Ph.D. Candidate Example Exam - MSBD 720: Survival Analysis and Regression Study Guide

Part 1: Multiple Choice - Survival Analysis Fundamentals

Question 1: Identification of Survival Analysis Characteristics
- Option a: "In a survival analysis, the outcome variable is dichotomous."
  - Status: Incorrect. While the event status is dichotomous ( $0$ or $1$ ), the outcome variable in survival analysis is the time until that event occurs, which is a continuous or discrete time-to-event variable.
- Option b: "In a survival analysis, the event is usually described by a (0, 1) variable."
  - Status: Correct. This binary indicator denotes whether the event occurred ( $1$ ) or the observation was censored ( $0$ ).
- Option c: "If the study ends before an individual has gotten the event, then his or her survival time is censored."
  - Status: Correct. This describes administrative censoring, where the observation period terminates before the event of interest is observed.
- Option d: "If, for a given individual, the event occurs before the person is lost to follow-up or withdraws from the study, then this person’s survival time is censored."
  - Status: Incorrect. If the event occurs while the participant is under observation, the survival time is considered a "complete" or "uncensored" observation.

Part 2: Practical Example – Linear Regression (Supermarket Scanner Data)

Context and Data Collection:
- Supermarkets utilize checkout scanners to evaluate the impact of pricing and promotional activities on product sales.
- Promotional Types:
  1. Flyers: Distributed outside the store or in newspapers; may or may not feature specific products.
  2. In-store Displays: Located at the end of aisles to capture customer attention.
- Variables Collected Weekly:
  - Sales: Number of units sold.
  - Price: Cost in dollars ().\n * **Flyer:** Binary variable (1 $if the product appeared in the flyer,$ 0 otherwise).\n * **Display:** Binary variable (1 $if a special in-store display was used,$ 0 otherwise).\n\n* **Fitted Model Analysis:**\n * A quadratic regression was applied to the data. Use the following structure for requirements:\n * **Model Equation:** The fitted model for sales should include linear and quadratic terms for price, plus terms for flyers and displays: \text{Sales} = \beta_0 + \beta_1 \cdot \text{Price} + \beta_2 \cdot \text{Price}^2 + \beta_3 \cdot \text{Flyer} + \beta_4 \cdot \text{Display} + \epsilon\n * **Coefficient of Determination (R^2):** This value represents the proportion of variance in Sales explained by the independent variables (Price, Price squared, Flyer, and Display).\n * **Test Statistic for Quadratic Term:** To determine if the quadratic term (\text{Price}^2 $) significantly differs from zero, the t-statistic is used:$ t = \frac{\hat{\beta}k}{SE(\hat{\beta}_k)} $. If the p-value associated with this t-statistic is less than the significance level (e.g.,$ 0.05), the quadratic term is significant.\n * **Prediction Calculation:** To find the average units sold when the price is \$1.10 $, substitute$ \text{Price} = 1.10 $and$ \text{Price}^2 = 1.21 into the fitted regression equation along with the means or specific values for Flyer and Display.\n * **Analysis of Variance (ANOVA):** The ANOVA table provides information regarding the overall significance of the model (F-test), comparing the variance explained by the regression model against the residual (error) variance.\n\n# Part 3: Practical Example – Logistic Regression (Framingham Heart Study)\n\n* **Study Overview:**\n * **Focus:** Sudden death as a lethal cardiovascular endpoint in women.\n * **Data Source:** Framingham Heart Study.\n * **Response Variable:** 2-year incidence of sudden death in females without prior coronary heart disease (CHD).\n * **Controlled Risk Factors:** Age, blood pressure (BP), cigarette smoking, and vital capacity.\n\n* **Statistical Model and Testing:**\n * **Equation:** \text{logit}(P) = \ln\left(\frac{P}{1-P}\right) = \alpha + \beta_1 \cdot \text{Age} + \beta_2 \cdot \text{BP} + \beta_3 \cdot \text{Smoking} + \beta_4 \cdot \text{Vital Capacity}\n * **Z-Statistic Calculation for Blood Pressure:**\n * Coefficient (b_1 $):$ 0.0019\n * Standard Error (SE(b_1) $.):$ 0.0070\n * Test Statistic: Z = \frac{b_1 - 0}{SE(b_1)} = \frac{0.0019}{0.0070} = 0.27\n * **Significance Decision:** At \alpha = 0.05 $, the critical value is$ Z{0.025} = 1.96 $. Since$ 0.27 < 1.96, BP is not a significant predictor in the presence of other variables.\n * **p-value:** 2P(Z \geq 0.27) = 0.7871.\n\n* **Interpretations and Odds Ratios (OR):**\n * **Age Coefficient:** Represents the change in the log-odds of sudden death for a one-year increase in age, adjusting for other factors.\n * **Vital Capacity Coefficient:** Represents the change in the log-odds of sudden death for each unit increase in vital capacity.\n * **OR for 1-year Age Increase:** OR = e^{\beta_{\text{age}}}.\n * **OR for 100-cl Decrease in Vital Capacity:** If the coefficient \beta_{\text{vc}} $is per 1 unit cl increase, the OR for a$ 100 $-unit decrease is$ e^{-100 \cdot \beta_{\text{vc}}}.\n * **95% Confidence Intervals for OR:** Calculated as e^{\hat{\beta} \pm 1.96 \cdot SE(\hat{\beta})}.\n\n# Part 4: Interpret Coefficients and Model Outputs (Loan Data)\n\n* **Model Specification:**\n * The model predicts the probability of a "good" loan based on FICO score and loan amount.\n * **Formula:** \text{logit(good)} = \beta_0 + \beta_1 \cdot \text{fico} + \beta_2 \cdot \text{loan_amnt}\n\n* **Model Coefficients Table:**\n | Term | Estimate (\beta $) | Std. Error | z value |$ Pr(>|z|) |\n | :--- | :--- | :--- | :--- | :--- |\n | (Intercept) | -7.367 \times 10^{0} $|$ 3.007 \times 10^{-1} $|$ -24.50 $|$ < 2 \times 10^{-16} |\n | fico | 1.319 \times 10^{-2} $|$ 4.306 \times 10^{-4} $|$ 30.62 $|$ < 2 \times 10^{-16} |\n | loan_amnt | -2.229 \times 10^{-5} $|$ 1.815 \times 10^{-6} $|$ -12.28 $|$ < 2 \times 10^{-16} |\n\n* **Interpretations:**\n * **FICO:** For every one-unit increase in FICO score, the log-odds of the loan being "good" increase by 0.01319, assuming the loan amount remains constant. This indicates a positive relationship between credit score and loan quality.\n * **Loan Amount:** For every one-dollar increase in the loan amount, the log-odds of the loan being "good" decrease by 0.00002229. This suggests larger loans are slightly less likely to be categorized as "good."\n\n* **Model Performance Metrics:**\n * **Null Deviance:** 35928 $on$ 42534 degrees of freedom. This represents how well the response is predicted by a model that includes only the intercept.\n * **Residual Deviance:** 34838 $on$ 42532 degrees of freedom. This measures the lack of fit of the model after including FICO and loan amount.\n * **Model Decision:** To determine if the model is a significant improvement over the null model, compute the difference in deviance: \Delta D = 35928 - 34838 = 1090 $. This difference follows a Chi-square distribution with$ \Delta df = 42534 - 42532 = 2 $. Since a chi-square value of$ 1090 $with$ 2 $degrees of freedom is highly significant ($ p < 0.05), we reject the null model in favor of the fitted model.\n\n# Part 5: Practical Example – Survival Analysis Calculations\n\n* **Requirements:**\n * Calculate survival probability for each group at every unique failure time.\n * Construct survival curves (Kaplan-Meier estimator) for Group 1 and Group 2.\n * **Kaplan-Meier Formula:** \hat{S}(t) = \prod_{t_i \le t} \left( 1 - \frac{d_i}{n_i} \right) $, where$ d_i $is the number of events at time$ t_i $and$ n_i is the number of individuals at risk.\n\n# Part 6: Optional – Evans County Study (1967–1980)\n\n* **Dataset Description:**\n * Survival times (in years) for two groups, n = 25 each.\n * **Group 1 (CHR = 0):** No history of chronic disease.\n * **Group 2 (CHR = 1):** Positive history of chronic disease.\n\n* **Raw Data (Survival Times in Years):**\n * **Group 1:** 1.8 $,$ 2.2 $,$ 2.5 $,$ 2.6 $,$ 3.0 $,$ 3.5 $,$ 3.8 $,$ 5.3 $,$ 5.4 $,$ 5.7 $,$ 6.6 $,$ 8.2 $,$ 8.7 $,$ 9.2 $,$ 9.2 $,$ 9.8 $,$ 10.0 $,$ 10.2 $,$ 10.7 $,$ 11.0 $,$ 11.1 $,$ 11.7 $,$ 12.1+ $(censored),$ 12.2+ $(censored),$ 12.3+ (censored).\n * **Group 2:** 1.4 $,$ 1.6 $,$ 1.8 $,$ 2.4 $,$ 2.8 $,$ 2.9 $,$ 3.1 $,$ 3.5 $,$ 3.6 $,$ 3.9 $,$ 4.1 $,$ 4.2 $,$ 4.7 $,$ 4.9 $,$ 5.2 $,$ 5.8 $,$ 5.9 $,$ 6.5 $,$ 7.8 $,$ 8.3 $,$ 8.4 $,$ 8.8 $,$ 9.1 $,$ 9.9 $,$ 11.4.\n\n* **Quantitative Metrics for Comparison:**\n * **Average Survival Time:** The arithmetic mean of the survival times (treating censored values as the time at which censoring occurred).\n * **Average Hazard Rate:** Calculated as the ratio of total observed events (D $) to the sum of person-time ($ T $):$ \text{Average Hazard} = \frac{\sum d_i}{\sum t_i}.\n * **Survival Prognosis Comparison:**\n 1. Determine which group has a higher survival probability before a specific time threshold (t^).\n 2. Determine which group has a better prognosis after t^ (potential crossing of curves).\n 3. **Median Survival Time:** The time point at which \hat{S}(t) = 0.5$$. Compare the two groups' medians to see which has a longer typical survival duration.