Applied Survival Analysis

Chapter 1 - Introduction

Lu Mao

lmao@biostat.wisc.edu

Department of Biostatistics & Medical Informatics

University of Wisconsin-Madison

Outline

1. Time-to-event data and examples
2. Censoring mechanisms and implications
3. Summarizing the raw data

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Data and Examples

What are time-to-event data?

• Common outcome type in medical studies
  • Starting point: Randomization, study entry, birth, etc.
  • Endpoint: Death, hospitalization, disease onset, etc.
  • In engineering: Machine failure times (reliability analysis)
• Right censoring
  • Event does not occur by study end or dropout
    
  • Only know event time \(>\) censoring time
• Survival analysis: Statistical methods for censored data

Example: Univariate event (I)

• German Breast Cancer (GBC) Study
  • Population: 686 patients with node-positive breast cancer
    
  • Objective: Assess if tamoxifen + chemo reduces mortality
    
  • Baseline info: Age, tumor size, hormone levels, menopausal status, etc.
    
  • Follow-up: Median 44 months
    • 171 deaths \(\to\) exact times known
      
    • 515 censored \(\to\) survival time \(>\) censoring time

Example: Univariate event (II)

• German Breast Cancer (GBC) Study

Example: Recurrent events (I)

• Chronic Granulomatous Disease (CGD) Study
  • Population: 128 patients in a randomized placebo-controlled trial
    
  • Objective: Assess gamma interferon effect on recurrent infections
    
  • Follow-up: Median 293 days
    • Infections: Min = 0, Max = 7
      
  • Challenge: Correlated events within individuals
  • Data in “long” format (multiple records per patient)

Example: Recurrent events (II)

• Chronic Granulomatous Disease (CGD) Study

Example: Multivariate/Clustered Events (I)

• Diabetic Retinopathy Study
  • Population: 197 high-risk diabetic patients in a randomized controlled trial
  • Objective: Determine if photocoagulation (a laser treatment) delays blindness onset
  • Design: One eye treated (by either xenon or argon), the other untreated (control)
  • Challenge: Correlation between eyes

Example: Multivariate/Clustered Events (II)

• Diabetic Retinopathy Study

Example: Competing Risks (I)

• Definition: Multiple types of events where one prevents the occurrence of the others
  • Natural example: different causes of death
• Competing risk vs censoring:
  • Both terminate follow-up
  • Competing risk: part of the outcome; inference based on its presence
  • Censoring: irrelevant to outcome; inference based on its absence
• Example: death from prostate cancer as main outcome
  • Death from other (metastasized) cancers \(\to\) competing risk
  • Death from traffic accidents \(\to\) censoring

Example: Competing Risks (II)

• Bone Marrow Transplant Study

  • population: 864 multiple-myeloma leukemia patients undergoing allogeneic haematopoietic cell transplantation (HCT)
  • Objective: Evaluate risk factors for treatment-related mortality (TRM) and relapse of leukemia
  • Competing risks: TRM defined as death in remission (i.e., before relapse); thus is precluded by relapse
  • Risk factors: cohort indicator (years 1995–2000 or 2001–2005), type of donor (unrelated or identical sibling), history of a prior transplant, time from diagnosis to transplantation (<24 months, or ≥ 24 months)

Example: Competing Risks (III)

• Bone Marrow Transplant Study
  • Why only one record per patient?

Example: More Complex Outcomes (Semi-competing risks)

• German Breast Cancer (GBC) Study
  • Nonfatal event + terminal event (death)

Example: More Complex Outcomes (with Longitudinal data)

• Anti-Retroviral Drug Trial
  • Repeated measures of CD4 cell count + death

Example: More Complex Outcomes (Multistate process)

• Breast Cancer Life History Study
  • Remission \(\to\) relapse \(\to\) metastasis \(\to\) death (can skip states)

Example: Composite Endpoints(I)

• Composite endpoint: one with multiple components
  • Recurrent/multivariate events
  • (Semi-)Competing risks
  • Longitudinal measurements
  • Multistate processes
• Analysis of complex outcomes
  • Marginal approach: models components separately
  • Conditional approach: models components jointly
  • Composite approach: combines components
    • Progression/relapse-free survival (time to the earlier of progression/relapse or death)

Example: Composite Endpoints(II)

• Advantages:
  • Concentrates information \(\to\) Statistical efficiency
  • No need for multiple testing adjustment
  • A single measure of overall effect size
• Preferred for primary analysis of Phase-III clinical trials by
  • US Food and Drug Administration (FDA)
  • ICH (International Council for Harmonisation for pharmaceuticals)
• Challenges:
  • Statistical efficiency (e.g., beyond first event)
  • Scientific relevance (e.g., relative importance of components)

Censoring mechanisms and implications

Censoring Mechanisms

• Two mechanisms
  • Study termination (administrative censoring)
  • Loss to follow-up (LTFU, e.g., withdrawal, death from other causes)

Caution about censoring

• Event/censoring time \(=\) time from starting point (e.g., randomization) to event/censoring (as opposed to time on the calendar)

• LTFU may not be independent of outcome (e.g., sicker patients withdraw early)

• Collect withdrawal reasons if possible

• Censoring or competing risk? \(\leftarrow\) Domain knowledge

Censoring Mechanisms: Illustration

• Calendar time vs time synchronized by starting point

Statistical Implications (I)

• Censored observation
  • Not completely missing!
  • Partial information: event time \(>\) censoring time
  • Ignoring partial information \(\to\) Bias in inference
• Naive approaches
  • Treat censoring as event \(\to\) Underestimates time to event
  • Exclude censored observations \(\to\) Underestimates time to event (longer event times more likely censored)

Statistical Implications (II)

• Notation

  • \(T\): Outcome event time
  • \(C\): Censoring time
  • Observed data: \(X=\min(T, C)\), \(\delta = I(T\leq C)\)
    • (𝑋, 𝛿) = (time, status) in previous data examples

• Estimation

  • Independent censoring assumption

  \[ C \indep T\]

  • Estimand: \(S(t)={\rm pr}(T > t)\), i.e., probability of subject “surviving” to time \(t\), using a random sample of \((X_i, \delta_i)\) \((i=1,\ldots, n)\)

Statistical Implications (III)

• Naive methods

  • Event-imputation empirical survival function: \[\hat S_{\rm imp}(t)=n^{-1}\sum_{i=1}^n I(X_i > t) \to {\rm pr}(X > t)\leq S(t)\]

  • Complete-case empirical survival function: \[\hat S_{\rm cc}(t)=\frac{\sum_{i=1}^n I(X_i > t, \delta_i = 1)}{\sum_{i=1}^n\delta_i} \to {\rm pr}(T > t\mid T\leq C)\leq S(t)\]

  • Both naïve methods underestimate the true survival function

Statistical Implications: Example

• German Breast Cancer (GBC) Study

Summarizing Raw Data

Importance of Descriptive Analysis

• Statistical models rely more or less on assumptions
• Good practice to summarize data descriptively as first step
  • Get to know the data
  • Informs subsequent analysis
  • Check balance of baseline characteristics between randomized arms
  • “Table 1” in medical research papers
• Two types of summary statistics
  • Subject-level characteristics (baseline variables, number of events per subject)
  • Event rates (over aggregate length of follow-up)

How to Calculate Event Rate (I)

• Length of follow-up is event-specific
  • If an event is “non-recurrent”, its occurrence means patient is no longer at risk for it

Note

Denominator is called person-year (or person-time) of follow-up.

How to Calculate Event Rate (II)

• Semi-competing risks

How to Calculate Event Rate (III)

• Recurrent events

Table One: Example

Conclusion

Chapter Summary

• Types of time-to-event outcomes
  • Univariate, recurrent, multivariate/clustered, (semi-)competing risks, repeated measures, multistate processes, and everything in between…
• Common feature: censoring
  • Arises if study ends or patient drops out prior to event
  • Must be handled with care to avoid false conclusion
• Importance of descriptive analysis
  • Event rate \(\to\) attention to denominator

HW1 (Due Feb 4)

• Choose one
  • Problem 1.2 (Recommended for PhD in Stats/BDS)
  • Problem 1.3
• Problem 1.9 (Attach you annotated code)
• (Extra credit) Problem 1.5

Guidelines for HW

• Present a readable and coherent text to report your methods and results
  • Include numerical/graphical results only if they contribute to your narrative
  • All tables and figures should be properly titled/captioned, with informative labels/legends
  • Use full names instead of abbreviations/acronyms
    • E.g., “meno” \(\to\) “Menopause (yes v no)”; “est” \(\to\) “Estrogen (fmol/mg)”
  • Specify the unit of variable, e.g., “Age (years)”
  • See Table 1.11 and Fig. 1.2 for examples
• Append the full code for diagnostic purposes