[R] Nonlinear logistic regression fitting

[Sebastien Bihorel]

Thank your, Pr. Nash, for your perspective on the issue.

Here is an example of binary data/response (resp) that were simulated and re-estimated assuming a non linear effect of the predictor (x) on the likelihood of response. For re-estimation, I have used gnlm::bnlr for the logistic regression. The accuracy of the parameter estimates is so-so, probably due to the low number of data points (8*nx). For illustration, I have also include a glm call to an incorrect linear model of x.

#install.packages(gnlm)
#require(gnlm)
set.seed(12345)

nx <- 10
x <- c(
  rep(0, 3*nx),
  rep(c(10, 30, 100, 500, 1000), each = nx)
)
rnd <- runif(length(x))
a <- log(0.2/(1-0.2))
b <- log(0.7/(1-0.7)) - a
c <- 30
likelihood <- a + b*x/(c+x)
p <- exp(likelihood) / (1 + exp(likelihood))
resp <- ifelse(rnd <= p, 1, 0)

df <- data.frame(
  x = x,
  resp = resp,
  nresp = 1- resp
)

head(df)

# glm can only assume linear effect of x, which is the wrong model
glm_mod <- glm(
  resp~x,
  data = df,
  family = 'binomial'
)
glm_mod

# Using gnlm package, estimate a model model with just intercept, and a model with predictor effect
int_mod <- gnlm::bnlr( y = df[,2:3], link = 'logit', mu = ~ p_a, pmu = c(a) )
emax_mod <- gnlm::bnlr( y = df[,2:3], link = 'logit',  mu = ~ p_a + p_b*x/(p_c+x),  pmu = c(a, b, c) )

int_mod
emax_mod

________________________________
From: J C Nash <profjcnash using gmail.com>
Sent: Tuesday, July 28, 2020 14:16
To: Sebastien Bihorel <Sebastien.Bihorel using cognigencorp.com>; r-help using r-project.org <r-help using r-project.org>
Subject: Re: [R] Nonlinear logistic regression fitting

There is a large literature on nonlinear logistic models and similar
curves. Some of it is referenced in my 2014 book Nonlinear Parameter
Optimization Using R Tools, which mentions nlxb(), now part of the
nlsr package. If useful, I could put the Bibtex refs for that somewhere.

nls() is now getting long in the tooth. It has a lot of flexibility and
great functionality, but it did very poorly on the Hobbs problem that
rather forced me to develop the codes that are 3/5ths of optim() and
also led to nlsr etc. The Hobbs problem dated from 1974, and with only
12 data points still defeats a majority of nonlinear fit programs.
nls() poops out because it has no LM stabilization and a rather weak
forward difference derivative approximation. nlsr tries to generate
analytic derivatives, which often help when things are very badly scaled.

Another posting suggests an example problem i.e., some data and a
model, though you also need the loss function (e.g., Max likelihood,
weights, etc.). Do post some data and functions so we can provide more
focussed advice.

JN

On 2020-07-28 10:13 a.m., Sebastien Bihorel via R-help wrote:
> Hi
>
> I need to fit a logistic regression model using a saturable Michaelis-Menten function of my predictor x. The likelihood could be expressed as:
>
> L = intercept + emax * x / (EC50+x)
>
> Which I guess could be expressed as the following R model
>
> ~ emax*x/(ec50+x)
>
> As far as I know (please, correct me if I am wrong), fitting such a model is to not doable with glm, since the function is not linear.
>
> A Stackoverflow post recommends the bnlr function from the gnlm (https://stackoverflow.com/questions/45362548/nonlinear-logistic-regression-package-in-r)... I would be grateful for any opinion on this package or for any alternative recommendation of package/function.
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