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The Problem

I have been working on an epidemiological meta-analysis of prevalence values, measured as proportions but analyzed in logit-space. The conditions I study are quite rare (<5%) so all estimates are close to 0. Due to the presence of heterogeneity across samples, it is important that I report both the mean as well as the prediction interval for my estimates. I am using a Bayesian approach, which means that I have a logit-transformed posterior representing the mean of the latent "true" prevalence distribution as well as a posterior representing variability within this latent "true" prevalence distribution from which individual study estimates are derived. This corresponds to a standard random-effects model. To calculate a confidence interval around the mean, I back-transform the posterior for the mean value into probability space (e.g., using the plogis function in R) and then calculate a highest density interval on the back-transformed distribution. To calculate a prediction interval, I generate a "new" study for each sample within my joint-posterior distribution using the appropriate mean and variance estimates. I then back-transform those values and calculate a highest density interval. This works fine and generally agrees with other standard approaches (e.g., the metafor in R).

However, in doing so I realized that the mean of the back-transformed predictive distribution no longer matches the back-transformed mean that I had estimated and used to generate the predictive distribution. The cause is easy to determine – it is the difference between taking the mean of a distribution on the logit scale or on the probability scale. It is a simple result of the back-transform changing the shape of the distribution overall and can be demonstrated (in R for the sake of simplicity) as follows:

set.seed(12345)
logit_var = rnorm(1000, -3,1) # Sample from a variable in logit-space
plogis(mean(logit_var))       # Mean on the logit-scale produces .05
mean(plogis(logit_var))       # Mean on the probability scale produces .07

Note: For those interested, this problem is most pronounced for logit- or probit-transformed variables that are close to the lower (p = 0) or upper (p = 1) boundary and becomes less pronounced as the variable approaches p = .5.

Generally, I have only ever seen people report the former. Specifically, they report an estimate of the back-transformed mean, its confidence interval and then the back-transformed prediction interval without recognizing that the expected means do not match.

The Question

  1. In a scenario similar to the one depicted above, is ever appropriate to report the mean of the prediction distribution in probability space either in lieu of or in combination with the more traditional mean estimate?
  2. If so, how could one best conceptualize the difference between each mean with respect to their interpretation?
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This is due to Jensen's inequality, that is, for a convex function $f()$, $$f(E(X)) \le E(f(X))$$ (and vice-versa for a concave function). For values below 0, plogis() is in fact convex and most values simulated here are below 0.

This is indeed often ignored in meta-analysis. The proper interpretation of the back-transformed value is that it is the median of the back-transformed distribution. To use your example:

set.seed(12345)
logit_var = rnorm(1000, -3,1)
plogis(median(logit_var))
median(plogis(logit_var))

Both yield the same value.

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  • $\begingroup$ Another informative and impressively concise answer. Thanks, @Wolfgang! $\endgroup$ – jmfawcet Feb 3 '17 at 4:10

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