, Univ. Bordeaux, ISPED, centre INSERM U897EpidémiologieBiostatistique, Bordeaux, F33000, FRANCE
, INSERM, ISPED, centre INSERM U897EpidémiologieBiostatistique, Bordeaux, F33000, FRANCE
, UniversityGrenoble 1, institut Albert Bonniot U823 team 11: outcome of mechanically ventilated patients and respiratory cancers, Grenoble, 38041, France
, UniversityGrenoble 1, teaching hospital Albert Michallon, Medical intensive care unit, Grenoble, 38043, France
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 criticallyill 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 multistate model with random effects is needed to obtain valid outcome estimates.
Methods
We show how a simple multistate frailty model can be used to study semicompeting 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
Results
We illustrate the use of our approach though the analysis of nosocomial infections (ventilatorassociated 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 multistate 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 multicentre study. We suggest a method and software that give accurate estimates and enables inference for any parameter or predictive quantity of interest.
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 multistate 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.
In this paper, we show how a simple multistate frailty model can be used to study semicompeting risks
Two distinct approaches are often used in multistate models. They differ in the definition of time
As discussed in the section on the
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 (ventilatorassociated 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 multicenter 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
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 = [413]).
The multistate approach and estimation
Multistate model
We consider the multistate model represented in Figure
where
The “disability Model”
The “disability Model”.
From the definition of the transition intensities, we focus on two kinds of models. First, we consider the
depends on
Intensity functions with a shared frailty term
Take the case of a study with
where
Remark
this definition is correct for the
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
In this work we will thus also consider some transitions jointly, in a joint frailty model setting as follows:
The random effects
In the traditional model, the assumption is that
Estimation procedure
First, in our study, we consider that the process (
In this paper, we use two different approaches. First, we use a parametric model for the multistate model specifying each baseline transition intensity by a Weibull distribution (
where is
Remark
In the presence of intensity functions with a joint frailty term in the model, a maximum penalized likelihood estimation is also used
Model choice
Several models with different approaches have been defined: a
where
where
Finally, to choose between the two estimators
Prediction
A posterior probability of any event can be easily derived from the multistate model and its parameters. This probability, which can be computed for a new subject using a given set of covariates at the current time, and given a postinclusion events, constitutes a dynamic tool of prediction. For instance, the aim may be to predict if VAP occurs between time
with,
The estimated posterior probabilities,
Results
Application revisited
The methodology exposed in the previous section is now applied to the OUTCOMEREA database. Table
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
Table
Three independent frailties for transitions 01, 02 and 03
01 (VAP)
02 (death)
03 (discharge)
exp(
95%CI
exp(
95%CI
exp(
95%CI
−: the corresponding covariate has not been selected for this transition by the descendant strategy based on model without frailties. SAPS II = Simplified Acute Physiology Score, version II at admission (33, 45, and 58 separate the four quartiles); ARDS = Acute Respiratory Distress Syndrome; Acute Respiratory failure: indicate the need for respiratory supportive therapy; Coma: admission for a neurological disease and a Glasgow coma score of less than 9; Shock: admission in the ICU with sign or symptoms of shock according to common definitions.
Sex (men=1)
1.51
(1.231.87)


0.85
(0.780.94)




0.95
(0.861.05)
33 <


1.62
(1.032.54)
0.66
(0.580.75)
45 <


2.70
(1.754.14)
0.56
(0.490.64)
58 <


4.83
(3.187.35)
0.40
(0.340.47)
Type of Admission :
Elective surgery
1


1
Emergency surgery
0.58
(0.410.83)


1.04
(0.901.20)
Medicine
0.89
(0.661.20)


0.96
(0.821.12)
Chronic diseases


1.37
(1.141.66)
0.84
(0.760.92)
Diabetes
1.48
(1.102.00)
1.30
(0.971.76)


ARDS
1.71
(1.162.54)




Trauma
2.52
(1.125.67)




Coma
1.23
(0.951.60)
2.90
(1.994.22)
1.06
(0.911.25)
Shock
1.21
(0.961.53)
2.12
(1.473.06)
0.73
(0.630.84)
Acute respiratory failure


1.79
(1.222.62)
0.65
(0.560.75)
Antimicrobials
0.61
(0.500.75)
0.66
(0.540.81)
0.86
(0.770.95)
Inotropes




0.74
(0.670.82)
Enteral nutrition
1.21
(0.971.50)
0.76
(0.600.95)
0.62
(0.550.71)
Parenteral nutrition




1.00
(0.871.14)
Variance of the frailty
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(
95%CI
exp(
95%CI


0.84
(0.661.07)
33 <
2.13
(1.054.31)


45 <
2.60
(1.275.31)


58 <
4.81
(2.389.72)


Parenteral nutrition

0.70
(0.510.97)
Variance of the frailty
0.04(0.06)
0.11(0.08)
A joint subjectspecific frailtyterm for transitions 12 and 13
12 (death with VAP)
13 (discharge with VAP)
exp(
95%CI
exp(
95%CI


0.78
(0.610.98)
33 <
2.22
(0.994.97)


45 <
2.65
(1.176.01)


58 <
5.19
(2.3111.65)


Parenteral nutrition

0.67
(0.490.90)
Common frailty variance
0.77 (0.13)
Power coefficient
0.16 (0.29)
We also fitted a semiMarkov model without taking into account the intracentre 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 1.071.79)), shock (Relative Risk= 1.28 (95%CI 1.011.62)) and the presence of an enteral nutritional therapy (Relative Risk= 1.24 (95%CI 1.001.53)) were incorrectly observed as significantly associated with the risk of VAP using the multistate model without frailty term.
The variance of the frailty is significantly different from 0 for the transitions 01, 03 and not significantly different from 0 for the other transitions (Table
Random centrespecific effect as estimated by posterior distribution for the different transitions
Random centrespecific effect as estimated by posterior distribution for the different transitions. The clusters have been ordered (ascending order) by the number of subjects in the cluster.
We previously evaluated the heterogeneity between centres, but considering a different random effect for each transition of the multistate model. We also fitted a joint frailty model for the two transitions 12 and 13, with a shared subjectspecific random effect for the two transitions
The variance of the joint random effect (
To illustrate the use of posterior probabilities of VAP between times
Predicted probabilities (see formula (7)) of a patient in state 0 developing VAP between
Predicted probabilities (see formula (7)) of a patient in state 0 developing VAP between
A Simulation study
Simulation details
In this section, we present a short simulation study, which in particular reveals the importance of taking into account the heterogeneity of the population for estimating the different intensity transition. We have considered the semiMarkov multistate model (with four states) described in Figure
Transition
01
02
03
12
13
The parameter
(1.3,15)
(1.3,35)
(1.25,15)
(1.3,45)
(1.25,41)
(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 semiMarkov model consists of: (i) for each subject generate 3 times
For simulation run, we estimated two parametric semimarkov 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
Simulation results
The results of the simulation studies using parametric estimation are summarized in Table
Mean
Mean
Mean
Empirical
Empirical
Empirical
Mean
Mean
Mean
SE(
SE(
SE(
SE(
SE(
SE(
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
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
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
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
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
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
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
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
Mean
Mean
Mean
Empirical
Empirical
Empirical
Mean
Mean
Mean
SE(
SE(
SE(
SE(
SE(
SE(
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
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
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
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
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
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
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
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., multicentre clinical trials, metaanalysis). Lack of software is a potential obstacle. We propose here a tractable model, semiMarkov as well as nonhomogenous Markov, with semiparametric or parametric estimates. The model can be readily derived with the R package
We provide a R code for simulating a data set and to analyze it: file name “supplementarymaterialRcode.R”.
Click here for file
Vital status and time of death or time of discharge in ICU are known exactly. However, there may be more complex schemes with intervalcensored times to events, i.e., the event occurs in a known time interval
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 timedependent 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 multistate models
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
Since the multistate model under consideration contains clusterspecific 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
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 multicentre study. We have suggested a method and software that gives accurate estimates and enables inference for any parameter or predictive quantity of interest.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
BL and VR developed the methodology, performed the simulation and the analysis on the dataset as well as wrote the manuscript. JFT collected and interpreted the dataset as well as wrote the description and the interpretation on the Application section. All authors read and approved the final manuscript.
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.
Prepublication history
The prepublication history for this paper can be accessed here: