When we conduct a meta-analysis, one of the most important questions we need to answer is whether the included studies are sufficiently homogeneous to justify combining their results. Tools such as the I2 statistic and Cochran's Q test give us a global answer to that question, but they do not tell us which specific studies are generating the heterogeneity. For that, we need more refined graphical tools. The Galbraith plot, also known as the radial plot, is one of the most useful and, unfortunately, least utilized options in meta-analytic practice.
Structure of the Galbraith plot
The Galbraith plot is a scatter diagram with two axes that transform each study's information in a particular way. The horizontal axis represents the precision of each study, defined as the inverse of its standard error (1/SE). This means that studies with larger samples and more precise estimates are located further to the right on the plot. The vertical axis represents the standardized effect, which is the ratio between the estimated effect size and its standard error (z = effect / SE). This transformation converts the effects from all studies to a common scale, regardless of the original metric used.
The logic behind this representation is elegant. If all studies were estimating exactly the same true population effect, then the relationship between the standardized effect and precision would be perfectly linear and would pass through the origin. The slope of that line would be exactly the combined effect size. In a real meta-analysis, the points do not fall exactly on the line, but if heterogeneity is low, they should be reasonably close to it.
The regression line and confidence bands
The central element of the Galbraith plot is the regression line that passes through the origin. This line is drawn so that its slope equals the combined effect size estimated by the meta-analysis. If we are working with standardized mean differences and the combined effect is d = 0.35, then the slope of the line will be 0.35. Each point on the plot should, under the assumption of homogeneity, be reasonably close to this line.
The confidence bands are drawn at a distance of plus or minus 2 units on the vertical axis from the regression line. Since the vertical axis represents standardized z scores, a distance of 2 corresponds approximately to the 95% confidence limit under a standard normal distribution. Studies that fall within these bands are consistent with the combined effect. Studies that fall outside the bands are candidates to be considered outliers or influential studies that deserve more detailed examination.
Detecting heterogeneity
The way points are distributed in the Galbraith plot gives us immediate information about the heterogeneity of the meta-analysis. If most points fall within the confidence bands and cluster reasonably tightly around the regression line, we can conclude that heterogeneity is low. The meta-analysis is capturing a relatively consistent effect across studies.
Conversely, if several points fall outside the bands, especially if they do so on both sides of the regression line, we have clear evidence of substantial heterogeneity. In that case, we should ask what characteristics of those outlier studies differentiate them from the rest. These could be differences in the population studied, in the intervention implemented, in the measurement instrument used, or in the methodological quality of the study. This is precisely the advantage of the Galbraith plot over global measures of heterogeneity: it allows us to identify which studies are responsible.
Comparison with the forest plot
The forest plot and the Galbraith plot offer complementary information. The forest plot shows individual effect sizes with their confidence intervals and the combined effect, all on the original measurement scale. It is intuitive, easy to read, and widely known. However, when there are many studies, it can be difficult to visually identify which ones contribute most to heterogeneity.
The Galbraith plot solves this problem because it standardizes the effects and represents them as a function of precision. This facilitates the visual identification of outliers. Furthermore, the relationship between a point's horizontal position (precision) and its distance from the regression line has a direct interpretation: highly precise studies (further to the right) that deviate from the line are particularly concerning, because their deviation cannot simply be attributed to the sampling variability of a small study.
Implementation in R
In R, the Galbraith plot can be easily generated with the metafor package. Once we have fitted our meta-analytic model using the rma() function, we can use the radial() function to generate the plot. The basic syntax is straightforward:
library(metafor)
model <- rma(yi, vi, data = dataset)
radial(model)The radial() function accepts several arguments for customization, such as main for the title, xlab and ylab for axis labels, and cex for controlling point size. If more advanced customization is preferred, we can construct the plot manually with ggplot2, first calculating the precision and standardized effect values from each study's estimates and standard errors.
When to use the Galbraith plot
The Galbraith plot should not be considered a replacement for the forest plot or the I2 statistic, but rather a complement that offers a different perspective. It is especially useful in the following situations: when the meta-analysis includes a moderate or large number of studies and we want to quickly identify outliers; when we have detected significant heterogeneity through I2 or the Q test and need to determine which studies are generating it; and when we want to conduct sensitivity analyses by removing potentially influential studies and need visual justification for selecting those studies.
In practice, the recommended procedure would be as follows: first, examine the forest plot to get a general overview of the results; second, evaluate overall heterogeneity with I2 and the Q test; third, if heterogeneity is substantial, use the Galbraith plot to identify the responsible studies; and fourth, conduct subgroup analyses or meta-regression to explore the possible sources of that heterogeneity. This sequential approach allows for a more rigorous and transparent meta-analytic investigation, and considerably improves the quality of the conclusions we can draw from the evidence synthesis.