# Sample Size proportion per control vs. experiment group

I am taking an online statistics course and I understand how to calculate the necessary sample size for a hypothesis test.

I am using an online calculator like http://www.evanmiller.org/ab-testing/sample-size.html or python like this https://stackoverflow.com/questions/15204070/is-there-a-python-scipy-function-to-determine-parameters-needed-to-obtain-a-ta

From what I understand, this gives me the minimum sample size for each group - control and treatment.

However, if I am designing a test and I have a total sample size of 30,000; how do I calculate how large the control vs. the treatment group should be.

I understand that the treatment group needs to be the minimum sample size I calculated before and I am reading that generally the 50/50 split leads to the highest statistical power, but how can I show this with a calculation. I have been googling it unsuccessfully, so even a link to the correct approach would be greatly appreciated.

This is the closest I found https://janhove.github.io/design/2015/11/02/unequal-sample-sized, but I wasn't able to extract the correct formula.

I found this helpful cross-validated answer Is a large control sample better than a balanced sample size when the treatment group is small? ; but I am still unsure how to calculate the best ratio between control and treatment group if I have a given total sample size. (or how to prove that the 50/50 split has the highest statistical power)

I also found this great answer Treatment and Control group, the sample size, but it applies to a different industry. The hypothesis test I am designing is in the industry of online user behavior psychology.

Thank you very much in advance for any hint in the right direction (even just the correct terminology I can search for).

First of all, your formula for necessary sample size looks suspicious, the part of the formula StdDev*(1-StdDev) doesn't make much sense, perhaps it's supposed to be proportion*(1-proportion) for cases when you have a binomial distribution with a sample proportion of successes.

But that formula is an aside from your main question: why does a 50/50 split of samples produce the highest power?

The hypothesis you are trying to test is that the mean of the experiment group $\mu_E$ is the same mean as the mean of the control $\mu_C$. Essentially you are testing if $\mu_E - \mu_C = 0$.

Suppose that the true variance (not sample variance) of the experiment group is $\sigma^2_E$ and that you have a sample size $n_E$. Likewise the control group variance and sample size is $\sigma^2_C$ and $n_C$.

From your samples you will be examining $\bar{X_E} - \bar{X_C}$ to test the hypothesis $\mu_E - \mu_C = 0$. For an unbiased sample the variance of the sample mean $\bar{X_E}$ is expected to be around $\frac{\sigma^2_E}{n_E}$. Likewise for the control group the variance of the sample mean is $\frac{\sigma^2_C}{n_C}$.

When you subtract one variable from another the resulting variable has a variance equal to the sum of the two variances. Therefore the variance of $\bar{X_E} - \bar{X_C}$ is $\frac{\sigma^2_E}{n_E} + \frac{\sigma^2_C}{n_C}$

Since you have no apriori reason to suspect that the variance of the control or the experiment group is larger we will just assume that they are equal. Therefore we assume $\sigma^2_E=\sigma^2_C=\sigma^2$, and the variance of $\bar{X_E} - \bar{X_C}$ is now $\frac{\sigma^2}{n_E} + \frac{\sigma^2}{n_C} = \sigma^2\left(\frac{1}{n_E} + \frac{1}{n_C}\right)$

To get the most powerful test we want to minimize the variance. If you have a total number of samples $N$ and a proportion $p$ of them are in the experiment group then $n_E=Np$ and $N_C=N(1-p)$.

The variance is $\sigma^2\left(\frac{1}{Np} + \frac{1}{N(1-p)}\right)= \frac{\sigma^2}{N} \left(\frac{1}{p} + \frac{1}{(1-p)}\right)$. You can see by plotting a graph of $\left( \frac{1}{p} + \frac{1}{(1-p)}\right)$ that the minimum occurs at $p=0.5$, alternatively you can use calculus to prove this minimum more rigorously.

• Thank you very much for your answer; May I confirm, that I understand it correctly: to get the most powerful test, we need to decrease the pooled Standard Error and we get the lowest possible one when we use a 50/50 split, since nE and nC are equal? – jeangelj Mar 5 '17 at 23:21
• If I am forced by the business requirements to have a smaller treatment group, would you recommend a power analysis or a t-test? – jeangelj Mar 5 '17 at 23:28
• @jeangelj You're right the we need to decrease the pooled standard error to get the most powerful test. I'm not sure you got the reason why nE and nC are equal, I've added a bit more to my answer. – Hugh Mar 5 '17 at 23:36
• @jeangeli It's quite a common problem that businesses can't afford a large treatment group. You should only need a power analysis in addition to a t-test if your treatment sample size is so small that the assumptions of the t-test are violated. I think you should post a new question specifically about this. – Hugh Mar 5 '17 at 23:38
• thank you very much; I greatly appreciate your time and that you elaborated further in your answer. I think I understand it now, that by using 0.5 for p, we get the lowest variance - I will plot it in python as advised. I will also post a new question as advised as well. Thank You – jeangelj Mar 5 '17 at 23:47

This post gives the results of simulations for several combinations of sample size, effect size and proportion of the sample that is assigned to the control group: https://www.markhw.com/blog/control-size

The key takeaways are:

1. Minimal losses in power occur when we shrink the control size to 40% [of the sample].

2. A 25% to 30% range is a good compromise, as this exposes 70% of the sample to the treatment, yet still does not harm power terribly.

3. You should not allocate less than 20% of the sample to the control condition, save for situations when you are looking for large effects (e.g., 8 point lifts) and/or using large samples (e.g., 15,000 participants).

The author also links to his code on GitHub that should allow you to run similar simulations using your expected effect size and sample size.