Dose-Response Model Coefficients

Question

I've been trying to fit my data to a logistic curve. I have come across 2 different packages that I have been working with. DRM and NLS, so starting with drm I am able to fit a model which is visualized as below.

Now my issue with this is the model summary.

Text:

Formula: percent_farm_tractor ~ SSlogis(year, Asym, xmid, scal)

Parameters:
     Estimate Std. Error t value Pr(>|t|)    
Asym    2.265      2.527   0.896   0.4207    
xmid 1975.306     17.589 112.305 3.77e-08 ***
scal    9.575      2.674   3.580   0.0232 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.03798 on 4 degrees of freedom

Number of iterations to convergence: 0 
Achieved convergence tolerance: 5.703e-06

Unless I am interpreting this incorrectly, the b coefficient represents the slope. How can this be negative when the model fit plot even shows a positive trend as year increases??

The nls model does return a positive number for the slope but I'm having trouble getting the nls package to work with the rest of my data.

Thanks

EDIT:

I used split to obtain multiple data.frames which each contain multiple records relating to a county and each record is a different year. So for example here a 2 different data.frame representing 2 different counties:

`Sample County 1

    year state county    name percent_farm_tractor stateAbb fips colorID
2   1925     1      3 BALDWIN           0.06760000       AL 1003       1
69  1930     1      3 BALDWIN           0.08679707       AL 1003       2
136 1940     1      3 BALDWIN           0.19938885       AL 1003       3
203 1950     1      3 BALDWIN           0.44627821       AL 1003       7
270 1954     1      3 BALDWIN           0.56669298       AL 1003       9
337 1964     1      3 BALDWIN           0.75094340       AL 1003      12
404 1969     1      3 BALDWIN           0.89988623       AL 1003      14

Sample County 2:

    year state county    name percent_farm_tractor stateAbb fips colorID
476 1925     5     13 CALHOUN          0.000000000       AR 5013       1
551 1930     5     13 CALHOUN          0.006680027       AR 5013       1
626 1940     5     13 CALHOUN          0.027145359       AR 5013       1
701 1950     5     13 CALHOUN          0.187435633       AR 5013       3
776 1954     5     13 CALHOUN          0.333333333       AR 5013       5
851 1964     5     13 CALHOUN          0.530150754       AR 5013       8
926 1969     5     13 CALHOUN          0.929824561       AR 5013      14

I am applying drm to each of these data.frames as demostrated below:

j <- 1
params <- data.frame()
for(j in 1:length(split_df)){
if(nrow(split_df[[j]]) != 1){
mL <- drm(percent_farm_tractor ~ year, data = 
as.data.frame(split_df[[j]]), fct = L.3(), type = "continuous")
params <- rbind(params, coef(mL))
}
}

Basically every county has a negative slope value, which seems very counter intuitive as basically every county also has a positive trend as year increases.

Answer 1

tl;dr Just flip the sign; it's an unusual parameterization where negative b corresponds to an increasing function. (I don't know why I never noticed this before; maybe because I've generally focused on the intercept/ED50 parameter ...)

This is the expression given for the generalized logistic in ?drc::logistic:

f(x) = c + \frac{d-c}{(1+\exp(b(x - e)))^f}          

For the three-parameter logistic, c=0, f=1, so we have

f(x) = \frac{d}{(1+\exp(b(x - e)))}

It's easy to see that d is the upper asymptote and the e is the half-maximum (when x=e, the equation reduces to \frac{d}{(1+\exp(0)} = d/2). b is indeed a slope, but the key is that for x>e the denominator is an increasing function of b; that means that the overall expression is an decreasing function of b.

This is in contrast to the more standard parameterization of the logistic which adds a negative sign before the slope, e.g. from plogis:

F(x) = 1 / (1 + exp(-(x-m)/s)) 

Note the - before (x-m)/s ! Here s is a scale, m is the half-max, 1/s is the slope ...