Is the binomial effect size display (BESD) a misleading representation of effect size? It's hard for me to accept that Donald Rubin would ever come up with a true lemon of a technique. Yet, that's my perception of the BESD [1, 2, 3].
The original paper by Rosenthal and Rubin (1982) claimed that there was value in showing "how to recast any product-moment correlation into such a [2x2] display, whether the original data are continuous or categorical."
The table below is from p. 451 of the 2nd link above:

This technique seems to overstate the magnitude of almost any effect size.  Here, $R^2$ from the original data = .01, but when "translated" into a 2x2 contingency table, we seem to be faced with a much stronger effect. I don't deny that, when the data are recast into categorical format in this way, $\phi$ indeed = .1, but I feel something has been very distorted in the translation.
Am I missing something truly valuable here? Also, I have the impression that in the past 10 or so years the statistical community has by and large rejected this as a legitimate method—am I wrong on that?
The equation to calculate experimental ($E$) and control ($C$) success rates ($sr$), respectively, is simply:
$E_{sr} = .50 + r/2$
and
$C_{sr} = .50 - r/2$

Reference:
Rosenthal, R., & Rubin, D.B. (1982). A simple general purpose display of magnitude of experimental effect. Journal of Educational Psychology, 74, 166–169.
 A: I can demonstrate that it is biased (I think), but I cannot explain why. I'm hoping someone can see my answer and help explain it more.
As in many meta-analyses and the image you posted, many people interpret the BESD as: If you were to median split both variables, you would accurately put people in the "right" cells of a 2 x 2 contingency table a given percent of the time. 
So if $.50 + r/2 = .70$, people might say, "Given this observed $r$, you can think of it like this: People above the median in X would also be above the median in Y 70% of the time." This is somewhat how Kraus (1995, p. 69) interprets it (he relies on a hypothetical situation where one variable is truly dichotomous, while the other is median split):

People have often used medical metaphors, too: "This $r$ corresponds to a difference in 40 percentage point between people in a control and experimental condition."
To see if the median-split-esque interpretation is biased, I simulated a population of 1,000,000 cases where the true population $r = .38$. I then drew 100 people from this population, calculated the BESD "correct rate" (i.e., $.50 + r/2$), and then calculated the actual median split cells for a 2 x 2 contingency table, like the one described above for categorizing people "correctly." I did this 10,000 times.
I then took the mean and standard deviation of each of these vectors of 10,000 in length. The code:
library(MASS)
# set population params
mu <- rep(0,2)
Sigma <- matrix(.38, nrow=2, ncol=2) + diag(2)*.62
# set seed
set.seed(1839)
# generate population
pop <- as.data.frame(mvrnorm(n=1000000, mu=mu, Sigma=Sigma))
# initialize vectors
besd_correct <- c()
actual_correct <- c()
# actually break up raw data by median split, see how it works
for (i in 1:10000) {
  samp <- pop[sample(1:1000000, 100),]
  besd_correct[i] <- round(100*(.50 + cor(samp)[1,2]/2),0)
  samp$V1_split <- ifelse(samp$V1 > median(samp$V1), 1, 0)
  samp$V2_split <- ifelse(samp$V2 > median(samp$V2), 1, 0)
  actual_correct[i] <- with(samp, table(V1_split==V2_split))[[2]]
}
# cells for BESD
mean(besd_correct)
100 - mean(besd_correct)
# cells for actual 2 x 2 table with median split
mean(actual_correct)
100 - mean(actual_correct)

Based on BESD, we get this table, where v1 and v2 refer to variables and low and high refer to below and above the median, respectively:
+---------+--------+---------+
|         | v2 low | v2 high |
+---------+--------+---------+
| v1 low  | 69     | 31      |
+---------+--------+---------+
| v1 high | 31     | 69      |
+---------+--------+---------+

Based on actually doing a median split with the raw data, we get this table:
+---------+--------+---------+
|         | v2 low | v2 high |
+---------+--------+---------+
| v1 low  | 62     | 38      |
+---------+--------+---------+
| v1 high | 38     | 62      |
+---------+--------+---------+

So while someone could argue, using BESD, that there is a "38 percentage point difference in control and experimental," the actual median split has this number at 24.
I'm not sure why this happens, or if it depends on sample size and correlation (one could easily do more simulations to figure out), I think this shows it is biased. I would love if someone could chime in with a mathematical—rather than computational—explanation.
A: Mark White's intuition is incorrect. The BESD is not actually modeling a median split. A median split is associated with real statistical information loss--it systematically attenuates relations (see http://psycnet.apa.org/record/1990-24322-001), which is why the median split values show a smaller accuracy than the BESD. The BESD is demonstrating classification accuracy as though the variables were truly dichotomous, not artificially dichotomized through a median split. To see this, compute the correlation on the median split data. You will see that it is smaller than the correlation for the original variables. If the variables were originally binary, the two methods would agree. By its nature, the BESD is displaying variables as though they were truly binary. When it is used for continuous variables, this necessarily represents an abstraction--there aren't really "success" and "failure" or "treatment" and "control" groups, but it is practically useful to show results in this way because it can be easier to understand two binary variables than two continuous variables. 
The BESD is not biased. It accurately reflects the impact of a particular treatment on classification accuracy if we were working with two binary variables. It is a useful display for demonstrating the potential practical value of a measure or treatment, and, yes, it does demonstrate that even effects with small variance accounted for statistics can be meaningfully important. The BESD is widely used in applied psychological and organizational practice, and it agrees strongly with other practical effect size displays (e.g., that top-down selecting a group using a measure with a validity correlation of r=.25 will lead to a .25 SD increase in outcome performance among the selected group versus an unselected group). 
Variance accounted for statistics consistently lead to misunderstandings and underestimates about the size of variable relations because the squaring operation is nonlinear. Many applied methodologists (e.g., https://us.sagepub.com/en-us/nam/methods-of-meta-analysis/book240589) strongly discourage their use in favor of their square roots (which more accurately convey the size of effects).
A: For a detailed answer, an analysis of when it makes a difference, and a better solution, please see
Exact Method for Computing Absolute Percentage Change in a Dichotomous Outcome from Meta-Analytic Effect Size: Improving Impact and Cost-Outcome Estimates, TR Miller, J Derzon, D Hendrie, Value in Health, 14:1, 144-151, 2011.
Here's the summary answer in the abstract of that article.
OBJECTIVES:
Meta-analyses typically compute a treatment effect size (Cohen's d), which is readily converted to another common measure, the binomial effect size display (BESD). BESD is the correlation coefficient and represents a percentage difference in outcome attributable to an intervention. Both d and BESD are in arbitrary units; neither measures the absolute change resulting from intervention. The method used to estimate absolute change from BESD assumes both a 50-50 split of the outcome and a balanced design. Consequently, inaccurate assumptions underpin most meta-analytic estimates of the gain resulting from an intervention (and of its cost effectiveness). This article develops an exact formula without these assumptions.
METHODS:
The formula is developed algebraically from 1) the formula for the correlation coefficient represented as a 2-by-2 contingency table constructed from the relative size of the treatment and control groups and the percentage of people who would have the condition absent intervention, and 2) the BESD correlation coefficient formula showing change in success probability with treatment.
RESULTS:
Simulation reveals that BESD only approximates the reduction in the outcome an intervention might well achieve when the problem outcome occurs in 35%-65% of cases. For less common outcomes, BESD substantially overestimates the impact of an intervention. Even when BESD accurately estimates the likely percentage change in outcome, it paints a misleading picture of the proportion of cases that will achieve a positive outcome.
