Effect size of a binomial test and its relation to other measures of effect size I would like to find the effect size of a binomial test and relate it to other measures of effect size such as Cohen’s d or Pearson’s r.
Say I have 13 successes for 18 trials, where the probability of obtaining one success is 0.333.  in R:
binom.test(13, 18, 0.333)

which gives the following output:
Exact binomial test

data:  13 and 18
number of successes = 13, number of trials = 18, p-value = 0.001526
alternative hypothesis: true probability of success is not equal to 0.333
95 percent confidence interval:
 0.4651980 0.9030508
sample estimates:
probability of success 
             0.7222222 

1) Can I calculate the effect size for this test as follows?  effect size:  0.722 – 0.333 = 0.389? 
2) How does this relate to Cohen’s d or Pearson’s r ?
 A: The effect size in this case is commonly described in terms of relative risk. Comparisons to Cohen's d or Pearson's r don't apply at all, because they deal with continuous data and your data is binary.
So the expected probability is $p_e = 1/3$ and you have 13 successes out of 18 trials, giving an observed probability of $p_o = 0.722$. The relative risk is thus $RR = 0.722/0.333 = 2.17$.
Since you talk about successes, I assume that a success is a good thing, and then it might be more useful to describe the failures as events, so that you have 5 failures out of 18 events, and $p_e = 2/3$:
binom.test(5,18,2/3)

    Exact binomial test

data:  5 and 18
number of successes = 5, number of trials = 18, p-value = 0.001529
alternative hypothesis: true probability of success is not equal to 0.6666667
95 percent confidence interval:
 0.09694921 0.53480197
sample estimates:
probability of success 
             0.2777778 

So the relative risk is calculated as $RR = 0.28/0.67 = 0.4166667$. Your treatment/procedure/whatever has a relative risk of 0.42 for the outcome being a failure. An often used effect measure in medicine is numbers needed to treat (NNT). I don't know if that applies in your field, but it means the number of units you need to apply your treatment/procedure/whatever to in order to get one less failure. It is calculated as $1 / (p_t - p_e)$ where $p_t$ is the estimated probability for your treatment/procedure/whatever and $p_e$ is the expected probability. Usually, the expected probability is calculated from what is observed in a control group, but if you have reliable data on the expected probability, it shouldn't be needed. If the number is negative, you take the absolute value and that is NNT and if the number is positive, you get NNH (number needed to harm) instead, which means the effect is negative. In your case: $1/(0.278-0.667) = -2.57$, which means that NNT = 2.57. You can apply the same procedure to the confidence intervals.
Note that if you used the results from the first case when calculating NNT, you would get  $1/(0.722-0.333) = 2.57$, so in this case NNH is calculated. NNH when the event is a success is equal to NNT when the event is a failure, just two different perspecives.
