Introduction to survival analysis (slides)

Introduction

• Health economic models often rely on estimates of time-to-event (TTE; known as survival analysis) for outcomes.
• TTE or survival analysis:
  • used to provide estimates of survivor functions and event rates that inform these models.
  • characterises event rates over an observed period of time, such as the duration of a clinical trial.
  • employed to extrapolate over lifetime.
• Cost-effectiveness estimates can be sensitive to the methods applied in modelling survival data.

Outline

• Fundamental concepts:
  • Time to event data
  • Censoring
  • Kaplan-Meier curve
• Parametric survival modelling
  • Introduction to standard parametric survival models

Survival analysis

• In survival analysis, we are interested in time to a specific event and how risk factors or treatments affect the time to that event.
  • Time: years, months etc. from the beginning of follow-up of an individual until an event occurs
  • Event: disease incidence, relapse or any designated experience of interest, MI, AIDS for HIV patients, tumor recurrence, death [thus, survival analysis]

Survival analysis

• Standard statistical methods are inappropriate for survival analysis:
  • Positive skew:
    • Survival data are non-negative and hence tend to be positively skewed
    • Normality assumption required for some statistical methods does not hold
  • Censoring

Survivor function

• \(S(t)\): Probability that a subject survives from follow-up (\(t=0\)) to some time beyond time \(t\) (not failing)

  Figure 1: Survivor function

Hazard function

• \(h(t)\): Instantaneous potential for the event to occur, given a patient has survived up to time \(t\) (failure), AKA conditional hazard

  Figure 2: Hazard function

Cumulative hazard function

• \(H(t)\): The accumulation of conditional hazards up until a particular point in time, i.e., area under the hazard function until \(t\)

  \[H(t)=\int_{0}^{t}h(u)du\]

Relationships between functions

• Survivor function can be written in terms of the (cumulative) hazard function and vice versa

\[S(t)=e^{-H(t)}\] \[H(t)=-ln(S(t))\]

Censoring

• Reasons censoring:
  • When we have some information about individual’s survival time, but exact survival time is not known
  • Individual does not experience the event before the study ends
  • Individual is lost to follow-up during the study period
  • Individual withdraws from the study (e.g. due to adverse drug reaction)
• Example: study ends while the patient is still alive, then that patient’s survival time is considered censored
  • We know the survival time is at least as long as the period that the person has been followed up for

Censoring

Figure 3: Censoring

Source: Dey, T., Mukherjee, A., & Chakraborty, S. (2020). A practical overview and reporting strategies for statistical analysis of survival studies. Chest, 158(1), S39-S48.

Truncation

• Exclusion of observations based on event time or due to restrictions in the selection process.
• Truncation is due to sampling bias that only those individuals whose lifetimes lie within a certain interval can be observed.
• Left Truncation:
  • Occurs when short survival times are missing from data, potentially introducing bias.
  • Occurs when the subjects have been at risk before entering the study (e.g., life insurance policy holders where the study starts on a fixed date, event of interest is age at death).
• Right Truncation:
  • Occurs when the entire study population has already experienced the event of interest (e.g., a historical survey of patients on a cancer registry)

Truncation

Figure 4: Truncation

• Different types of censoring and truncation in calendar time (left panel) and analysis time (right panel).
• Dots are events and arrows indicate censoring.

Source: https://duyngocnguyen.files.wordpress.com/2022/06/image-24.png?w=656

Kaplan-Meier Method

Method to compute survival time:

• Order event (survival) times (from earliest to latest)
• number of subjects at risk of experiencing event just before time \(t\) of interval \(j\): \(n_j\)
  • Will depend on number of patients censored prior to time \(t\) of interval \(j\)
• number of failures (events) at time \(t_j\): \(d_j\)
• probability of surviving beyond time \(t_j\): \(\frac{n_{j}-d_{j}}{n_{j}}\)
  • between \(t_{j}\) and just before \(t_{j+1}\) there are zero events
  • censored observations deemed to occur just after \(t_{j}\) \[\hat{S}(t)=\prod_{t_{i}\le t}(1-\frac{d_{i}}{n_{i}})\]

Kaplan-Meier Survivor Function

• Step function:
  • Survivor function constant between event times, decreasing at each event time
  • Survivor function undefined where \(t>t^{max=censored}\)

Figure 5: Kaplan-Meier Curve

KM Life Table Example: Time to treatment discontinuation

Table 1: Life table

X	A	B	C	D	E	F	G
Row #	Time_days	Number At Risk	Treatment discontinuation	Censor	Proportion event free in interval	Cumulative Survival St	Formula
1	0					1
2	2	100	0	1	1	1	=(B2-C2)/B2
3	6	99	1	0	0.98989899	0.98989899	=(B3-C3)/B3*F2
4	8	98	1	0	0.989795918	0.97979798
5	14	97	2	1	0.979381443	0.95959596
6	21	94	1	0	0.989361702	0.949387492
7	22	93	3	1	0.967741935	0.918762089
8	27	89	2	0	0.97752809	0.89811575
9	30	87	0	1	1	0.89811575
10	35	86	1	1	0.988372093	0.887672544
11	40	84	2	0	0.976190476	0.866537483

KM example

Example data for surviving not looking at phones:

id	time	Event	sex
1	15	1	1
2	60	1	1
3	25	1	1
4	40	1	1
5	10	0	1
6	26	1	2
7	45	1	2
8	30	0	2
9	5	1	2
10	55	1	2

Results:

Call: npsurv(formula = Surv(time, Event) ~ 1, data = km_workshop)

      n events median 0.95LCL 0.95UCL
[1,] 10      8     40      25      NA

Comparison of groups

• The Kaplan-Meier function depicts the survival function for a single group
  • Plotting the KM for multiple groups we begin to compare across groups
  • Comparison is often of treatment arms in RCTs, but could be by risk groups
• Hypotheses testing for differences between groups
  • Log rank - equal weighting for all failure times
  • Alternatives: Wilcoxon - weighted by risk set

Comparison of groups

• Cox proportional hazards
  • Semi-parametric
  • Linear component makes no assumption about underlying functional form \[h(t|x_{j})=h_{0}(t)e^{x_{j}\beta_{x}}\]

KM example: curve (total)

library(survminer)

# ggsurvplot(fit)
km_fit_total <- survfit(Surv(time, Event)~1, data=km_workshop)
km_total <- ggsurvplot(km_fit_total, conf.int=TRUE, risk.table=TRUE, 
                       risk.table.height=.30)
km_total

Figure 5: Kaplan-Meier Curve for Surviving This Workshop by Not Looking at Phone

KM example: curve (by sex)

library(survminer)

# ggsurvplot(fit)
km_fit_by_sex <- survfit(Surv(time, Event)~sex, data=km_workshop)
km_by_sex <- ggsurvplot(km_fit_by_sex, conf.int=TRUE, pval=TRUE, risk.table=TRUE, 
                        legend.labs=c("Male", "Female"), legend.title="Sex",  
                        palette=c("dodgerblue2", "orchid2"), 
                        risk.table.height=.30)
km_by_sex

Figure 6: Kaplan-Meier Curve for Surviving This Workshop by Not Looking at Phone

Parametric survival modelling

• A parametric survival model can be fitted to the survival data
  • Assumes survival time follows a distribution
  • Enables prediction of survival times beyond the follow-up of a clinical trial
  • Unbiased estimate of mean survival (accounts for censoring)
  • Generates survival curves more consistent with theoretical example than KM curve

Parametric survival modelling

Figure 6: Fitted parametric survival function

Types of parametric survival models

Main model types:

• Proportional hazards (PH): Models the hazard

  • Assume the treatment (or other covariate) effect is a constant applied to the hazard function

• Accelerated failure-time (AFT): Models log time

  • Assume the treatment (or other covariate) effect impacts the time-to-event (rather than the hazard of the event)

PH models

• Hazard at time \(t\) is the product of two quantities:
  • Baseline hazard, \(h_{0}(t)\)
  • Exponential expression of linear sum of explanatory variables (\(X\)) and coefficients \((\beta)\)
• Baseline hazard is a function of \(t\) but not \(X\)
• Linear predictor includes \(X\) but not \(t\)

PH models

Table 2: Examples of parametric PH models

Model	Hazard	Parameter
Exponential	Constant	\(\lambda\)
Weibull	Monotonic	\(\lambda\), \(p\)
Gompertz	Monotonic	\(\lambda\), \(\theta\)

Figure 7: PH models (Stata manual)

PH models assumptions

Hazards are proportional if ratios are independent of time

• Cox PH model assumes the HR comparing two groups is constant

\[h_{t}(t)=\theta h_{0}(t)\] \[\theta = \frac{h_{t}(t)}{h_{0}(t)}\] - \(\theta\) is constant over time

PH assumption violated if the graph of the hazards for treatment and control cross

• Assumption can still be violated even if the hazard functions don’t cross

AFT models

The effect of treatment is interpreted in terms of its effect on the time-to-event, relative to control

• Relative treatment effect referred to as a time ratio (TR)
  • Difference in the log event times between treatment arms is the log TR
• Acceleration factor (AF) is a simple transformation of TR

Exponential and Weibull can be used as both PH and AFT models

AFT models

Table 3. Examples of parametric AFT models

Model	Hazard	Parameters
Weibull	Monotonic	\(\lambda\), \(p\)
Lognormal	Non-monotonic	\(\sigma\), \(\mu\), \(\phi\)
Log-logistic	Non-monotonic	\(\gamma\), \(\lambda\)
Generalised gamma	Non-monotonic	\(\kappa\), \(\mu\), \(\theta\)

Figure 8: AFT models (Stata manual)

Time ratios and acceleration factors

The probability of surviving beyond time 𝑡 on treatment is equal to the probability of surviving beyond \(\frac{t}{TR}\) on control.

\[S_{t}(t)=S_{0}(\frac{t}{TR})\] \[TR = \frac{S_{0}(t)}{S_{t}(t)}\] AF is the inverse of TR:

\[AF=\frac{1}{TR}=\frac{S_{t}(t)}{S_{0}(t)}\]

AFT assumptions and associated test

Assumptions:

• Relative treatment effect acts multiplicatively on the time-to-event (i.e. time-to-event x TR)
• Relative treatment effect is constant over time

Tested by plotting the survival time quantiles for treatment against those for control (quantile-quantile [QQ] plot):

• Plot of survival at times \(t_{q}\) for equally spaced apart quartiles \(q\)
• Straight lines indicate a multiplicative effect of treatment on time, validating TRs as an appropriate measure of relative treatment effect.

NICE DSU Technical Support Unit 14: Survival Analysis for Economic Evaluations Alongside Clinical Trials

The document discusses:

• Survival Benefit Estimation Challenges: Censored survival data in interventions impact accurate estimation of benefits, leading to underestimation of survival gains and Quality Adjusted Life Years (QALYs) gained.
• Extrapolation Techniques: Various methods like Exponential, Weibull, Gompertz, log-logistic, or log-normal models, along with more complex models, are available for estimating survival benefits.
• Variability in Survival Estimates: Different methods yield varying survival estimates, emphasizing the need to justify chosen extrapolation approaches for decision-makers’ confidence.
• Model Justification and Validation: Justification of chosen survival models is crucial, requiring statistical tests for model comparison, plausibility assessment of extrapolated segments, and reliance on external data, biological plausibility, or clinical expertise.
• Analysis of NICE Technology Appraisals: Reveals a lack of systematic survival analysis in most cases, with varying degrees of justification for chosen methods, often suboptimal.
• Recommendations for Systematic Approach: Proposed algorithmic approach involves fitting and testing various survival models, assessing internal and external validity, aiming for better model selection and more robust economic evaluations.

Conditions of use of the methods discussed in the TSD

• Patient level data are available.
• If only summary statistics are available, methods introduced by Guyot el al. (2012) to recreate patient level data must be used.
• If evidence synthesis is required to include all relevant comparators within a TA, the methods discussed in the TSD should not be utilized.

Survival analysis modelling methods

• Two main approaches to survival analysis modelling: parametric and non-parametric.
  • Parametric models assume that the survival distribution follows a specific parametric form, such as the Weibull or exponential distribution.
  • Non-parametric models do not make any assumptions about the form of the survival distribution and estimate the survival function directly from the data.
• The choice of model depends on the nature of the data and the research question.

Parametric Survival Analysis Modelling

• Parametric models are often used when there is a clear theoretical basis for the assumed distribution.
• They can be more efficient than non-parametric models when the data is well-fitted by the assumed distribution.
• However, parametric models can be sensitive to violations of their assumptions.

Non-Parametric Survival Analysis Modelling

• Non-parametric models are more flexible than parametric models and can be used with any type of data.
• They are less sensitive to violations of assumptions than parametric models.
• However, non-parametric models can be less efficient than parametric models.

Common Parametric Survival Analysis Models

• The exponential distribution is the simplest parametric model as it incorporates a hazard function that is constant over time, and therefore it has only one parameter, \(\lambda\).
• The Weibull distribution can be parameterised either as a PH model or an AFT model.
• Similar to the Weibull distribution the Gompertz has two parameters – a shape parameter and a scale parameter. Also similar to the Weibull distribution the hazard in the Gompertz distribution increases or decreases monotonically.
• The log-normal distribution is another flexible parametric model that can be used to fit survival curves that are skewed.

Common Non-Parametric Survival Analysis Models

• The Kaplan-Meier estimator is a non-parametric estimator of the survival function that is based on the observed survival times.
• The Nelson-Aalen estimator is another non-parametric estimator of the cumulative hazard function.
• The Cox proportional hazards model is a semi-parametric model that can be used to adjust for covariates.

Choosing a Survival Analysis Modelling Method

• The choice of survival analysis modelling method depends on the nature of the data, the research question, and the assumptions that can be made about the survival distribution.
• It is important to consider the strengths and weaknesses of each method before making a decision.

Data-related Factors

• Data type: Is it continuous (e.g., time to event), binary (e.g., alive/dead), or something else? Each type requires different models.
• Censored data: Are there instances where the event didn’t occur within the observation period? Can censoring bias the results? Choose models that handle this effectively.
• Data completeness: Are there missing data points or outliers? Address these before attempting modeling.

Research Question

• Comparison of survival curves: Parametric or semi-parametric models like Weibull or Cox proportional hazards might be suitable.
• Estimating cumulative hazard (risk over time): Non-parametric models like Kaplan-Meier or Nelson-Aalen could be better choices.
• Predicting individual survival times: Specialized models like frailty models might be appropriate.

Model Assumptions

• Parametric models: Check if the data fits the assumed distribution (e.g., Weibull, exponential). Misfits can lead to unreliable results.
• Non-parametric models: No strict assumptions about the distribution, but may be less efficient than parametric models if the data fits a known form.

Goodness-of-Fit:

• Graphical methods: Plot observed vs. predicted survival curves, check for deviations and trends.
• Statistical tests: Chi-square tests, Kolmogorov-Smirnov tests, etc., check for statistically significant differences between observed and predicted survival.
• Information criteria: Akaike Information Criterion (AIC) or Bayesian Information Criterion (BIC) compare different models based on their complexity and fit.

Extrapolation Ability:

• Validity beyond observed data: If used for prediction outside the study period, assess if the model’s assumptions hold true in that extended range.
• Validation datasets: Test the model performance on other data sets similar to the original one.
• Sensitivity analysis: See how results change with slight variations in model parameters or data assumptions.

Interpreting Results:

• Survival curves: Visualize and compare the probability of event-free survival over time for different groups or treatments.
• Hazard rates: Analyze the instantaneous risk of an event occurring at any given time point.
• Confidence intervals: Understand the range of uncertainty surrounding the estimated model parameters.
• Sensitivity analysis: Explain how robust the results are to potential changes in assumptions or data.