First, clarify the test setup: two-sided z-test for proportions at alpha=0.05 and 80% power. Then, compute the required per-arm sample size using the standard formula, explicitly stating whether you assume equal variances (pooled) or unequal variances (unpooled), and show the calculation with the given rates.
Pro tip: Mention that in practice, you'd use a power analysis tool or simulation to account for continuity corrections and exact tests, but the z-test approximation is a good starting point. Also, note that Amazon often deals with large-scale tests, so even small differences like 0.2% can be practically significant if the sample size is large enough.
State the null and alternative hypotheses: H0: p1 = p2 vs H1: p1 ≠ p2. Set alpha=0.05 (two-sided) and power=0.80, so beta=0.20. Let p1=0.02 and p2=0.018.
Decide whether to assume equal variances (pooled) or unequal variances (unpooled). For sample size calculation, the unpooled approach is common because it directly uses the alternative proportions, but some prefer pooled for a conservative estimate.
Use the formula for two proportions: n = ( (z_{1-alpha/2} * sqrt(2 * p_bar * (1-p_bar)) + z_{1-beta} * sqrt(p1*(1-p1) + p2*(1-p2)) )^2 ) / (p1 - p2)^2, where p_bar = (p1+p2)/2 if pooled, or use unpooled version. Plug in z-values: z_{0.975}=1.96, z_{0.80}=0.84.
Calculate the numerical value. With p1=0.02, p2=0.018, the difference is 0.002. Using unpooled: p1(1-p1)=0.0196, p2(1-p2)=0.017676, sum=0.037276, sqrt=0.19307. p_bar=0.019, 2*p_bar*(1-p_bar)=2*0.019*0.981=0.037278, sqrt=0.19308. Then n = ( (1.96*0.19308 + 0.84*0.19307)^2 ) / (0.002)^2 = ( (0.37844 + 0.16218)^2 ) / 0.000004 = (0.54062^2)/0.000004 = 0.29227/0.000004 = 73067.5 per arm. Round up to 73068 per arm.
State that approximately 73,068 users per arm are needed. Discuss that this assumes no continuity correction, large sample normality, and that the test is two-sided. Mention that if using pooled variance, the sample size might be slightly different (e.g., pooled p_bar=0.019 gives similar result).
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First, compute the sample proportions and their difference. Then calculate the standard error of the difference using the pooled proportion for the hypothesis test and the unpooled proportions for the confidence interval. Finally, construct the 95% confidence interval and compute the z-statistic and p-value for the two-proportion z-test.
Pro tip: Always clarify whether the confidence interval should be based on the unpooled or pooled standard error; for a confidence interval, use unpooled, but for the hypothesis test, use pooled. Also, mention that with such large sample sizes, the normal approximation is valid and the p-value will be extremely small, so you might report it as <0.0001.
Compute the proportion of spam in control (p1 = 0.021) and treatment (p2 = 0.0185). The absolute difference is p1 - p2 = 0.0025 (or 0.25 percentage points).
Use the unpooled standard error formula: SE = sqrt( p1*(1-p1)/n1 + p2*(1-p2)/n2 ). With n1 = n2 = 500,000, calculate SE ≈ sqrt(0.021*0.979/500000 + 0.0185*0.9815/500000) ≈ 0.000284.
The 95% CI for the difference is (p1 - p2) ± 1.96 * SE. This yields 0.0025 ± 1.96*0.000284 ≈ (0.00194, 0.00306). So we are 95% confident the true difference is between 0.194 and 0.306 percentage points.
For the hypothesis test, use the pooled proportion: p_pool = (x1 + x2)/(n1 + n2) = (10500 + 9250)/1,000,000 = 0.01975. Then SE_pooled = sqrt( p_pool*(1-p_pool)*(1/n1 + 1/n2) ) ≈ 0.000278. The z-statistic is (0.0025 - 0)/0.000278 ≈ 8.99. The two-sided p-value is approximately 2.4e-19, essentially 0.
The confidence interval does not include zero, and the p-value is far below 0.05, indicating a statistically significant reduction in spam rate in the treatment group. The effect size is small (0.25 percentage points) but may be practically significant given the large volume.
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Seasonality I had a decent answer for, stratify by day-of-week or time window and randomize within strata.
Start by defining class imbalance drift and traffic seasonality as threats to internal and external validity, then explain how stratification ensures representative train/test splits and blocking controls for temporal confounds. Finally, discuss practical implementation in an A/B testing framework, emphasizing trade-offs between bias reduction and variance.
Pro tip: Frame your answer around Amazon's leadership principles, such as 'Customer Obsession' (ensuring the classifier works for all customers) and 'Dive Deep' (understanding data drift). Mention that you would monitor drift post-deployment and set up alerts, showing proactive ownership.
Explain how class imbalance drift (changing spam-to-ham ratio over time) and traffic seasonality (e.g., holiday spikes) can cause the classifier to learn spurious patterns and fail to generalize, threatening both internal and external validity.
Describe how stratification by class and time (e.g., stratified sampling by week) ensures that training, validation, and test sets have similar class distributions and temporal coverage, reducing bias in performance estimates.
Discuss how blocking (e.g., grouping by time periods or user segments) can control for seasonality by comparing treatments within homogeneous blocks, isolating the treatment effect from temporal confounds.
Propose a combined approach: use stratified blocking in experiment design, then validate with time-based cross-validation. Mention tools like scikit-learn's StratifiedKFold and time series split.
Emphasize the need for continuous monitoring of class balance and traffic patterns post-deployment, with automated alerts and retraining triggers to maintain validity.
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Explain that peeking and stopping early inflates Type I error because each look is a hypothesis test, and multiple testing increases the chance of false positives. Then describe corrections like alpha spending functions (e.g., O'Brien-Fleming), group sequential designs, or always-valid p-values, and mention practical alternatives like fixed-horizon tests or Bayesian methods.
Pro tip: Emphasize that the inflation depends on the number and timing of peeks, and that corrections should be pre-specified to maintain validity. Also note that in practice, companies like Amazon often use sequential testing frameworks to allow continuous monitoring without inflating error rates.
Clarify that peeking and stopping early when p<0.05 is a form of optional stopping, which leads to multiple comparisons and inflates the family-wise error rate (FWER).
Describe how each peek is an additional test, and the probability of at least one false positive accumulates over looks, similar to the multiple comparisons problem.
Mention that with many peeks, Type I error can approach 1, and that the exact inflation depends on the number of peeks and correlation between test statistics.
Discuss group sequential designs with alpha spending (e.g., O'Brien-Fleming, Pocock), always-valid p-values, or Bayesian approaches that allow continuous monitoring without inflation.
Note that corrections may require larger sample sizes or reduce power, and that pre-specification of the analysis plan is crucial to avoid p-hacking.
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Start by defining precision at fixed recall as a ratio of two dependent sample means (TP/(TP+FP) at a threshold chosen to achieve the target recall), then explain how the delta method uses a first-order Taylor expansion to approximate the variance of this ratio, while bootstrapping resamples the data to empirically estimate the sampling distribution. Compare their assumptions, computational costs, and performance in small samples or with complex metrics, and conclude with when each is preferable in an A/B testing context at Amazon.
Pro tip: Emphasize that the delta method requires the threshold to be fixed (or the recall constraint to be handled carefully) and that bootstrapping must resample at the appropriate unit (e.g., user or session) to respect the data's dependence structure; this shows you understand practical pitfalls beyond textbook formulas.
Express precision at fixed recall as a ratio of two sample means: precision = TP / (TP + FP) where the threshold is chosen so that recall = TP / (TP + FN) equals the target. Clarify that both numerator and denominator are functions of the same threshold and thus dependent.
Use the delta method: compute the gradient of the ratio with respect to the means of TP, FP, and FN (or directly with respect to the confusion matrix counts), then apply the multivariate delta method to obtain an approximate variance and confidence interval. Mention that this requires estimates of the covariance matrix of the counts.
Describe a bootstrap procedure: resample the data (e.g., users or sessions) with replacement, recompute the threshold to achieve the fixed recall on each resample, then compute precision. Repeat many times to get the empirical distribution, and form a percentile or BCa confidence interval.
Contrast the delta method's reliance on asymptotic normality and correct variance estimation (which may fail with small samples or extreme probabilities) with bootstrapping's flexibility but higher computational cost and potential bias if the resampling unit is wrong.
Conclude that the delta method is preferable for large samples, simple metrics, and when speed is critical (e.g., real-time monitoring), while bootstrapping is better for small samples, complex metrics, or when the asymptotic approximation is questionable. In A/B testing, consider using both for validation.
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