Random assignment provides a justification not just for estimates of effects but also for estimates of uncertainty about effects. The basic approach, due to Neyman, is to estimate the variance in estimates of the difference between outcomes in treatment and in control outcomes using the variability that can be observed among units in control and units in treatment. It’s an ingenious approach and dispenses with the need to make any assumptions about the shape of statistical distributions or about asymptotics. The problem though is that it can sometimes be **upwardly biased**, meaning that it might lead you to maintain null hypotheses when you should be rejecting them. We use design diagnosis to get a handle on how great this problem is and how it matters for different estimands.

In many experiments, random assignment is performed at the level of clusters. Researchers are conscious that in such cases they cannot rely on the usual standard errors and they should take account of this feature by clustering their standard errors. Another, more subtle, risk in such designs is that if clusters are of different sizes, clustering can actually introduce bias, *even if all clusters are assigned to treatment with the same probability*. Luckily, there is a relatively simple fix that you can implement at the design stage.

We usually think that the bigger the study the better. And so huge studies often rightly garner great publicity. But the ability to generate more precise results also comes with a risk. If study designs are at risk of bias and readers (or publicists!) employ a statistical significance filter, then big data might not remove threats of bias and might actually make things worse.

Cluster-robust standard errors are known to behave badly with too few clusters. There is a great discussion of this issue by Berk Özler “Beware of studies with a small number of clusters” drawing on studies by Cameron, Gelbach, and Miller (2008). See also this nice post by Cyrus Samii and a recent treatment by Esarey and Menger (2018). A rule of thumb is to start worrying about sandwich estimators when the number of clusters goes below 40. But here we show that diagnosis of a canonical design suggests that some sandwich approaches fare quite well even with fewer than 10 clusters.

In many experiments, different groups of units get assigned to treatment with different probabilities. This can give rise to misleading results unless you properly take account of possible differences between the groups. How best to do this? The go-to approach is to “control” for groups by introducing “fixed-effects” in a regression set-up. The bad news is that this procedure is prone to bias. The good news is that there’s an even simpler and more intuitive approach that gets it right: estimate the difference-in-means within each group, then average over these group-level estimates weighting according to the size of the group. We’ll use design declaration to show the problem and to compare the performance of this and an array of other proposed solutions.