AB test sample size calculation by hand

Question

Evan Miller has created a well-known online AB test sample size calculator. For the sake of being able to program and modify this formula, I would like to know how to calculate sample size Evan Miller-style by hand.

Personally, I'll calculate such a metric by working backwards from how we calculate a 95% confidence interval with the z-test of proportions around the difference in conversion between the two variations ($\hat{d}$) by setting it zero.

I'll define/assume:

• $\alpha$ = .05, $\beta$ = .2
• a 50/50 split between the control and experiment, i.e. $n\_exp$ = $n\_control$
• the control conversion rate, i.e. the base rate before the experiment = $c$
• $p$ = pooled conversion rate = (number of exp conversions + number of control conversions/ (n_control + n_experiment)) -> in this context -> $(nc+n(c+\hat{d}))/2n$ = $(2c+\hat{d})/2$

Now time to solve for $n$ ...

$$ \hat{d} + Z_{(1+\alpha)/2} * StandardError = 0 $$ $$ \hat{d} + 1.96 * StandardError = 0 $$ $$ \hat{d} + 1.96 * \sqrt{p(1-p)(\frac{1}{n\_exp} + \frac{1}{n\_control})} = 0$$ $$ \hat{d} + 1.96 * \sqrt{p(1-p)(\frac{2}{n})} = 0$$ $$ \sqrt{p(1-p)(\frac{2}{n})} =\frac{-\hat{d}}{1.96}$$

with more simplifying we get to:

$$ \frac{(1.96^2) 2p(1-p)}{\hat{d}^2} = n $$ $$ \frac{(1.96^2) (2c+2c\hat{d}-2c^2+\frac{3}{2}\hat{d}^2)}{\hat{d}^2} = n $$

At the moment though, my calculation doesn't incorporate power (1-$\beta$) but Evan Miller's does.

What should I think about as next steps to incorporate power into my sample size calculation?

(Feel free to also point out other errors in my calculation or assumptions!)

Answer 1

try this: $$ n=\frac{(Z_{\alpha/2}\sqrt{2p_1 (1-p_1)}+Z_{\beta}\sqrt{p_1(1-p_1)+p_2(1-p_2)})^2}{|p_2-p_1|^2} $$ where:

• $p_1$ is the "Baseline conversion rate"

• $p_2$ is the conversion rate lifted by Absolute "Minimum Detectable Effect", which means $p_1+\text{Absolute Minimum Detectable Effect}$

• $\alpha$ is the "Significance level $\alpha$"

• $\beta$ is the $\beta$ in "Statistical power $1−\beta$"

• $Z_{\alpha/2}$ means Z Score from the z table that corresponds to $\alpha/2$

• $Z_{\beta}$ means Z Score from the z table that corresponds to $\beta$

in Sample Size Calculator (Evan’s Awesome A/B Tools)

I found the formula in A/B测试系列文章之怎么计算实验所需样本量

when I choose:

• $p_1=20\%$
• $p_2=p_1+\text{Absolute Minimum Detectable Effect}=20\%+5\%=25\%$
• $\alpha = 5\% $
• $\beta = 20\% $
• $Z_{\alpha/2}=Z_{5\%/2}=-1.959963985$
• $Z_{\beta}=Z_{20\%}=-0.841621234$

I got 1030.219283 using this formula, which is 1030 in Size Calculator (Evan’s Awesome A/B Tools)

Answer 2

I can confirm the chosen answer by looking at the code.

function num_subjects(alpha, power_level, p, delta) {
    if (p > 0.5) {
        p = 1.0 - p;
    }
    var t_alpha2 = ppnd(1.0-alpha/2);
    var t_beta = ppnd(power_level);

    var sd1 = Math.sqrt(2 * p * (1.0 - p));
    var sd2 = Math.sqrt(p * (1.0 - p) + (p + delta) * (1.0 - p - delta));

    return (t_alpha2 * sd1 + t_beta * sd2) * (t_alpha2 * sd1 + t_beta * sd2) / (delta * delta);
}

where ppnd is used to calculate the 𝑍 values.

Answer 3

This even simpler code gets me the exact same results as Evan Miller's calculator. It's an N-1 variant of the two proportion test (normally the N-1 is used for chi squared tests but it's mathematically equivalent anyway):

n = 2z^2 p (1 - p) / d^2 + 1/2

z^2 is the sum of z for alpha + beta.