I am reading Tutz & Schmid "Modeling Discrete Time-to-Event Data" (2016) chapter 4 Evaluation and Model Choice section 4.3.3 Discrimination Measures. On p. 94 estimators for sensitivity and specificity in presence of censoring are defined: $$ \widehat{\text{sens}}(c,t) = \frac{ \frac{ \sum_j \delta_j^\mathcal{T} I(\hat\eta_j^\mathcal{T}>c \ \cap \ t_j^\mathcal{T}=t) }{ \hat G_j(t_j^\mathcal{T}-1) }} { \frac{ \sum_j \delta_j^\mathcal{T} I( t_j^\mathcal{T}=t) }{ \hat G_j(t_j^\mathcal{T}-1) } } $$ and $$ \widehat{\text{spec}}(c,t) = \frac{ \sum_j I(\hat\eta_j^\mathcal{T}\leq c \ \cap \ t_j^\mathcal{T}>t) }{ \sum_j I( t_j^\mathcal{T}>t) } $$ where

  • $c$ is a threshold value,
  • $t$ is time to event (failure),
  • $\delta_j^\mathcal{T}$ is an indicator that the observation from the test set (hence the superscript $\mathcal{T}$) on which the estimate is based was not censored,
  • $\hat\eta_j^\mathcal{T}$ is a predictor (e.g. $\eta=x^\top\gamma$ for some covariates $x$ and a parameter vector $\gamma$) that is transformed into a probability estimate by the inverse link function,
  • $\hat G_j$ is a censoring-specific survival function (in the sense of surviving without failure so that the observation may or may not be censored; contrast this to a failed observation that could not be censored).

This is certainly a lot of notation, but it should be possible to follow my question that comes next without understanding it thoroughly.

Question: The definition of $\widehat{\text{sens}}(c,t)$ accounts for censoring by inverse probability weighting, i.e. division by censoring-specific survival probability $\hat G_j(t_j^\mathcal{T}-1)$. Meanwhile, the definition of $\widehat{\text{spec}}(c,t)$ does not do that. Why?

enter image description here

  • $\begingroup$ Sensitivity and specificity apply to retrospectively sample data such as with case-control studies. Using them for a prospective cohort study muddies the waters, Look at the conditioning involved with sens and spec; you'll see that it conditions on the unknown to predict what is already known. Not very helpful. $\endgroup$ Aug 10, 2022 at 14:40
  • 1
    $\begingroup$ @FrankHarrell, is this related to In Machine Learning Predictions for Health Care the Confusion Matrix is a Matrix of Confusion? I read it a couple of times before but forgot quite a bit, as I did not work on any related problems since then. $\endgroup$ Aug 10, 2022 at 14:45

1 Answer 1


If we indeed follow Uno et al. (2007), we find on p. 16 equations (3.1) and (3.2) that the estimator of specificity does include inverse probability weighting due to censoring – just like the estimator of sensitivity. So this seems like a mistake in Tutz & Schmid (2016).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.