Calculate variance for user retention

Question

Given the following scenario about revenue per session, calculate the the sample size needed.

Codemia · Accepted Answer

To calculate the sample size needed for user retention analysis at HashiCorp, we need to set up a mathematical framework that defines our parameters clearly. Let:

- $ p_1 $: The proportion of users retained after a session in the control group.
- $ p_2 $: The proportion of users retained after a session in the treatment group.
- $ \alpha $: The significance level (commonly 0.05).
- $ eta $: The power of the test (commonly set at 0.8, meaning an 80% chance of correctly rejecting the null hypothesis).
- $ d $: The minimum detectable effect size, which is defined as $ p_2 - p_1 $.

Our goal is to find the sample size $ n $ needed to detect a difference in user retention rates with a given confidence level.

To calculate the sample size, we can use the following formula derived from the two-proportion z-test:

$$ n = \frac{(Z_{\alpha/2} + Z_{\beta})^2 \cdot (p_1(1-p_1) + p_2(1-p_2))}{d^2} $$

Where:
- $ Z_{\alpha/2} $ is the z-score associated with the desired significance level (for $ \alpha = 0.05 $, $ Z_{0.025} \approx 1.96 $).
- $ Z_{\beta} $ is the z-score corresponding to the desired power (for $ \beta = 0.2 $, $ Z_{0.2} \approx 0.84 $).

Assuming $ p_1 = 0.6 $ and $ p_2 = 0.7 $ (a 10% increase), we can calculate the pooled proportion, $ p = \frac{p_1 + p_2}{2} $.

Let's calculate the sample size using the values defined:

1. Calculate $ p $:  
   $$ p = \frac{0.6 + 0.7}{2} = 0.65 $$

2. Calculate $ p_1(1-p_1) $ and $ p_2(1-p_2) $:  
   $$ p_1(1-p_1) = 0.6(0.4) = 0.24 $$  
   $$ p_2(1-p_2) = 0.7(0.3) = 0.21 $$

3. Now, plug these values into the sample size formula:
   $$ n = \frac{(1.96 + 0.84)^2 \cdot (0.24 + 0.21)}{(0.1)^2} $$
   $$ n = \frac{(2.8)^2 \cdot 0.45}{0.01} $$
   $$ n = \frac{7.84 \cdot 0.45}{0.01} = 352.8 $$

Thus, you would need approximately 353 participants in each group.

The result indicates that to detect a statistically significant difference in user retention with a 10% increase, HashiCorp needs to recruit 353 users in both the control and treatment groups. This sample size ensures that the results will be reliable and robust, minimizing the risk of Type I and Type II errors. It is crucial to recognize that while the sample size is adequate for statistical significance, practical significance should also be considered, especially in terms of the business impact of the retention rates.

This problem illustrates the importance of understanding sample sizes in experimental design. Related concepts include:
- **Power Analysis**: A deeper exploration of how varying $ \alpha $ and $ \beta $ affects sample size.
- **Confounding Variables**: Identifying and controlling for variables that might affect user retention, such as session length or user demographics.
- **Bayesian vs. Frequentist Methods**: Considering a Bayesian approach could provide a different perspective on estimating sample size based on prior beliefs about user retention and how they are updated with new data.

Calculate variance for user retention

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