1. 毒性测试数据^[1]

下载数据：Cd_test_data_tan2011EST.csv

2. 拟合模型

数据拟合采用“2参数log-logistic”模型：

3. 分别拟合两组数据

3.1 R 语言代码

#读取数据(附件Cd_test_data_tan2011EST)
data<-read.csv(file.choose())
#查看数据表头
head(data)
#作图，将两组实测数据画出
plot(r1~x,data=data,pch=1,log="x",xlab="Concentration",ylab="Mortality (%)",xlim=c(1,max(data$x)))
points(r2~x,data=data,col="red",pch=1)
#用（2参数）log-logistic模型拟合第1组数据
mod.1<-nls(r1 ~ 100/(1+exp(b*(log10(x/x0)))), data=data, start=list(x0=10,b=-1))
#查看拟合结果和置信区间
summary(mod.1)
confint(mod.1)
#作图，将第1组数据的拟合曲线画出
x<-seq(1,300,1)
y<-100/(1+exp(summary(mod.1)$coefficients[2]*(log10(x/summary(mod.1)$coefficients[1]))))
lines(x,y,col="grey50",lwd=1)
 
#用（2参数）log-logistic模型拟合第2组数据
mod.2<-nls(r2 ~ 100/(1+exp(b*(log10(x/x0)))), data=data, start=list(x0=10,b=-1))
#查看拟合结果和置信区间
summary(mod.2)
confint(mod.2)
#作图，将第2组数据的拟合曲线画出
x<-seq(1,300,1)
y<-100/(1+exp(summary(mod.2)$coefficients[2]*(log10(x/summary(mod.2)$coefficients[1]))))
lines(x,y,col="red",lwd=1)
#加网格线，美化
grid()

3.2 拟合结果

（1）“Mortality@pH 7.0”数据拟合结果

参数值：

> summary(mod.1)

Formula: r1 ~ 100/(1 + exp(b * (log10(x/x0))))

Parameters:

   Estimate Std. Error t value Pr(>|t|)    

x0  55.7923     0.7644   72.99 2.11e-07 ***

b   -8.7897     0.4997  -17.59 6.14e-05 ***

---

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.326 on 4 degrees of freedom

Number of iterations to convergence: 8 

Achieved convergence tolerance: 5.34e-07

置信区间： 

> confint(mod.1)

Waiting for profiling to be done...

        2.5%    97.5%

x0  53.72977 58.00931

b  -10.45632 -7.59122

（2）“Mortality@pH 8.2”数据拟合结果

参数值：

> summary(mod.2)

Formula: r2 ~ 100/(1 + exp(b * (log10(x/x0))))

Parameters:

   Estimate Std. Error t value Pr(>|t|)   

x0  20.7279     1.9107  10.848  0.00041 ***

b   -3.6334     0.5151  -7.053  0.00213 **

---

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.173 on 4 degrees of freedom

Number of iterations to convergence: 9

Achieved convergence tolerance: 7.255e-06

置信区间： 

> confint(mod.2)

Waiting for profiling to be done...

        2.5%     97.5%

x0 15.701914 26.782837

b  -5.364096 -2.497817

3.3 作图

4. 一并拟合两组数据（共用参数b）

4.1 R 语言代码

#读取数据(附件Cd_test_data_tan2011EST)
data<-read.csv(file.choose())
#查看数据表头
head(data)
#作图，将两组实测数据画出
plot(r1~x,data=data,pch=1,Cd_test_data_tan2011EST.csvlog="x",xlab="Concentration",ylab="Mortality (%)",xlim=c(1,max(data$x)))
points(r2~x,data=data,col="red",pch=1)
#有几个不同浓度？
n1 <- length(data$r1)
n2 <- length(data$r2)
#合并两组数据
y <- c(data$r1,data$r2)
x <- c(data$x,data$x)
mcon1 <- rep(c(1,0), c(n1,n2))
mcon2 <- rep(c(0,1), c(n1,n2))
#一并拟合两组数据，各自使用不同x0值，共用相同b值
mod.common<-nls(y ~ 100/(1+exp(b*(log10(x/(x0_1*mcon1+x0_2*mcon2))))), start = list(b=-1,x0_1=10,x0_2=10))
#查看拟合结果和置信区间
summary(mod.common)
confint(mod.common)
#作图，将第2组数据的拟合曲线画出
x<-seq(1,300,1)
y1<-100/(1+exp(summary(mod.common)$coefficients[1]*(log10(x/summary(mod.common)$coefficients[2]))))
lines(x,y1,col="grey50",lwd=1)
y2<-100/(1+exp(summary(mod.common)$coefficients[1]*(log10(x/summary(mod.common)$coefficients[3]))))
lines(x,y2,col="red",lwd=1)
grid()

4.2 拟合结果

 参数值： 

> summary(mod.common)

Formula: y ~ 100/(1 + exp(b * (log10(x/(x0_1 * mcon1 + x0_2 * mcon2)))))

Parameters:

     Estimate Std. Error t value Pr(>|t|)   

b     -5.2707     0.9739  -5.412 0.000996 ***

x0_1  55.2566     6.6770   8.276 7.34e-05 ***

x0_2  21.1349     2.6295   8.038 8.85e-05 ***

---

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 8.566 on 7 degrees of freedom

Number of iterations to convergence: 10

Achieved convergence tolerance: 4.701e-06

置信区间： 

> confint(mod.common)

Waiting for profiling to be done...

          2.5%     97.5%

b    -9.082322 -3.556506

x0_1 42.037993 71.640210

x0_2 15.208174 29.657370

4.3 作图

毒性测试数据来源：

[1] Tan, Q. G., & Wang, W. X. (2011). Acute toxicity of cadmium in daphnia magna under different calcium and ph conditions: importance of influx rate. Environmental Science & Technology, 45(5), 1970-1976.