Spanish ICU data
We used a multi-center data base from the Spanish surveillance network HELICS-ENVIN (http://hws.vhebron.net/envin-helics/), embedded in the HELICS project (Hospitals in Europe Link for Infection Control through Surveillance) [20]. We included ICUs which contributed to the registry between January 2006 and December 2011 and we included only patients who stayed at least two days in an ICU due to the definition of hospital-acquired infections. We excluded ICUs which contributed fewer than 500 patient admissions to the cohort to ensure a sufficient amount of patient time at risk for each ICU. The study population contains 81 intensive care units with 84,843 admissions (693,180 admission-days).
Statistical methods
In the Additional file 1 is a Lexis diagram [21] of individual patient data from one selected ICU over 100 days in calendar time. It demonstrates how the data depend on the two time scales. In the following, we compare the two time scales in several steps. For the ICU time scale, the time origin is the time of admission. For the calendar time scale, patient admissions entered the model with staggered or delayed entry with left-truncation occurring at the time of admission.
Overall hazard rates
We used a penalized likelihood approach [22] to estimate the overall hazard rates λk(t) separately for each event k: for ICU-acquired MRSA infection (the event of interest), death and discharge without MRSA. The overall hazard rates depend both on ICU or calendar time. The variation of the overall hazards due to different ICUs was accounted by using a shared frailty model [22]. More formally, let c represent the calendar time and c0 the truncation time, i.e., the admission time in calendar time scale. Thus, for each competing event k (MRSA, death without MRSA, discharge without MRSA) and the i-th ICU, the event-specific hazard rate with a shared frailty term \({Z^{k}_{i}}\) is defined as
$$\text{ICU time: } \lambda^{k}(c-c_{0}|{Z^{k}_{i}})={Z^{k}_{i}} {\lambda_{0}^{k}}(c-c_{0}) $$
$$\text{calendar time: } \tilde{\lambda}^{k}(c|c_{0},\tilde{Z}^{k}_{i})=\tilde{Z}^{k}_{i} \tilde{\lambda}_{0}^{k}(c|c_{0}) $$
with baseline hazard \({\lambda _{0}^{k}}(.)\) (or \(\tilde {\lambda }_{0}^{k}(.)\)) for event k; the term |c0 denotes the left-truncation time. The frailty term \({Z^{k}_{i}}\) (or \(\tilde {Z}^{k}_{i}\)) is a random effect which varies across ICUs (patients within ICU share the same frailty) and is assumed to be Gamma distributed with shape parameter 1/θ
k
and inverse scale parameter 1/θ
k
, thus E(Zk)=1 and Var(Zk)=θ
k
. Large values of θ
k
signify a closer positive relationship between patients within ICU and greater heterogeneity across ICUs.
Patient-level risk factors
We used event-specific Cox proportional hazards models (rate metric) and a Fine & Gray model [23] (risk metric) to explore covariate effects of vector X (gender, age, type of diagnosis, antibiotic treatment 48 h before and/or after ICU admission, trauma, days in hospital before ICU admission, APACHE II (Acute Physiology And Chronic Health Evaluation) score) comparing results between models where the timescale was calendar time and ICU time. The assumption of proportional hazards was checked via the inspection of the Schoenfeld residuals [24]; note that proportionality due to the rate metric does not lead to proportionality due to the risk metric but even if proportionality is not fulfilled the hazard ratio has the meaningful interpretation of an time-averaged effect [25]. We then model both times together by including the second time scale as a covariate. We stratified for ICU in order to allow the hazard to be different across ICUs, and hence we did not use the frailty terms.
Models with one time scale
Model 1a: ICU time as basic time scale
$$\lambda^{k}(c-c^{0}|X)=\lambda_{0i}^{k}\left(c-c^{0}\right)\text{exp}\left(\sum_{j} {\beta^{k}_{j}} X_{j}\right) $$
This is an event-specific Cox model with ICU time as the basic time scale. The exponential of the regression coefficients \({\beta ^{k}_{j}}\) are corresponding hazard ratios of variable X
j
and event k.
Model 2a: calendar time as basic time scale (the real-time approach)
$$\tilde{\lambda}^{k}(c|c^{0},X)=\tilde{\lambda}_{0i}^{k}(c|c^{0})\text{exp}\left(\sum_{j} \tilde{\beta}^{k}_{j} X_{j}\right) $$
This is an event-specific Cox model with calendar time as the basic time scale and staggered or delayed entry with left-truncation occurring at the time of admission.
Model 3a: subdistribution for MRSA, basic time scale is ICU time
In a competing risks setting, the cumulative incidence function for event k (CIFk(t)) depends on all event-specific hazards [26]. This can be seen with following formula with t=c−c0:
$$\begin{aligned} \text{CIF}^{k}(t)={\int_{0}^{t}} \left(\text{exp}\left(-\sum_{\text{all events }i} {\int_{0}^{u}} \lambda^{i} (v) dv \right)\right) \times \lambda^{k} (u) du \end{aligned} $$
We are basically interested in MRSA infection as our event of interest. In simple words, the formula above is the product of the time-dependent probability of staying alive at-risk on ICU and the conditional probability of acquiring a MRSA infection. Note that the probability of staying alive at-risk on ICU depends on the competing events hazards (discharge or dying without MRSA) in addition to the MRSA hazard.
Fine & Gray [23] defined the subdistribution hazard which is in our setting the probability of the MRSA infection given that a patient has stayed in ICU up to time t without a MRSA infection or has had the competing event (death ore discharge) prior to time t [27]. Thus, the risk sets for the subdistribution hazard are unnatural (discharged and died patients remain technically at-risk). However, unlike the event-specific hazard, the subdistribution hazard is directly linked to the corresponding cumulative incidence function of MRSA infection. Based on the subdistribution hazard, Fine & Gray proposed a proportional hazards model to study risk factors on the risk metric. The resulting subdistribution hazard ratios of an exposure can be interpreted as effects which can be seen when plotting cumulative incidence functions, grouped by exposure categories.
Models with two time scales
Model 1b: model 1a plus year of admission as a covariate
$$\lambda^{k}(c-c^{0}|X)=\lambda_{0i}^{k}\left(c-c^{0}\right)\text{exp}\left(\sum_{j} {\beta^{k}_{j}} X_{j}+\gamma f\left(c^{0}\right)\right) $$
with f(c0) as the calendar year of admission. If necessary, other more detailed functions (which, for instance, includes also the calendar month of admission) can be chosen.
Model 2b: model 2a plus length of stay at-risk as a time-dependent covariate
$$\tilde{\lambda}^{k}(c|c^{0},X)=\tilde{\lambda}_{0i}^{k}\left(c|c^{0}\right)\text{exp}\left(\sum_{j} \tilde{\beta}^{k}_{j} X_{j}+\gamma g\left(c-c^{0}\right)\right) $$
with g(c−c0) as a function to categorize ICU time in 0-4, 5-9, 10-14 and 15+ days. If necessary, other categorizations can be chosen.
Model 3b: model 3a plus year of admission as covariate.
