Mike Lawrence wrote:
Hi all,
Where f(x) is a logistic function, I have data that follow:
g(x) = f(x)*.5 + .5
How would you suggest I modify the standard glm(..., family='binomial')
function to fit this? Here's an example of a clearly ill-advised attempt to
simply use the standard glm(..., family='binomial') approach:
########
# First generate some data
########
#define the scale and location of the modified logistic to be fit
location = 20
scale = 30
# choose some x values
x = runif(200,-200,200)
# generate some random noise to add to x in order to
# simulate real-word measurement and avoid perfect fits
x.noise = runif(length(x),-10,10)
# define the probability of success for each x given the modified logistic
prob.success = plogis(x+x.noise,location,scale)*.5 + .5
# obtain y, the observed success/failure at each x
y = rep(NA,length(x))
for(i in 1:length(x)){
y[i] = sample(
x = c(1,0)
, size = 1
, prob = c(prob.success[i], 1-prob.success[i])
)
}
#show the data and the source modified logistic
plot(x,y)
curve(plogis(x,location,scale)*.5 + .5,add=T)
########
# Now try to fit the data
########
#fit using standard glm and plot the result
fit = glm(y~x,family='binomial')
curve(plogis(fit$coefficients[1]+x*fit$coefficients[2])*.5+.5,add=T,col='red',lty=2)
# It's clear that it's inappropriate to use the standard
"glm(y~x,family='binomial')"
# method to fit the modified logistic data, but what is the alternative?
A custom-made fit function using nlm?
Below, in the second line the logistic model has been re-parameterized
to include p[1]=level of the predictor which predicts a 50% probability
of success, and p[2]=level of the predictor which predicts a 95%
probability of success. You can change the model in this line to your
version. In the 4th line (negative log-likelihood) mat is a vector of 0s
and 1s representing success and imat is a vector of 1s and 0s
representing failure. The fit is for grouped data.
fn <- function(p){
prop.est <- 1/(1+exp(log(1/19)*(l-p[1])/(p[2]-p[1]))); # estimated
proportion of success
iprop.est <- 1-prop.est; # estimated
proportion of failure
negloglik <- -sum(count*(mat*log(prop.est)+imat*log(iprop.est)));
#binomial likelihood, prob. model
}
prop.lik <- nlm(fn,p=c(25.8,33.9),hessian=TRUE)
HTH
Rubén
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