Define ARPU as the ratio of sample means R̄ and Ā, then apply the delta method to approximate its variance using the first-order Taylor expansion. Include the covariance term because Revenue and Active Users are correlated (ρ=0.6). Finally, explain how to estimate the required inputs (means, variances, covariance) from the experimental data.
Pro tip: Emphasize that the delta method variance is only valid for large samples; for small samples or heavy-tailed revenue, consider bootstrap or other robust methods. Also, clarify that the covariance term can be positive or negative, and its sign affects the variance estimate.
State that ARPU = R / A, where R is the sample mean revenue and A is the sample mean active users per bucket. Note that both are random variables with means μ_R and μ_A.
Use the first-order Taylor expansion around (μ_R, μ_A) to approximate Var(ARPU) ≈ (1/μ_A^2) Var(R) + (μ_R^2/μ_A^4) Var(A) - 2(μ_R/μ_A^3) Cov(R, A).
Replace Cov(R, A) with ρ * sqrt(Var(R) * Var(A)), where ρ = 0.6, to explicitly include the correlation in the variance formula.
Compute sample means (R̄, Ā), sample variances (s_R^2, s_A^2), and sample covariance (s_{RA}) from the experiment bucket data. Use these as plug-in estimates for the delta method variance.
Plug the estimates into the formula to get an approximate variance for ARPU. Discuss how the correlation term affects the variance and any assumptions (e.g., large sample, finite variance).
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Start by clarifying the goal: ARPU is a ratio metric, so direct testing can be noisy and sensitive to outliers. Then compare the two approaches in terms of statistical power, interpretability, and assumptions, highlighting when each is preferable. Conclude with a recommendation based on the experiment's context and practical considerations.
Pro tip: Mention that log-transforming revenue and using log-active users as a covariate can improve normality and reduce variance, but it changes the interpretation to a multiplicative effect on ARPU. Also note that ANCOVA can adjust for baseline differences in active users, but it requires the covariate to be unaffected by treatment.
Clarify that ARPU is revenue divided by active users, and the goal is to detect a treatment effect on ARPU. Discuss the implications of ratio metrics.
Explain that testing ARPU directly uses a t-test or similar on the ratio. It's simple and interpretable, but can have high variance and be sensitive to outliers in revenue or active users.
Describe modeling log(revenue) with log(active users) as a covariate. This can improve normality, reduce variance, and adjust for baseline differences. However, it assumes a linear relationship on the log scale and that the covariate is not affected by treatment.
Contrast the two approaches: direct ARPU is straightforward but may lack power; ANCOVA can be more powerful but requires stronger assumptions and careful interpretation (e.g., treatment effect on log-revenue given active users).
Suggest choosing based on data characteristics, business context, and whether active users is a post-treatment variable. If active users is affected by treatment, ANCOVA may be invalid; consider alternative methods like delta method or bootstrap.
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Start by outlining the assumptions of the standard t-test (independence, normality, homoscedasticity) and explain when they hold for ARPU, such as with large samples due to the CLT or when the metric is not heavily skewed. Then discuss when to switch to a bootstrap (e.g., small samples, extreme skewness, or when the CLT is unreliable) and how to handle heavy tails via winsorization or robust estimators, emphasizing the trade-offs.
Pro tip: Mention that at Meta, ARPU is often heavy-tailed due to power users, so you'd typically use a bootstrap or a robust method like trimmed means, but always validate with a simulation or sensitivity analysis to ensure the chosen method controls Type I error and maintains power.
Explain that the t-test assumes independent observations, normally distributed data (or large sample for CLT), and equal variances. For ARPU, these may be violated if the distribution is skewed or has outliers.
The t-test is acceptable when sample sizes are large enough (e.g., >30 per group) for the CLT to apply, and the data is not extremely skewed. Also, if ARPU is roughly symmetric and variances are similar, it's fine.
Switch to bootstrap when sample sizes are small, the distribution is highly skewed or heavy-tailed, or when you need to estimate other statistics (e.g., median). Bootstrap makes fewer assumptions and can provide more reliable inference.
Discuss options: winsorization (capping extreme values at a percentile) to reduce influence of outliers, or using robust estimators like trimmed mean or median. Also consider transformations (e.g., log) but note interpretation challenges.
Acknowledge that winsorization introduces bias but reduces variance; robust estimators may be less efficient if data is normal. Recommend simulation or sensitivity analysis to choose the best method for the specific context.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.