Cox-Proportional-Hazards-Regression (1).pptx

Regression analysis

1. Cox Proportional-Hazards Regression A Practical Guide with Numerical Example This presentation provides a comprehensive overview of Cox Proportional-Hazards Regression, a powerful statistical method for survival analysis. We'll break down the complex concepts into digestible components and walk through a numerical example to demonstrate its practical application. By the end, you'll understand how to interpret Cox regression results and apply this technique to your own research questions.

2. What is Cox Proportional-Hazards Regression? A statistical model designed specifically for survival analysis that examines how multiple variables simultaneously influence the time until an event of interest occurs. Unlike simpler methods, Cox regression can handle: Multiple predictors at once Both categorical and continuous variables Censored observations (incomplete follow-up) The model's key feature is its assumption of proportional hazards — the effect of each predictor remains constant over time. Cox regression produces hazard ratios that quantify how predictors influence the risk of the event occurring.

3. Why Use Cox Regression? Handles Complex Data Simultaneously analyzes multiple predictors while adjusting for confounding variables like age, gender, or treatment type. Manages Censored Data Effectively uses information from subjects who didn't experience the event before study completion or were lost to follow-up. More Flexible Analysis Surpasses Kaplan-Meier or Log-Rank tests which can only analyze one factor at a time or require stratification. No Distribution Assumptions As a semi-parametric model, it makes fewer assumptions about the underlying survival distribution than parametric models. These advantages make Cox regression the gold standard for analyzing time-to-event data in clinical research, epidemiology, and reliability studies.

4. The Cox Model Formula Where: h(t) : hazard at time t for an individual h (t) : baseline hazard (when all predictors = 0) b 1 regression coefficients measuring effect size X 1 ----- : predictor values The Hazard Ratio (HR) is calculated as: This represents the multiplicative change in hazard for each unit increase in the predictor. The Cox model doesn't estimate the baseline hazard function h_0(t) directly. Instead, it focuses on the relative effects of predictors, making it a "semi-parametric" approach.

5. Numerical Example Setup Let's examine a clinical study tracking survival time (in months) of patients after diagnosis of a disease with two key predictors: Drug Type Drug A (coded as 0) Drug B (coded as 1) Patient Age Measured in years Continuous variable Patient Time (months) Event (1=death, 0=censored) Drug Age 1 5 1 0 (Drug A) 60 2 8 1 (Drug B) 55 3 12 1 1 (Drug B) 70 4 4 1 0 (Drug A) 65 This sample dataset will illustrate how Cox regression analyzes the effects of drug choice and age on patient survival.

6. Running Cox Regression on Example Data After fitting the Cox model to our example data, we obtain the following results: Variable Coefficient (b) Hazard Ratio (HR) p-value Drug (B vs A) -0.8 0.45 0.03 Age (per year) 0.05 1.05 0.02 Calculating the Hazard Ratios: For Drug B: HR = \exp(-0.8) \approx 0.45 For Age: HR = \exp(0.05) \approx 1.05 The model equation becomes: Where Drug is 0 for Drug A and 1 for Drug B, and Age is measured in years.

7. Interpreting the Results 55% Risk Reduction Drug B reduces the hazard (risk of death) by 55% compared to Drug A at any given time point 5% Risk Increase Per Year Each additional year of patient age increases the hazard by 5% These effects are multiplicative . For example, a 70-year-old patient on Drug B would have a hazard of: The proportional hazards assumption means these effects remain constant over the entire follow-up period.

8. Visualizing Survival Curves The survival curves illustrate the probability of survival over time for different patient groups. Key Observations: Drug B shows a higher survival probability at all time points compared to Drug A The gap between curves represents the drug effect (HR = 0.45) Age shifts both curves downward (older) or upward (younger) The proportional hazards assumption means the curves never cross These curves are adjusted for the effects of other variables in the model, showing the isolated impact of each predictor. For our numerical example, we can calculate and plot separate curves for different combinations of drug type and age to visualize their combined effects on patient survival.

9. Key Assumptions & Diagnostic Checks 1 Proportional Hazards The effect of predictors (hazard ratios) must remain constant over time. Check: Schoenfeld residuals should show no pattern when plotted against time. 2 Linearity of Continuous Predictors Continuous variables like age should have a linear relationship with the log hazard. Check: Plot martingale residuals against continuous covariates to detect non-linearity. 3 No Influential Observations Results shouldn't be driven by a few extreme data points. Check: Examine deviance residuals and delta-beta values to identify influential observations. In our numerical example, we would verify these assumptions to ensure the validity of our conclusion that Drug B reduces hazard by 55% and each additional year of age increases hazard by 5%.

10. Summary & Practical Applications Key Takeaways Cox regression simultaneously models multiple predictors' effects on survival time Hazard ratios quantify risk changes per unit increase in predictors Our example showed Drug B reduces death risk by 55% compared to Drug A Each year of age increases risk by 5% Proportional hazards assumption is crucial for valid interpretation Applications Clinical trials comparing treatment efficacy Epidemiological studies of disease progression Reliability analysis in engineering Customer churn prediction in business Recidivism studies in criminal justice Identify Question Define the event of interest and potential predictors Run Cox Model Fit the model and check assumptions Interpret Results Translate hazard ratios into clinical or practical meaning With the knowledge from this presentation, you're now equipped to apply Cox Proportional-Hazards regression to your own time-to-event data analysis challenges.