Relate coefficients under linear feature transformation
Company: Databricks
Role: Data Scientist
Category: Statistics & Math
Difficulty: easy
Interview Round: Technical Screen
Answer both parts.
## Part 1: Coefficients under feature transformation
You have original predictors \(x_1, x_2\) and define transformed predictors:
- \(z_1 = x_1 + x_2\)
- \(z_2 = x_1 - x_2\)
Consider two linear regression specifications (with the same response \(y\)):
- **Model A:** \(y = \alpha_0 + \alpha_1 x_1 + \alpha_2 x_2 + \varepsilon\)
- **Model B:** \(y = \beta_0 + \beta_1 z_1 + \beta_2 z_2 + \varepsilon\)
1. What is the relationship between \((\alpha_1, \alpha_2)\) and \((\beta_1, \beta_2)\)?
2. Are Model A and Model B **equivalent** (i.e., do they represent the same set of functions of \(x_1, x_2\))? State any assumptions.
## Part 2: Multicollinearity in a wage/gender regression
You want to estimate/assess the relationship between **gender** and **income** using a regression model with additional covariates such as:
- age
- education level
- years since receiving degree
1. Explain whether multicollinearity could be present and why.
2. Describe how you would **diagnose** it.
3. If it exists, explain multiple ways to **address** it, and discuss trade-offs (interpretability vs variance vs bias).
Quick Answer: This question evaluates understanding of linear regression parameter relationships under linear feature transformations, model equivalence and identifiability, and detection and consequences of multicollinearity, testing competencies in linear algebra, statistical inference, and regression modeling.