Calculate expected value for user retention

Question

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

Codemia · Accepted Answer

To calculate the expected value for user retention at Jump Trading, we need to define the analytical question clearly: "What is the required sample size to detect a significant difference in user retention rates after a new feature is implemented?" Specifically, we want to analyze how the revenue per session impacts user retention.

**Data Needed:**
1. Historical user retention rates (e.g., percentage of users retained after 30 days).
2. Revenue per session associated with retained users.
3. Estimated effect size (the minimum difference in retention rates we want to detect).
4. Desired statistical power (commonly set at 0.8) and significance level (commonly set at 0.05).

To determine the sample size needed, we can use the formula for sample size calculation for comparing two proportions:

$$ n = \frac{(Z_{\alpha/2} + Z_{\beta})^2 \cdot (p_1(1-p_1) + p_2(1-p_2))}{(p_1 - p_2)^2} $$  
Where:
- Z_{\alpha/2} is the Z-score for the desired significance level (1.96 for 0.05).
- Z_{\beta} is the Z-score for the desired power (0.84 for 0.8).
- p_1 is the baseline retention rate.
- p_2 is the expected retention rate after the intervention.

For this specific problem, let’s assume:
- p_1 (baseline retention) = 0.60
- p_2 (expected retention after intervention) = 0.70

Using the above formula, we can plug in the numbers to calculate the required sample size.

Using the assumed values:
- Z_{\alpha/2} = 1.96 (for 0.05 significance level)
- Z_{\beta} = 0.84 (for 0.8 power)
- p_1 = 0.60
- p_2 = 0.70

Now, we calculate:
$$ n = \frac{(1.96 + 0.84)^2 \cdot (0.60(1-0.60) + 0.70(1-0.70))}{(0.60 - 0.70)^2} $$
$$ n = \frac{(2.80)^2 \cdot (0.60 \cdot 0.40 + 0.70 \cdot 0.30)}{(-0.10)^2} $$
$$ n = \frac{7.84 \cdot (0.24 + 0.21)}{0.01} $$
$$ n = \frac{7.84 \cdot 0.45}{0.01} $$
$$ n = \frac{3.528}{0.01} = 352.8 $$
Thus, we would need approximately 353 participants in each group (control and treatment) to achieve our desired power and significance.

In the context of Jump Trading, the calculated sample size of 353 users per group (control and treatment) indicates that to reliably assess the impact of the new feature on user retention, we need to have this minimum number of users in each group. If the experiment is conducted with insufficient sample size, we risk either failing to detect a true effect (Type II error) or detecting a spurious effect (Type I error). Therefore, by adhering to this sample size, we can confidently interpret any differences in retention rates as being a result of the new feature rather than random chance.

Common pitfalls in this type of analysis include:
1. **Ignoring Confounding Variables**: Failing to account for other factors that might influence retention, such as seasonality, user demographics, or external market conditions.
2. **Underestimating Variability**: Assuming that the retention rates will remain stable without considering potential fluctuations due to external factors.
3. **Inaccurate Effect Size Estimation**: Overestimating the expected difference in retention rates (p_2), which can lead to underpowered tests and insufficient sample size.
4. **Not Using Cohort Analysis**: Neglecting to analyze retention in different user cohorts, which could provide more granular insights into how different segments respond to the new feature.

Calculate expected value for user retention

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