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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?

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  • $\begingroup$ 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
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    $\begingroup$ @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

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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).

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