Logistic Regression in R: Optimization Issues concerning Initial Guess - r

Question

I need to implement a logistic regression manually, using the Score/GMM approach, without the use of GLM. This is because at later stages the model will be much more complicated. Currently I am running into a problem where for the logistic regression, the optimization procedures are very initial point dependent.To illustrate, here is my code using an online dataset. More details about the procedure are in the comments:
library(data,table)
library(nleqslv)
library(Matrix)
mydata <- read.csv("https://stats.idre.ucla.edu/stat/data/binary.csv")
data_analysis<-data.table(mydata)
data_analysis[,constant:=1]
#Likelihood function for logit
#The logistic regression will regress the binary variable
#admit on a constant and the variable gpa
LL <- function(beta){
beta=as.numeric(beta)
data_temp=data_analysis
mat_temp2 = cbind(data_temp$constant,
data_temp$gpa)
one = rep(1,dim(mat_temp2)[1])
h = exp(beta %*% t(mat_temp2))
choice_prob = h/(1+h)
llf <- sum(data_temp$admit * log(choice_prob)) + (sum((one-data_temp$admit) * log(one-choice_prob)))
return(-1*llf)
}
#Score to be used when optimizing using LL
#Identical to the Score function below but returns negative output
Score_LL <- function(beta){
data_temp=data_analysis
mat_temp2 = cbind(data_temp$constant,
data_temp$gpa)
one = rep(1,dim(mat_temp2)[1])
h = exp(beta %*% t(mat_temp2))
choice_prob = h/(1+h)
resid = as.numeric(data_temp$admit - choice_prob)
score_final2 = t(mat_temp2) %*% Diagonal(length(resid), x=resid) %*% one
return(-1*as.numeric(score_final2))
}
#The Score/Deriv/Jacobian of the Likelihood function
Score <- function(beta){
data_temp=data_analysis
mat_temp2 = cbind(data_temp$constant,
data_temp$gpa)
one = rep(1,dim(mat_temp2)[1])
h = exp(beta %*% t(mat_temp2))
choice_prob = as.numeric(h/(1+h))
resid = as.numeric(data_temp$admit - choice_prob)
score_final2 = t(mat_temp2) %*% Diagonal(length(resid), x=resid) %*% one
return(as.numeric(score_final2))
}
#Derivative of the Score function
Score_Deriv <- function(beta){
data_temp=data_analysis
mat_temp2 = cbind(data_temp$constant,
data_temp$gpa)
one = rep(1,dim(mat_temp2)[1])
h = exp(beta %*% t(mat_temp2))
weight = (h/(1+h)) * (1- (h/(1+h)))
weight_mat = Diagonal(length(weight), x=weight)
deriv = t(mat_temp2)%*%weight_mat%*%mat_temp2
return(-1*as.array(deriv))
}
#Quadratic Gain function
#Minimized at Score=0 and so minimizing is equivalent to solving the
#FOC of the Likelihood. This is the GMM approach.
Quad_Gain<- function(beta){
h=Score(as.numeric(beta))
return(sum(h*h))
}
#Derivative of the Quadratic Gain function
Quad_Gain_deriv <- function(beta){
return(2*t(Score_Deriv(beta))%*%Score(beta))
}
sol1=glm(admit ~ gpa, data = data_analysis, family = "binomial")
sol2=optim(c(2,2),Quad_Gain,gr=Quad_Gain_deriv,method="BFGS")
sol3=optim(c(0,0),Quad_Gain,gr=Quad_Gain_deriv,method="BFGS")
When I run this code, I get that sol3 matches what glm produces (sol1) but sol2, with a different initial point, differs from the glm solution by a lot. This is something happening in my main code with the actual data as well. One solution is to create a grid and test multiple starting points. However, my main data set has 10 parameters and this would make the grid very large and the program computationally infeasible. Is there a way around this problem?

Answer 1

Your code seems overly complicated. The following two functions define the negative log-likelihood and negative score vector for a logistic regression with the logit link:
logLik_Bin <- function (betas, y, X) {
eta <- c(X %*% betas)
- sum(dbinom(y, size = 1, prob = plogis(eta), log = TRUE))
}
score_Bin <- function (betas, y, X) {
eta <- c(X %*% betas)
- crossprod(X, y - plogis(eta))
}
Then you can use it as follows:
# load the data
mydata <- read.csv("https://stats.idre.ucla.edu/stat/data/binary.csv")
# fit with optim()
opt1 <- optim(c(-1, 1, -1), logLik_Bin, score_Bin, method = "BFGS",
y = mydata$admit, X = cbind(1, mydata$gre, mydata$gpa))
opt1$par
# compare with glm()
glm(admit ~ gre + gpa, data = mydata, family = binomial())
Typically, for well-behaved covariates (i.e., expecting to have a coefficients in the interval [-4 to 4]), starting at 0 is a good idea.