# 65

(The same content is broken up into three posts and given a very slightly different presentation on my blog.)

# Overview

GiveWell models the cost-effectiveness of its top charities. Because the input parameters are uncertain (How much moral weight should we give to increasing consumption? What is the current income of a typical GiveDirectly recipient?), the resulting cost-effectiveness estimates are also fundamentally uncertain. By performing uncertainty analysis, we get a better sense of just how uncertain the results are. Uncertainty analysis is also the first step on the route to sensitivity analysis. Sensitivity analysis reveals which input parameters each charity's cost-effectiveness estimate is most sensitive to. That kind of information helps us target future investigations (i.e. uncertainty reduction). The final step is to combine the individual charity cost-effectiveness estimates into one giant model. By performing uncertainty and sensitivity analysis on this giant model, we get a better sense of which input parameters have the most influence on the relative cost-effectiveness of GiveWell's top charities—i.e. how the charities rank against each other.

A key feature of the analysis outlined above and performed below is that it requires the analyst to specify their uncertainty over each input parameter. Because I didn't want all of the results here to reflect my idiosyncratic beliefs, I instead pretended that each input parameter is equally uncertain. This makes the results "neutral" in a certain sense, but it also means that they don't reveal much about the real world. To achieve real insight, you need to adjust the input parameters to match your beliefs. You can do that by heading over to the Jupyter notebook, editing the parameters in the second cell, and clicking "Runtime > Run all". This limitation means that all the ensuing discussion is more akin to an analysis template than a true analysis.

# Uncertainty analysis of GiveWell's cost-effectiveness estimates

## Section overview

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.

## 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 you may have heard of 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.

### Model

The Python source code implementing GiveWell's models can be found on GitHub. The core models can be found in cash.py, nets.py, smc.py, worms.py and vas.py.

### Inputs

For the purposes of the uncertainty analysis that follows, it doesn't make much sense to infect the results with my own idiosyncratic views on the appropriate value of the input parameters. Instead, what I have done is uniformly taken GiveWell's best guess and added and subtracted 20%. These upper and lower bounds then become the 90% confidence interval of a log-normal distribution. For example, if GiveWell's best guess for a parameter is 0.1, I used a log-normal with a 90% CI from 0.08 to 0.12.

While this approach screens off my influence it also means that the results of the analysis will primarily tell us about the structure of the computation rather than informing us about the world. Fortunately, there's a remedy for this problem too. I have set up a Jupyter notebook with the all the input parameters to the calculation which you can manipulate and rerun the analysis. That is, if you think the moral weight given to increasing consumption ought to range from 0.8 to 1.5 instead of 0.8 to 1.2, you can make that edit and see the corresponding results. Making these modifications is essential for a realistic analysis because we are not, in fact, equally uncertain about every input parameter.

It's also worth noting that I have considerably expanded the set of input parameters receiving special scrutiny. The GiveWell cost-effectiveness analysis is (with good reason—it keeps things manageable for outside users) fairly conservative about which parameters it highlights as eligible for user manipulation. In this analysis, I include any input parameter which is not tautologically certain. For example, "Reduction in malaria incidence for children under 5 (from Lengeler 2004 meta-analysis)" shows up in the analysis which follows but is not highlighted in GiveWell's "User inputs" or "Moral weights" tab. Even though we don't have much information with which to second guess the meta-analysis, the value it reports is still uncertain and our calculation ought to reflect that.

### Results

Finally, we get to the part that you actually care about, dear reader: the results. Given input parameters which are each distributed log-normally with a 90% confidence interval spanning ±20% of GiveWell's best estimate, here are the resulting uncertainties in the cost-effectiveness estimates: Probability distributions of value per dollar for GiveWell's top charities

For reference, here are the point estimates of value per dollar using GiveWell's values for the charities:

GiveWell's cost-effectiveness estimates for its top charities

Charity Value per dollar
GiveDirectly 0.0038
The END Fund 0.0222
Deworm the World 0.0738
Schistosomiasis Control Initiative 0.0378
Sightsavers 0.0394
Malaria Consortium 0.0326
Helen Keller International 0.0223
Against Malaria Foundation 0.0247

I've also plotted a version in which the results are normalized—I divided the results for each charity by that charity's expected value per dollar. Instead of showing the probability distribution on the value per dollar for each charity, this normalized version shows the probability distribution on the percentage of that charity's expected value that it achieves. This version of the plot abstracts from the actual value per dollar and emphasizes the spread of uncertainty. It also reëmphasizes the earlier point that—because we use the same spread of uncertainty for each input parameter—the current results are telling us more about the structure of the model than about the world. For real results, go try the Jupyter notebook! Probability distributions for percentage of expected value obtained with each of GiveWell's top charities

## Section recap

Our preliminary conclusion is that all of GiveWell's top charities cost-effectiveness estimates have similar uncertainty with GiveDirectly being a bit more certain than the rest. However, this is mostly an artifact of pretending that we are exactly equally uncertain about each input parameter.

# Sensitivity analysis of GiveWell's cost effectiveness estimates

## Section overview

In the previous section, we introduced GiveWell's cost-effectiveness analysis which uses a spreadsheet model to take point estimates of uncertain input parameters to point estimates of uncertain results. We adjusted this approach to take probability distributions on the input parameters and in exchange got probability distributions on the resulting cost-effectiveness estimates. But this machinery lets us do more. Now that we've completed an uncertainty analysis, we can move on to sensitivity analysis.

The basic idea of sensitivity analysis is, when working with uncertain values, to see which input values most affect the output when they vary. For example, if you have the equation and each of and varies uniformly over the range from 5 to 10, is much more sensitive to then . A sensitivity analysis is practically useful in that it can offer you guidance as to which parameters in your model it would be most useful to investigate further (i.e. to narrow their uncertainty).

Visual (scatter plot) and delta moment-independent sensitivity analysis on GiveWell's cost-effectiveness models show which input parameters the cost-effectiveness estimates are most sensitive to. Preliminary results (given our input uncertainty) show that some input parameters are much more influential on the final cost-effectiveness estimates for each charity than others.

## Visual sensitivity analysis

The first kind of sensitivity analysis we'll run is just to look at scatter plots comparing each input parameter to the final cost-effectiveness estimates. We can imagine these scatter plots as the result of running the following procedure many times: sample a single value from the probability distribution for each input parameter and run the calculation on these values to determine a result value. If we repeat this procedure enough times, it starts to approximate the true values of the probability distributions.

(One nice feature of this sort of analysis is that we see how the output depends on a particular input even in the face of variations in all the other inputs—we don't hold everything else constant. In other words, this is a global sensitivity analysis.)

(Caveat: We are again pretending that we are equally uncertain about each input parameter and the results reflect this limitation. To see the analysis result for different input uncertainties, edit and run the Jupyter notebook.)

### Direct cash transfers

#### GiveDirectly Scatter plots showing sensitivity of GiveDirectly's cost-effectiveness to each input parameter

The scatter plots show that, given our choice of input uncertainty, the output is most sensitive (i.e. the scatter plot for these parameters shows the greatest directionality) to the input parameters:

Highlighted input factors to which result is highly sensitive

Input Type of uncertainty Meaning/importance
value of increasing ln consumption per capita per annum Moral Determines final conversion between empirical outcomes and value
transfer as percent of total cost Operational Determines cost of results
return on investment Opportunities available to recipients Determines stream of consumption over time
baseline consumption per capita Empirical Diminishing marginal returns to consumption mean that baseline consumption matters

### Deworming

Some useful and non-obvious context for the following is that the primary putative benefit of deworming is increased income later in life.

#### The END Fund Scatter plots showing sensitivity of the END Fund's cost-effectiveness to each input parameter

Here, it's a little harder to identify certain factors as more important. It seems that the final estimate is (given our input uncertainty) the result of many factors of medium effect. It does seem plausible that the output is somewhat less sensitive to these factors:

Highlighted input factors to which result is minimally sensitive

Input Type of uncertainty Meaning/(un)importance
num yrs between deworming and benefits Forecast Affects how much discounting of future income streams must be done
duration of long-term benefits Forecast The length of time for a which a person works and earns income
expected value from leverage and funging Game theoretic How much does money donated to the END Fund shift around other money

#### Deworm the World Scatter plots showing sensitivity of the Deworm the World's cost-effectiveness to each input parameter

Again, it's a little harder to identify certain factors as more important. It seems that the final estimate is (given our input uncertainty) the result of many factors of medium effect. It does seem plausible that the output is somewhat less sensitive to these factors:

Highlighted input factors to which result is minimally sensitive

Input Type of uncertainty Meaning/(un)importance
num yrs between deworming and benefits Forecast Affects how much discounting of future income streams must be done
duration of long-term benefits Forecast The length of time for a which a person works and earns income
expected value from leverage and funging Game theoretic How much does money donated to Deworm the World shift around other money

#### Schistosomiasis Control Initiative Scatter plots showing sensitivity of the Schistosomiasis Control Initiative's cost-effectiveness to each input parameter

Again, it's a little harder to identify certain factors as more important. It seems that the final estimate is (given our input uncertainty) the result of many factors of medium effect. It does seem plausible that the output is somewhat less sensitive to these factors:

Highlighted input factors to which result is minimally sensitive

Input Type of uncertainty Meaning/(un)importance
num yrs between deworming and benefits Forecast Affects how much discounting of future income streams must be done
duration of long-term benefits Forecast The length of time for a which a person works and earns income
expected value from leverage and funging Game theoretic How much does money donated to Schistosomiasis Control Initiative shift around other money

#### Sightsavers Scatter plots showing sensitivity of the Sightsavers' cost-effectiveness to each input parameter

Again, it's a little harder to identify certain factors as more important. It seems that the final estimate is (given our input uncertainty) the result of many factors of medium effect. It does seem plausible that the output is somewhat less sensitive to these factors:

Highlighted input factors to which result is minimally sensitive

Input Type of uncertainty Meaning/(un)importance
num yrs between deworming and benefits Forecast Affects how much discounting of future income streams must be done
duration of long-term benefits Forecast The length of time for a which a person works and earns income
expected value from leverage and funging Game theoretic How much does money donated to Sightsavers shift around other money

### Seasonal malaria chemoprevention

#### Malaria Consortium Scatter plots showing sensitivity of Malaria Consortium's cost-effectiveness to each input parameter

The scatter plots show that, given our choice of input uncertainty, the output is most sensitive (i.e. the scatter plot for these parameters shows the greatest directionality) to the input parameters:

Highlighted input factors to which result is highly sensitive

Input Type of uncertainty Meaning/importance
direct mortality in high transmission season Empirical Fraction of overall malaria mortality during the peak transmission season and amenable to SMC
internal validity adjustment Methodological How much do we trust the results of the underlying SMC studies
external validity adjustment Methodological How much do the results of the underlying SMC studies transfer to new settings
coverage in trials in meta-analysis Historical/methodological Determines how much coverage an SMC program needs to achieve to match studies
value of averting death of a young child Moral Determines final conversion between empirical outcomes and value
cost per child targeted Operational Affects cost of results

### Vitamin A supplementation

#### Helen Keller International Scatter plots showing sensitivity of the Helen Keller International's cost-effectiveness to each input parameter

The scatter plots show that, given our choice of input uncertainty, the output is most sensitive to the input parameters:

Highlighted input factors to which result is highly sensitive

Input Type of uncertainty Meaning/importance
relative risk of all-cause mortality for young children in programs Causal How much do VAS programs affect mortality
cost per child per round Operational Affects cost of results
rounds per year Operational Affects cost of results

### Bednets

#### Against Malaria Foundation Scatter plots showing sensitivity of Against Malaria Foundation's cost-effectiveness to each input parameter

The scatter plots show that, given our choice of input uncertainty, the output is most sensitive (i.e. the scatter plot for these parameters shows the greatest directionality) to the input parameters:

Highlighted input factors to which result is highly sensitive

Input Type of uncertainty Meaning/importance
num LLINs distributed per person Operational Affects cost of results
cost per LLIN Operational Affects cost of results
deaths averted per protected child under 5 Causal How effective is the core activity
lifespan of an LLIN Empirical Determines how many years of benefit accrue to each distribution
net use adjustment Empirical Determines benefits from LLIN as mediated by proper and improper use
internal validity adjustment Methodological How much do we trust the results of the underlying studies
percent of mortality due to malaria in AMF areas vs trials Empirical/historical Affects size of the problem
percent of pop. under 5 Empirical Affects size of the problem

## Delta moment-independent sensitivity analysis

If eyeballing plots seems a bit unsatisfying to you as a method for judging sensitivity, not to worry. We also have the results of a more formal sensitivity analysis. This method is called delta moment-independent sensitivity analysis.

(the delta moment-independent sensitivity indicator of parameter ) "represents the normalized expected shift in the distribution of [the output] provoked by [that input]". To make this meaning more explicit, we'll start with some notation/definitions. Let:

1. be the random variables used as input parameters
2. so that is a function from to describing the relationship between inputs and outputs—i.e. GiveWell's cost-effectiveness model
3. be the density function of the result —i.e. the probability distributions we've already seen showing the cost-effectiveness for each charity
4. be the conditional density of Y with one of the parameters fixed—i.e. a probability distribution for the cost-effectiveness of a charity while pretending that we know one of the input values precisely

With these in place, we can define . It is:

.

The inner can be interpreted as the total area between probability density function and probability density function . This is the "shift in the distribution of provoked by " we mentioned earlier. Overall, then says:

• pick one value for and measure the shift in the output distribution from the "default" output distribution
• do that for each possible and take the expectation

Some useful properties to point out:

• ranges from 0 to 1
• If the output is independent of the input, for that input is 0
• The sum of for each input considered separately isn't necessarily 1 because there can be interaction effects

In the plots below, for each charity, we visualize the delta sensitivity (and our uncertainty about that sensitivity) for each input parameter.

### Direct cash transfers

#### GiveDirectly Delta sensitivities for each input parameter in the GiveDirectly cost-effectiveness calculation

Comfortingly, this agrees with the results of our scatter plot sensitivity analysis. For convenience, I have copied the table from the scatter plot analysis describing the most influential inputs:

Highlighted input factors to which result is highly sensitive

Input Type of uncertainty Meaning/importance
value of increasing ln consumption per capita per annum Moral Determines final conversion between outcomes and value
transfer as percent of total cost Operational Affects cost of results
return on investment Opportunities available to recipients Determines stream of consumption over time
baseline consumption per capita Empirical Diminishing marginal returns to consumption mean that baseline consumption matters

### Deworming

#### The END Fund Delta sensitivities for each input parameter in the END Fund cost-effectiveness calculation

Comfortingly, this again agrees with the results of our scatter plot sensitivity analysis. For convenience, I have copied the table from the scatter plot analysis describing the least influential inputs:

Highlighted input factors to which result is minimally sensitive

Input Type of uncertainty Meaning/(un)importance
num yrs between deworming and benefits Forecast Affects how much discounting of future income streams must be done
duration of long-term benefits Forecast The length of time for a which a person works and earns income
expected value from leverage and funging Game theoretic How much does money donated to the END Fund shift around other money

#### Deworm the World Delta sensitivities for each input parameter in the Deworm the World cost-effectiveness calculation

For convenience, I have copied the table from the scatter plot analysis describing the least influential inputs:

Highlighted input factors to which result is minimally sensitive

Input Type of uncertainty Meaning/(un)importance
num yrs between deworming and benefits Forecast Affects how much discounting of future income streams must be done
duration of long-term benefits Forecast The length of time for a which a person works and earns income
expected value from leverage and funging Game theoretic How much does money donated to Deworm the World shift around other money

#### Schistosomiasis Control Initiative Delta sensitivities for each input parameter in the Schistosomiasis Control Initiative cost-effectiveness calculation

For convenience, I have copied the table from the scatter plot analysis describing the least influential inputs:

Highlighted input factors to which result is minimally sensitive

Input Type of uncertainty Meaning/(un)importance
num yrs between deworming and benefits Forecast Affects how much discounting of future income streams must be done
duration of long-term benefits Forecast The length of time for a which a person works and earns income
expected value from leverage and funging Game theoretic How much does money donated to Schistosomiasis Control Initiative shift around other money

#### Sightsavers Delta sensitivities for each input parameter in the Sightsavers cost-effectiveness calculation

For convenience, I have copied the table from the scatter plot analysis describing the least influential inputs:

Highlighted input factors to which result is minimally sensitive

Input Type of uncertainty Meaning/(un)importance
num yrs between deworming and benefits Forecast Affects how much discounting of future income streams must be done
duration of long-term benefits Forecast The length of time for a which a person works and earns income
expected value from leverage and funging Game theoretic How much does money donated to Sightsavers shift around other money

#### Deworming comment

That we get substantially identical results in terms of delta sensitivities for each deworming charity is not surprising: The structure of each calculation is the same and (for the sake of not tainting the analysis with my idiosyncratic perspective) the uncertainty on each input parameter is the same.

### Seasonal malaria chemoprevention

#### Malaria Consortium Delta sensitivities for each input parameter in the Malaria Consortium cost-effectiveness calculation

Again, there seems to be good agreement between the delta sensitivity analysis and the scatter plot sensitivity analysis though there is perhaps a bit of reordering in the top factor. For convenience, I have copied the table from the scatter plot analysis describing the most influential inputs:

Highlighted input factors to which result is highly sensitive

Input Type of uncertainty Meaning/importance
internal validity adjustment Methodological How much do we trust the results of the underlying SMC studies)
direct mortality in high transmission season Empirical Fraction of overall malaria mortality during the peak transmission season and amenable to SMC
cost per child targeted Operational Afffects cost of results
external validity adjustment Methodological How much do the results of the underlying SMC studies transfer to new settings
coverage in trials in meta-analysis Historical/methodological Determines how much coverage an SMC program needs to achieve to match studies
value of averting death of a young child Moral Determines final conversion between outcomes and value

### Vitamin A supplementation

#### Hellen Keller International Delta sensitivities for each input parameter in the Helen Keller International cost-effectiveness calculation

Again, there's broad agreement between the scatter plot analysis and this one. This analysis perhaps makes the crucial importance of the relative risk of all-cause mortality for young children in VAS programs even more obvious. For convenience, I have copied the table from the scatter plot analysis describing the most influential inputs:

Highlighted input factors to which result is highly sensitive

Input Type of uncertainty Meaning/importance
relative risk of all-cause mortality for young children in programs Causal How much do VAS programs affect mortality
cost per child per round Operational Affects the total cost required to achieve effect
rounds per year Operational Affects the total cost required to achieve effect

### Bednets

#### Against Malaria Foundation Delta sensitivities for each input parameter in the Against Malaria Foundation cost-effectiveness calculation

Again, there's broad agreement between the scatter plot analysis and this one. For convenience, I have copied the table from the scatter plot analysis describing the most influential inputs:

Highlighted input factors to which result is highly sensitive

Input Type of uncertainty Meaning/importance
num LLINs distributed per person Operational Affects the total cost required to achieve effect
cost per LLIN Operational Affects the total cost required to achieve effect
deaths averted per protected child under 5 Causal How effective is the core activity
lifespan of an LLIN Empirical Determines how many years of benefit accrue to each distribution
net use adjustment Empirical Affects benefits from LLIN as mediated by proper and improper use
internal validity adjustment Methodological How much do we trust the results of the underlying studies
percent of mortality due to malaria in AMF areas vs trials Empirical/historical Affects size of the problem
percent of pop. under 5 Empirical Affects size of the problem

## Section recap

We performed visual (scatter plot) sensitivity analyses and delta moment-independent sensitivity analyses on GiveWell's top charities. Conveniently, these two methods generally agreed as to which input factors had the biggest influence on the output. For each charity, we found that there were clear differences in the sensitivity indicators for different inputs.

This suggests that certain inputs are better targets than others for uncertainty reduction. For example, the overall estimate of the cost-effectiveness of Helen Keller International's vitamin A supplementation program depends much more on the relative risk of all-cause mortality for children in VAS programs than it does on the expected value from leverage and funging. If the cost of investigating each were the same, it would be better to spend time on the former.

An important caveat to remember is that these results still reflect my fairly arbitrary (but scrupulously neutral) decision to pretend that we equally uncertain about each input parameter. To remedy this flaw, head over to the Jupyter notebook and tweak the input distributions.

# Uncertainty and sensitivity analysis of GiveWell's ranking

## Section overview

In the last two sections, we performed uncertainty and sensitivity analyses on GiveWell's charity cost-effectiveness estimates. Our outputs were, respectively:

• probability distributions describing our uncertainty about the value per dollar obtained for each charity and
• estimates of how sensitive each charity's cost-effectiveness is to each of its input parameters

One problem with this is that we are not supposed to take the cost-effectiveness estimates literally. Arguably, the real purpose of GiveWell's analysis is not to produce exact numbers but to assess the relative quality of each charity evaluated.

Another issue is that by treating each cost-effectiveness estimate as independent we underweight parameters which are shared across many models. For example, the moral weight that ought to be assigned to increasing consumption shows up in many models. If we consider all the charity-specific models together, this input seems to become more important.

Our solution to these problems will be to use distance metrics on the overall charity rankings. By using distance metrics across these multidimensional outputs, we can perform uncertainty and sensitivity analysis to answer questions about:

• how uncertain we are about the overall relative cost-effectiveness of the charities
• which input parameters this overall relative cost-effectiveness is most sensitive to

## Metrics on rankings

Our first step on the path to a solution is to abstract away from particular values in the cost-effectiveness analysis and look at the overall rankings returned. That is we want to transform:

GiveWell's cost-effectiveness estimates for its top charities

Charity Value per \$10,000 donated
GiveDirectly 38
The END Fund 222
Deworm the World 738
Schistosomiasis Control Initiative 378
Sightsavers 394
Malaria Consortium 326
Against Malaria Foundation 247
Helen Keller International 223

into:

Givewell's top charities ranked from most cost-effective to least

• Deworm the World
• Sightsavers
• Schistosomiasis Control Initiative
• Malaria Consortium
• Against Malaria Foundation
• Helen Keller International
• The END Fund
• GiveDirectly

But how do we usefully express probabilities over rankings (rather than probabilities over simple cost-effectivness numbers)? The approach we'll follow below is to characterize a ranking produced by a run of the model by computing its distance from the reference ranking listed above (i.e. GiveWell's current best estimate). Our output probability distribution will then express how far we expect to be from the reference ranking—how much we might learn about the ranking with more information on the inputs. For example, if the distribution is narrow and near 0, that means our uncertain input parameters mostly produce results similar to the reference ranking. If the distribution is wide and far from 0, that means our uncertain input parameters produce results that are highly uncertain and not necessarily similar to the reference ranking.

### Spearman's footrule

What is this mysterious distance metric between rankings that enables the above approach? One such metric is called Spearman's footrule distance. It's defined as:

where:

• and are rankings,
• varies over all the elements of the rankings and
• returns the integer position of item in ranking .

In other words, the footrule distance between two rankings is the sum over all items of the (absolute) difference in positions for each item. (We also add a normalization factor so that the distance varies ranges from 0 to 1 but omit that trivia here.)

So the distance between A, B, C and A, B, C is 0; the (unnormalized) distance between A, B, C and C, B, A is 4; and the (unnormalized) distance between A, B, C and B, A, C is 2.

### Kendall's tau

Another common distance metric between rankings is Kendall's tau. It's defined as:

where:

• and are again rankings,
• and are items in the set of unordered pairs of distinct elements in and
• if and are in the same order (concordant) in and and otherwise (discordant)

In other words, the Kendall tau distance looks at all possible pairs across items in the rankings and counts up the ones where the two rankings disagree on the ordering of these items. (There's also a normalization factor that we've again omitted so that the distance ranges from 0 to 1.)

So the distance between A, B, C and A, B, C is 0; the (unnormalized) distance between A, B, C and C, B, A is 3; and the (unnormalized) distance between A, B, C and B, A, C is 1.

### Angular distance

One drawback of the above metrics is that they throw away information in going from the table with cost-effectiveness estimates to a simple ranking. What would be ideal is to keep that information and find some other distance metric that still emphasizes the relationship between the various numbers rather than their precise values.

Angular distance is a metric which satisfies these criteria. We can regard the table of charities and cost-effectiveness values as an 8-dimensional vector. When our output produces another vector of cost-effectiveness estimates (one for each charity), we can compare this to our reference vector by finding the angle between the two.

## Results

### Uncertainties

To recap, what we're about to see next is the result of running our model many times with different sampled input values. In each run, we compute the cost-effectiveness estimates for each charity and compare those estimates to the reference ranking (GiveWell's best estimate) using each of the tau, footrule and angular distance metrics. Again, the plots below are from running the analysis while pretending that we're equally uncertain about each input parameter. To avoid this limitation, go to the Jupyter notebook and adjust the input distributions. Probability distributions of value per dollar for each of GiveWell's top charity and probability distributions for the distance between model results and the reference results

We see that our input uncertainty does matter even for these highest level results—there are some input values which cause the ordering of best charities to change. If the gaps between the cost-effectiveness estimates had been very large or our input uncertainty had been very small, we would have expected essentially all of the probability mass to be concentrated at 0 because no change in inputs would have been enough to meaningfully change the relative cost-effectiveness of the charities.

### Visual sensitivity analysis

We can now repeat our visual sensitivity analysis but using our distance metrics from the reference as our outcome of interest instead of individual cost-effectiveness estimates. What these plots show is how sensitive the relative cost-effectiveness of the different charities is to each of the input parameters used in any of the cost-effectiveness models (so, yes, there are a lot of parameters/plots). We have three big plots, one for each distance metric—footrule, tau and angle. In each plot, there's a subplot corresponding to each input factor used anywhere in the GiveWell's cost-effectiveness analysis.

Scatter plots showing sensitivity of the footrule distance with respect to each input parameter Scatter plots showing sensitivity of the tau distance with respect to each input parameter Scatter plots showing sensitivity of the angular distance with respect to each input parameter (The banding in the tau and footrule plots is just an artifact of those distance metrics returning integers (before normalization) rather than reals.)

These results might be a bit surprising at first. Why are there so many charity-specific factors with apparently high sensitivity indicators? Shouldn't input parameters which affect all models have the biggest influence on the overall result? Also, why do so few of the factors that showed up as most influential in the charity-specific sensitivity analyses from last section make it to the top?

However, after reflecting for a bit, this makes sense. Because we're interested in the relative performance of the charities, any factor which affects them all equally is of little importance here. Instead, we want factors that have a strong influence on only a few charities. When we go back to the earlier charity-by-charity sensitivity analysis, we see that many of the input parameters we identified as most influential where shared across charities (especially across the deworming charities). Non-shared factors that made it to the top of the charity-by-charity lists—like the relative risk of all-cause mortality for young children in VAS programs—show up somewhat high here too.

But it's hard to eyeball the sensitivity when there are so many factors and most are of small effect. So let's quickly move on to the delta analysis.

### Delta moment-independent sensitivity analysis

Again, we'll have three big plots, one for each distance metric—footrule, tau and angle. In each plot, there's an estimate of the delta moment-independent sensitivity for each input factor used anywhere in the GiveWell's cost-effectiveness analysis (and an indication of how confident that sensitivity estimate is).

Delta sensitivities for each input parameter in footrule distance analysis Delta sensitivities for each input parameter in the tau distance analysis Delta sensitivities for each input parameter in angular distance analysis So these delta sensitivities corroborate the suspicion that arose during the visual sensitivity analysis—charity-specific input parameters have the highest sensitivity indicators.

The other noteworthy result is which charity-specific factors are the most influential depends somewhat on which distance metric we use. The two rank-based metrics—tau and footrule distance—both suggest that the final charity ranking (given these inputs) is most sensitive to the worm intensity adjustment and cost per capita per annum of Sightsavers and the END Fund. These input parameters are a bit further down (though still fairly high) in the list according to the angular distance metric.

#### Needs more meta

It would be nice to check that our distance metrics don't produce totally contradictory results. How can we accomplish this? Well, the plots above already order the input factors according to their sensitivity indicators... That means we have rankings of the sensitivities of the input factors and we can compare the rankings using Kendall's tau and Spearman's footrule distance. If that sounds confusing hopefully the table clears things up:

Using Kendall's tau and Spearman's footrule distance to assess the similarity of sensitivity rankings generated under different distance metrics

Delta sensitivity rankings compared Tau distance Footrule distance
Tau and footrule 0.358 0.469
Tau and angle 0.365 0.516
Angle and footrule 0.430 0.596

So it looks like the three rankings have middling agreement. Sensitivities according to tau and footrule agree the most while sensitivities according to angle and footrule agree the least. The disagreement probably also reflects random noise since the confidence intervals for many of the variables' sensitivity indicators overlap. We could presumably shrink these confidence intervals and reduce the noise by increasing the number of samples used during our analysis.

To the extent that the disagreement isn't just noise, it's not entirely surprising—part of the point of using different distance metrics is to capture different notions of distance, each of which might be more or less suitable for a given purpose. But the divergence does mean that we'll need to carefully pick which metric to pay attention to depending on the precise questions we're trying to answer. For example, if we just want to pick the single top charity and donate all our money to that, factors with high sensitivity indicators according to footrule distance might be the most important to pin down. On the other hand, if we want to distribute our money in proportion to each charity's estimated cost-effectiveness, angular distance is perhaps a better metric to guide our investigations.

## Section recap

We started with a couple of problems with our previous analysis: we were taking cost-effectiveness estimates literally and looking at them independently instead of as parts of a cohesive analysis. We addressed these problems by redoing our analysis while looking at distance from the current best cost-effectiveness estimates. We found that our input uncertainty is consequential even when looking only at the relative cost-effectiveness of the charities. We also found that input parameters which are important but unique to a particular charity often affect the final relative cost-effectiveness substantially.

Finally, we have the same caveat as last time: these results still reflect my fairly arbitrary (but scrupulously neutral) decision to pretend that we equally uncertain about each input parameter. To remedy this flaw and get results which are actually meaningful, head over to the Jupyter notebook and tweak the input distributions.

# Recap

GiveWell models the cost-effectiveness of its top charities with point estimates in a spreadsheet. We insisted that working with probability distributions instead of point estimates more fully reflects our state of knowledge. By performing uncertainty analysis, we got a better sense of how uncertain the results are (e.g. GiveDirectly is the most certain given our inputs). After uncertainty analysis, we proceeded to sensitivity analysis and found that indeed there were some input parameters that were more influential than others. The most influential parameters are likely targets for further investigation and refinement. The final step we took was to combine the individual charity cost-effectiveness estimates into one giant model. By looking at how far (using three different distance metrics) these results deviated from the current overall cost-effectiveness analysis, we accomplished two things. First, we confirmed that our input uncertainty is indeed consequential—there are some plausible input values which might reorder the top charities in terms of cost-effectiveness. Second, we identified which input parameters (given our uncertainty) have the highest sensitivity indicators and therefore are the best targets for further scrutiny. We also found that this final sensitivity analysis was fairly sensitive to which distance metric we use so it's important to pick a distance metric tailored to the question of interest.

Finally, throughout, I reminded you that this is more of a template for an analysis than an actual analysis because we pretended to be equally uncertain about each input parameter. To get a more useful analysis, you'll have to edit the input uncertainty to reflect your actual beliefs and run the Jupyter notebook.

# Appendix

## Sobol indices for per-charity cost-effectiveness

I also did a variance-based sensitivity analysis with Sobol indices. Those plots follow.

The variable order in each plot is from the input parameter with the highest sensitivity to the input parameter with the lowest sensitivity. That makes it straightforward to compare the ordering of sensitivities according to the delta moment-independent method and according to the Sobol method. We see that there is broad—but not perfect—agreement between the methods. Sobol sensitivities for each input parameter in the GiveDirectly cost-effectiveness calculation Sobol sensitivities for each input parameter in the END Fund cost-effectiveness calculation Sobol sensitivities for each input parameter in the Deworm the World cost-effectiveness calculation Sobol sensitivities for each input parameter in the Schistosomiasis Control Initiative cost-effectiveness calculation Sobol sensitivities for each input parameter in the Sightsavers cost-effectiveness calculation Sobol sensitivities for each input parameter in the Malaria Consortium cost-effectiveness calculation Sobol sensitivities for each input parameter in the Helen Keller International cost-effectiveness calculation Sobol sensitivities for each input parameter in the Against Malaria Foundation cost-effectiveness calculation

## Sobol indices for relative cost-effectiveness of charities

The variable order in each plot is from the input parameter with the highest sensitivity to the input parameter with the lowest sensitivity. That makes it straightforward to compare the ordering of sensitivities according to the delta moment-independent method and according to the Sobol method. We see that there is broad—but not perfect—agreement between the different methods.

Sobol sensitivities for each input parameter in footrule distance analysis Sobol sensitivities for each input parameter in tau distance analysis Sobol sensitivities for each input parameter in angular distance analysis 1. Unfortunately, the code implements the 2019 V4 cost-effectiveness analysis instead of the most recent V5 because I just worked off the V4 tab I'd had lurking in my browser for months and didn't think to check for a new version until too late. I also deviated from the spreadsheet in one place because I think there's an error (Update: The error will be fixed in GiveWell's next publically-released version). ↩︎

2. Log-normal strikes me as a reasonable default distribution for this task: because it's support is (0, +∞) which fits many of our parameters well (they're all positive but some are actually bounded above by 1); and because "A log-normal process is the statistical realization of the multiplicative product of many independent random variables" which also seems reasonable here. ↩︎

3. When you follow the link, you should see a Jupyter notebook with three "cells". The first is a preamble setting things up. The second has all the parameters with lower and upper bounds. This is the part you want to edit. Once you've edited it, find and click "Runtime > Run all" in the menu. You should eventually see the notebook produce a serious of plots. ↩︎

4. This is, in fact, approximately what Monte Carlo methods do so this is a very convenient analysis to run. ↩︎

5. I swear I didn't cheat by just picking the results on the scatter plot that match the delta sensitivities! ↩︎

6. If we just look at the probability for each possible ranking independently, we'll be overwhelmed by the number of permutations and it will be hard to find any useful structure in our results. ↩︎

7. The angle between the vectors is a better metric here than the distance between the vectors' endpoints because we're interested in the relative cost-effectiveness of the charities and how those change. If our results show that each charity is twice as effective as in the reference vector, our metric should return a distance of 0 because nothing has changed in the relative cost-effectiveness of each charity. ↩︎