2024-05-01
What do we do when outcome is right censored survival time?
For a given threshold \(c\), the time-dependent sensitivity (Se) and specificity (Sp) can defined respectively by
\(Se(c,t)=P(S_i>c|D_i(t)=1)\)
\(Sp(c,t)=P(S_i\leq c|D_i(t)=0)\)
At any time \(t\), \[AUC(t)=\int_{-\infty}^{\infty}Se(c,t)d[1-Sp(c,t)]\]
At each time point \(t\), each individual is classified as a case or control. A case is defined as an individual with an event up to time \(t\), while a control is an individual remaining event-free at time \(t\).
\[AUC^{C/D}(t)=P(S_i>S_j|T_i\leq t,T_j>t), i\neq j.\]
Why it is biased?
\(\hat{S}(t)\) is the estimated survival function, \(\hat{S}(t|S_i>c)\) is the estimated conditional survival function for the subset defined by \(S > c\).
It is more appropriate to apply the C/D definitions when there is a specific time of interest that is used to discriminate between individuals experiencing the event and those event-free prior to the specific time.
However, since some individuals may contribute as controls at an earlier time and then contribute as cases later, this definition uses redundant information in separating cases and controls.
A case is defined as an individual with an event right at time \(t\), while the control is an event-free individual at time \(t\).
\[AUC^{I/D}(t)=P(S_i>S_j|T_i= t,T_j>t), i\neq j.\]
In this definition, there are individuals neither a control nor case (when the event time is less than the target time, i.e. \(T_i < t\)).
Each individual who had an event may play the role of control at the earlier time (when the event time is greater than target time, i.e., \(T_i > t\)) but then contributes as a case at the later incident time (when the event time is the same as the target time, i.e., \(T_i = t\)).
The I/D is more appropriate when the exact event time is known and we want to discriminate between individuals experiencing the event and those event-free at a given event-time, i.e. \(T_i = t\).
This is essentially dichotomizing the risk set at time \(t\) into cases and controls, a natural companion to hazard models.
This also allows time-averaged concordance measure \(C\)-index. This is a special advantage of the I/D definition, since in many applications no a prior time \(t\) is identified, thus a global accuracy summary is usually desired.
First, assume that the survival time \(X\) is actually observed without any censoring, i.e., \(\delta_i=1\) (\(i=1, 2, \ldots, n\)). Upon randomly drawing a pair of subjects, say \((i, j)\), \(i\neq j\), we may have five types of pairs between the survival time \(X\) and the predictive score \(Y\).
These five possibilities for a random pair are comprehensive and mutually exclusive, and therefore \[\Pi_c+\Pi_d+\Pi_{tX}+\Pi_{tY}+\Pi_{tXY}=1.\]
Kim’s measure \(d_{X\cdot Y}\), \[d_{X\cdot Y}=\dfrac{\Pi_c-\Pi_d}{\Pi_c+\Pi_d+\Pi_{tY}}=\dfrac{\Pi_c-\Pi_d}{1-\Pi_{tX}-\Pi_{tXY}}\] is the probability of a concordance minus the probability of a discordance, both conditioned on the occurrence of distinct values of outcome \(X\), for quantifying the degree of relationship between \(X\) and \(Y\).
Define \(sign\) and \(csign\) (\(sign\) with censoring) functions as below, \[ sign\left( Y_i, Y_j\right) = I(Y_i > Y_j) - I(Y_i < Y_j)\] \[ csign\left(X_i, \delta_i, X_j, \delta_j\right) = I(X_i \geq X_j)\delta_j - I(X_i \leq X_j)\delta_i \]
The order of two survival times \(X_i\) and \(X_j\) can be unambiguously determined if and only if \(csign\left(X_i, \delta_i, X_j, \delta_j\right)\neq 0\).
\[ P(csign\left(X_i, \delta_i, X_j, \delta_j\right)sign\left( Y_i, Y_j\right)=1|csign\left(X_i,\delta_i, X_j, \delta_j\right)\neq0)+\\\frac{1}{2}P(sign(Y_i,Y_j)=0|csign\left(X_i,\delta_i, X_j, \delta_j\right)\neq 0)\]
\[C_{XY}^g = \dfrac{\Pi_c^g+\frac{1}{2}\Pi_{tY}^g}{\Pi_c^g+\Pi_d^g+\Pi_{tY}^g}=\dfrac{\Pi_c^g+\frac{1}{2}\Pi_{tY}^g}{1-\Pi_{tX}^g-\Pi_{tXY}^g}.\]
\[C_{XY}^g = \frac{1}{2}\left(\dfrac{\Pi_c^g-\Pi_d^g}{\Pi_c^g+\Pi_d^g+\Pi_{tY}^g}+1\right)\]
Notice that,
\[\dfrac{\Pi_c^g-\Pi_d^g}{\Pi_c^g+\Pi_d^g+\Pi_{tY}^g}=\dfrac{\Pi_c^g-\Pi_d^g}{1-\Pi_{tX}^g-\Pi_{tXY}^g}\\=\dfrac{E[csign\left(X_i, \delta_i, X_j, \delta_j\right)sign\left( Y_i, Y_j\right)]}{E[csign\left(X_i, \delta_i, X_j, \delta_j\right)^2]}\]
Consider these two quantities,
\[\dfrac{1}{n(n-1)}\sum_i\sum_{j\neq i}csign\left(X_i, \delta_i, X_j, \delta_j\right)sign\left( Y_i, Y_j\right)\] \[\dfrac{1}{n(n-1)}\sum_i\sum_{j\neq i}csign\left(X_i, \delta_i, X_j, \delta_j\right)^2\]