Time to Event Models

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Configuring R

Functions from these packages will be used throughout this document:

library(conflicted) # check for conflicting function definitions
# printr is not loaded here: it inserts help-file output into markdown output
library(rmarkdown) # Convert R Markdown documents into a variety of formats.
library(pander) # format tables for markdown
library(ggplot2) # graphics
library(ggfortify) # help with graphics
library(dplyr) # manipulate data
library(tibble) # `tibble`s extend `data.frame`s
library(magrittr) # `%>%` and other additional piping tools
library(haven) # import Stata files
library(knitr) # format R output for markdown
library(tidyr) # Tools to help to create tidy data
library(plotly) # interactive graphics
library(dobson) # datasets from Dobson and Barnett 2018
library(parameters) # format model output tables for markdown
library(haven) # import Stata files
library(latex2exp) # use LaTeX in R code (for figures and tables)
library(fs) # filesystem path manipulations
library(survival) # survival analysis
library(survminer) # survival analysis graphics
library(KMsurv) # datasets from Klein and Moeschberger
library(parameters) # format model output tables for
library(webshot2) # convert interactive content to static for pdf
library(forcats) # functions for categorical variables ("factors")
library(stringr) # functions for dealing with strings
library(lubridate) # functions for dealing with dates and times
# Summarizes key information about statistical objects in tidy tibbles
library(broom)
# Provides suite of functions to work with regression model
# `broom::tidy()` tibbles
library(broom.helpers)

Here are some R settings I use in this document:

rm(list = ls()) # delete any data that's already loaded into R

conflicts_prefer(dplyr::filter)
ggplot2::theme_set(
  ggplot2::theme_bw() +
    # ggplot2::labs(col = "") +
    ggplot2::theme(
      legend.position = "bottom",
      text = ggplot2::element_text(size = 12, family = "serif")
    )
)

knitr::opts_chunk$set(message = FALSE)
options(digits = 6)

panderOptions("big.mark", ",")
pander::panderOptions("table.emphasize.rownames", FALSE)
pander::panderOptions("table.split.table", Inf)
# use the `filter()` function from dplyr() by default
conflicts_prefer(dplyr::filter)
legend_text_size <- 9
run_graphs <- TRUE

In many health sciences applications, binary outcomes are incompletely observed. For example, if we are studying whether cancer patients experience a relapse after an initial remission, we may not be able to follow patients to the end of their lives; instead, we may only know whether each patient has relapsed before the end of the study. If a patient has not relapsed by that point, we might not know if they will relapse at some other date or if they will stay cancer-free for the rest of their lives. ¹ Their recurrence status at end-of-life is missing data. If some study participants withdraw from a study before the scheduled end date, additional missing data arise. Standard logistic regression cannot handle such right-censored data directly without introducing bias.

However, these outcome observations are not entirely missing. We know that those patients stayed relapse-free at least until the time point when they were last observed. If we also know the time-to-event for participants who experienced events while under study, we can analyze the time-to-event-or-study-exit alongside an event indicator using survival analysis. The survival analysis framework provides the foundation for the topics covered in this module.

1 Why censoring matters

ExmExample 1 (Six patients followed for relapse)

Example 1 (Six patients followed for relapse) Suppose a study follows six cancer patients in remission for up to 12 months, and we want to estimate the 12-month relapse risk, \(1 - {\operatorname{S}\mathopen{}\left(12\right)\mathclose{}}\), where \(\operatorname{S}\) is the survival function of time to relapse (in months). Table 1 shows each patient’s observed follow-up time and event indicator (1 = relapse observed; 0 = right-censored).

Table 1: Hypothetical follow-up data for six patients

Patient	Follow-up time (months)	Event indicator	Relapse status at 12 months
A	3	1	relapsed
B	5	0	unknown (withdrew)
C	7	1	relapsed
D	9	0	unknown (withdrew)
E	12	0	relapse-free
F	12	0	relapse-free

Naive approach 1: treat censored patients as event-free. Counting patients B and D as relapse-free gives \(2/6 \approx 0.333\). This approach can only underestimate the risk, because any relapses that B or D experienced after withdrawal are counted as non-relapses.

Naive approach 2: drop patients with unknown status. Discarding patients B and D gives \(2/4 = 0.5\). This approach throws away the information that B stayed relapse-free for 5 months and D for 9 months, so in this example it overestimates the risk.

Using the censored observations. The Kaplan-Meier estimator keeps each censored patient in the risk set until their censoring time [1, Ch. 2]. Relapses occur at 3 months (6 patients at risk) and at 7 months (4 patients at risk: C, D, E, F), so

\[ \begin{aligned} {\mathop{\hat{\operatorname{S}}}\nolimits_{\text{KM}}\mathopen{}\left(12\right)\mathclose{}} &= \mathopen{}\left(1 - \frac{1}{6}\right)\mathclose{} \cdot \mathopen{}\left(1 - \frac{1}{4}\right)\mathclose{} && \text{(product over the event times 3 and 7)} \\ &= \frac{5}{6} \cdot \frac{3}{4} && \text{(subtract inside each factor)} \\ &= \frac{15}{24} && \text{(multiply numerators and denominators)} \\ &= 0.625 && \text{(divide)} \end{aligned} \]

and the estimated 12-month risk is \(1 - 0.625 = 0.375\). That estimate lies between the two naive estimates, and it is valid under non-informative censoring [1, Ch. 1].

2 Module roadmap

This module has three chapters.

1. Introduction to Survival Analysis defines the basic quantities for time-to-event data: right censoring, the survival function, the hazard function, and the cumulative hazard. It then covers nonparametric methods: the Kaplan-Meier estimator, the log-rank test, and the Nelson-Aalen estimator.

2. Proportional Hazards Models introduces the Cox proportional hazards model, which relates covariates to a baseline hazard through hazard ratios, and estimates its coefficients using the partial likelihood. It also introduces competing risks.

3. Parametric Survival Models assumes a specific distributional family for the event times, such as the exponential regression model and the Weibull distribution, and introduces accelerated failure time models.

References

[1]

D. G. Kleinbaum and M. Klein, Survival analysis: A self-learning text, 3rd ed. Springer, 2012. Available: https://link.springer.com/book/10.1007/978-1-4419-6646-9

Footnotes

1. Binary outcomes are typically defined for a specific time-point. It is important to clearly define whether we are interested in outcome status at end of study, at end of life, or at some other time.