Calculate expected value for user retention

Question

Given the following scenario about revenue per session, calculate the the p-value.

Codemia · Accepted Answer

To calculate the expected value for user retention at Compass, we need to define the analytical question: **What is the expected revenue generated from retained users over a specified period?** We will focus on the average revenue per session (ARPS) and the retention rate of users.

**Data Required:**  
1. Historical data on user sessions, including the number of sessions per user and the revenue generated per session.  
2. Data on user retention rates, specifically how many users return for subsequent sessions within a defined time frame (e.g., 30 days).  
3. Control group data to compare users who experienced a retention-focused intervention versus those who did not.

To calculate the p-value for user retention, we will use a two-sample t-test to compare the average revenue per session for the control group (users without the retention intervention) to the treatment group (users with the intervention). This will allow us to assess whether the intervention had a statistically significant effect on revenue per session.

**Steps:**  
1. Calculate the average revenue per session (ARPS) for both the control group and treatment group.  
2. Compute the t-statistic using the formula:  
   $$ t = \frac{\bar{X}_1 - \bar{X}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} $$  
   where $ \bar{X}_1 $ and $ \bar{X}_2 $ are the means, $ s_p $ is the pooled standard deviation, and $ n_1 $ and $ n_2 $ are the sample sizes.  
3. Determine the p-value corresponding to the calculated t-statistic.

Assume the following data:  
- Control Group: 100 users, ARPS = $10, standard deviation = $2  
- Treatment Group: 100 users, ARPS = $12, standard deviation = $3

**Calculating Means and Sample Sizes:**  
- $ \bar{X}_1 = 10, n_1 = 100 $  
- $ \bar{X}_2 = 12, n_2 = 100 $

**Pooled Standard Deviation Calculation:**  
$$ s_p = \sqrt{\frac{(n_1-1) s_1^2 + (n_2-1) s_2^2}{n_1+n_2-2}} = \sqrt{\frac{(100-1)(2^2) + (100-1)(3^2)}{100 + 100 - 2}} = 2.576 $$

**t-statistic Calculation:**  
$$ t = \frac{10 - 12}{2.576 \sqrt{\frac{1}{100} + \frac{1}{100}}} = -4.1 $$

**Finding the p-value:**  
Using a t-distribution table for 198 degrees of freedom, a t-statistic of -4.1 gives a p-value < 0.0001.

The p-value obtained (< 0.0001) indicates that there is a statistically significant difference in the average revenue per session between the control and treatment groups. This suggests that the retention intervention positively impacted user retention, leading to increased revenue. In the context of Compass, this result could inform strategic decisions regarding retention efforts and resource allocation to enhance user engagement and revenue generation.

Common pitfalls in this type of analysis include:  
1. **Ignoring Confounding Variables:** Ensure that other factors (e.g., seasonality, marketing efforts) are controlled for, as they may skew the results.  
2. **Sample Size Miscalculation:** A small sample size can lead to unreliable p-values; ensure sufficient power in the experiment.  
3. **Misinterpretation of P-value:** A low p-value does not measure the size of the effect; consider confidence intervals for effect size.  
4. **Overlooking Long-term Effects:** Focus only on short-term metrics may miss critical insights about long-term retention and revenue trends.

Calculate expected value for user retention

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