# Why account for censoring in estimating sensitivity but not specificity?

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?

• 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. Aug 10, 2022 at 14:40
• @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. Aug 10, 2022 at 14:45