There are two approaches to combining results in a meta-analysis. They are called the fixed effects model and the random effects model. The fixed effects model effectively weights each study by the sample size, or by a measurement that is closely related to the sample size, such as the inverse of the standard error of the estimate. A random effects meta-analysis, in contrast, will assume that an estimate from a single study has two sources of error. One error is the same as in the fixed effects analysis and varies by the sample size of the study. The other error is a random component that is independent of the sample size and represents uncertainties due to conditions in this particular study that differ from conditions in other studies.
The random effects analysis will provide different weights for each study. The weights for studies with large sample sizes are smaller, sometimes only slightly smaller, but sometimes markedly smaller, depending on how variable that second random component is. Studies with smaller sample sizes will get larger weights under a random effects meta-analysis compared to a fixed effects meta-analysis.
Someone on the epidemio-l list claimed that this was “unfair”. Certainly it is counterintuitive at first glance. I would still argue that the random effects meta-analysis still makes sense.
If you indeed believe that there are two sources of error that tells you that each study has more uncertainty than a fixed effects model might assume. This seems logical enough. There more uncertainty than mere sampling error at work here. When you account for this extra uncertainty, it has a larger relative effect on studies with large sample sizes than studies with small sample sizes. In a study with a large sample size, the second random component can dominate the calculations. In a study with a small sample size, the sampling error is still likely to be the major contributor to uncertainty. This differential impact leads to a weighting system that does not place as much emphasis on the largest studies.
If you really feel strongly that studies should be weighted only by their sample size, you are elevating a general principle to an inviolable standard. It’s almost as if this were an edict from the Supreme Court like “one person-one vote”.
But I suspect most people are not so upset when random effects are used in a randomized trial to account for the center effect in a multi-center trial. This model also is “unfair” because the weights given to each center are not directly proportional to the number of patients at each center. I will admit that there is some level of controversy here as well, but if you accept random effects in a multi-center trial but don’t accept them in a meta-analysis, I would ask you what is the difference?
Suppose you have a linear regression model with measurement errors in the independent variable. Would you trust a least squares estimate because least squares always produces the “best” fit to the data? I would argue that clinging to weights that are proportional to the sample size in the face of heterogeneity is no different than clinging to the least squares principle in the face of measurement error.
Another common criticism of random effects meta-analysis is the assumption that you are making an unrealistic assumption that the second random component is normally distributed. This can indeed lead to bad inferences, but which is worse. Pretending that a non-normally distributed variable is actually normally distributed or pretending that a non-normally distributed variable is uniformly zero?
Now I realize that people smarter than me have criticized the random effects meta-analysis. That’s worth remembering and I would not blithely recommend this approach to a client without considerable discussion of the controversies. But I would still argue that the random effects meta-analysis is the more intuitive approach, even though at first glance it does indeed look counter-intuitive.
You can find an earlier version of this page on my original website.