Issues related to Gamma Regression - r

Question

I want to derive coefficients of Gamma regression by iterated reweighted method (manually). When I run this code with out for{} loop it works properly but with loop it produce NaN. My code is:
n<-10
y <- rgamma(n, 10, 0.1)
x1 <- rnorm(n, -1,1)
x2 <- rnorm(n, -1,1)
x3 <- rnorm(n, -1,1)
x<-as.matrix(cbind(1,x1,x2,x3))
reg <-glm(y~x1+x2+x3, family=Gamma(link = "inverse"))
### step 1
W<-G<-matrix(0,ncol=length(y),nrow=length(y))
b<-rep(0,4)
for(i in 1:50) {
### step 2
eta<-x%*%b
mu<-pnorm(eta)
diag(G)<-1/dnorm(eta)
z<-eta + G%*%(y - mu)
diag(W)<-(dnorm(eta)^2)/(mu*(1-mu))
### step 3
b <- solve(t(x)%*%W%*%x)%*%t(x)%*%W%*%z
}
Kindly help. My 2nd question is related to glm(). Is there any way which describe that how many iterations has glm() used?
Regards.
Updates
with help of this I update this code but its not working.
library(gnlm)
# custom link / inverse
inv <- function(eta) -1/(eta)
n<-10
y <- rgamma(n, 10, 0.1)
x1 <- rnorm(n, -1,1)
x2 <- rnorm(n, -1,1)
x3 <- rnorm(n, -1,1)
x<-as.matrix(cbind(1,x1,x2,x3))
reg <-glm(y~x1+x2+x3, family=Gamma(link = "inverse"))
library(gnlm)
reg1<- gnlr(y=y,
distribution = "gamma",
mu = ~ inv(beta0 + beta1*x1 + beta2*x2 + beta3*x3),
pmu = list(beta0=1, beta1=1, beta2=1, beta3=1),
pshape=0.1
)
I want to derive reg and reg1 same results.
Kindly help.

Answer 1

For the first code chunk, the algorithm is for probit regression, not gamma. To perform the iterations manually using glm's default of no weights and no offset for family = Gamma(link = "inverse"), update the code as follows.
n <- 10
y <- rgamma(n, 10, 0.1)
x1 <- rnorm(n, -1,1)
x2 <- rnorm(n, -1,1)
x3 <- rnorm(n, -1,1)
x <- as.matrix(cbind("(Intercept)" = 1,x1,x2,x3))
reg <- glm(y~x1+x2+x3, family = Gamma(link = "inverse"))
### step 1
eta <- 1/y
for(i in 1:reg$iter) {
tX <- t(X <- x/eta)
b <- drop(solve(tX%*%X)%*%tX%*%(2 - y*eta))
eta <- drop(x %*% b)
}
reg$iter is the number of iterations performed by the glm function. Check that b is equal to the coefficients given by glm:
all.equal(reg$coefficients, b)
#> [1] TRUE

Answer 2

Your inverse function is negative. Take away the minus sign.
Also, change pshape to 1.0.
I'm setting a seed for reproducibility.
Initial values for small datasets is key. Setting them using glm results is a common approach if you can get a similar enough link. Another approach would be that in the answer by #jblood94. Yet another one would be to use nls() for (rough) initial estimates.
argument trace=TRUE in glm() will show how many iterations
set.seed(111)
library(gnlm)
# custom link / inverse
inv <- function(eta) 1/(eta)
n<-10
y <- rgamma(n, 10, 0.1)
x1 <- rnorm(n, -1,1)
x2 <- rnorm(n, -1,1)
x3 <- rnorm(n, -1,1)
x<-as.matrix(cbind(1,x1,x2,x3))
reg <-glm(y~x1+x2+x3, family=Gamma(link = "inverse"), trace=TRUE)
library(gnlm)
reg1<- gnlr(y=y,
distribution = "gamma",
mu = ~ inv(beta0 + beta1*x1 + beta2*x2 + beta3*x3),
pmu = c(0.002, -0.002, -0.001, -0.001), ## or set to reg$coeff,
pshape=1
)
cbind(c(reg$coeff,NA), reg1$coeff)
Which gives:
> cbind(c(reg$coeff,NA), reg1$coeff)
[,1] [,2]
(Intercept) 0.0033899338 0.0033914440
x1 -0.0037481699 -0.0037476263
x2 -0.0007462714 -0.0007463346
x3 -0.0014941431 -0.0014936034
NA 2.8592334563
An example of different link and using nls to get starting values:
nls.init3 <-
nls(y ~ beta0 + 1/(beta1+1)*x1 + sqrt(beta2)*x2 + beta3^2*x3,
data=data.frame(y=y, x1=x1, x2=x2, x3=x3),
start=list(beta0=1,beta1=.1,beta2=.1,beta3=.1)
)
summary(nls.init3)$coefficients[,1]
reg3<- gnlr(y=y,
distribution = "gamma",
mu = ~ beta0 + 1/(beta1+1)*x1 + sqrt(beta2)*x2 + beta3^2*x3,
pmu = summary(nls.init3)$coefficients[,1],
pshape=1
)
reg3$coeff
And another
nls.init4 <-
nls(y ~ exp(beta0 + 1/(beta1+1)*x1),
data=data.frame(y=y, x1=x1),
start=list(beta0=0, beta1=0)
)
summary(nls.init4)$coefficients[,1]
reg4<- gnlr(y=y,
distribution = "gamma",
mu = ~ exp(beta0 + 1/(beta1+1)*x1),
pmu = summary(nls.init4)$coefficients[,1],
pshape=1
)
reg4$coeff