How to incorporate weights into a likelihood function?

Question

I want to incorporate weights into the likelihood to do what the svyglm does with weights.

According to Jeremy Miles and elsewhere, svyglm function uses weights to "weight the importance of each case to make them representative (to each other, ...)".

Here is my data:

(dat <- data.frame(
  A = c(1, 1, 0, 0), B = c(1, 0, 1, 0),
  Pass = c(278, 100, 153, 79), Fail = c(743, 581, 1232, 1731), Weights= c(3, 1, 12, 3)
))

Here is my likelihood function:

ll <- function (b0, b1, b2, b3) {
  odds <- exp(b0) * (1 + b1 * dat$A + b2 * dat$B + b3 * dat$A * dat$B)
  -sum(dbinom(
    x = dat$Pass, size = rowSums(dat[, 3:4]),
    prob = odds / (1 + odds), log = TRUE))
}

Answer 1

As it is said in your referred answers, different weights are used differently in different contexts. In your current example I don't really see any populations, while the goal is pretty clear: for each observation to have a specified "importance". Then the weighted maximum likelihood would simply use

ll <- function (b0, b1, b2, b3) {
  odds <- exp(b0) * (1 + b1 * dat$A + b2 * dat$B + b3 * dat$A * dat$B)
  -sum(dat$Weights * dbinom(
    x = dat$Pass, size = rowSums(dat[, 3:4]),
    prob = odds / (1 + odds), log = TRUE))
}

That can indeed be interpreted as including the i-th observation to the sample w_i times (this interpretation, of course, is exactly true only for integer weights, while the method works with any weights):