Suppose you want to estimate how tall Albert Einstein was. You ask a friend of yours, who suggests $5^{\prime }{9}^{\prime \prime }$, though they have no particular reason to know.

$5^{\prime }{9}^{\prime \prime }$ doesn't sound unreasonable. Of course you're still very uncertain. Say you're $95\%$ sure your friend, like any random American, would guess Einstein's height within $\pm 10$ inches.

You'd like a more precise estimate, so you do a survey. You contact $100$ million people in the US and get them to estimate Einstein's height for you; the average of the survey responses is $5^{\prime }{8.453}^{\prime \prime }$. You also visit the Library of Congress and Bern Historical Museum to find an immigration form and a Swiss passport giving his height as $5^{\prime }{7}^{\prime \prime }$ and $175\text{cm}$ (near $5^{\prime }{9}^{\prime \prime }$) respectively.

How tall do you think Einstein was? How confident are you?

Overcomplicating the problem with math

Let's formalize everything in a few equations. Call Einstein's height $h$. From the survey, we have $100$ million estimates of the form $$y_i=h+{ϵ}_i$$ where we assume the error ${ϵ}_i$ has a standard deviation of $5$ inches (corresponding to the $95\%$ probability of being within $10$ inches we had above). For convenience, assume all the errors are normally distributed, though that's not essential to the qualitative argument. We couldn't predict whether $y_i$ would be too high or too low, so assume ${ϵ}_i$ has mean zero.

We also have two estimates $y_{LOC}$ and $y_{BHM}$ from the Library of Congress and museum. The formal documents should be more reliable than an average person's guesswork. Let's assume their errors have a standard deviation of $0.5$ inches. With our assumption of normal distributions, that means being $95\%$ sure Einstein's passport had his height correct to within $\pm 1$ inch.

Blindly weighted averages

One more assumption and we'll be ready to calculate how tall Einstein is. Suppose all of our errors are independent, as is often assumed in meta-analyses of scientific studies. Then the standard best thing to do is take an average of the estimates with inverse variance (precision) weights; the new precision will be the sum of the previous precisions. Don't worry about the details; they aren't particularly important.

Using just your friend's guess, we'd estimate $${\hat{h}}_{friend}=5^{\prime }{9}^{\prime \prime }\pm 5^{\prime \prime }.$$

The uncertainty there is $\pm 1$ standard deviation, i.e. half the width of a $95\%$ confidence interval. Using just the Library of Congress immigration form, we'd estimate $${\hat{h}}_{LOC}=5^{\prime }{7}^{\prime \prime }\pm {0.5}^{\prime \prime }.$$ To put them together, we give ${\hat{h}}_{LOC}$ $100\times$ higher weight because it has $10\times$ smaller standard error. That leads to a combined estimate $${\hat{h}}_{friend+LOC}=5^{\prime }{7.02}^{\prime \prime }\pm {0.498}^{\prime \prime }$$ So far so good. We basically ignored the unreliable estimate from your friend in favor of the better one from the Library of Congress. I only included the mostly-meaningless extra digits to show there is a tiny change. I won't include them again.

The Swiss passport seems just as reliable as the immigration form, so they each get equal weight when averaged: $${\hat{h}}_{LOC+BHM}=5^{\prime }{8}^{\prime \prime }\pm {0.4}^{\prime \prime }$$ That seems ok too.^[1] What if we average in our survey responses in the same way? $${\hat{h}}_{LOC+BHM+survey}=5^{\prime }{8.453}^{\prime \prime }\pm {0.0005}^{\prime \prime }$$ Our model is $95\%$ confident it knows Einstein's height to the nearest thousandth of an inch, around the diameter of a single skin cell or human hair. Its estimate is entirely based on the survey; the documents are basically ignored.

Oops.

Bayes and bias

Where did the analysis go wrong? We made many assumptions in setting up the statistical model. Several were suspect. To find a better set of assumptions, let's take two intuitions about the conclusions our inference method should draw.

1. The first estimate from a friend would convey some information to an alien who had no idea how big people are.
2. Once you have a few such estimates, adding a million more doesn't tell you much of anything.

The analysis above satisfies 1 but not 2. Throwing out all the survey responses because they must be biased satisfies 2 but not 1. Is there an approach in between those extremes?

The first thing we'll do is change to a Bayesian perspective. We didn't start totally ignorant about Einstein's height. We could either invent a prior distribution for $h$ that feels reasonable or look for references like this paper claiming the height distribution for European men born at the end of the nineteenth century has a mean of $5^{\prime }{7.5}^{\prime \prime }$ and standard deviation ${2.5}^{\prime \prime }$.

However, just adding a prior to the model above doesn't help at all. A standard deviation of ${2.5}^{\prime \prime }$ is twice as small as the standard deviation we postulated for survey respondents, so the prior contributes the same amount to the posterior as four survey respondents out of $100$ million. It would be ignored too.

The more important change will be to restructure our model. Let's explicitly allow for the possibility that our survey responses are systematically biased by some amount $\beta$:

$$y_i=h+\beta +{ϵ}_i$$

Note the lack of subscript on $\beta$. This isn't about unknown individual biases that average out to zero. We're trying to drop the assumption that an infinitely large survey would give us an infinitely precise estimate.^[2]

Separately, let ${\beta }_{GOV}$ be the average height bias in government documents. To complete the model, we'll need to put priors on $\beta$, ${\beta }_{GOV}$, and the random errors. Suppose we arbitrarily decide that about half our uncertainty is due to systematic effects ($\beta$ and ${\beta }_{GOV}$) and half is due to random errors that could average out (the ${ϵ}_i$).^[3]

Now, we conclude from the survey that $$h+\beta =5^{\prime }{8.453}^{\prime \prime }\pm {0.0005}^{\prime \prime }.$$ We're still extremely confident in the survey outcome. But because we're acknowledging the possibility of bias, that no longer translates into confidence about Einstein's height. Instead, $${\hat{h}}_{survey}=5^{\prime }8\pm 2^{\prime \prime }.$$ The survey adds moves us very little from our prior. Adding in the documents, we get $${\hat{h}}_{LOC+BHM+survey}=5^{\prime }{8}^{\prime \prime }\pm {0.4}^{\prime \prime }.$$ That's what we would have concluded by throwing out the survey, but we didn't need to tell your friend their guess has no value in any circumstances.

Meta-analyses

After all, a good Bayesian should be able to say “Well, I got some impressive results, but my prior for psi is very low, so this raises my belief in psi slightly, but raises my belief that the experiments were confounded a lot.”

Scott Alexander, The Control Group is Out of Control

This post isn't really about Einstein. Meta-analyses that matter share the same potential problem we saw with height, that a sufficiently large quantity of not-very-good estimates can swamp better estimates.

Standard statistical approaches to meta-analysis, like Dersimonian and Laird with its 33000 citations, do nothing to allow for unknown bias that's consistent across studies.^[4] This isn't necessarily wrong. If you don't have many studies and the statistical uncertainty is large, the possibility of bias may not affect your conclusions much. Or maybe you're CERN and you understand the noise in your experiment so well that you believe the ten digits of precision you might get.

Consider, however, observing the scientific study of a phenomenon from its beginning. First, an exploratory study comes out that suggests $X$. It's soon followed by more small studies in favor of $X$; after a couple years, the literature has a huge number of studies generally supporting $X$. Only a few have obvious fatal flaws. The rest appear to be reasonable, honest, small-scale efforts that individually wouldn't be too compelling. Eventually, someone does one more thorough and better-conducted study. Perhaps it's an RCT where the others were observational; perhaps the authors looked up Einstein's passport instead of surveying friends. It concludes not-$X$.

Many people's belief in $X$ would go up a little with the first study, go up a bounded amount more as the field grows and continues to find $X$, and then crash after the higher quality study regardless of how many low quality studies preceded it or how thoroughly they were critiqued individually. Adding a literature-wide bias term like we did above is a way to translate that qualitative thought process into a Bayesian meta-analysis.^[5]

Importantly, with that approach you are not going through the studies and looking for specific causes of bias to model (as here, for example). Instead, you're acknowledging that even when you can't explicitly identify flaws your confidence in the study-generating process is finite. At the same time, you're not totally throwing out potentially-biased studies. You don't need to give up if no studies are perfect. You just increase your uncertainty.

When you read your next meta-analysis, consider how much you believe its implicit prior on the bias of the field as a whole. Please do keep in mind that this essay is descriptive more than prescriptive. The model discussed here won't always be appropriate; I am not dealing with any of the difficulties in when to use it or how to nail down the details; I've been quick and glib with making up priors. There will be more work to do in any particular case.

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1. Perhaps a little overconfident. ↩︎

2. We implicitly assumed that when declaring errors to be independent. ↩︎

3. To me this feels generous to the survey. ↩︎

4. "Consistent across studies" is the important part here. Dersimonian and Laird do model bias in individual studies. Their method should give reasonable results if you treat the survey as one study with $100$ million samples rather than $100$ million studies. ↩︎

5. See here (paywalled) for a more formal writeup of a similar idea. It doesn't look widely used. ↩︎