Calculate probability for user retention

Question

Given the following scenario about user conversion, calculate the the sample size needed.

Codemia · Accepted Answer

To determine the sample size needed for ensuring user retention analysis at Vercel, we first define the parameters of the study. Let:\n\n1. **p1** = the proportion of retained users in the control group (e.g., current retention rate).\n2. **p2** = the proportion of retained users in the experimental group (e.g., after implementing a new feature).\n3. **α** = significance level (commonly set at 0.05).\n4. **β** = power of the test (commonly set at 0.80, indicating an 80% chance of detecting a true effect).\n5. **d** = the minimum detectable effect size, defined as |p2 - p1|.\n\nWe can model this problem using proportions and apply a two-proportion z-test for hypothesis testing.

1. **Calculate the effect size (d)**: Identify the expected change in retention rate. For example, if p1 = 0.60 and we expect p2 = 0.70, then d = 0.70 - 0.60 = 0.10.\n2. **Use the sample size formula** for two proportions:  
   $$ n = \left( \frac{(Z_{\alpha/2} \sqrt{2p(1-p)} + Z_{\beta} \sqrt{p1(1-p1) + p2(1-p2)})^2}{(p2 - p1)^2} \right]  $$  
   where $ p = \frac{p1 + p2}{2} $ is the average proportion.\n3. **Find the z-scores**: For α = 0.05, Z_{0.025} = 1.96 and for β = 0.20, Z_{0.20} = 0.84.\n4. **Substitute values** into the formula to solve for n.

Assuming p1 = 0.60 and p2 = 0.70, we calculate:\n1. **Calculate p**:  $$ p = \frac{0.60 + 0.70}{2} = 0.65 $$ \n2. **Calculate variances**: $$ \text{Var}_{p1} = 0.60(1 - 0.60) = 0.24, \text{Var}_{p2} = 0.70(1 - 0.70) = 0.21 $$ \n3. **Calculate z-scores**:  $$ Z_{0.025} = 1.96, Z_{0.20} = 0.84 $$ \n4. **Plug values into the formula**:  $$ n = \frac{(1.96 \sqrt{2 \times 0.65(1-0.65)} + 0.84 \sqrt{0.24 + 0.21})^2}{(0.10)^2} $$  \n5. **Calculate**: Solve the above equation to get the required sample size.

The result of the sample size calculation tells us how many users we need to involve in the study to confidently detect the difference in retention rates between the control and experimental groups. For instance, if the calculation yields n = 385, this means we need 385 users in each group to ensure that we can detect a 10% increase in retention rate with 80% power at a 5% significance level. This is crucial for making evidence-based decisions on feature implementation.

This problem connects to several related concepts such as: 1. **Bayesian Inference**: Discuss how a Bayesian approach might provide a different perspective on user retention by incorporating prior information. 2. **Power Analysis**: Explore the implications of power in the context of user retention and how it affects the interpretation of results. 3. **Multiple Testing Corrections**: If testing multiple features, discuss how Bonferroni or False Discovery Rate adjustments might be necessary to control for Type I error rates. 4. **Confounding Factors**: Identify potential confounders such as user demographics or seasonal trends, and discuss how to control for these in the analysis.

Calculate probability for user retention

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