First, clarify the data structure and definitions: identify the strata (city × week), the variants, and the conversion metric. Then, for each stratum and variant, compute the conversion rate as conversions/visitors, and calculate the 95% confidence interval using the normal approximation (or Wilson score) for a proportion. Finally, present the results in a clear table and discuss any notable patterns or caveats.
Pro tip: Always check the sample size and conversion counts per stratum; if counts are small, use the Wilson interval or exact methods instead of the normal approximation, and mention that you would consider multiple testing corrections if making comparisons across many strata.
Confirm the strata (e.g., City A Week 1, City A Week 2, City B Week 1, City B Week 2), the variants (control and treatment), and the conversion metric (e.g., click-through rate, sign-up rate). Ensure you know the numerator (conversions) and denominator (visitors or users) for each cell.
For each stratum and variant, calculate the conversion rate as the number of conversions divided by the number of visitors. Present these rates clearly, perhaps in a table.
For each rate, compute the standard error using sqrt(p*(1-p)/n) and the 95% CI as p ± 1.96*SE. If sample sizes are small or rates extreme, use the Wilson score interval for better coverage.
Summarize the rates and CIs, noting any overlapping intervals or significant differences. Discuss whether the stratified analysis reveals consistent effects across cities and weeks, and mention any limitations (e.g., multiple comparisons, small sample sizes).
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This is where I started sweating a little.
First, explain that the Mantel-Haenszel test assesses whether the treatment effect is consistent across strata by estimating a common odds ratio. Then, outline the steps to compute the MH odds ratio, its confidence interval, and the p-value, emphasizing the assumptions and interpretation in the context of A/B testing.
Pro tip: Mention that the MH test is most appropriate when strata are not too sparse and the treatment effect is expected to be homogeneous; otherwise, consider a test for interaction or a random-effects model.
Explain that the Mantel-Haenszel test estimates a common odds ratio across strata, assuming a consistent treatment effect. Discuss when to use it, such as when analyzing A/B test results across different user segments.
Describe the formula for the MH odds ratio: sum of (a_i * d_i / n_i) divided by sum of (b_i * c_i / n_i), where a_i, b_i, c_i, d_i are the cell counts in each 2x2 table for stratum i.
Use the standard error of the log odds ratio: sqrt( sum of ( (a_i+d_i)/n_i^2 * (a_i*d_i) + (b_i+c_i)/n_i^2 * (b_i*c_i) ) ) or the Robins-Breslow-Greenland variance estimator. Then compute the CI as exp(log(OR_MH) ± 1.96 * SE).
Perform the Mantel-Haenszel chi-square test: sum of (a_i - E[a_i]) / sqrt(sum of Var[a_i]), where E[a_i] = (a_i+b_i)(a_i+c_i)/n_i and Var[a_i] = (a_i+b_i)(c_i+d_i)(a_i+c_i)(b_i+d_i)/(n_i^2 (n_i-1)). Square the statistic and compare to a chi-square distribution with 1 degree of freedom.
Report the common odds ratio, its 95% CI, and p-value. Discuss whether the treatment effect is statistically significant and consistent across strata, and relate to business impact.
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I blanked on the Breslow-Day formula specifically.
Start by explaining that the Breslow-Day test (or an equivalent interaction test) checks whether the treatment effect is consistent across different strata or subgroups. Then describe what a significant result means: there is evidence of effect heterogeneity, meaning the treatment effect varies by subgroup. Finally, discuss the implications for decision-making, such as the need for further investigation or tailored strategies.
Pro tip: Emphasize that a non-significant result does not prove homogeneity; it may simply lack power to detect interactions, especially with small sample sizes per stratum. Always consider practical significance and the context of the experiment.
Explain that the Breslow-Day test assesses whether the odds ratio (or treatment effect) is homogeneous across strata. It is an interaction test between treatment and subgroup.
State that the null hypothesis is that the treatment effect is the same across all strata (no interaction), while the alternative is that at least one stratum has a different effect.
If the test is significant (p < alpha), there is evidence of effect heterogeneity; the treatment effect differs by subgroup. If not significant, there is insufficient evidence to conclude heterogeneity.
A significant result suggests the overall effect may not be generalizable; investigate which subgroups drive the difference and consider whether to tailor interventions or run further experiments.
Mention that the test has assumptions (e.g., large sample sizes) and that multiple testing can inflate false positives; adjust for multiple comparisons if needed.
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First, compute the overall conversion rates by pooling all users across strata, ignoring any segmentation. Then, compare the pooled rates to the stratum-specific rates to check if the direction of the effect reverses, which would indicate Simpson's paradox. Explain the paradox by highlighting how confounding variables (like device type or user activity) can distort the pooled comparison.
Pro tip: Always consider whether the strata are balanced between treatment and control; if not, the pooled analysis can be misleading. Mention that Simpson's paradox is a sign of confounding and that stratified analysis or regression adjustment is needed for valid inference.
Aggregate the data across all strata to calculate the overall conversion rate for each variant (e.g., control and treatment). This gives the naive pooled rates.
Calculate conversion rates within each stratum (e.g., by device, region, or user segment) for both variants to understand the effect within homogeneous groups.
Check if the direction of the difference between variants reverses when moving from pooled to stratified analysis. If it does, Simpson's paradox is present.
Examine whether the distribution of strata differs between variants. If one variant has more users in a stratum with inherently higher conversion, it can distort the pooled rate.
Discuss why Simpson's paradox matters for decision-making and recommend using stratified analysis, regression, or propensity scores to adjust for confounders.
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Start by clearly stating your final recommendation based on the stratified analysis, then explain why the naive pooled result is misleading due to confounding or Simpson's paradox. Reconcile the two by discussing the importance of controlling for key covariates and the validity of stratified results for decision-making.
Pro tip: Always check for Simpson's paradox by examining segment-level results; it's a common pitfall in A/B testing that can lead to wrong decisions. Mention that you would validate the stratified findings with a regression model or interaction tests to ensure robustness.
Begin with a clear, decisive recommendation on which email variant to choose, based on the stratified analysis. For example, 'I recommend variant B because it performs better within key user segments.'
Describe why the naive pooled result differs from the stratified result, likely due to confounding variables or Simpson's paradox. Highlight that the pooled analysis may be biased by unequal distribution of segments across variants.
Argue that the stratified analysis is more reliable because it controls for important covariates, providing a truer estimate of the treatment effect within relevant subpopulations.
Acknowledge limitations such as multiple testing or reduced power in strata, and suggest validation methods like regression adjustment or interaction tests to confirm findings.
Tie the recommendation back to business goals, explaining how the chosen variant will improve key metrics and why the stratified insight is actionable for LinkedIn.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.