ISYE 6414 Regression Analysis Final_Part1_Solutions: Georgia Institute Of Technology

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ISYE 6414 Regression Analysis Final_Part1_Solutions Final Exam – Part 1 Solutions 1. We should always use mean squared error to determine the best value of lambda in lasso regression. a. True b. False Sol: False. The criterion used is a choice we make.

2. Standard linear regression is an example of a generalized linear model where the response is normally distributed and the link is the identity function. a. True b. False Sol: True. See Unit 4.4.1.

3. Goodness-of-fit assessment for logistic regression involves checking for the independence, constant variance, and normality of the deviance residuals. a. True b. False Sol: False. We don’t have constant variance in binomial regression.

4. You are interested in understanding the relationship between stress level and exercise, with stress as the response. In your model, the number of hours a person spends exercising per week would be considered an explanatory variable while the person’s age would be a controlling variable. a. True b. False Sol: True. Time spent exercising is part of the relationship you are trying to understand while age could act as a confounding variable that you need to control.

5. The hypothesis test for goodness-of-fit using Pearson residuals and the test using deviance residuals will always reach the same conclusion. a. True b. False Sol: False. One test may conclude plausibly good fit while the other rejects it.

6. A logistic regression model with high goodness of fit can have low predictive power. a. True b. False Sol: True. See Unit 4.2.3.

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7. If we apply a Poisson regression model using a small sample size, the estimators of the regression coefficient may not follow an approximate Normal distribution, affecting the reliability of the statistical inference on the coefficients. a. True b. False Sol: True. See Unit 4.2.1.

8. You fit a regression model using three predictors. You notice the estimated coefficient for predictor X1 is an order of magnitude larger than the estimated coefficient for predictor X2. It is correct to conclude that X1 has a greater effect on the response than X2. a. True b. False Sol: False. We do not know that the variables are on the same scale in order to directly compare them. We can only conclude that a 1-unit change in X1 is associated with a greater change in the response than a 1-unit change in X2 holding other variables constant.

9. Regularized regression with a lambda value equal to 0 is equivalent to regression model estimation without penalization. a. True b. False Sol: True. See Unit 5.2.2.

10. In a Poisson regression model, the difference between the null deviance and residual deviance follows a normal distribution. a. True b. False Sol: False. It follows a Chi-square distribution (this is used to check the overall regression significance).

11. You want to examine the relationship between study time and score on exams. You create five exams and recruit 50 participants. For each participant in your study, you record their time studying for and grade on each of those five exams. If you were to use all the data you recorded to build a simple linear regression model, you would violate the independence assumption. a. True b. False Sol: True. Because you are collecting 5 observations from each person, the observations coming from the same person would be correlated. Similarly, all observations from the same test may be correlated.

12. Maximum likelihood estimation produces unbiased coefficient estimates for logistic and Poisson regression. a. True b. False

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Sol: True. Coefficient estimators follow an approximate normal distribution with the true coefficient as the mean.

13. If considering only BIC for the model selection criterion, a model with lower BIC is preferred over a model with higher BIC. a. True b. False Sol: True. A lower BIC corresponds to a lower penalized prediction risk, and the statement specified that is the only thing we are considering.

14. If we do not penalize the training risk, our variable selection method would always prefer more complex models. a. True b. False Sol: True. See Unit 5.1.3.

15. One goal of variable selection is to balance the bias-variance tradeoff when making predictions. a. True b. False Sol: True. See Unit 5.1.4.

16. Forward stepwise selection is computationally more expensive than backward stepwise selection because it takes more iterations to terminate. a. True b. False Sol: False. Backwards is more expensive because it fits larger models.

17. When testing for the significance of a subset of predictors, the null hypothesis is that all coefficients for variables in that subset are 0 and the alternative is that all those coefficients are not 0. a. True b. False Sol: False. The alternative is that at least one variable is significantly nonzero.

18. Classification error estimated from leave-one-out cross validation tends to have higher bias but lower variability than classification error estimated from 2-fold cross validation. a. True b. False Sol: False. It has lower bias but higher variability.

19. One reasonable method for variable selection is to fit the full model and then drop all variables with a high p-value in that full model. a. True b. False

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Sol: False. Significance should be interpreted conditionally. Correlations between predictors may cause multiple predictors to have high p-values despite at least one being useful to the model.

By dropping them all at once, you cannot evaluate how one performs without the others.

20. The logit link is the only link function that yields s-shaped curves. a. True b. False Sol: False. Other link functions, such as probit and cloglog, yield s-shaped curves.

21. Which of the following is TRUE about adjusted R-squared? a. The adjusted R2 can be used to compare models, and its value will always be less than or equal to that of R2. b. The adjusted R2 cannot be used to compare models, and its value will always be less than or equal to that of R2. c. The adjusted R2 can be used to compare models, and its value will always be greater than or equal to that of R2. d. The adjusted R2 cannot be used to compare models, and its value will always be greater than or equal to that of R2.

Sol: a. Negative values can occur with a sufficiently small R-square value and large number of predictors. It cannot be greater than R-squared (you can either recall this from previous examples or work it out: 𝑎𝑑𝑗 𝑅 2 = 1 − 2

𝑅 =𝑅

2

(𝑛−1)(1−𝑅2 )

𝑛−𝑘−1 𝑘 𝑘 2 2 − 𝑛−𝑘−1 (1 − 𝑅 ) ≤ 𝑅 − 𝑛−𝑘−1 ≤ 𝑅 2).

𝑘

(𝑛−1)𝑅2

𝑘

𝑘𝑅2

= − 𝑛−𝑘−1 + 𝑛−𝑘−1 = − 𝑛−𝑘−1 + 𝑛−𝑘−1 +

22. Mary has a dataset with height (in inches), weight (in lbs), and math_score (final exam score out of 100) of 300 students in an undergraduate math course. She creates another field called BMI (Body Mass Index) calculated as BMI=703*(weight/height2). She wants to examine if math_score is related to height, weight and BMI. She plans to use a linear regression model math_score ~ height+weight+BMI to study this relationship.

Leonard hears about Mary’s plan and tells Mary that BMI should not be used in her experiment because it is created from the height and weight variables which are already included in the model. He says this leads to an issue called multicollinearity in linear regression.

Which of the below options is TRUE? a. Leonard is right; retaining height, weight, BMI in the model will definitely lead to multicollinearity. b. Leonard is wrong because BMI is not a linear combination of weight and height. c. Leonard is wrong; it is impossible to say whether multicollinearity is a problem in a proposed model without first fitting the model. d. Leonard is right, but the correct name for this issue is homoscedasticity.

Sol: b. For multicollinearity to exist, there needs to be linear dependence among the predictors, meaning predictors are linear combinations of the others. There is no linear dependence created by the BMI variable.

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23. When working to create a logistic regression model, an analyst is considering two models: Model one with only one predicting variable A. Model two includes predicting variable B in addition to variable A. The analyst notices that the sign of the estimated coefficient for A is negative in model one and positive in model two. This is most likely because: a. A is significant in Model two. b. B is significant in Model two. c. A is a confounding variable. d. B is correlated with A. Sol: d. B being significant is not enough to explain the change in direction of the relationship between A and the response. Calling A a confounding variable implies that A is not the variable whose effect we are interested in but rather one we may need to control for, which is not implied by the problem description.

24. The regression glm(Y~X, family=poisson) was fitted to count data, resulting in the estimate of β0 to be 16 and the estimate of β1 to be 9.4. For a one unit increase in X, a. the rate increases by 9.4 units. b. the log rate increases by 9.4 units. c. the log rate increases by exp(9.4) units. d. the rate increases by exp(9.4) percent.

Sol: b. See Unit 4.4.3.

25. Which of the following is correct? a. Lasso regression uses the L1 norm. b. Ridge regression uses the L0 norm. c. Ridge regression performs variable selection. d. Elastic net does not perform variable selection.

Sol: a. See Unit 5.2.2. Ridge uses the L2 norm and does not perform variable selection. Elastic net does perform selection.

26. A data point with predictor and response values three orders of magnitude higher than the corresponding values of all other observations is for certain a. an outlier with respect to the data b. an influential point but not an outlier c. an erroneous data point d. None of the above Sol: a. The point must be an outlier because it is far from the other observations.

27. In simple linear regression, what is the relation between the correlation coefficient ρ and Rsquared? a. ρ = R-squared b. ρ = sqrt(R-squared) c. ρ = R-squared + Adjusted R-squared d. There is no relation between the two

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Sol: b. See Unit 1.2.4.

28. High VIF (> 10) value for predictors in linear regression suggests a. Homoscedasticity. b. Multicolinearity. c. Autocorrelation. d. non-Linear relation between dependent and independent variables.

Sol: b. See Unit 3.3.3.

29. Which of the following is true regarding logistic regression? a. Logistic regression can also be replaced by standard linear regression if with repetitions. b. Logistic regression can only be used with continuous predicting variables. c. Logistic regression can only be used when the response variable is binary. d. The logit link is the canonical link function.

Sol: d. See Unit 4.2.3. Logisitc regression can be used with binomial data so (c) is false. Logistic regression and standard linear regression do not have the same model form, so they are not interchangeable, even with a large number of successes per observation.

30. Which of the following is correct? a. Overdispersion is a concern for Poisson regression but not for logistic regression. b. For both logistic and Poisson regression, the variance of the response equals the expectation of the response given the predicting variables. c. Overdispersion affects the reliability of our statistical inferences if not modeled correctly. d. Under overdispersion, the observed variance is smaller than the variance implied by our model.

Sol: c. See Unit 4.5.3. Overdispersion means the observed variance is larger than implied by the model. For logistic regression, variance is 𝑛𝑖 𝑝𝑖 (1 − 𝑝𝑖 ) while the mean would be 𝑛𝑖 𝑝𝑖 .

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