No convergence of the estimates of the parameters of the Richards model in the "nls" package of r even with good kick start

Question

I am estimating parameters of nonlinear regression models sigmoidais, only the Richards model returns the error:

fator de passos 0.000488281 reduzido abaixo de 'minFactor' de 0.000976562

found the initial kicks by the method of graphical analysis, the curve was very close to the points but it was not enough for the method to converge.

researching some similar problems, the apparent solution I found is to modify the minFactor and maxiter to respectively 1/10 6 and 100000, even so the problem repeats again.

Database:

x<-c(60,90,120,150)
y<-c(1.14, 4.22, 19.3, 22.3)
dados<-data.frame(DAP=x, MSTP=y)

Graph with the initial kicks:

richar<-function(x){22.3/((1+exp(18.7-0.167*x))^(1/2.1))}

library(ggplot2)
ggplot(dados)+
  geom_point(aes(x=DAP, y=MSTP))+
  stat_function(fun=richar)

pet:

richards<-nls(y~a/((1+exp(b-c*x))^(1/d)),start = list(a=22.3,b=18.7,c=0.165,d=2.1))
summary(richards)

Answer 1

In general, by adjusting a model to a data set, we seek to adjust the simplest possible model in the largest data set available. Remember this when reading my answer below.

The problem with your code is in charge richards<-nls(y~a/((1+exp(b-c*x))^(1/d)),start = list(a=22.3,b=18.7,c=0.165,d=2.1)), which does not inform the R what is the data set in which the function nls should adjust the curve.

x<-c(60,90,120,150)
y<-c(1.14, 4.22, 19.3, 22.3)
dados<-data.frame(DAP=x, MSTP=y)

richar<-function(x){22.3/((1+exp(18.7-0.167*x))^(1/2.1))}

library(ggplot2)
ggplot(dados)+
    geom_point(aes(x=DAP, y=MSTP))+
    stat_function(fun=richar)

richards<-nls(y~a/((1+exp(b-c*x))^(1/d)),
                            start = list(a=22.3,b=18.7,c=0.165,d=2.1), 
                            data = dados)
#> Warning in min(x): no non-missing arguments to min; returning Inf
#> Warning in max(x): no non-missing arguments to max; returning -Inf
#> Error in nls(y ~ a/((1 + exp(b - c * x))^(1/d)), start = list(a = 22.3, : step factor 0.000488281 reduced below 'minFactor' of 0.000976562
summary(richards)
#> Error in summary(richards): object 'richards' not found

^{Created on 2020-07-16 by the reprex package (v0.3.0)}

Note that I have now received another error message. By definition, a nonlinear regression model with m parameters can be adjusted to a set with n observations, provided that m ≤ n. In your case, m = 4 = n. This can lead to several problems, such as numerical instabilities. See how the solution to your problem is found using a simulated data set with only three points more than the original problem.

x <- seq(60, 150, length.out = 7)
y <- richar(x) + rnorm(7, sd = 1)
dados2<-data.frame(DAP=x, MSTP=y)

ggplot(dados2)+
    geom_point(aes(x=DAP, y=MSTP))+
    stat_function(fun=richar)

richards<-nls(y~a/((1+exp(b-c*DAP))^(1/d)),
                            start = list(a=22.3,b=18.7,c=0.165,d=2.1), 
                            data = dados2)
summary(richards)
#> 
#> Formula: y ~ a/((1 + exp(b - c * DAP))^(1/d))
#> 
#> Parameters:
#>   Estimate Std. Error t value Pr(>|t|)    
#> a  20.9785     0.7074  29.655 8.42e-05 ***
#> b  28.3889    16.9568   1.674    0.193    
#> c   0.2451     0.1407   1.742    0.180    
#> d   4.0256     2.9087   1.384    0.260    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 0.9551 on 3 degrees of freedom
#> 
#> Number of iterations to convergence: 6 
#> Achieved convergence tolerance: 9.361e-06

^{Created on 2020-07-16 by the reprex package (v0.3.0)}

I find it very difficult that your problem has any numerical solution with such a complex model (4 parameters) and so few points (4 observations). From here there are at least two possible approaches:

1. Use a simpler function such as logistical function, which has only two parameters to adjust. The problem with this approach, I believe, is that Richards' function is being used because, in the original problem, each of its four parameters must have an appropriate meaning for the real problem being considered here.

2. Use more data. The problem with this approach is just getting this data. This costs time and money and in some cases it is not even possible to obtain data from the same experiment, either because there is no money or because the bodies used in the trials were destroyed.