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Survival Modelling

Variable studies is the time until an event occurs

Predict when some event will happen

Focus on right-censored data

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Always finite time period: start & end time period

Cases

Treatment Event knowledge of true survival duration
within time period within time period ✅
within time period within time period
but not noticed		 ❌
within time period not within time period ❌
within time period Discontinued study ❌
within time period Unexpected event ❌

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Challenges to classification

  • Less training data
  • Pessimistic estimates due to choice of window

Challenges to regression

  • Non-gaussian: \(T\) is non-negative; may want long tails
  • If we just naively remove censored events, model will be biased predict lower survival, since the people who got removed due to censoring did not get diabetes (ever/yet)

Notation & Formalization

Let

  • Data
  • \(X=\) features
  • \(t=\) time
  • \(b(t)=\{ 0, 1 \} =\) binary indicator denoting whether time is of censoring/event occurrence
  • \(f(t)=P(t) =\) probability of death at time \(t\)
  • \(S(t) =\) probability of individual surviving beyond time \(t\)
  • \(T_d=\) time of death
\[ \begin{aligned} s(t) &= P(T_d>t) &&= 1-P(T_d<t) \\ &= \int_{t}^\infty f(x) \cdot dx &&= 1-\int_{0}^t f(x) \cdot dx \end{aligned} \]

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Methods

Kaplan-Meier Estimator

Graphical representation of survival function

  • Non-parametric model
  • Good for unconditional density estimation

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  • \(d(t)=\) no of events at time \(t\)
  • \(n(t)=\) no of individuals alive and uncensored at time \(t\)
\[ \hat s_{k-m}(t) = \prod_{k: y(k) \le t} \left( 1-\dfrac{d(k)}{n(k)} \right) \]

Log-Rank Test

Compares the time until and event occurs of 2/more independent samples

“Is there a significant difference between the 2 curves”

Hypothesis test

  • \(H_0:\) identical distribution curves
  • \(H_1:\) different distribution curves

\(p < \alpha \implies\) reject \(H_0\)

Cox Proportional Hazards Model

Identify if there are other parameters that affect the curve

Evaluation

C-Statistic/Concordance-Index

Evaluate model’s ability to predict relative survival times

Equivalent to AUC for binary classification without censoring $$ \hat c = \dfrac{1}{n} \sum_{i: b_i=0} \sum_{j: y_i < y_j} I \Big[ S(\hat y_j \vert X_j) > S(\hat y_i \vert X_i) \Big] $$ image-20240530134221231

Mean-Squared Error for uncensored individuals

Held-out censored likelihood

Binary Classifier

Derive binary classifier from learnt model and check calibration

Last Updated: 2024-05-14 ; Contributors: AhmedThahir

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