RDS 286: Case #5A Utility Assessment

Part I.

1. What are the patient’s utilities for the four outcomes ( R), (S), (A), and (D)?
By definition, the best outcome ( R) has a utility of 1, and the worst outcome (D) has a utility of 0. By standard gamble, the patient took 40% risk of death to avoid amputation for sure, so the amputation status has a utility of 0.6.

By gambling between the intermediate outcome (S) for sure and ( R) vs (A), the patient took 0.2 risk of (A). This can be solved by the chained gamble method taking (A) as the anchor health state that has been tested against both the best and the worst utilities. By averaging out from a combined tree (Weinstein et al, 1980), it should be 0.8 + 0.2 * 0.6 = 0.92.

Thus, ( R) 1, (S) 0.92, (A) 0.60, and (D) 0.

  1. How would this patient respond to the following question if his utilities are internally consistent?

Let p be the probability of (S), which has a utility value of 0.92. To balance between (A) for sure, which has a utility value of 0.6, p has to meet the below condition.
0.92 * p = 0.6
Thus, p = 0.6 / 0.92 = 0.6522.

Part II.

i. Plot this patient’s utility function for survival.

First, survival for 25 years is given utility of 1, and immediate death is given utility of 0 as reference points.

Then, this can be solved using “The Certainty Equivalent of a Gamble” (p 210, Weinstein et al, 1980). For this person, the certainty equivalent of a gamble between 50% chance of the longest survival and immediate death is 5 years from A. The utility from gamble A (1 + 0) / 2 = 0.5. Thus, survival for 5 years has a utility of 0.5. In the gamble between immediate death (utility = 0) and survival for 5 years (utility = 0.5), the patient chose 2 years, thus the patient's utility for survival for 2 years is (0 + 0.5) / 2 = 0.25. Similarly, in the gamble between survival for 5 years (utility = 0.5) and survival for 25 years (utility = 1, by definition), the patient chose 12 years, giving it a utility of 0.75. Therefore, the graph can be plotted as below.

dat.p2i <- data.frame(Longevity = c(0, 2, 5, 12, 25), Utility = c(0, 0.25, 0.5, 
    0.75, 1))
dat.p2i

##   Longevity Utility
## 1         0    0.00
## 2         2    0.25
## 3         5    0.50
## 4        12    0.75
## 5        25    1.00


ggplot(dat.p2i) + geom_line(aes(x = Longevity, y = Utility)) + geom_abline(slope = 1/25, 
    intercept = 0, linetype = 2, color = "gray")

plot of chunk unnamed-chunk-1

ii. Is this patient risk-averse, risk-neutral, or risk-seeking with respect to life years?

The patient is asking for less time for sure than the expected survival from the gambling. Thus, the patient rather has a shorter time of survival for sure than gamble, which on average gives the patient more time to live. “If the certainty equivalent is less than the expected value, the individual is said to be risk averse with respect to longevity” according to Weinstein et al, 1980. Therefore, the patient is risk-averse.

iii. Change the bold-faced numbers in the statements above to describe the preferences of a patient who is risk-neutral with respect to life years.

In a risk-neutral patient, the expected survival from the gamble should equal the time the patient wants for sure instead of the gamble.

iv. Change the bold-faced numbers in the above statements to describe the preferences of a patient who is risk-seeking with respect to life years.

A risk-seeking patient would go for the gamble even if the time which the patient can get for sure instead of gambling is the same as the expected value for the gambling.