- Research article
- Open Access
- Open Peer Review
Investigating hospital heterogeneity with a multi-state frailty model: application to nosocomial pneumonia disease in intensive care units
- Benoit Liquet^{1, 2}Email author,
- Jean-François Timsit^{3, 4} and
- Virginie Rondeau^{1, 2}
https://doi.org/10.1186/1471-2288-12-79
© Liquet et al.; licensee BioMed Central Ltd. 2012
- Received: 6 February 2012
- Accepted: 10 May 2012
- Published: 15 June 2012
Abstract
Background
Multistate models have become increasingly useful to study the evolution of a patient’s state over time in intensive care units ICU (e.g. admission, infections, alive discharge or death in ICU). In addition, in critically-ill patients, data come from different ICUs, and because observations are clustered into groups (or units), the observed outcomes cannot be considered as independent. Thus a flexible multi-state model with random effects is needed to obtain valid outcome estimates.
Methods
We show how a simple multi-state frailty model can be used to study semi-competing risks while fully taking into account the clustering (in ICU) of the data and the longitudinal aspects of the data, including left truncation and right censoring. We suggest the use of independent frailty models or joint frailty models for the analysis of transition intensities. Two distinct models which differ in the definition of time t in the transition functions have been studied: semi-Markov models where the transitions depend on the waiting times and nonhomogenous Markov models where the transitions depend on the time since inclusion in the study. The parameters in the proposed multi-state model may conveniently be computed using a semi-parametric or parametric approach with an existing R package FrailtyPack for frailty models. The likelihood cross-validation criterion is proposed to guide the choice of a better fitting model.
Results
We illustrate the use of our approach though the analysis of nosocomial infections (ventilator-associated pneumonia infections: VAP) in ICU, with “alive discharge” and “death” in ICU as other endpoints. We show that the analysis of dependent survival data using a multi-state model without frailty terms may underestimate the variance of regression coefficients specific to each group, leading to incorrect inferences. Some factors are wrongly significantly associated based on the model without frailty terms. This result is confirmed by a short simulation study. We also present individual predictions of VAP underlining the usefulness of dynamic prognostic tools that can take into account the clustering of observations.
Conclusions
The use of multistate frailty models allows the analysis of very complex data. Such models could help improve the estimation of the impact of proposed prognostic features on each transition in a multi-centre study. We suggest a method and software that give accurate estimates and enables inference for any parameter or predictive quantity of interest.
Keywords
- Intensive Care Unit
- Transition Intensity
- Frailty Model
- Frailty Term
- Shared Frailty Model
Background
Multistate models have become increasingly useful to understand complicated biological processes and to evaluate the relations between different types of events. These methods have been developed to study simultaneously several competing causes of failure (e.g. competing risks of death) or to study the evolution of a patient’s state over time (e.g. admission in intensive care units (ICU), infections, alive discharge or death in ICU) and the focus is in the process of going from one state to another.
Furthermore, many studies include clustering of survival times. For instance, in critically ill patients, data come from different ICUs and because observations are clustered into groups (or units), the observed outcomes cannot be considered as independent. Thus a flexible multi-state model with random effects is needed to obtain valid outcome estimates. Ignoring the existence of clustering may underestimate the variance of regression coefficients specific to each group, leading to incorrect inferences.
Ripatti et al. [1] proposed a three-state frailty model to model age at onset of dementia and death in Swedish twins. The intra-pairs correlations and the other parameters were estimated using hierarchical bayesian model formulation and Gibbs sampling, both of which can be time-consuming. Katsahian et al. [2, 3] extended Fine and Gray’s [4] model to the case of clustered data, by including random effects in the subdistribution hazards. They first used the residual maximum likelihood then the penalized partial log-likelihood to estimate the parameters. However, the estimation approach does not directly yield estimators of the transition intensities, which often have a meaningful interpretation in epidemiological studies. Most of the time, the baseline intensity estimate is based on Breslow’s estimate leading to a piecewise-constant baseline hazard function or unspecified baseline hazard function.
In this paper, we show how a simple multi-state frailty model can be used to study semi-competing risks [5] while fully taking into account the clustering (in ICU) of the data and the longitudinal aspects of the data, including left truncation and right censoring. We suggest the use of independent frailty models or joint frailty models for the analysis of transition intensities. This approach is of interest for several reasons. First, it allows to deal with heterogeneity between ICUs. We do this by including cluster-specific random effects or frailties in the multi-state model. Frailties represent the unmeasured covariates at the cluster level, which may affect the rate of occurrence of each of the events. Moreover, this approach allows us to evaluate different prognostic effects of covariates on each transition probability. Another interesting and perhaps underrated advantage of multi-state models is the possibility to use them to predict clinical prognosis whereby a patient will be in a given health state at time u given a particular history at time t. This work extends previous research by dealing with clustered competing risks and by giving smoothed estimates of the transition rates. In addition the joint approach allows the analysis of two processes that evolve with time leading to more accurate estimates.
Two distinct approaches are often used in multi-state models. They differ in the definition of time t in the transition functions. In the first approach, the transition probability between two states depends only on the waiting times, the clock is reset to zero every time a patient enters a new state and a semi-Markov model is used. In the second approach, the transition depends only on the time since inclusion in the study, and nonhomogenous Markov models are used. The proposed approach can deal with both situations and is illustrated in this article. The choice between one the two approaches depends on the clinical knowledge of the events of interest. If it is expected that the transition probability (for instance toward death) will not change as a function of the time since randomization or inclusion, the analysis can be based solely on the semi-Markov model and it can thus be studied how the transition probability evolves after an event has taken place (for instance after nosocomial infections). However, if it is difficult to choose clinically between the semi-Markov or the nonhomogeneous Markov approach, one can use statistical criteria [6].
As discussed in the section on the estimation procedure method, an important advantage of our proposed approach is that the parameters in the multi-state model may conveniently be computed using a semi-parametric or parametric approach with an existing R package FrailtyPack for frailty models.
The paper is organized as follows. First, the ICU data is briefly presented. The next section describes the statistical multistate frailty model for clustered data with estimation procedure. Then, the model is applied to the analysis of nosocomial infections (ventilator-associated pneumonia infections) in ICUs, with “alive discharge” and “death” in ICU as other endpoints. Results from a simulation study are reported. Finally, a concluding discussion is presented.
Methods
Motivating example
Data Source
We conducted a prospective observational study using data from the multi-center Outcomerea database between November 1996 to April 2007. The database contains data from 16 French ICUs, among which data on admission features and diagnosis, daily disease severity, iatrogenic events, nosocomial infections, and vital status. Every year, the data of a subsample of at least 50 patients per ICU were entered in the database; patients had to be older than 16 years and to have stayed in ICU for more than 24 hours. To define this random subsample, each participating ICU selected either consecutive admissions to specific ICU beds throughout the year or consecutive admissions to all ICU beds over a single month.
Data collection
Database quality measures were taken such as the continuous training of investigators in each ICU or regular data quality checks (see [7]). A one day coding course was organized annually with the study investigators and research monitors. In all ICUs, as previously reported [8, 9], VAP was suspected based on the development of persistent pulmonary infiltrates on chest radiographs combined with purulent tracheal secretions, and/or body temperature ≥ 38.5°C or ≤ 36.5°C, and/or peripheral blood leukocyte count ≥ 10 × 10^{9} / L(Giga/liter) or ≤ 4 × 10^{9} / L. The definite diagnosis of VAP required a positive culture result from a protected specimen brush (≥ 10^{3} cfu/ml), plugged telescopic catheter specimen (≥ 10^{3} cfu/ml), BAL fluid specimen (≥ 10^{4} cfu/ml), or quantitative endotracheal aspirate (≥ 10^{5} cfu/ml). Investigators systematically performed bacteriological sampling before changing antimicrobials.
Study population
We considered death in ICU and discharge to be absorbing state and VAP as a non absorbing state. Patients were included in the study if they had stayed in the ICU for at least 48 hours and had received mechanical ventilation (MV) within 48 hours after ICU admission. We obtained 2871 patients, corresponding to 37395 ICU days. The median MV duration was 7 days with inter quartile range (IQR = [4-13]).
The multi-state approach and estimation
Multi-state model
Intensity functions with a shared frailty term
where $\left(\right)close="">{\alpha}_{0,\mathrm{hk}}^{{\theta}_{\mathrm{hk}}}\left(t\right)$ is the baseline transition intensity specific to the transition hk (with specific parameter θ _{ hk } for parametric model) and the random ICU effects ${u}_{\mathrm{hk}}^{i}$ are also specific to the transition hk, independent and follow a Gamma distribution (${u}_{\mathrm{hk}}^{i}\sim \Gamma \left(\frac{1}{{\gamma}_{\mathrm{hk}}},\frac{1}{{\gamma}_{\mathrm{hk}}}\right)$, $E\left[{u}_{\mathrm{hk}}^{i}\right]=1$, $\mathrm{Var}\left[{u}_{\mathrm{hk}}^{i}\right]={\gamma}_{\mathrm{hk}}$). The variance γ _{ hk } of the ${u}_{\mathrm{hk}}^{i}$ represents the heterogeneity between ICU of the overall underlying baseline risk for the transition hk.
Remark
this definition is correct for the non-homogeneous Markov model. We get a similar definition for the semi-Markov model by replacing the current time t in the equation (4) by the time spent in the current state t − T _{1}(for the transition 12 and 13).
Intensity functions with a joint frailty term
In the model (4) we assume that the different times to transitions are independent. In some cases this assumption may be violated, for instance in our motivating example, the transition times to death with the VAP and the transition times to discharge with VAP may be dependent. This dependency should be accounted for in the joint modelling of these two survival endpoints. There can be many reasons to use joint models of two survival endpoints, including giving a general description of the data, correcting for bias in survival analysis due to dependent dropout or censoring, and improving efficiency of survival analysis due to the use of auxiliary information [11].
The random effects ω _{ j } (frailties) are assumed independent. Mainly for reasons of mathematical convenience, the frailty terms are often assumed to follow a gamma distribution. The gamma frailty density is adopted here with unit mean and variance η. This choice and other possibilities such as log-normal, positive stable distributions are discussed in several papers [12, 13]. The common frailty parameter ω _{ j } will take into account the heterogeneity of the data, associated with unobserved covariates.
In the traditional model, the assumption is that ζ = 0 in (5), that is α _{ hk }(t) does not depend on ${\alpha}_{{\mathrm{hk}}^{\prime}}\left(t\right)$, and thus the two intensity functions α _{ hk }(t) and ${\alpha}_{{\mathrm{hk}}^{\prime}}\left(t\right)$ are not associated, conditional on covariates.When ζ = 1, the effect of the frailty is identical for the two transition times. When ζ > 0, the two transition times are positively associated; higher frailty will result for instance in a higher risk of discharge and a higher risk of death; while ζ < 0 implies a negative association. This means that unobserved individual factors produce higher transition rates to death and smaller transition rates to discharge, or inversely. We can think that for sicker patients, the mortality will be high but with a low discharge rate, conversely for healthy patients, the discharge rates will be high with a low death rate. The interpretation of the value of ζ only makes sense in case of heterogeneity, i.e. when the variance of the random effects is significantly different from zero. However, in this model we assume that there is no intra-cluster correlation anymore after having taken into account prognostic factors and after adjusting for a subject specific random effect term.
Estimation procedure
First, in our study, we consider that the process (X(t))_{ t ≥ 0} is observed in continuous time, i.e., we know at each time t the state of the process for each subjects. Secondly, for the model with shared frailties terms, we specify that each transition intensity has its own set of parameters. In other words, the parameter θ _{ hk }, β _{ hk } and γ _{ hk } are specific to the transition hk. Such an assumption is common when dealing with multi-state models and [14] has shown that this assumption allows us to consider the problem of estimating (parametrically or semi-parametrically) the transition intensities separately (that is transition by transition). Thus, each transition intensity can be evaluated by estimation methods used in survival analysis in the presence of right-censored data only (for instance when semi-Markov models are used) or in the presence of right-censored and left-truncated data (for instance when non-homogeneous Markov models are used, the transition intensities 12 and 13 are evaluated using delayed entry).
where is l(β _{ hk }, γ _{ hk }, α _{0,hk }(·)) the full log likelihood, ${\alpha}_{0,\mathrm{hk}}^{\mathrm{\prime \prime}}\left(t\right)$ is the second derivate of the baseline intensity function, and κ _{ hk } is a positive smoothing parameter that controls the trade-off between the data fit and the smoothness of the functions. Maximization of (6) defines the maximum penalized likelihood estimators (MPnLEs). The estimator (MPnLEs) cannot be calculated explicitly but can be approximated on the basis of splines and the smoothing parameters κ _{ hk } can be chosen by maximizing a likelihood cross-validation criterion as in Joly et al. [18]. The maximum penalized likelihood method is also implemented in the FrailtyPack R package [16].
Remark
In the presence of intensity functions with a joint frailty term in the model, a maximum penalized likelihood estimation is also used [11] and implemented in the FrailtyPack R package.
Model choice
where ${\mathcal{\mathcal{L}}}_{{\mathcal{O}}_{i}}^{{P}_{1}^{\widehat{\zeta}-i}}$ represents the likelihood of the observation ${\mathcal{O}}_{i}$ based on the estimator ${P}_{1}^{\widehat{\zeta}-i}$ defined without this observation.
Finally, to choose between the two estimators ${P}_{1}^{\widehat{\zeta}}$ and ${P}_{2}^{\widehat{\zeta}}$, we compute some differences of Expected Kullbak-Leibler risk (based on difference of LCV criterion). Commenges et al. [6] give an interpretation of the magnitude of these risks: values of 10^{−1}, 10^{−2}, 10^{−3}, 10^{−4} are respectively qualified “large”, “moderate”, “small” and “negligible”. The FrailtyPack R package provides the LCV criterion for survival analysis (for us a transition hk). As we can estimate each transition separately, it is straightforward to get the LCV criterion for a particular Multi-state model. Concerning the selection of the covariates in each model, we use a manual selection procedure motivated by epidemiological/clinical knowledge and also based on statistical significance of hazard ratios.
Prediction
The estimated posterior probabilities, ${\widehat{\Pi}}_{0k}^{i}({t}^{\ast},{t}^{\ast}+h|{U}_{i},{Z}_{i})\phantom{\rule{2.77695pt}{0ex}}i=1,\dots ,N$ can be obtained by substituting the maximum penalized likelihood estimates of parameters β _{0k }, γ _{0k }, α _{0,0k }, the empirical Bayes estimates for ${u}_{0k}^{i}$ and the individual information for the covariates ${Z}_{0k}^{i}$ by equation (7).
Results
Application revisited
Number of patients from the OUTCOMEREA database present in each transition
Transition | No events | Events | (%) | Total |
---|---|---|---|---|
01 (VAP) | 2438 | 433 | 0.15 | 2871 |
02 (death without VAP) | 2401 | 470 | 0.16 | 2871 |
03 (discharge without VAP) | 903 | 1968 | 0.69 | 2871 |
12 (death with VAP) | 314 | 119 | 0.27 | 433 |
13 (discharge with VAP) | 119 | 314 | 0.73 | 433 |
Result of the semi-markov model with penalized likelihood estimation incorporating frailty terms: HR (exp( β )) estimates and corresponding confidences intervals for the different transition intensities
Three independent frailties for transitions 01, 02 and 03 | 01 (VAP) | 02 (death) | 03 (discharge) | |||
---|---|---|---|---|---|---|
exp(β) | 95%CI | exp(β) | 95%CI | exp(β) | 95%CI | |
Sex (men=1) | 1.51 | (1.23-1.87) | - | - | 0.85 | (0.78-0.94) |
Age ≥ 62 | - | - | - | - | 0.95 | (0.86-1.05) |
33 < SAPSII ≤ 45 | - | - | 1.62 | (1.03-2.54) | 0.66 | (0.58-0.75) |
45 < SAPSII ≤ 58 | - | - | 2.70 | (1.75-4.14) | 0.56 | (0.49-0.64) |
58 < SAPSII | - | - | 4.83 | (3.18-7.35) | 0.40 | (0.34-0.47) |
Type of Admission : | ||||||
Elective surgery | 1 | - | - | 1 | ||
Emergency surgery | 0.58 | (0.41-0.83) | - | - | 1.04 | (0.90-1.20) |
Medicine | 0.89 | (0.66-1.20) | - | - | 0.96 | (0.82-1.12) |
Chronic diseases | - | - | 1.37 | (1.14-1.66) | 0.84 | (0.76-0.92) |
Diabetes | 1.48 | (1.10-2.00) | 1.30 | (0.97-1.76) | - | - |
Diagnostic or symptoms on ICU admission: | ||||||
ARDS | 1.71 | (1.16-2.54) | - | - | - | - |
Trauma | 2.52 | (1.12-5.67) | - | - | - | - |
Coma | 1.23 | (0.95-1.60) | 2.90 | (1.99-4.22) | 1.06 | (0.91-1.25) |
Shock | 1.21 | (0.96-1.53) | 2.12 | (1.47-3.06) | 0.73 | (0.63-0.84) |
Acute respiratory failure | - | - | 1.79 | (1.22-2.62) | 0.65 | (0.56-0.75) |
Use in the first 24 h of ICU admission : | ||||||
Antimicrobials | 0.61 | (0.50-0.75) | 0.66 | (0.54-0.81) | 0.86 | (0.77-0.95) |
Inotropes | - | - | - | - | 0.74 | (0.67-0.82) |
Enteral nutrition | 1.21 | (0.97-1.50) | 0.76 | (0.60-0.95) | 0.62 | (0.55-0.71) |
Parenteral nutrition | - | - | - | - | 1.00 | (0.87-1.14) |
Variance of the frailty γ (SE) | 0.19(0.11) | 0.09(0.06) | 0.15(0.07) | |||
Two independent frailties for transitions 12 and 13 | 12 (death with VAP) | 13 (discharge with VAP) | ||||
exp(β) | 95%CI | exp(β) | 95%CI | |||
Age ≥ 62 | - | - | 0.84 | (0.66-1.07) | ||
33 < SAPSII ≤ 45 | 2.13 | (1.05-4.31) | - | - | ||
45 < SAPSII ≤ 58 | 2.60 | (1.27-5.31) | - | - | ||
58 < SAPSII | 4.81 | (2.38-9.72) | - | - | ||
Parenteral nutrition | - | 0.70 | (0.51-0.97) | |||
Variance of the frailty γ (SE) | 0.04(0.06) | 0.11(0.08) | ||||
A joint subject-specific frailtyterm for transitions 12 and 13 | 12 (death with VAP) | 13 (discharge with VAP) | ||||
exp( β ) | 95%CI | exp( β ) | 95%CI | |||
Age ≥ 62 | - | - | 0.78 | (0.61-0.98) | ||
33 < SAPSII ≤ 45 | 2.22 | (0.99-4.97) | - | - | ||
45 < SAPSII ≤ 58 | 2.65 | (1.17-6.01) | - | - | ||
58 < SAPSII | 5.19 | (2.31-11.65) | - | - | ||
Parenteral nutrition | - | 0.67 | (0.49-0.90) | |||
Common frailty variance η (SE) | 0.77 (0.13) | |||||
Power coefficient ζ (from η ^{ ζ }) | -0.16 (0.29) |
We also fitted a semi-Markov model without taking into account the intra-centre correlation (results not shown). Using this model some factors were wrongly significantly associated. For instance, the effects of coma (Relative Risk= 1.39 (95%CI 1.07-1.79)), shock (Relative Risk= 1.28 (95%CI 1.01-1.62)) and the presence of an enteral nutritional therapy (Relative Risk= 1.24 (95%CI 1.00-1.53)) were incorrectly observed as significantly associated with the risk of VAP using the multi-state model without frailty term.
We previously evaluated the heterogeneity between centres, but considering a different random effect for each transition of the multi-state model. We also fitted a joint frailty model for the two transitions 12 and 13, with a shared subject-specific random effect for the two transitions [11]. This approach allows us to simultaneously evaluate the prognostic effects of covariates on the two survival endpoints, discharge or death with a VAP. This joint frailty model accounts for the dependency between the two outcomes, and corrects for bias in the analysis due to dependency. Results are exposed at the bottom of Table 2. When comparing the joint model for transitions 12 and 13 to the reduced shared frailty models for the same transitions, covariates effects are similar, while the hazard ratio of SAPSII is slightly greater in the joint frailty model. This illustrates that ignoring the dependence between time to death and time to discharge may result in biases in the reduced shared frailty model compared to the joint model.
The variance of the joint random effect ($\widehat{\eta}=0.77$, one-sided Wald test = 0.77/0.13 = 5.92 > 1.64) is significantly different from 0 but the power coefficient ζ is not significantly different from 0 (two-sided Wald test = 0.16/0.29 = 0.55 < 1.96). This shows that times of deaths and discharge are correlated, but with a slightly non-signicant negative association ($\widehat{\zeta}=-0.16$).
A Simulation study
Simulation details
Simulation parameters of the semi-markov model
Transition | |||||
---|---|---|---|---|---|
01 | 02 | 03 | 12 | 13 | |
θ _{ hk }=(a _{ hk },b _{ hk }) | (1.3,15) | (1.3,35) | (1.25,15) | (1.3,45) | (1.25,41) |
${\beta}_{\mathrm{hk}}=({\beta}_{\mathrm{hk}}^{1},{\beta}_{\mathrm{hk}}^{2})$ | (0.8,1.0) | (0.6,1.2) | (1.3,0.3) | (0.7,1.1) | (0.6,1.2) |
Briefly, a simulation of a semi-Markov model consists of: (i) for each subject generate 3 times T _{01},T _{02} and T _{03} (representing respectively the occurrence of VAP, Death or Discharge) according to the intensities transitions defined in (4); (ii) if T _{01} ≠ min(T _{01},T _{02},T _{03}) then the subject dies at time T _{02} (if T _{02} = min(T _{01},T _{02},T _{03})) or transits into the state Discharge (if T _{03} = min(T _{01},T _{02},T _{03})) and then back to (i) for a new subject; (iii) if T _{01} = min(T _{01},T _{02},T _{03}) then the subject contracts VAP at time T _{01}and then generates 2 new times (T _{12},T _{13}) representing respectively the occurrence of Death or Discharge for a subject with VAP; (iv) if T _{12} = min(T _{12},T _{13}) then the subject die at time T _{01} + T _{12} else the subject transits in the state Discharge at time T _{01} + T _{13}; and then back to (i) for a new subject.
For simulation run, we estimated two parametric semi-markov models: 1) with specific frailty term in each transition and 2) without frailty term. For the two models, we computed the mean, the empirical standard errors (SEs), i.e. the SEs of estimates and the mean of the estimated SEs for ${\widehat{\beta}}_{\mathrm{hk}}=({\widehat{\beta}}_{\mathrm{hk}}^{1},{\widehat{\beta}}_{\mathrm{hk}}^{2})$, and ${\widehat{\gamma}}_{\mathrm{hk}}$.
Simulation results
Estimates and standard errors (SE) according to the number of clusters G and the number of patients per cluster ( n _{ i } ) for the parametric semi-Markov model integrating or not random effects (for M=500 simulated samples, γ = 0.15 and for simulation parameters explained in Table 3)
Mean | Mean | Mean | Empirical | Empirical | Empirical | Mean | Mean | Mean | |
---|---|---|---|---|---|---|---|---|---|
h→k | $\widehat{\mathbf{\gamma}}$ | ${\widehat{\mathbf{\beta}}}^{\mathbf{1}}$ | ${\widehat{\mathbf{\beta}}}^{\mathbf{2}}$ | SE($\widehat{\mathbf{\gamma}}$S) | SE(${\widehat{\mathbf{\beta}}}^{\mathbf{1}}$) | SE(${\widehat{\mathbf{\beta}}}^{\mathbf{2}}$) | SE($\widehat{\mathbf{\gamma}}$) | SE(${\widehat{\mathbf{\beta}}}^{\mathbf{1}}$) | SE(${\widehat{\mathbf{\beta}}}^{\mathbf{2}}$) |
γ = 0.15, G = 30n _{ i } = 75 | |||||||||
With frailties | |||||||||
01 | 0.139 | 0.798 | 1.002 | 0.042 | 0.064 | 0.081 | 0.042 | 0.062 | 0.082 |
02 | 0.134 | 0.599 | 1.204 | 0.052 | 0.089 | 0.093 | 0.049 | 0.089 | 0.091 |
03 | 0.133 | 1.300 | 0.291 | 0.055 | 0.100 | 0.103 | 0.052 | 0.098 | 0.096 |
12 | 0.141 | 0.697 | 1.094 | 0.055 | 0.100 | 0.094 | 0.053 | 0.096 | 0.099 |
13 | 0.138 | 0.597 | 1.200 | 0.049 | 0.083 | 0.098 | 0.047 | 0.080 | 0.092 |
Without frailties | |||||||||
01 | 0.763 | 0.955 | 0.065 | 0.083 | 0.061 | 0.038 | |||
02 | 0.587 | 1.175 | 0.090 | 0.095 | 0.088 | 0.055 | |||
03 | 1.267 | 0.290 | 0.099 | 0.105 | 0.097 | 0.058 | |||
12 | 0.669 | 1.055 | 0.098 | 0.093 | 0.095 | 0.065 | |||
13 | 0.570 | 1.143 | 0.083 | 0.101 | 0.078 | 0.053 | |||
γ = 0.15, G = 30n _{ i } = 150 | |||||||||
With frailties | |||||||||
01 | 0.141 | 0.800 | 1.003 | 0.041 | 0.042 | 0.077 | 0.039 | 0.044 | 0.077 |
02 | 0.139 | 0.595 | 1.198 | 0.044 | 0.067 | 0.084 | 0.043 | 0.063 | 0.083 |
03 | 0.137 | 1.300 | 0.299 | 0.046 | 0.073 | 0.087 | 0.044 | 0.069 | 0.085 |
12 | 0.137 | 0.699 | 1.100 | 0.046 | 0.067 | 0.095 | 0.044 | 0.067 | 0.085 |
13 | 0.139 | 0.600 | 1.203 | 0.042 | 0.054 | 0.087 | 0.041 | 0.056 | 0.082 |
Without frailties | |||||||||
01 | 0.766 | 0.961 | 0.043 | 0.078 | 0.043 | 0.027 | |||
02 | 0.584 | 1.168 | 0.068 | 0.086 | 0.063 | 0.039 | |||
03 | 1.267 | 0.296 | 0.072 | 0.094 | 0.068 | 0.041 | |||
12 | 0.673 | 1.064 | 0.069 | 0.097 | 0.066 | 0.045 | |||
13 | 0.572 | 1.146 | 0.055 | 0.091 | 0.055 | 0.037 | |||
γ = 0.15, G = 100n _{ i } = 75 | |||||||||
With frailties | |||||||||
01 | 0.147 | 0.801 | 1.001 | 0.023 | 0.035 | 0.044 | 0.024 | 0.034 | 0.044 |
02 | 0.149 | 0.598 | 1.201 | 0.026 | 0.051 | 0.052 | 0.029 | 0.049 | 0.050 |
03 | 0.146 | 1.299 | 0.293 | 0.031 | 0.052 | 0.054 | 0.031 | 0.053 | 0.052 |
12 | 0.146 | 0.697 | 1.099 | 0.031 | 0.054 | 0.056 | 0.030 | 0.053 | 0.053 |
13 | 0.147 | 0.602 | 1.202 | 0.029 | 0.042 | 0.050 | 0.027 | 0.044 | 0.049 |
Without frailties | |||||||||
01 | 0.765 | 0.953 | 0.036 | 0.044 | 0.034 | 0.020 | |||
02 | 0.585 | 1.168 | 0.051 | 0.052 | 0.048 | 0.029 | |||
03 | 1.262 | 0.291 | 0.054 | 0.056 | 0.053 | 0.030 | |||
12 | 0.670 | 1.059 | 0.053 | 0.055 | 0.052 | 0.034 | |||
13 | 0.573 | 1.139 | 0.044 | 0.052 | 0.043 | 0.028 | |||
γ=0.15, G=100n _{ i }=150 | |||||||||
With frailties | |||||||||
01 | 0.148 | 0.800 | 0.997 | 0.022 | 0.024 | 0.041 | 0.022 | 0.024 | 0.042 |
02 | 0.147 | 0.601 | 1.200 | 0.025 | 0.036 | 0.047 | 0.024 | 0.034 | 0.045 |
03 | 0.146 | 1.302 | 0.297 | 0.025 | 0.039 | 0.045 | 0.026 | 0.038 | 0.046 |
12 | 0.148 | 0.702 | 1.103 | 0.025 | 0.037 | 0.049 | 0.025 | 0.037 | 0.047 |
13 | 0.147 | 0.602 | 1.204 | 0.024 | 0.031 | 0.045 | 0.024 | 0.031 | 0.045 |
Without frailties | |||||||||
01 | 0.763 | 0.949 | 0.025 | 0.041 | 0.024 | 0.014 | |||
02 | 0.588 | 1.168 | 0.036 | 0.049 | 0.034 | 0.021 | |||
03 | 1.265 | 0.295 | 0.039 | 0.051 | 0.037 | 0.021 | |||
12 | 0.674 | 1.063 | 0.037 | 0.049 | 0.036 | 0.024 | |||
13 | 0.573 | 1.139 | 0.032 | 0.048 | 0.030 | 0.020 |
Estimates and standard errors (SE) according to the number of clusters G and the number of patients per cluster ( n _{ i } ) for the parametric semi-Markov model integrating or not random effects (for M=500 simulated samples, γ =0.30 and for simulation parameters explained in Table 3 )
Mean | Mean | Mean | Empirical | Empirical | Empirical | Mean | Mean | Mean | |
---|---|---|---|---|---|---|---|---|---|
h→k | $\widehat{\mathbf{\gamma}}$ | ${\widehat{\mathbf{\beta}}}^{\mathbf{1}}$ | ${\widehat{\mathbf{\beta}}}^{\mathbf{2}}$ | SE($\widehat{\mathbf{\gamma}}$) | SE(${\widehat{\mathbf{\beta}}}^{\mathbf{1}}$) | SE(${\widehat{\mathbf{\beta}}}^{\mathbf{2}}$) | SE($\widehat{\mathbf{\gamma}}$) | SE(${\widehat{\mathbf{\beta}}}^{\mathbf{1}}$) | SE(${\widehat{\mathbf{\beta}}}^{\mathbf{2}}$) |
γ = 0.30, G = 30n _{ i } = 75 | |||||||||
With frailties | |||||||||
01 | 0.281 | 0.800 | 1.004 | 0.079 | 0.064 | 0.114 | 0.076 | 0.063 | 0.109 |
02 | 0.285 | 0.611 | 1.201 | 0.087 | 0.091 | 0.125 | 0.087 | 0.089 | 0.118 |
03 | 0.276 | 1.302 | 0.296 | 0.093 | 0.096 | 0.126 | 0.090 | 0.098 | 0.121 |
12 | 0.276 | 0.697 | 1.106 | 0.093 | 0.098 | 0.128 | 0.088 | 0.097 | 0.123 |
13 | 0.277 | 0.599 | 1.207 | 0.084 | 0.082 | 0.117 | 0.082 | 0.081 | 0.117 |
Without frailties | |||||||||
01 | 0.734 | 0.917 | 0.068 | 0.114 | 0.061 | 0.037 | |||
02 | 0.586 | 1.141 | 0.092 | 0.128 | 0.088 | 0.053 | |||
03 | 1.235 | 0.297 | 0.097 | 0.135 | 0.096 | 0.057 | |||
12 | 0.646 | 1.032 | 0.101 | 0.133 | 0.094 | 0.063 | |||
13 | 0.547 | 1.096 | 0.086 | 0.126 | 0.079 | 0.052 | |||
γ = 0.30, G = 30n _{ i } = 150 | |||||||||
With frailties | |||||||||
01 | 0.281 | 0.803 | 0.999 | 0.077 | 0.044 | 0.107 | 0.073 | 0.044 | 0.106 |
02 | 0.278 | 0.599 | 1.197 | 0.078 | 0.065 | 0.116 | 0.077 | 0.062 | 0.109 |
03 | 0.280 | 1.304 | 0.300 | 0.082 | 0.073 | 0.124 | 0.080 | 0.069 | 0.112 |
12 | 0.274 | 0.699 | 1.103 | 0.078 | 0.069 | 0.115 | 0.078 | 0.068 | 0.112 |
13 | 0.285 | 0.601 | 1.200 | 0.080 | 0.056 | 0.116 | 0.077 | 0.057 | 0.111 |
Without frailties | |||||||||
01 | 0.737 | 0.912 | 0.051 | 0.107 | 0.044 | 0.026 | |||
02 | 0.576 | 1.138 | 0.067 | 0.121 | 0.062 | 0.038 | |||
03 | 1.236 | 0.291 | 0.078 | 0.139 | 0.068 | 0.040 | |||
12 | 0.648 | 1.027 | 0.071 | 0.123 | 0.066 | 0.045 | |||
13 | 0.546 | 1.090 | 0.062 | 0.127 | 0.056 | 0.037 | |||
γ = 0.30, G = 100n _{ i } = x75 | |||||||||
With frailties | |||||||||
01 | 0.293 | 0.800 | 1.003 | 0.045 | 0.034 | 0.060 | 0.043 | 0.034 | 0.059 |
02 | 0.294 | 0.596 | 1.196 | 0.047 | 0.047 | 0.060 | 0.049 | 0.049 | 0.063 |
03 | 0.296 | 1.304 | 0.308 | 0.053 | 0.054 | 0.067 | 0.051 | 0.053 | 0.066 |
12 | 0.293 | 0.699 | 1.100 | 0.050 | 0.052 | 0.067 | 0.050 | 0.053 | 0.066 |
13 | 0.293 | 0.600 | 1.200 | 0.049 | 0.046 | 0.063 | 0.047 | 0.044 | 0.064 |
Without frailties | |||||||||
01 | 0.732 | 0.912 | 0.037 | 0.062 | 0.034 | 0.020 | |||
02 | 0.570 | 1.130 | 0.049 | 0.065 | 0.048 | 0.028 | |||
03 | 1.230 | 0.305 | 0.056 | 0.074 | 0.052 | 0.030 | |||
12 | 0.644 | 1.017 | 0.055 | 0.073 | 0.051 | 0.033 | |||
13 | 0.544 | 1.081 | 0.048 | 0.068 | 0.043 | 0.028 | |||
γ=0.30, G=100n _{ i }=150 | |||||||||
With frailties | |||||||||
01 | 0.291 | 0.799 | 1.004 | 0.041 | 0.023 | 0.057 | 0.041 | 0.024 | 0.057 |
02 | 0.292 | 0.600 | 1.201 | 0.043 | 0.036 | 0.063 | 0.044 | 0.034 | 0.060 |
03 | 0.292 | 1.301 | 0.302 | 0.046 | 0.037 | 0.065 | 0.045 | 0.037 | 0.061 |
12 | 0.294 | 0.697 | 1.099 | 0.044 | 0.037 | 0.063 | 0.045 | 0.037 | 0.062 |
13 | 0.294 | 0.601 | 1.202 | 0.043 | 0.030 | 0.058 | 0.043 | 0.031 | 0.060 |
Without frailties | |||||||||
01 | 0.733 | 0.913 | 0.028 | 0.059 | 0.024 | 0.014 | |||
02 | 0.574 | 1.135 | 0.037 | 0.068 | 0.034 | 0.020 | |||
03 | 1.230 | 0.298 | 0.039 | 0.071 | 0.037 | 0.021 | |||
12 | 0.642 | 1.017 | 0.038 | 0.068 | 0.036 | 0.024 | |||
13 | 0.543 | 1.079 | 0.032 | 0.067 | 0.031 | 0.020 |
Discussion
We have described a multistate model with frailty terms to account for heterogeneity between clusters on each transition. Such models appear promising in the setting of competing risk analyses using clustered data (i.e., multi-centre clinical trials, meta-analysis). Lack of software is a potential obstacle. We propose here a tractable model, semi-Markov as well as non-homogenous Markov, with semi-parametric or parametric estimates. The model can be readily derived with the R package FrailtyPack a simple and free approach, which does not require any time-consuming calculation. We also proposed a measure of models selection which evaluates the relative goodness of fit among a collection of models. To give an example, we provided a R code for simulating a data set and to analyze it (see Additional file 1).
Vital status and time of death or time of discharge in ICU are known exactly. However, there may be more complex schemes with interval-censored times to events, i.e., the event occurs in a known time interval L,R. The semi-Markov or non-homogenous Markov models discussed in this paper do not allow the direct treatment of these interval-censored data. It would be interesting to extend the multistate frailty model in the case of interval-censoring [22]. This would lead to numerical integration in the estimation process due to the lack of a closed-form solution of the multiple integrals, and this can be very time-consuming especially when the number of states rises. Also for large numbers of states, it is clear that substantial datasets are required for frailty variance estimation.
The proposed approach can also be used to predict probabilities of future events, given a patient’s history, covariates, and random effets, using parameter estimates and the estimates of corresponding baseline hazards and survival functions. Open research questions include prediction assessment with time-dependent prognostic factors. The aim would be to develop an updating mechanism which would allow dynamic updating of the predictions for a given patient in case of important changes in biomarkers.
A recent article discussed the identifiability and the (im)possibilities of frailties in multi-state models [23] but without considering covariates. They also compared predictive accuracy of different multi-state models with or without frailties using a k-fold cross validated partial log-likelihood [24]. They obtained that frailty models showed the best predictive performance in the comparison.
VAP represents an important and challenging example in which multistate frailty models should be used. This nosocomial infection is very frequent in ICU and is associated with an increase in ICU mortality, length of stay and cost. Many risk factors have been described in the past, and some new preventive interventions have been tested with often conflicting results [25]. This result could be due to the absence of control of appropriate confounding factors, to the heterogeneity of effects according to ICU subpopulations or to discrepancies in the diagnostic procedure. Indeed the diagnosis is based on the physician’s behaviour when clinical, biological and radiological signs of pneumonia occur. Even when diagnostic criteria are fixed in randomized control trials, the reported incidences vary from nine to 70%. Even if all fixed covariates are taken into account, and definitions carefully followed, the incidence densities still vary from 9.7 to 26.1 per 1000 mechanical-ventilation days [26]. The use of frailty terms in multistate models might then be important to unmask residual sources of variability (centre effect) when looking for risk factors of VAP. It may also be useful to compare incidences of VAP between hospitals if used as a quality indicator. It may also be important to explain the huge variability in the estimation of the attributable mortality of the disease.
Since the multi-state model under consideration contains cluster-specific random effects, the definition of the predictions is not straightforward. The proposed posterior prediction probabilities may be used to predict survival functions of subjects from existing clusters. In this cluster focus, the random effects ${u}_{\mathrm{hk}}^{i}$ are themselves of interest, and they are parameters to be estimated. In contrast, when the interest is on predicting survival for patients from new clusters, a marginal approach is better and corresponds to a population focus rather than a cluster focus. In this population focus, each conditional survival function is replaced by its marginal version. The marginal or observable survival function for the shared gamma frailty model (4) is $S\left(t\right|{Z}_{\mathrm{hk}}^{\mathrm{ij}})={E}_{u}[S\left(t\right|{Z}_{\mathrm{hk}}^{\mathrm{ij}},{u}_{\mathrm{hk}}^{i}\left)\right]=$ $\frac{1}{{(1+{\gamma}_{\mathrm{hk}}{\Lambda}_{\mathrm{hk}}^{\mathrm{ij}}\left(t\right))}^{1/{\gamma}_{\mathrm{hk}}}}$ with ${\Lambda}_{\mathrm{hk}}^{\mathrm{ij}}\left(t\right)$ the cumulative transition intensity function specific to the transition hk.
Conclusions
The use of multistate frailty models allows the simple analysis of very complex data. Such models could help improve the estimation of the impact of proposed prognostic features on each transition in a multi-centre study. We have suggested a method and software that gives accurate estimates and enables inference for any parameter or predictive quantity of interest.
Declarations
Acknowledgements
This work was supported by the ANR grant 2010 PRSP 006 01 for the MOBIDYQ project (Dynamical Biostatistical models). We would like to thank the members of the Outcomerea Study Group for sharing their database.
Authors’ Affiliations
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