Calculate variance for user retention

Question

Given the following scenario about user conversion, calculate the the p-value.

Codemia · Accepted Answer

To analyze user retention for Dropbox, we start by defining our variables. Let $ X_i $ be the number of users retained after a certain period, where $ i $ ranges from 1 to $ n $, the total number of users. We can represent our data as a sample of user retention rates across different cohorts. Our goal is to test if the observed retention rates significantly differ from a hypothesized retention rate ($ p_0 $). We will use a binomial model here, with $ p $ representing the true retention probability. The null hypothesis is defined as $ H_0: p = p_0 $, and the alternative hypothesis is $ H_1: p \neq p_0 $.

To calculate the p-value, we will perform the following steps:
1. Collect the sample data of retained users and total users.
2. Calculate the sample proportion of retained users $ \hat{p} = \frac{X}{n} $.
3. Use the binomial test to determine the p-value based on the observed and expected proportions. We will apply the formula for the binomial test:
   $$ p = P(X \geq k) = \sum_{x=k}^{n} \binom{n}{x} p_0^x (1-p_0)^{n-x} $$
   where $ k $ is the observed number of successes (retained users).
4. Determine if the p-value is less than a significance level (commonly $ \alpha = 0.05 $).

Suppose we have a sample of 1000 users, and 450 of them retained after the trial period. Thus, $ X = 450 $ and $ n = 1000 $. We hypothesize that the expected retention rate is $ p_0 = 0.40 $.
1. Calculate the sample proportion:  $ \hat{p} = \frac{450}{1000} = 0.45 $.
2. Calculate the p-value using the binomial test:
   - $ k = 450 $
   - Using a binomial formula or statistical software, we find the p-value. For example, using Python's `scipy.stats.binom_test`, we might find $ p-value \approx 0.04 $.
3. Compare this to $ \alpha = 0.05 $. Since 0.04 < 0.05, we reject the null hypothesis.

The calculated p-value of 0.04 suggests that there is statistically significant evidence to conclude that the retention rate of users differs from the hypothesized rate of 0.40. In practical terms, this indicates that Dropbox's user retention strategies may have positively influenced user retention, warranting further investigation into which factors contributed to this improvement. However, it's crucial to note that statistical significance does not imply practical significance; further analysis is required to assess the impact of this change on overall business outcomes.

This analysis can be extended by considering potential confounders that might affect retention, such as user demographics or engagement levels. Additionally, one might explore the use of Bayesian inference for a more nuanced understanding of retention probabilities and to incorporate prior information. The Central Limit Theorem can also be invoked if we gather larger samples, allowing us to use normal approximations for the binomial distribution. Lastly, causal inference methods may be employed to determine the effectiveness of specific retention strategies.

Calculate variance for user retention

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