MS-DS Master of Data science Statistical Inference & Regression Models 5 — Questions and Answers
Question 1: In multiple regression, the partial F-test is used to:
- Test each individual coefficient against zero
- Compare a reduced model to a full model by testing a subset of predictors simultaneously (Correct answer)
- Detect outliers in the response variable
- Test for heteroscedasticity in the residuals
Correct answer: Compare a reduced model to a full model by testing a subset of predictors simultaneously
The partial F-test evaluates whether a group of predictors collectively adds significant explanatory power beyond what the reduced model already explains.
Question 2: A Q-Q plot of regression residuals that shows heavy tails indicates:
- Heteroscedasticity in the model
- Departure from normality with more extreme values than a normal distribution (Correct answer)
- Perfect normality of residuals
- Multicollinearity among predictors
Correct answer: Departure from normality with more extreme values than a normal distribution
Heavy tails in a Q-Q plot mean the residuals have a leptokurtic distribution with more extreme outliers than expected under normality.
Question 3: Which scenario correctly describes an interaction term in multiple regression?
- The effect of X₁ on Y depends on the value of X₂ (Correct answer)
- X₁ and X₂ are correlated with each other
- The model includes both X₁² and X₂² as polynomial terms
- The regression coefficients for X₁ and X₂ have opposite signs
Correct answer: The effect of X₁ on Y depends on the value of X₂
An interaction term (X₁ × X₂) captures the situation where the marginal effect of one predictor on the response changes depending on another predictor's value.
Question 4: The bootstrap method is used in regression to:
- Increase the sample size by duplicating observations
- Estimate the sampling distribution of an estimator by resampling with replacement (Correct answer)
- Remove outliers from the dataset iteratively
- Regularize coefficients to prevent overfitting
Correct answer: Estimate the sampling distribution of an estimator by resampling with replacement
Bootstrap resampling creates many pseudo-samples from the observed data to empirically estimate the sampling distribution and standard errors of estimators.
Question 5: In generalized linear models (GLMs), the variance function describes:
- How the variance of the response depends on its mean (Correct answer)
- The link between the linear predictor and the mean response
- The penalty term applied to regression coefficients
- The dispersion of predictor variables
Correct answer: How the variance of the response depends on its mean
The variance function V(μ) in a GLM specifies how the variance of the response relates to its mean, distinguishing different members of the exponential family.
Question 6: Cook's distance for observation i is used to measure:
- The leverage of observation i in predictor space
- The overall influence of observation i on all fitted values (Correct answer)
- The standardized residual of observation i
- The correlation between observation i and the predicted response
Correct answer: The overall influence of observation i on all fitted values
Cook's distance combines leverage and residual size to quantify how much the regression estimates change if observation i is removed.
Question 7: The Neyman-Pearson lemma establishes that the most powerful test for a simple vs. simple hypothesis test is based on:
- The p-value from the chi-square distribution
- The likelihood ratio statistic (Correct answer)
- The sample mean divided by its standard error
- The posterior odds ratio
Correct answer: The likelihood ratio statistic
The Neyman-Pearson lemma proves that the likelihood ratio test (rejecting when L(H₁)/L(H₀) > k) is the uniformly most powerful test at level α.
MS-DS Master of Data science Statistical Inference & Regression Models 5
In multiple regression, the partial F-test is used to: