This is where I spent most of my mental energy.
Frame the problem as right-censored survival analysis where conversion time is the event time and the 14-day conversion rate is the cumulative incidence at day 14. Propose a nonparametric estimator that combines the empirical conversion-time distribution from historical cohorts with the current cohort's partially observed data, using inverse probability of censoring weighting (IPCW) or a Kaplan-Meier-type estimator to debias the current cohort.
Pro tip: Emphasize that the key assumption is that the delay distribution is stable over time; validate this by comparing historical cohorts and testing for drift. Also, mention that you would use the current cohort's early conversions to adjust the historical delay curve if needed, e.g., via a multiplicative shift.
Clarify that the 14-day conversion rate is the probability of conversion within 14 days of impression. For each user i, we observe impression time T_i, conversion time C_i (if converted), and censoring time U_i = min(C_i, 14 days). Data: current cohort has (T_i, U_i, δ_i) where δ_i=1 if converted by 14 days, else 0; historical cohorts have full conversion times for users who converted within some window.
Using historical cohorts with complete follow-up (e.g., 30 days), compute the empirical distribution of conversion delays D = C - T. Estimate the survival function S(d) = P(D > d) nonparametrically via Kaplan-Meier or empirical CDF. This requires historical data with known conversion times for all converters.
For the current cohort, each observed conversion by day 14 is weighted by 1 / (1 - S(14 - age_i)), where age_i is the time since impression. The 14-day conversion rate estimator is the weighted sum of conversions divided by the total number of impressions. Formula: \hat{p} = (1/N) * Σ_{i: δ_i=1} [1 / (1 - S(14 - age_i))], where N is total impressions.
Alternatively, treat the current cohort as right-censored data and compute the Kaplan-Meier estimate of the conversion time distribution, then evaluate at day 14. This uses only current cohort data but may be unstable if few conversions. Formula: \hat{S}_{KM}(t) = Π_{j: t_j ≤ t} (1 - d_j / n_j), where d_j is conversions at time t_j and n_j is at risk. Then \hat{p} = 1 - \hat{S}_{KM}(14).
Check that the historical delay distribution is stable over time (e.g., compare S(d) across cohorts). If stable, the IPCW estimator is consistent. If not, consider a hybrid: use current cohort's early conversions to update S(d) via a parametric or semi-parametric model. Report uncertainty via bootstrap.
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I knew Greenwood's formula in principle but applying the delta method on top of it for a transformed CVR was a bit awkward to write out under pressure.
Start by defining CVR14 as a cumulative incidence function (CIF) derived from Kaplan-Meier estimates, then apply Greenwood's formula to obtain the variance of the survival function and use the delta method to transform it to the CVR14 scale. Finally, discuss assumptions and how to adjust intervals if the delay distribution shifts over time.
Pro tip: Emphasize that CVR14 is a cumulative incidence metric and that ignoring censoring or competing risks can bias the estimate; always check the proportional hazards assumption and consider time-varying effects.
Clarify that CVR14 is the cumulative incidence of conversion within 14 days, estimated via 1 - S(14) from Kaplan-Meier, accounting for censoring.
Use Greenwood's formula: Var(S(t)) = S(t)^2 * sum_{t_i <= t} d_i / (n_i (n_i - d_i)), where d_i and n_i are events and at-risk at each event time.
Since CVR14 = 1 - S(14), the variance is Var(CVR14) = Var(S(14)) by the delta method (derivative -1). Construct a 95% CI as CVR14 ± 1.96 * sqrt(Var(CVR14)).
Assumptions include independent censoring, no competing risks, correct specification of the survival model, and that the delta method approximation is valid (large sample).
If the delay distribution shifts, use a time-varying coefficient model or stratified Kaplan-Meier, and widen intervals via bootstrapping or robust variance estimation to account for model misspecification.
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Honestly the cleanest part of the interview for me.
Start by clarifying the metric CVR14 (conversion rate within 14 days) and the data available: conversions, impressions, and delays. Then propose a conservative bound using distribution-free methods: apply the DKW inequality to bound the delay distribution, or use Clopper-Pearson for observed conversions and a worst-case bound for unresolved impressions. Finally, show the computation from raw counts, emphasizing minimal assumptions and conservativeness.
Pro tip: Emphasize that the bound is conservative and distribution-free, which is crucial for trust in A/B testing where false positives are costly. Also, mention that you would validate the bound with historical data or simulations to ensure it's not overly pessimistic.
Define CVR14 as the proportion of impressions that convert within 14 days. Identify the raw counts available: total impressions, conversions observed so far, and the time since each impression (to know which are unresolved).
Select either the DKW inequality to bound the empirical CDF of conversion delays, or Clopper-Pearson for observed conversions combined with a worst-case bound for unresolved impressions. Justify the choice based on minimal assumptions.
For DKW: compute the empirical CDF of delays for resolved impressions, then use the DKW inequality to get a confidence band. For Clopper-Pearson: compute the lower confidence bound on conversion rate for resolved impressions, then assume all unresolved impressions convert (or not) to get a conservative bound.
Combine the bounds appropriately (e.g., take the minimum for a lower bound or maximum for an upper bound) to get a conservative estimate of CVR14. Clearly state the confidence level and assumptions.
Mention how to handle edge cases (e.g., zero conversions), the trade-off between conservativeness and tightness, and how to communicate the bound to stakeholders.
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I talked through PSI for categorical features and KS for continuous ones, then mentioned stratifying the KM estimator by device or traffic source if shift is detected.
Start by framing the problem as a distribution shift detection task, then outline a systematic diagnostic process: compare historical and current delay distributions using statistical tests and drift metrics, segment by key covariates (traffic mix, device type, day-of-week), and finally propose actionable remediation if shift is detected. Emphasize the importance of both global and segment-level monitoring to catch subtle shifts that could impact model performance.
Pro tip: Don't just run tests—quantify the impact of shift on your specific downstream metric (e.g., ETA accuracy) and prioritize segments by business impact. Also, consider using a holdout set from recent data to validate whether retraining or recalibration actually improves performance before deploying changes.
Clearly specify the delay metric (e.g., delivery time minus estimated time) and compute its distribution for historical and current periods. Use summary statistics, histograms, and quantile plots to visually inspect differences.
Apply statistical tests like Kolmogorov-Smirnov (KS) or Population Stability Index (PSI) on the overall delay distribution to detect any significant shift. Set thresholds based on business tolerance and monitor over time.
Break down the data by traffic mix (e.g., source channel), device type, day-of-week, and other relevant dimensions. Compute PSI or KS for each segment to identify where the shift is concentrated.
If shift is detected, drill down to understand why (e.g., new traffic source, device update, seasonal effect). Assess the impact on model performance by evaluating prediction errors on recent data.
Decide on actions: retrain the model on recent data, recalibrate predictions, or adjust for segment-specific shifts. Implement continuous monitoring with alerts and A/B test the fix before full rollout.
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Start by clarifying the metric definition and the data limitations, then propose a bounding approach using the relationship between same-day CVR and CVR14. Use assumptions like stable conversion delay distribution or stationarity to derive bounds, and discuss how to validate or refine with additional data.
Pro tip: Acknowledge that without user-level data, CVR14 is not identifiable, but you can provide useful bounds by leveraging known patterns from similar products or historical data. Always state your assumptions explicitly and discuss sensitivity.
Define CVR14 as conversions within 14 days of impression divided by impressions. Confirm that you only have daily aggregated impressions and same-day conversions, so you cannot track users over time.
Recognize that CVR14 requires knowing conversions on days 1-13 after impression, which are not directly observed. The same-day conversions only give you CVR0.
Use assumptions to bound CVR14. For example, if you assume conversions only happen within 14 days, then CVR14 >= CVR0. If you assume a maximum conversion rate per day, you can set an upper bound. Alternatively, use a conversion delay distribution from historical data to estimate.
To make CVR14 identifiable, you need assumptions like: the conversion delay distribution is stable over time, or the ratio of CVR14 to CVR0 is constant. Discuss how these assumptions might be validated.
Suggest ways to validate assumptions, such as using holdout data or A/B tests where user-level data is available. Perform sensitivity analysis on the bounds to show robustness.
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