This one tripped me up more than I expected.
Recognize that sampling children introduces size-biased sampling: the probability of selecting a child from a family is proportional to family size. Use the observed distribution of family sizes among sampled children to estimate the underlying family-size distribution via a weighted estimator, then construct a confidence interval for the proportion of each family type using the delta method or bootstrap.
Pro tip: Explicitly state the size-biased sampling assumption and show how ignoring it leads to overestimating larger families. Mention that the estimator is essentially a weighted average with weights inversely proportional to family size.
Clarify that the sample is of children, not families, so each child is sampled with probability proportional to their family size. Assume families are independent and the sample is random.
Let p1, p2, p3 be the true proportions of one-, two-, and three-child families. The probability a randomly selected child comes from a family of size k is k*pk / (p1+2p2+3p3). Use the observed counts of children from each family size to solve for p1, p2, p3.
Given the sample counts n1, n2, n3 of children from one-, two-, and three-child families, compute the estimated proportions: p1_hat = (n1/100) / (n1/100 + 2*n2/100 + 3*n3/100) * (1/1)? Actually, solve the system: n1/100 = p1 / (p1+2p2+3p3), n2/100 = 2p2 / (p1+2p2+3p3), n3/100 = 3p3 / (p1+2p2+3p3). Then p1_hat = (n1/100) / (n1/100 + n2/100 + n3/100) * (1/1)? Wait, the denominator is the average family size. More precisely, let S = n1/100 + 2*(n2/100) + 3*(n3/100)? No, the observed proportions of children from size k families are qk = nk/100. Then q1 = p1 / (p1+2p2+3p3), q2 = 2p2 / (p1+2p2+3p3), q3 = 3p3 / (p1+2p2+3p3). So p1 = q1 * (p1+2p2+3p3), p2 = q2/2 * (p1+2p2+3p3), p3 = q3/3 * (p1+2p2+3p3). Since p1+p2+p3=1, we have (p1+2p2+3p3) = 1 + p2 + 2p3. But easier: Let D = p1+2p2+3p3. Then p1 = q1 D, p2 = q2 D /2, p3 = q3 D /3. Sum: D(q1 + q2/2 + q3/3) = 1 => D = 1 / (q1 + q2/2 + q3/3). Then p1 = q1 / (q1 + q2/2 + q3/3), p2 = (q2/2) / (q1 + q2/2 + q3/3), p3 = (q3/3) / (q1 + q2/2 + q3/3).
Use the delta method to approximate the variance of the estimated proportions, or use bootstrap resampling of children to obtain empirical confidence intervals. For the delta method, compute the gradient of the estimator with respect to the observed proportions and use the multinomial covariance matrix.
Check that the estimated proportions sum to 1 and lie in [0,1]. Discuss potential biases if the sampling scheme is not truly random or if family sizes are correlated with other factors.
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Knew the mechanical answer (OLS minimizes vertical vs horizontal residuals) but the causal direction part is where it gets interesting.
Start by explaining the mathematical asymmetry: OLS minimizes vertical errors, so swapping X and Y changes the objective and yields different slopes. Then connect this to causality: the regression slope reflects the direction of conditioning, and only a correctly specified causal model (e.g., via instrumental variables or experiments) can identify the true causal effect. Emphasize that regression alone cannot determine causal direction; it only quantifies association under a given conditioning set.
Pro tip: Mention that the product of the two slopes equals the squared correlation (r²), which shows they are not reciprocals and highlights the asymmetry. Also note that in a true causal model with no confounding, regressing Y on X gives the causal effect, while regressing X on Y gives the inverse of the causal effect only if the relationship is deterministic and invertible—otherwise it's biased.
Clarify that OLS minimizes the sum of squared residuals in the direction specified (Y on X vs X on Y). This difference in loss function leads to different slope estimates.
Explain that the slope for Y on X is Cov(X,Y)/Var(X), while for X on Y it is Cov(X,Y)/Var(Y). Since variances differ, the slopes differ, and their product equals r².
Discuss that regression captures association, not causation. The slope's interpretation depends on which variable is treated as the outcome and which as the predictor, reflecting an assumed causal direction.
Explain that without controlling for confounders or using causal inference methods (e.g., instrumental variables, experiments), neither regression reveals the true causal effect. The direction of regression should be guided by causal assumptions.
Conclude that data scientists must be explicit about causal assumptions and choose regression direction based on the causal question, not just statistical fit.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by acknowledging that this is a common scenario in regression modeling, often due to multicollinearity, high dimensionality, or overfitting. Explain the statistical reasons behind the discrepancy between individual coefficient significance and overall predictive performance, then outline a systematic approach to diagnose and address the issue, focusing on model validation and feature selection.
Pro tip: Emphasize that predictive performance and inferential significance serve different goals; in practice, you might prioritize one over the other depending on the business objective. Mention that at Upstart, where interpretability and regulatory compliance matter, you'd balance both by using techniques like regularization and careful feature engineering.
Check for multicollinearity using VIF, examine the number of predictors relative to sample size, and assess potential overfitting by comparing training and validation performance.
Use cross-validation to confirm that the strong predictive performance is not due to overfitting and holds on unseen data.
Look beyond p-values: assess overall model fit (e.g., R-squared, RMSE) and predictive metrics (e.g., AUC, lift) to understand the model's utility.
If multicollinearity is present, use regularization (ridge, lasso), feature selection, or dimensionality reduction. If overfitting, simplify the model or gather more data.
Explain to stakeholders that the model may be good for prediction but not for inference, and align the modeling approach with the business goal.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.