,
Bc>0, Bw>0,
Bw>0, Bb>0,
Bb>0, sig>0;
sig>0;
if
then
do;
if biotype
biotype=
’w'
then
if
biotype
= =’w'
’w'
then
do; do;
I
=
Iw;
B=Bw;
I == Iw;
Iw;B=Bw;
B=Bw;
end;
end;
end;
else
if biotype
=
then
do;
else
= 'c'
'c'
thenthen
do; do;
else if
ifbiotype
biotype
=
'c'
I
=
Ic;
B=Bc;
I
I == Ic;
Ic;B=Bc;
B=Bc;
end;
end;
end;
else
else if
if biotype
biotype =
= 'b'
'b' then
then do;
do;
else if I
biotype
=
'b'
then
do;
=
Ib;
B=Bb;
I = Ib; B=Bb;
end;
I = Ib; B=Bb;
end;
end;
if
if dose
dose =
= 0
0 then
then mu
mu =
= D;
D;
else
mu
=
C
+
(D-C)/(1
+
if dose
else
mu = C
0 +
then
(D-C)/(1
mu = D;
+ exp(B*(ldose-log(I))));
exp(B*(ldose-log(I))));
model
adj_wt
~
normal(mu,
sig**2);
else mu
model
adj_wt
= C +~(D-C)/(1
normal(mu,
+ exp(B*(ldose-log(I))));
sig**2);
model adj_wt ~ normal(mu, sig**2);
contrast
'Iwvsvs
Iw-Ic;
contrast 'Iw
Ic'Ic'
Iw-Ic;
contrast
'Iwvsvs
Iw-Ib;
contrast 'Iw
Ic'Ib'
Ib'
Iw-Ic;
Iw-Ib;
contrast 'Ib
'Iw
Ib'Ic'
Ic'
Iw-Ib;
Ib-Ic;
contrast
'Ibvsvs
Ib-Ic;
contrastmu
predict
'Ib
out=fitted;
vs Ic' Ib-Ic;
predict mu out=fitted;
NLMIXED Example: Treatment Structure
Parameter Estimates
Parm
Estimate
C
D
Iw
Ib
Ic
Bw
Bb
Bc
sig
0.02757
0.1734
0.1188
0.2221
1.8937
4.9642
2.9461
1.2230
0.02626
Standard
Error
0.003604
0.004979
0.01605
0.01863
0.3140
1.3211
1.1567
0.2016
0.001450
DF
156
156
156
156
156
156
156
156
156
t Value Pr > |t| Alpha
7.65
34.82
7.40
11.92
6.03
3.76
2.55
6.07
18.11
<.0001
<.0001
<.0001
<.0001
<.0001
0.0002
0.0118
<.0001
<.0001
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
Lower
0.02045
0.1635
0.08710
0.1853
1.2736
2.3548
0.6613
0.8248
0.02339
Contrasts
Label
Iw
Iw
Ib
Bw
Bw
Bb
vs
vs
vs
vs
vs
vs
Ic
Ib
Ic
Bc
Bb
Bc
Num
DF
Den
DF
F Value
Pr > F
1
1
1
1
1
1
156
156
156
156
156
156
32.64
21.25
28.98
8.27
1.63
2.32
<.0001
<.0001
<.0001
0.0046
0.2030
0.1300
Upper
0.03469
0.1832
0.1505
0.2589
2.5139
7.5737
5.2309
1.6213
0.02912
NLMIXED Example: Treatment Structure
Predicted weight (g)
0.18
0.16
0.14
0.12
0.10
Biotype
0.08
b
c
w
0.06
0.04
0.02
0.010
0.100
Dose (kg ai/ha)
1.000
10.000
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).
NLMIXED Example: Poisson data
100
90
Variety C
80
70
60
50
40
30
20
10
0
0.0001
0.0010
0.0100
0.1000
Harmony Dose (lb ai/A)
1.0000
NLMIXED Example: Poisson data
• Assume the number of branches to be distributed as
a Poisson variable.
• In the Poisson distribution, mean = variance = mu.
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;
NLMIXED Example: Poisson data
Parameter Estimates
Parm
Standard
Estimate
Error
DF
t Value
C
I
B
D
11.6834
0.01075
1.9860
69.1586
38
38
38
38
11.53
8.41
6.12
20.40
1.0136
0.0013
0.3246
3.3894
Pr > |t| Alpha
<.0001
<.0001
<.0001
<.0001
0.05
0.05
0.05
0.05
Lower
Upper
9.6315
0.0081
1.3289
62.2972
13.7352
0.0133
2.6432
76.0201
NLMIXED Example: Poisson data
100
90
80
Variety C
70
60
50
40
30
20
10
0
0.0001
0.0010
0.0100
Harmony Dose (lb ai/A)
0.1000
1.0000
NLMIXED: Alternative Models
Example:
• Log-logistic Model
yij = C + (D - C) / (1 + exp(B(dosei - I)))
• Exponential Model
yij = (a-c) exp(-bdose) + c
Exponential Model for Pea Biomass
• A linear pattern of data on a log scale.
• Implies an exponential model, e.g.
Biomass = (a-c) exp(-bdose) + 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;
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
0.8
Biomass (g/plant)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0001
0.0010
0.0100
Harmony Dose (lb ai/A)
0.1000
Log-logistic Model for Pea Biomass
Parameter Estimates
Parm
D
I
B
C
sig
Estimate
1.2751
0.02792
0.1261
-0.7546
0.09240
Standard
Error
DF
t Value
Pr > |t|
Alpha
Lower
Upper
4.3499
0.4771
0.6405
5.9311
0.01060
38
38
38
38
38
0.29
0.06
0.20
-0.13
8.72
0.7710
0.9536
0.8449
0.8994
<.0001
0.05
0.05
0.05
0.05
0.05
-7.5308
-0.9379
-1.1704
-12.7614
0.07094
10.0810
0.9937
1.4227
11.2522
0.1139
Correlation Matrix of Parameter Estimates
Row
1
2
3
4
5
Parameter
D
I
B
C
sig
D
I
B
C
sig
1.0000
0.6108
-0.9903
-0.9568
-0.00074
0.6108
1.0000
-0.7133
-0.8146
-0.00306
-0.9903
-0.7133
1.0000
0.9873
0.001188
-0.9568
-0.8146
0.9873
1.0000
0.001668
-0.00074
-0.00306
0.001188
0.001668
1.0000
Exponential Model for Pea Biomass
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);
Exponential Model for Pea Biomass
Parameter Estimates
Estimate
Standard
Error
DF
t Value
Pr > |t|
Alpha
Lower
Upper
0.5260
106.50
0.2026
0.09861
0.03375
51.3184
0.03622
0.01131
38
38
38
38
15.59
2.08
5.59
8.72
<.0001
0.0448
<.0001
<.0001
0.05
0.05
0.05
0.05
0.4577
2.6113
0.1293
0.07571
0.5943
210.39
0.2760
0.1215
Parm
a
b
c
sig
Correlation Matrix of Parameter Estimates
Row
1
2
3
4
Parameter
a
b
c
sig
a
b
c
sig
1.0000
0.6473
0.2864
0.000060
0.6473
1.0000
0.6696
0.000091
0.2864
0.6696
1.0000
0.000062
0.000060
0.000091
0.000062
1.0000
Additional Estimates
Label
I50
Estimate
Standard
Error
DF
t Value
Pr > |t|
Alpha
Lower
Upper
0.01576
0.006829
38
2.31
0.0265
0.05
0.001937
0.02958
Exponential Model for Pea Biomass
0.8
Biomass (g/plant)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0001
0.0010
0.0100
Harmony Dose (lb ai/A)
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.
Nonlinear Mixed Models
• 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.
Nonlinear Mixed Models
• Fixed effects model:
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;
Nonlinear Mixed Models
• Fixed effects model:
Fit Statistics
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
1593.6
1599.6
1599.8
1609.5
Parameter Estimates
Parm
Estimate
Standard
Error
DF
t Value
Pr > |t|
Alpha
Lower
Upper
I
B
c
0.2387
-1.6369
0.2567
0.01117
0.1708
0.01610
200
200
200
21.38
-9.59
15.94
<.0001
<.0001
<.0001
0.05
0.05
0.05
0.2167
-1.9736
0.2249
0.2607
-1.3002
0.2884
Fixed Effects Model
20
10
0
0.01
0.10
Rapeseed Meal (mg)
Nonlinear Mixed Models
• 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;
Nonlinear Mixed Models
• Random effects model
Fit Statistics
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
Parm
Standard
Estimate
Error
I
B
C
sigC
0.2580
-1.7834
0.2861
0.1981
0.009979
0.1668
0.06377
0.04511
1182.9
1190.9
1191.1
1192.1
Parameter Estimates
DF
t Value
Pr > |t|
Alpha
Lower
Upper
9
9
9
9
25.85
-10.69
4.49
4.39
<.0001
<.0001
0.0015
0.0017
0.05
0.05
0.05
0.05
0.2354
-2.1606
0.1419
0.09609
0.2805
-1.4062
0.4304
0.3002
Random Effects Model
20
10
0
0.01
0.10
Rapeseed Meal (mg)
Random and Fixed Effects Models
20
10
Fixed Effects
Random Effects
0
0.01
0.10
Rapeseed Meal (mg)
Nonlinear Mixed Models
• 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 / Comments
http://www.uidaho.edu/ag/statprog