Survival for each strategy
Define probabilities
Probablities of being in each stage.
stage.probabilities <- c(IIA = 0.8, IIIA1 = 0.15, IIIA2 = 0.05)
5-year survival probabilities given radiotherapy (RT) or optimal therapy.
survival.with.RT <- c(IIA = 0.92, IIIA1 = 0.8, IIIA2 = 0.7)
survival.with.optimal.Tx <- c(IIA = 0.92, IIIA1 = 0.9, IIIA2 = 0.85)
Probability of complication from laparotomy.
laparotomy.complication.probability <- 0.005
Calculate expected 5-year survival probability for radiotherapy without laparotomy strategy.
Element-by-element multiplication, then summation.
stage.probabilities * survival.with.RT
## IIA IIIA1 IIIA2
## 0.736 0.120 0.035
expected.survival.no.laparotomy <- sum(stage.probabilities * survival.with.RT)
expected.survival.no.laparotomy
## [1] 0.891
Calculate expected 5-year survival probability for for laparotomy-then-optimal therapy strategy.
Element-by-element multiplication, summation, then multiplication by no complication probability.
stage.probabilities * survival.with.optimal.Tx
## IIA IIIA1 IIIA2
## 0.7360 0.1350 0.0425
expected.survival.laparotomy <- sum((stage.probabilities * survival.with.optimal.Tx)) *
(1 - laparotomy.complication.probability)
expected.survival.laparotomy
## [1] 0.9089
a. Utility of diagnostic laparotomy
As 0.9089 > 0.891, performing laparotomy to accurately stage the patient's disease the worth the risk, and increases the expected 5-year survival of the patient.
b. Oneway sensitivity analysis for operative mortality
If laparotomy had no risk, the expected survival from the laparotomy-first strategy would be 0.9135. Let \( X \) be the value of complication probability at which both strategies have the same expected survival, then 0.9135 * (1 - X) = 0.891. Thus, X = 1 - 0.891 / 0.9135 = 0.0246. Therefore, if the complication probability is less than 2.46%, the laparotomy-first strategy is more beneficial.
Expected survival in two strategies given a range of operative mortality
sequence.of.complication.probabilities <- seq(from = 0, to = 1, by = 0.001)
laparotomy.complication.probability <- sequence.of.complication.probabilities
expected.survival.laparotomy.seq <- sum((stage.probabilities * survival.with.optimal.Tx)) *
(1 - laparotomy.complication.probability)
survival.probabilities <- data.frame(laparotomy.compl.probab = rep(laparotomy.complication.probability,
2), group = rep(c("no laparotomy", "laparotomy"), c(1001, 1001)), survival = c(rep(expected.survival.no.laparotomy,
1001), expected.survival.laparotomy.seq))
ggplot(survival.probabilities) + geom_line(aes(x = laparotomy.compl.probab,
y = survival, color = group)) + scale_x_continuous(name = "Probability of complication from laparotomy",
limits = c(0, 0.2)) + scale_y_continuous(limits = c(0.5, 1))
By looking at data:
library(reshape)
survival.probabilities.cast <- cast(data = survival.probabilities,
formula = laparotomy.compl.probab ~ group, value = "survival")
survival.probabilities.cast <- within(survival.probabilities.cast,
{
higher <- laparotomy > `no laparotomy`
higher <- factor(higher, levels = c("FALSE", "TRUE"), labels = c("no laparotomy",
"laparotomy"))
})
survival.probabilities.cast[20:30, ]
## laparotomy.compl.probab laparotomy no laparotomy higher
## 20 0.019 0.8961 0.891 laparotomy
## 21 0.020 0.8952 0.891 laparotomy
## 22 0.021 0.8943 0.891 laparotomy
## 23 0.022 0.8934 0.891 laparotomy
## 24 0.023 0.8925 0.891 laparotomy
## 25 0.024 0.8916 0.891 laparotomy
## 26 0.025 0.8907 0.891 no laparotomy
## 27 0.026 0.8897 0.891 no laparotomy
## 28 0.027 0.8888 0.891 no laparotomy
## 29 0.028 0.8879 0.891 no laparotomy
## 30 0.029 0.8870 0.891 no laparotomy
c. EVCI of diagnostic laparotomy
The expected value of clinical information (EVCI) is by definition the difference between the averated-out outcome value with the test and the averaged-out outcome value without the test (Hunink, 2001, page 173). Thus, it is expected survival of laparotomy-first stragegy - expected survival of no-laparotomy strategy = 0.0179 (net EVCI). If no risk of laparotomy is considered, it is 0.0225 (gross EVCI).
For other information: http://rpubs.com/kaz_yos/
If you find errors: kazky AT mac.com