Multiple Logistic Regression with Interaction between Quantitative and Qualitative Explanatory Variables

Question

As a follow up to this question, I fitted the Multiple Logistic Regression with Interaction between Quantitative and Qualitative Explanatory Variables. MWE is given below:

Answer 1

Your link function of choice (\eta= X\hat\beta) has variance for a new observation (x_0): V_{x_0}=x_0^T(X^TWX)^{-1}x_0

So, for a set of candidate doses, we can predict the expected percentage of deaths using the inverse function:

newdata= data.frame(Type=rep(x=LETTERS[1:3], each=5),
                    Conc=rep(x=seq(from=0, to=40, by=10), times=3))
mm <- model.matrix(fm1, newdata)

# get link on link terms (could also use predict)
eta0 <- apply(mm, 1, function(i) sum(i * coef(fm1)))

# inverse logit function
ilogit <- function(x) return(exp(x) / (1+ exp(x)))

# predicted probs
ilogit(eta0)


# for comfidence intervals we can use a normal approximation
lethal_dose <- function(mod, newdata, alpha) {
  qn <- qnorm(1 - alpha /2)
  mm <- model.matrix(mod, newdata)
  eta0 <- apply(mm, 1, function(i) sum(i * coef(fm1)))
  var_mod <- vcov(mod)

  se <- apply(mm, 1, function(x0, var_mod) {
    sqrt(t(x0) %*% var_mod %*% x0)}, var_mod= var_mod)

  out <- cbind(ilogit(eta0 - qn * se),
               ilogit(eta0),
               ilogit(eta0 + qn * se))
  colnames(out) <- c("LB_CI", "point_est", "UB_CI")

  return(list(newdata=newdata,
              eff_dosage= out))
}

lethal_dose(fm1, newdata, alpha= 0.05)$eff_dosage
$eff_dosage
       LB_CI point_est     UB_CI
1  0.2465905 0.3418240 0.4517820
2  0.4361703 0.5152749 0.5936215
3  0.6168088 0.6851225 0.7462674
4  0.7439073 0.8166343 0.8722545
5  0.8315325 0.9011443 0.9439316
6  0.1863738 0.2685402 0.3704385
7  0.3289003 0.4044270 0.4847691
8  0.4890420 0.5567386 0.6223914
9  0.6199426 0.6990808 0.7679095
10 0.7207340 0.8112133 0.8773662
11 0.1375402 0.2112382 0.3102215
12 0.3518053 0.4335213 0.5190198
13 0.6104540 0.6862145 0.7531978
14 0.7916268 0.8620545 0.9113443
15 0.8962097 0.9469715 0.9736370

Rather than doing this manually, you could also manipulate:

predict.glm(fm1, newdata, se=TRUE)$se.fit