CVS Health·Data Scientist·Take-home Assignment·Intermediate

Intermediate

May 2026

Summary

CVS Health data scientist interview that was basically a coding session disguised as a take-home. Two fairly involved problems: one on implementing R-squared from scratch and another on PCA with and without standardization. More math-heavy than I expected for a health company role.

Questions Asked (2)

Implement an r2_score function using only NumPy (no scikit-learn) that handles edge cases like perfect predictions, zero-variance targets, and division by zero. Test it on a standard case and two edge cases where all true values are identical.

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

The normal case was fine, SS_res over SS_tot, nothing crazy.

Create a free account to read the full note

AI HintsAI Generated

Suggested Approach

Start by writing the r2_score formula from scratch using NumPy operations, then explicitly handle edge cases like perfect predictions and zero-variance targets. Test the function on a standard dataset and two edge cases where all true values are identical, demonstrating correct behavior and discussing trade-offs.

Pro tip: Mention that scikit-learn's r2_score returns 1.0 for perfect predictions and 0.0 for constant predictions (when the model predicts the mean), but raises a warning for zero-variance targets; replicate this behavior to show attention to detail.

1. Define the R² formula

Write the mathematical definition: R² = 1 - (SS_res / SS_tot), where SS_res = sum((y_true - y_pred)^2) and SS_tot = sum((y_true - y_true_mean)^2).

2. Implement with NumPy

Use NumPy arrays and vectorized operations to compute SS_res and SS_tot efficiently, avoiding loops.

3. Handle edge cases

Check for perfect predictions (SS_res = 0) and zero-variance targets (SS_tot = 0). Decide on return values: 1.0 for perfect predictions, and for zero-variance, either 0.0 if predictions are perfect else -inf or raise an error, mirroring scikit-learn's behavior.

4. Test on standard and edge cases

Create test cases: a standard case with varying y_true, and two edge cases where all y_true are identical (e.g., all zeros and all fives). Verify outputs and discuss expected behavior.

5. Discuss trade-offs and robustness

Explain why handling division by zero is important, and how your implementation compares to scikit-learn's, including any warnings or exceptions.

Key Points to Mention

The mathematical definition of R² and its interpretation as proportion of variance explained.
The importance of handling SS_tot = 0 to avoid division by zero, and the ambiguity when y_true is constant.
Scikit-learn's behavior: returns 1.0 for perfect predictions, 0.0 for constant predictions (if predictions equal mean), and warns for zero-variance targets.
Use of NumPy's vectorized operations for efficiency and numerical stability.
Testing strategy: include a standard case and two edge cases with constant y_true to validate edge case handling.
Potential trade-offs: whether to return a value, raise an error, or issue a warning for zero-variance targets.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.

Given a 6x3 matrix with columns on very different scales, run PCA twice: once on raw data and once on column-standardized data. For each, compute the covariance matrix via NumPy, sort eigenvectors by eigenvalue, report explained variance ratio for the first two components, and discuss why scaling changes the principal components.

Algorithms & Data StructuresTechnical Trade-offsData Modeling

Author's notes

The second column in that matrix is in the hundreds while the third is under 1, so raw PCA is completely dominated by column 2.

Create a free account to read the full note

AI HintsAI Generated

Suggested Approach

Start by explaining the importance of scaling for PCA when features have different units or variances. Then outline the computational steps: center the data, compute covariance matrix, eigen-decomposition, sort eigenvalues, and compute explained variance ratios. Finally, compare the results and discuss why standardization changes the principal components.

Pro tip: Emphasize that PCA is sensitive to scale because it seeks directions of maximum variance; without standardization, features with larger scales dominate. Mention that in practice, standardization is almost always recommended unless features are already on comparable scales.

1. Data Preparation and Centering

Load or generate the 6x3 matrix. For raw PCA, center each column by subtracting its mean. For standardized PCA, additionally divide each column by its standard deviation (z-score normalization).

2. Covariance Matrix Computation

Compute the covariance matrix using NumPy's np.cov on the centered data (raw) and standardized data. Ensure the correct orientation (rows as observations, columns as features).

3. Eigen-decomposition and Sorting

Perform eigen-decomposition on each covariance matrix using np.linalg.eigh. Sort eigenvalues in descending order and reorder the corresponding eigenvectors accordingly.

4. Explained Variance Ratio Calculation

Compute the explained variance ratio for each principal component by dividing each eigenvalue by the sum of all eigenvalues. Report the ratios for the first two components for both raw and standardized PCA.

5. Interpretation and Discussion

Compare the principal components and explained variance ratios. Discuss how standardization equalizes the influence of each feature, leading to different principal directions that capture correlations rather than raw variance.

Key Points to Mention

PCA is sensitive to feature scaling because it maximizes variance along principal components.
Standardization (z-score normalization) ensures each feature contributes equally by giving them unit variance.
Covariance matrix computation: np.cov(data, rowvar=False) for features in columns.
Eigen-decomposition: use np.linalg.eigh for symmetric matrices, which returns eigenvalues in ascending order; sort descending.
Explained variance ratio: eigenvalue divided by sum of eigenvalues; indicates the proportion of total variance captured.
Without scaling, features with larger variances dominate the principal components, potentially obscuring meaningful patterns.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.