Prediction after Hurdle model

Question

I am estimating the effect of future expectations of profits on hospital admissions and days of hospital stays. The situation is as follows: Hospitals are paid a fixed tariff for the first x number of days of hospital stay, at x+1 days the tariff increases considerably. I am testing whether the relative increase in tariffs (or the ratio of tariff_1 and tariff_2) affects the decision whether to admit the patient today and how long the patient will be kept in the hospital.

My dataset includes number of hospital stays, tariff1 and tariff2 along with a complete set of individual health characteristics.

Due to the large number of "zero" hospital days, I would like to run a hurdle model with negative binomial distribution.

Here is a play example:

days <- c(rep(0, 30) , sample(0:100, 70, replace=TRUE))
tariff1 <- sample(1000:2000, 100, replace=TRUE) 
tariff2 <- sample(2000:4000, 100, replace=TRUE) 

dta <- data.frame(days=days, tariff1=tariff1, tariff2=tariff2)

dta$rate <- (dta$tariff2 - dta$tariff1)/dta$tariff1

library(pscl)

model <- hurdle(days ~ tariff1 + rate , data = dta, dist = "negbin")
summary(model)

After this I would like to estimate what this distribution would look like if tariff1=tariff2.

# Prediction:
new <-dta
new$rate <- 0

pred_old <- predict(model,data=dta, type = "response")
pred_new <- predict(model,data=new, type = "response")
pred <-cbind(pred_old, pred_new) # the two predictions are equal

My questions are as follows:

1. I was told that this kind of adjustment is incorrect. I would like to understand why this is.
2. What would be a correct way to evaluate a situation of tariff1=tariff2?
3. If correct, then why am I getting the same predictions for both?

Thank you.

Answer 1

I will answer your questions in order.

1.) Disclaimer, this first question I am less certain about, but here is my best estimate:

In including both the Tariff 1 and the ratio between Tariff 1 and the change to Tariff 2, you are likely including some collinearity, considering rate includes Tariff 1 in the data. In inputting both Tariff 1 and rate into your model equation, you are examining how Tariff 1 is effecting the count of days, while controlling for the rate between Tariff 1 and Tariff 2. It may be beneficial for you to examine either Tarrif1 controlling for Tariff2 (or vice versa), or rate by itself. If you try running:

model <- hurdle(days ~ tariff1+tariff2, data = dta, dist = "negbin")
summary(model)

You will see that both variables will return values for estimate, standard error, z value and p-value. If you run:

model <- hurdle(days ~ tariff1 + rate , data = dta, dist = "negbin")
summary(model)

you will get NAs for rate value. If rate is run by itself, you will get values for all.

model <- hurdle(days ~ rate, data=dta,dist='negbin')
summary(model)

2.) A correct way to examine Tariff1=Tariff2 would be to create categorical variables for your observations, like so:

dta$cat=ifelse(dta$tariff1==dta$tariff2,1,0)
model <- hurdle(days ~ cat, data = dta, dist = "negbin")
summary(model)

3.) You are likely getting the same predictions for both because you are defining the model in your prediction calls that includes this model:

model <- hurdle(days ~ tariff1 + rate , data = dta, dist = "negbin")

where

dta

is your data in that model. Redefining your data in your prediction call will not alter the model you created earlier. You should define and predict separate models like so:

model <- hurdle(days ~ tariff1 + rate , data = dta, dist = "negbin")
model2 <- hurdle(days ~ tariff1 + rate , data = new, dist = "negbin")
pred_old <- predict(model, type = "response")
pred_new <- predict(model2, type = "response")

Again, you should change your model formula to not include a tariff and rate in the same model.