Table 1 Summary of current methods for each definition

C/D

t = 0

\( \begin{array}{l} S{e}^C\left( c, t\right)= P\left({X}_i> c\Big|{T}_i\le t\right)\\ {} S{p}^D\left( c, t\right)= P\left({X}_i\le c\Big|{T}_i> t\right)\end{array} \)

CD1

survivalROC

Pro: Easy Cons:

a) Produce non-monotone sensitivity and specificity

b) Not robust to marker-dependent censoring

Pro: Clinically relevant since many clinical experiments aim to discriminate individuals with disease prior to specific time and healthy individual beyond that time

Con: Use redundant information in separating cases and controls

CD2

survivalROC

Pros:

a) Produce monotone sensitivity and specificity

b) Allow censoring to depend on marker

Con: Does involve smoothing parameter

CD3

Programme code

Pro: Does not involve any smoothing parameter

Cons:

a) Does involve recursive computation

b) Produce non-monotone specificity

CD4

survAUC

Pros:

a) Produce monotone sensitivity and specificity if the score is produced from a survival model

b) Allow censoring to depend on marker

Con: Not invariant to an increasing transformation of the marker

CD5

timeROC

Pros:

a) Produce monotone sensitivity and specificity

b) More robust than CD1 and CD3

Con: Does not robust to marker dependent censoring

CD6

timeROC

Pros:

a) Produce monotone sensitivity and specificity

b) Robust to marker-dependent censoring

c) More less biased than CD2 when censoring strongly depends on marker

CD8, VL Cox

survival

Pro: Straightforward to implement

VL Aalen

timereg

VL KM

prodlim

C/D

A longitudinal time point

\( \begin{array}{l} S{e}^C\left( c, t\right)= P\left({Y}_{i k}> c\Big|{T}_i-{s}_{i k}\le t\right)\\ {} S{p}^D\left( c,{t}^{*}\right)= P\left({Y}_{i k}\le c\Big|{T}_i-{s}_{i k}> t\right)\end{array} \)

AD4 (ECD2)

Pros:

a) Produce monotone sensitivity and specificity

b) Allow censoring to depend on marker

Con: Does involve smoothing parameter

Pro: Use the most recent marker value prior to prediction time

Con: Just use a marker value at a particular time instead of using all serial of marker value

I/D

t = 0

\( \begin{array}{l} S{e}^I\left( c, t\right)= P\left({X}_i> c\Big|{T}_i= t\right)\\ {} S{p}^D\left( c, t\right)= P\left({X}_i\le c\Big|{T}_i> t\right)\end{array} \)

ID1

risksetROC

Pros: Produce consistent sensitivity and specificity if the control set is unbiased

Pro: Allow time-averaged summaries that directly relate to a familiar concordance measures such as Kendall’s tau or c-index

Con: Require an exact time of interest which often just a few individual has an event at a particular point

ID2

Pro: Potentially more robust than ID1

Con: Computationally intensive

ID3

Programme code

Pros:

a) Easier especially when involve a large number of marker

b) Understandable since it is a “regression-type” model

I/S

t = 0

\( \begin{array}{l} S{e}^I\left( c, t\right)= P\left({Y}_i> c\Big|{T}_i= t\right)\\ {} S{p}^S\left( c,{t}^{*}\right)= P\left({Y}_i\le c\Big|{T}_i>{t}^{*}\right)\end{array} \)

None

I/S

A longitudinal time point

\( \begin{array}{l} S{e}^I\left( c, t\right)= P\left({Y}_{i k}> c\Big|{T}_i-{s}_{i k}= t\right)\\ {} S{p}^S\left( c,{t}^{*}\right)= P\left({Y}_{i k}\le c\Big|{T}_i-{s}_{i k}>{t}^{*}\right)\end{array} \)

IS1

Pro: Provides unbiased estimates of model parameters of sensitivity and specificity

Con: Computationally intensive since involve spline functions

Pro: Use the most recent marker value prior to prediction time

Con: Just use a most recent of marker value instead of all marker values

IS2

Pro: Use the most recent marker value prior to prediction time

Con: not a natural companion to hazard models

Other

All longitudinal time points

ROC(t, p) = S[a ₀(T _ik ) + a ₁(T _ik )S ^− 1(p)]

AD1

Programme code