To the Editor—With great interest we read the article by Carter et alReference Carter, Langley, Kuhle and Kirkland ¹ investigating risk factors for central line–associated bloodstream infections (CLABSI). For estimating the proportion without CLABSI depending on time, Kaplan-Meier (KM) curves were calculated to account for the at-risk time from insertion until the occurrence of CLABSI or removal of the catheter (which is treated as censoring in the model). If a bloodstream infection occurs during the use of a central venous catheter it is considered to be catheter associated and it is rather unlikely to develop a CLABSI 48 hours after removal of the catheter.Reference O’Grady, Alexander and Dellinger ² However, KM models assume that the hazard of CLABSI remains unchanged when a censoring event occurs. This censoring assumption is clearly not fulfilled in the case of removal of the catheter since removal leads to a reduction of risk. Hence, removal of the catheter without CLABSI should be considered as a competing event for CLABSI.Reference Wolkewitz, Cooper, Bonten, Barnett and Schumacher ³

Using standard KM models in the presence of competing events leads to overestimation of the cumulative risk.Reference Wolkewitz, Cooper, Bonten, Barnett and Schumacher ³ This can be seen in figure 3 of Carter et al.Reference Carter, Langley, Kuhle and Kirkland ¹ The KM curve lies at approximately 80% without CLABSI, which corresponds to a risk of CLABSI of approximately 20%. But considering the actual number of patients with CLABSI this leads to a risk of ${\rm CLABSI=}{{385} \over {5648}}=6.8\,\%\,$ .

To illustrate the bias in this setting, we analyzed simulated data of a simplified competing event setting based on values from the article of Carter et alReference Carter, Langley, Kuhle and Kirkland ¹ (code is available upon request). We consider 2 constant competing event hazards, λ₁ for CLABSI and λ₂ for removal without CLABSI. Hence, the cumulative incidence function (CIF) of CLABSI and the CIF of removal without CLABSI are given by this formulaReference Grambauer, Schumacher, Dettenkofer and Beyersmann ⁴ :

(1) $$CIF_{1} \left( t \right)={{\lambda _{1} } \over {\lambda _{1} {\plus}\lambda _{2} }}{\rm {\times}}(1{\minus}\exp ({\minus}\left( {\lambda _{1} {\plus}\lambda _{2} } \right){\rm {\times}}t))$$

(2) $$CIF_{2} \left( t \right)={{\lambda _{2} } \over {\lambda _{1} {\plus}\lambda _{2} }}{\rm {\times}}(1{\minus}\exp ({\minus}\left( {\lambda _{1} {\plus}\lambda _{2} } \right){\rm {\times}}t))$$

With λ_i being the hazard for event i, i=1; 2. Formulas 1 and 2 illustrate that the CIFs of the respective events depend on both the hazard for the event of interest and the hazard for the competing event. The right terms of formulas 1 and 2 represent the probabilities that any event occurs at time t. The left terms ${{\lambda _{i} } \over {(\lambda _{1} {\plus}\lambda _{2} )}},{\rm }({\rm i }={\rm }1;{\rm }2)$ display the probabilities that the occurring event at time t is event i.

As seen in the formulas above, there is a direct connection between the overall risk of CLABSI and the rates of both eventsReference Beyersmann, Gastmeier and Schumacher ⁵ : CIF ₁ (t) approximates the overall CLABSI risk ${{\lambda _{1} } \over {(\lambda _{1} {\plus}\lambda _{2} )}}$ for large time points, which is estimated by ${{\,\#{\rm CLABSI}} \over {\,\#{\rm patients}}}={{385{\rm }} \over {5648}}=6.8\%\,$ . Analogously, the overall probability of removal without CLABSI is ${{5648\,{\minus}\,385{\rm }} \over {5648}}=93.2\%\,$ .

The constant hazard rate λ₁ is estimated by ${{\,\#\,{\rm CLABSI}} \over {\,\#{\rm line}{\minus}{\rm days\, at\, risk}}}$ . Note that line-days at risk are line-days the patients are actually at risk—that is, line-days from insertion until removal without infection or until CLABSI. If D_i is the individual line-days contribution of patient i, λ₁ can also be written as

(3) $$\lambda _{1} ={{\,\#\,CLABSI} \over {\sum_{\left\{ {i\,=\,1} \right\}}^N D_{i} }}={{\,\#\,CLABSI} \over {N{\rm {\times}}{1 \over N}\sum_{\left\{ {i\,=\,1} \right\}}^N D_{i} }}={{\,\#\,CLABSI} \over N}{\rm {\times}}{1 \over {\bar{D}}}$$

with N being the number of patients and $\bar{D}$ being the mean line-days at risk. Similarly, the hazard for removal without CLABSI can be calculated by

(4) $$\lambda _{1} ={{\,\#removal\,w/o\,\,CLABSI} \over N}{\rm {\times}}{1 \over {\bar{D}}}$$

Motivated by the data of Carter et alReference Carter, Langley, Kuhle and Kirkland ¹ we simulated data. The number of events (#CLABSI=385) and number of patients (N=5648) are given. For the formulas of the hazard rates 3 and 4 the mean line-days at risk is required. Since there is no information on the line-days of the first central venous catheter given in Carter et alReference Carter, Langley, Kuhle and Kirkland ¹ we considered $\overline{{D }} =50$ . This value lies somewhere in between the mean line-days considering either in-hospital line-days or total line-days as given by Carter et alReference Carter, Langley, Kuhle and Kirkland ¹ : $\left( {{{\,\#{\rm in\,hospital \,line}\,{\minus}\,{\rm days}} \over {\,\#CVC}}\,\approx\,20\,{\rm and}\,$ ${{\,\#{\rm total\,line}\,{\minus}\,{\rm days}} \over {\,\#CVC}}\,\approx\,100} \right)$ . In Figure 1 we compare CIFs for the 2 events and 1 minus the KM curve in order to demonstrate the overestimation of the risk of developing an infection by using standard KM models mentioned above.

FIGURE 1

Cumulative incidence functions to study the bias caused by not accounting for the competing event (removal of central venous catheter [CVC] without central line–associated bloodstream infection [CLABSI]). Comparison of cumulative incidence functions (correct approach) with 1 minus the Kaplan-Meier curve (incorrect approach) for both events, CLABSI and removal without CLABSI.

In Figure 1 the CIFs (correct approach) for each event, CLABSI and removal without CLABSI, are plotted. In addition, 1 minus the KM curve for both events (incorrect approach, 1− KM) is shown. At every time point the probabilities of CLABSI, removal without CLABSI, and remaining under risk should add up to 100%. Considering the curves of the KM approach at day 220 (dashed line) the probabilities of CLABSI (≈20%) and of removal without CLABSI (≈90%) already add up to more than 100% (≈110%). The incorrect approach ignoring the presence of competing events leads to an overestimation of the occurrence of the event of interest. Using the correct approach the CIF for CLABSI reaches 6.8%, which equals the actual risk of CLABSI as seen above.

In a further analysis Carter et alReference Carter, Langley, Kuhle and Kirkland ¹ used a Cox proportional hazards model in order to investigate the association between several covariates and CLABSI. This is a suitable approach but it is incomplete if it is performed only for the event of interest. Risk factors can be indirectly associated with the event of interest if they are associated with the competing event. Therefore, a Cox proportional hazards model for the competing events should be performed in addition in order to understand direct and indirect effects (cf. Wolkewitz et alReference Wolkewitz, Cooper, Bonten, Barnett and Schumacher ³ ). Furthermore, investigation of the cumulative risk is necessary. This can be performed by the subdistribution hazard approach via a Fine and Gray model.Reference Fine and Gray ⁶

We hope this letter provides a constructive contribution to future risk factor analyses in this kind of setting.