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The Frechet-Hoeffding bounds provide a limit on values of a joint distribution given the observed marginal distributions.
Consider the case of trial results for a new drug, like the Emilia trial. From the trial we observe the distribution of outcomes (survival probabilities) for those on Kadcyla and the distribution of outcomes for those on the standard chemotherapy. If the trial was a large perfect randomized controlled trial, the observed distributions would provide a reasonable estimate of the distribution of potential outcomes for patients taking Kadcyla and the distribution of potential outcomes for those on the standard chemotherapy. With this information we could determine the Average Treatment Effect. But the average treatment effect only tells us that one person is better off with the new drug, it doesn't tell us how many people are better off. The Frechet-Hoeffding bounds allow us to use the distribution of potential outcomes for each treatment to bound the proportion of people who are better off with the new treatment.
From the chart above we see that about 65% of patients on Kadcyla are alive at 24 months, while only 52% of patients on the standard chemotherapy are alive at this point. From this can we determine that at least some people live longer on Kadcyla? Yes. Can we determine how many? Sort of.
We can use the Frechet-Hoeffding bounds to place range restrictions on the proportion of people who would live longer than 24 months on Kadcyla but less than 24 months on the standard treatment. From the diagram, we know that 48% of patients would live less than 24 months on the standard of treatment. How many of those people, would live longer than 24 months on Kadcyla?
To determine this we remove the people who would live less than 24 months on Kadcyla. We don't know exactly how many this is, but we can bound it. At a minimum it is zero. That is, all the people who live longer than 24 months on Kadcyla also would have died prior to the two year mark on the standard of care. So at the top end, 48% of patients would live longer than 24 months on Kadcyla but pass away prior to that mark on the standard chemo.
The maximum value for how many people would live less than 24 months on Kadcyla and less than 24 months on the standard chemo is 35%. That is, every person who passes away prior to 24 months on Kadcyla could have also passed away prior to 24 months on the standard care. Taking this percentage from our 48% number we get 13%.
The proportion of patients that would live longer than 24 months on Kadcyla but pass away prior to that mark on the standard chemotherapy lies between 13% and 48%.
Using these types of arguments (and assumptions on conduct of the clinical trial) we can use the chart above to determine that at least 13% of patients would live longer on Kadcyla than on the standard chemotherapy offered in the trial.
An R Example
An R Example
It is straightforward to calculated bounds on the proportion of patients who are better off under a treatment using a Kolmogorov-Smirnov Test in R.
Consider we have two distributions where 500 patients have outcomes that are distributed N(0,2) (normal distribution with zero mean and a standard deviation of 2), and 500 patients have outcomes that are distributed N(0.5,1). This example is due to John Mullahy.
Below is the R code. Note that you should get bounds that are around 0.10 to 0.25.
x1 <- rnorm(500, mean = 0, sd = 2) # creates pseudo-random numbers drawn from a normal distribution.
x2 <- rnorm(500, mean = 0.5, sd = 1)
lower_bound <- ks.test(x1, x2, alternative = "less") # runs a version of the Kolmogorov-Smirnov test.
lower_bound # just prints out the result
upper_bound <- ks.test(x1, x2, alternative = "greater")
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