我想了解如何为逻辑回归估计生成预测区间。

有人建议我遵循 Collett 的Modeling Binary Data，第 2 版 p.98-99 中的程序。在实现了这个过程并将其与 R 进行比较之后predict.glm，我实际上认为这本书展示了计算置信区间的过程，而不是预测区间。

Collett 的过程的实现，以及与 的比较predict.glm，如下所示。

我想知道：我如何从这里生成预测区间而不是置信区间？

#Derived from Collett 'Modelling Binary Data' 2nd Edition p.98-99
#Need reproducible "random" numbers.
seed <- 67

num.students <- 1000
which.student <- 1

#Generate data frame with made-up data from students:
set.seed(seed) #reset seed
v1 <- rbinom(num.students,1,0.7)
v2 <- rnorm(length(v1),0.7,0.3)
v3 <- rpois(length(v1),1)

#Create df representing students
students <- data.frame(
    intercept = rep(1,length(v1)),
    outcome = v1,
    score1 = v2,
    score2 = v3
)
print(head(students))

predict.and.append <- function(input){
    #Create a vanilla logistic model as a function of score1 and score2
    data.model <- glm(outcome ~ score1 + score2, data=input, family=binomial)

    #Calculate predictions and SE.fit with the R package's internal method
    # These are in logits.
    predictions <- as.data.frame(predict(data.model, se.fit=TRUE, type='link'))

    predictions$actual <- input$outcome
    predictions$lower <- plogis(predictions$fit - 1.96 * predictions$se.fit)
    predictions$prediction <- plogis(predictions$fit)
    predictions$upper <- plogis(predictions$fit + 1.96 * predictions$se.fit)


    return (list(data.model, predictions))
}

output <- predict.and.append(students)

data.model <- output[[1]]

#summary(data.model)

#Export vcov matrix 
model.vcov <- vcov(data.model)

# Now our goal is to reproduce 'predictions' and the se.fit manually using the vcov matrix
this.student.predictors <- as.matrix(students[which.student,c(1,3,4)])

#Prediction:
this.student.prediction <- sum(this.student.predictors * coef(data.model))
square.student <- t(this.student.predictors) %*% this.student.predictors
se.student <- sqrt(sum(model.vcov * square.student))

manual.prediction <- data.frame(lower = plogis(this.student.prediction - 1.96*se.student), 
    prediction = plogis(this.student.prediction), 
    upper = plogis(this.student.prediction + 1.96*se.student))

print("Data preview:")
print(head(students))
print(paste("Point estimate of the outcome probability for student", which.student,"(2.5%, point prediction, 97.5%) by Collett's procedure:"))
manual.prediction
print(paste("Point estimate of the outcome probability for student", which.student,"(2.5%, point prediction, 97.5%) by R's predict.glm:"))    
print(output[[2]][which.student,c('lower','prediction','upper')])