## Uncertainty analysis of GiveWell's cost-effectiveness analysis

GiveWell produces cost-effectiveness models of its top charities. These models take as inputs many uncertain parameters. Instead of representing those uncertain parameters with point estimates—as the cost-effectiveness analysis spreadsheet does—we can (should) represent them with probability distributions. Feeding probability distributions into the models allows us to output explicit probability distributions on the cost-effectiveness of each charity.

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### GiveWell’s cost-effectiveness analysis

GiveWell, an in-depth charity evaluator, makes their detailed spreadsheets models available for public review. These spreadsheets estimate the value per dollar of donations to their 8 top charities: GiveDirectly, Deworm the World, Schistosomiasis Control Initiative, Sightsavers, Against Malaria Foundation, Malaria Consortium, Helen Keller International, and the END Fund. For each charity, a model is constructed taking input values to an estimated value per dollar of donation to that charity. The inputs to these models vary from parameters like “malaria prevalence in areas where AMF operates” to “value assigned to averting the death of an individual under 5”.

Helpfully, GiveWell isolates the input parameters it deems as most uncertain. These can be found in the “User inputs” and “Moral weights” tabs of their spreadsheet. Outsiders interested in the top charities can reuse GiveWell’s model but supply their own perspective by adjusting the values of the parameters in these tabs.

For example, if I go to the “Moral weights” tab and run the calculation with a 0.1 value for doubling consumption for one person for one year—instead of the default value of 1—I see the effect of this modification on the final results: deworming charities look much less effective since their primary effect is on income.

### Uncertain inputs

GiveWell provides the ability to adjust these input parameters and observe altered output because the inputs are fundamentally uncertain. But our uncertainty means that picking any particular value as input for the calculation misrepresents our state of knowledge. From a subjective Bayesian point of view, the best way to represent our state of knowledge on the input parameters is with a probability distribution over the values the parameter could take. For example, I could say that a negative value for increasing consumption seems very improbable to me but that a wide range of positive values seem about equally plausible. Once we specify a probability distribution, we can feed these distributions into the model and, in principle, we’ll end up with a probability distribution over our results. This probability distribution on the results helps us understand the uncertainty contained in our estimates and how literally we should take them.

#### Is this really necessary?

Perhaps that sounds complicated. How are we supposed to multiply, add and otherwise manipulate arbitrary probability distributions in the way our models require? Can we somehow reduce our uncertain beliefs about the input parameters to point estimates and run the calculation on those? One candidate is to take the single most likely value of each input and using that value in our calculations. This is the approach the current cost-effectiveness analysis takes (assuming you provide input values selected in this way). Unfortunately, the output of running the model on these inputs is necessarily a point value and gives no information about the uncertainty of the results. Because the results are probably highly uncertain, losing this information and being unable to talk about the uncertainty of the results is a major loss. A second possibility is to take lower bounds on the input parameters and run the calculation on these values, and to take the upper bounds on the input parameters and run the calculation on these values. This will produce two bounding values on our results, but it’s hard to give them a useful meaning. If the lower and upper bounds on our inputs describe, for example, a 95% confidence interval, the lower and upper bounds on the result don’t (usually) describe a 95% confidence interval.

#### Computers are nice

If we had to proceed analytically, working with probability distributions throughout, the model would indeed be troublesome and we might have to settle for one of the above approaches. But we live in the future. We can use computers and Monte Carlo methods to numerically approximate the results of working with probability distributions while leaving our models clean and unconcerned with these probabilistic details. Guesstimate is a tool that works along these lines and bills itself as “A spreadsheet for things that aren’t certain”.

### Analysis

We have the beginnings of a plan then. We can implement GiveWell’s cost-effectiveness models in a Monte Carlo framework (PyMC3 in this case), specify probability distributions over the input parameters, and finally run the calculation and look at the uncertainty that’s been propagated to the results.

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