Two-part model for ventilator-free days in a cluster randomized cross-over clinical trial

Abstract

Background

Ventilator-free days, which combine mortality and duration of mechanical ventilation into a single measure, are often considered as a primary endpoint in clinical trials involving critically ill patients. Despite the composite nature, ventilator-free days are commonly analyzed as continuous or count data with no distinction between a zero score from a patient who dies and a zero score from a patient who is alive but still on ventilator. In this study, we propose a two-part statistical model to compare the effects of two airway management strategies on mortality and duration of mechanical ventilation among patients with out-of-hospital cardiopulmonary arrest in a cluster randomized cross-over clinical trial.

Methods

In the proposed two-part model, failure to achieve return of spontaneous circulation (ROSC), death after ROSC, and survival are modeled in the first part; the number of ventilator-free days conditional on survival is modeled in the second part. To account for the cluster randomized cross-over design, each part also includes a random cluster effect that is assumed to be either shared or independent across the two parts. We conducted a simulation study to evaluate type I error rates and power of the two-part shared random cluster effect model and the mis-specified two-part model with independent random cluster effects in detecting an overall intervention effect.

Results

We found that parameter estimates were similar whether the random cluster effects were assumed to be shared or independent across the two parts whereas the shared random cluster effect approach showed higher log-likelihood, but lower Akaike information criterion (AIC) and Bayesian information criterion (BIC). Initial laryngeal tube insertion reduced odds of failing to achieve ROSC and marginally decreased odds of death after ROSC compared with initial endotracheal intubation in Part 1, whereas initial laryngeal tube insertion was not associated with duration of mechanical ventilation among patients alive in Part 2. The shared random cluster effect approach showed higher odds of death associated with lower odds of being ventilator-free. This confirms the expectation that a patient who is less likely to achieve ROSC and survive is more likely to require prolonged mechanical ventilation if the patient indeed survives during hospitalization. Our simulation studies found that the two-part model with a shared random cluster effect yielded type I error rates close to the nominal level. The two-part shared random cluster effect model has better power to detect an overall intervention effect when intervention effects are present in both parts rather than in only one of the two part.

Conclusions

The proposed two-part model provides a more comprehensive assessment of intervention effects on ventilator-free days in critical care trials. Researchers and clinicians can obtain greater insights with this approach about the direction and magnitude of the intervention effects on mortality, ROSC, and duration of mechanical ventilation.

Background

Ventilator-free days (VFDs) introduced by Bernard et al. (1994) [1] have been commonly used as the primary endpoint in clinical care trials. The number of VFDs is defined as the number of days alive and free of mechanical ventilation during the measurement period. In the case of 28-day period, a patient is given a value of 0 if the patient dies on or before day 28 or is still on mechanical ventilator on day 28 [2]. Hence, VFDs combine mortality and duration of mechanical ventilation into a single measure.

Despite the composite nature, VFDs are often reported as a continuous or count variable with no distinction between a zero score from a patient who dies and a zero score from a patient who is alive but still on ventilator. VFDs are then analyzed by a t-test, the Wilcoxon rank-sum test, Poisson regression, or negative binomial regression to assess treatment efficacy. Excessive zeros from patients who die are likely to violate the distributional assumption for the t-test. When there are a substantial number of ties due to excessive zeros, the normal approximation to the null probability distribution of the rank-sum becomes questionable. Excessive zeros causing over-dispersion also invalidate analysis based on the Poisson or negative binomial distribution. Interpreting treatment effects from any of these tests that do not account for excessive zeros from mortality can lead to misleading conclusions. Several trials have reported that VFDs can lead to false conclusions that an intervention reduces both mortality and duration on mechanical ventilator when in fact the intervention affect only ventilator duration rather than mortality [3, 4].

Although both outcomes are significant, mortality is generally considered more consequential than prolonged mechanical ventilation. Assigning equal weights for death and prolonged mechanical ventilation may not reflect patient, family, clinician, or social values and beliefs [5]. Novack et al. (2020) [6] proposed a hierarchical outcome through modification of VFDs by treating death as a worse outcome than prolonged ventilation. Following Finkelstein and Schoenfeld (1999) [7], each patient is compared to every other patient in the trial and assigned a score (win = +1; lose = −1; tie = 0) for each pairwise comparison. Scores from all pairwise comparisons are summed for each patient and compared between two trial arms using the Wilcoxon rank-sum test. However, statistically significant results from the win-lose-tie approach cannot inform investigators as to the magnitude, nature, and direction of the intervention effects on mortality and duration of mechanical ventilation. Novack and colleagues suggested reporting mortality and ventilator-free days among survivors separately by treatment group along with the probability that an individual randomly selected from the study population will have a superior outcome if assigned to a given treatment arm.

Yehya et al. (2019) [4] considered mortality and achieving unassisted breathing as competing events and applied the Fine-Gray sub-distribution hazard model to assess the relative effect of an intervention on time to achieve unassisted breathing, accounting for the occurrence of death as a competing risk. Competing risks arise in clinical research when there are a number of potential failure events and the occurrence of one event prevents the occurrence of other events. While competing risk regression allows researchers and clinicians to account for the competing nature of death, Schoenfeld (2006) [8] advocated that endpoints such as 28-day or 60-day hospital mortality should be compared mainly during hospitalization using binary analysis methods such as tests or logistic regression. He argued that survival methods such as Cox proportional hazards regression or competing risk regression are not appropriate because prolonged survival in patients who die during their hospitalization does not benefit the patients and, therefore, should not be measured in trials involving critically ill patients.

Verghis et al. (2023) [2] considered VFDs as zero-inflated count data and suggested modeling them using hurdle (or two-part) models [9] or zero-inflated models [10]. In a hurdle model, zeros from patients who die or patients alive but still on mechanical ventilator and non-zeros from patients alive and free of mechanical ventilation are analyzed separately. Logistic regression can be used to model the probability of having non-zero ventilator-free days in the first part and conditional on being non-zero, a truncated Poisson or negative binomial regression can be used to model the number of ventilator-free days among survivors achieving unassisted breathing in the second part. In a zero-inflated model, zeros are modeled as a mixture of structural (or true) zeros from patients who die and random (or sampling) zeros from patients alive but still on mechanical ventilator under the assumption that structural zeros are latent and not directly observable. Logistic regression can be applied to estimate the probability that an observed zero is a structural zero while a Poisson or negative binomial regression can be applied to model the number of ventilator-free days among patients not classified as structural zeros. Nevertheless, in the context of ventilator-free days, structural zeros due to death are observed rather than latent as its timing is known. Failing to explicitly model the source of these zeros either by treating zeros due to death as latent (as in zero-inflated models) or by assuming that all zeros, whether structural or random, come from a single generating process (as in hurdle models) can obscure important distinctions in treatment effects and lead to biased or misleading conclusions.

As VFDs combine mortality and duration of mechanical ventilation into a single measure, we propose a two-part model in which mortality is modeled in the first part and conditional on survival, the number of ventilator-free days is modeled in the second part. This approach aligns more closely with clinical interpretation of VFDs and avoids conflating deaths with poor but surviving outcomes from patients who survive but remian on a ventilator for the entire period. An indicator of death or survival is incorporated into the outcome variable VFDs to facilitate the proposed two-part modeling. Unlike the hurdle or zero-inflated model, the proposed two-part model not only provides a more comprehensive assessment of intervention effects on mortality and duration of mechanical ventilation but also helps investigators determine the magnitude and direction of intervention effects on mortality and duration of mechanical ventilation. To account for the cluster randomized cross-over design of the clinical trial that motivated this paper, each part of the two-part model also includes a random cluster effect that is assumed to be either shared, to account for a common latent cluster-specific quantity or trait that affects both the likelihood of mortality and duration of mechanical ventilation, or independent across the two parts. We conduct a simulation study to evaluate type I error rates and power of the different methods to detect an overall intervention effect. We illustrate the two-part model using data from the Pragmatic Airway Resuscitation Trial (PART) [11], a cluster randomized cross-over clinical trial to compare two airway management strategies for patients with out-of-hospital cardiac arrest.

PART trial

The PART trial was designed to compare the effectiveness of initial endotracheal intubation and laryngeal tube insertion on 72-hour survival in adults with out-of-hospital cardiopulmonary arrest (OHCA) [12]. More than 350,000 cardiac arrests occur outside of the hospital each year in the United States according to the American Heart Association. Endotracheal intubation is commonly performed by emergency medical services (EMS) professionals in which an endotracheal tube is inserted into the trachea to create an airway during cardiopulmonary resuscitation. Alternatively a laryngeal tube can be inserted into the pharynx and above the vocal cords. The PART trial included 27 EMS agencies from the Birmingham (Alabama), Dallas-Fort Worth (Texas), Milwaukee (Wisconsin), Pittsburgh (Pennsylvania), and Portland (Oregon) communities of the Resuscitation Outcomes Consortium [11]. These 27 EMS agencies were grouped into 13 clusters. Patients with OHCA were assigned to either initial endotracheal intubation (ETI) or initial laryngeal tube (LT) insertion based on a cluster randomized cross-over design in which each EMS agency alternated between interventions at 3-5-month intervals. There were 2 time periods in 2 clusters, 3 time periods in 3 clusters, 4 time periods in 6 clusters, and 5 time periods in 2 clusters. Patients with OHCA in each cluster within the same period received the same intervention, and randomization determined the order of interventions within a cluster. Further details on the PART trial are given in Wang et al. (2018) [12] and Wang et al. (2022) [11].

In this secondary analysis of the PART trial, the ventilator-free day outcome was defined as the number of days during the first 30 rather than 28 days of hospitalization that the patient was alive and free of mechanical ventilation. A patient with OHCA was given a value of 0 if the patient died without achieving return of spontaneous circulation (ROSC), died after achieving ROSC on or before day 30, or was still on mechanical ventilator on day 30. ROSC was defined as the presence of palpable pulses on emergency department arrival. Patients discharged alive from the hospital prior to 30 days were assumed alive and not on mechanical ventilator on day 30.

Of 3,004 patients with OHCA enrolled from December 1, 2015, to November 4, 2017, 1505 (51.1%) patients were randomized to initial LT and 1499 patients were randomized to initial ETI. The median age (interquartile range) was 65 (53-76) years and 1829 (60.9%) patients were men. For details on characteristics of these 3,004 patients by interventions, see Table 1 in Wang et al. (2018) [12]. Ventilator-free days were available for 2,947 (98.1%) patients in this secondary analysis. Of the 2,947 patients, 1831 (62.1%) patients [947 (64.8%) initial ETI and 884 (59.5%) initial LT] died without achieving ROSC, 884 (30.0%) patients [423 (28.9%) initial ETI and 461 (31.0%) initial LT] died after achieving ROSC on or before day 30, and 232 (7.9%) patients [92 (6.3%) initial ETI and 140 (9.5%) initial LT] were alive on day 30.

Table 1.

Two-part shared random cluster effect model for ventilator-free days in 2947 patients from the PART trial

Parameter	Estimate	Standard error	P-value
Part 1: multinomial logit model for odds of death without ROSC
Intercept	2.62	0.24	<0.001
Period 2	−0.25	0.20	0.244
Period 3	−0.02	0.21	0.939
Period 4	−0.50	0.28	0.096
Period 5	−0.37	0.81	0.655
Initial LT	−0.54	0.16	0.006
Part 1: multinomial logit model for odds of death after ROSC
Intercept	1.82	0.25	<0.001
Period 2	−0.35	0.21	0.131
Period 3	−0.04	0.22	0.844
Period 4	−0.25	0.29	0.396
Period 5	−0.01	0.82	0.995
Initial LT	−0.36	0.17	0.054
Part 2: beta-binomial model for the number of ventilator-free days in survivors
Intercept	1.60	0.23	<0.001
Period 2	−0.24	0.19	0.230
Period 3	−0.08	0.20	0.694
Period 4	0.22	0.29	0.453
Period 5	0.05	0.78	0.954
Initial LT	−0.05	0.15	0.765
	−0.87	0.23	0.003
	0.17	0.02
	0.46	014
log-likelihood	−3077.3
AIC	6196.5
BIC	6489.6

Of the 232 patients alive, 4 (1.7%) patients (4 initial ETI and 0 initial LT) were still on mechanical ventilator on day 30 whereas 189 (81.5%) patients [75 (39.7%) initial ETI and 114 (60.3%) initial LT] were discharged after 7 or less days of hospitalization. The mean ± SD number of ventilator-free days among survivors was 24.8 ± 6.7 in 92 patients initial ETI and 25.6± 4.6 in 140 patients initial LT. The histogram of the ventilator-free days in 232 patients alive is displayed in Fig. 1. From inspection of the figure, it appears that the distribution of the number of days alive and free of mechanical ventilation is towards the right side, leaving the left side heavily tailed. The binomial, Poisson, or negative binomial distribution is not likely to fit the observed left-skewed distribution of ventilator-free days well. Unlike the binomial distribution, the beta-binomial distribution allows the probability of success to vary from individual to individual, following a beta distribution. With two shape parameters from the beta distribution, the beta-binomial is flexible in fitting distributions that are flat, left-skewed, right-skewed, bell-shaped, U-shaped, or bimodal. The beta-binomial distribution can be used to fit the number of ventilator-free days in 232 patients alive.

Fig. 1.

Histogram of number of ventilator-free days in 232 patients achieving return of spontaneous circulation and alive on day 30

Two-part model

Let denote the number of ventilator-free days during the measurement period and be a response variable that can take on one of three possible values: 1 if the patient fails to achieve ROSC, 2 if the patient dies after ROSC, or 3 if the patient is alive for patient within period within cluster . The probability function of is given by

Part 1: Let , , and that are modeled through multinomial logistic regression equations:

1

and

2

where indicates initial ETI or LT received by patient k within period j within cluster i, and , , , and are unknown parameters. Fixed period effects are denoted by and with and . Cluster-specific random effects are denoted by that is assumed to follow a normal distribution with mean 0 and variance . A larger value of implies that patients being treated in cluster i tend to have a higher probability of death without or after ROSC.

Part 2: Model the number of ventilator-free days among patients alive. Conditional on , is assumed to follow a binomial distribution, with the probability of ventilator-free, , assumed to follow a beta distribution, . Integrating with respect to , we find that follows a beta-binomial distribution with probability mass function given by

3

where , , and n is the length of the measurement period. For example, n is 30 in the PART trial.

The conditional mean and variance of are given respectively as

4

and

5

where is the mean of the probability from the beta distribution and is the dispersion parameter. We assume that the odds of a patient who fails to achieve ROSC or dies after ROSC (in Part 1) and number of days the patient is alive and free of mechanical ventilation (in Part 2) are correlated through a common random cluster effect . A logit link function is used to model as follows:

6

where indicates initial ETI or LT received by patient k within period j within cluster i; , , and are unknown parameters; and are fixed period effects with . is a real-valued association parameter that describes the dependency between Part 1 and Part 2, suggesting a common latent cluster-specific quantity attributing to a patient’s probabilities of both death and ventilator-free. If , patients in a cluster with a higher latent cluster-specific quantity are more likely to be free of mechanical ventilation whereas if , patients in a cluster with a higher latent cluster-specific quantity are less likely to be free of mechanical ventilation. If , Part 1 and Part 2 are independent and can be modeled separately with each part having its own cluster-specific random effect. Note that the two-part model with a shared random cluster effect is a simplified version of the two-part model with correlated random cluster effects.

Let be 1 if and 0 otherwise, and be 1 if and 0 otherwise. Under the conditional independence assumption, the marginal likelihood for the kth patient within period j within cluster i is

7

where , and are given in (1) and (2), is given in (3), and is the density function for . The likelihood to be maximized is

8

The likelihood in (8) involves integrating a nonlinear function over the shared random effect given in (7). Quasi-Newton methods along with adaptive Gauss-Hermite quadrature to approximate the integral can be applied to find the maximum likelihood estimator for . This approach can be implemented in statistical software packages such as R, SAS or Stata.

On a final note, there are three effects associated with the intervention in the two-part model. One is the intervention effect on odds of death without ROSC measured by in (1), another one is the intervention effect on odds of death after ROSC measured by in (2), and the last one is the intervention effect on odds of being ventilator-free measured by in (6). The likelihood ratio test or the Wald test can be used to determine whether there is an overall intervention effect through testing the null hypothesis of .

Results and simulations

To demonstrate its usefulness, we applied the proposed two-part model with a shared random cluster effect in (7) and (8) to fit the number of ventilator-free days in 2947 patients with OHCA from the PART trial. The parameter estimates and their corresponding standard errors are given in Table 1. The likelihood ratio test indicated that there was an overall intervention effect (). Initial LT reduced odds of death without ROSC () and was marginally associated with decreased odds of death after ROSC () compared with initial ETI in Part 1 whereas initial LT was not associated with duration of mechanical ventilation () in 232 patients alive in Part 2. Time periods were not associated with odds of death without or after ROSC in Part 1 or duration of mechanical ventilation in Part 2. The estimated shared random cluster effect standard deviation was with a standard error (SE) of 0.14. The estimated dependency parameter between Part 1 and Part 2, , was negative and significantly different from zero (), suggesting lower latent cluster-specific quantities associated with lower odds of death without or after ROSC in Part 1 but higher odds of being ventilator-free leading to shorter duration of mechanical ventilation in Part 2. The log-likelihood was −3077.3 with the Akaike information criterion (AIC) of 6196.5 and the Bayesian information criterion (BIC) of 6489.6.

In practice, an independence assumption about random effects is often made for computational feasibility as the likelihood components for odds of death without or after ROSC and odds of ventilator-free become separable and maximization of the likelihood is computational much simpler and faster. The PART data were also fitted to a two-part model with independent cluster-specific random effects under the assumption that Part 1 and Part 2 are independent and modeled separately. The parameter estimates and their corresponding standard errors are given in Table 2. Comparisons of the parameter estimates given in Table 2 with those given in Table 1 indicate that there were no substantial changes in these parameter estimates when Part 1 and Part 2 were assumed independent and modeled separately. The estimated random cluster effect standard deviation was (SE = 0.12) in Part 1 and (SE = 0.15) in Part 2. Both standard deviation estimates were approximately equal to the estimated dependency between Part 1 and Part 2 times the estimated shared random cluster effect standard deviation, 0.87 0.46 = 0.4002, from the two-part model with a shared random cluster effect. The log-likelihood was −3082.5 with AIC= 6207.0 and BIC = 6500.1.

Table 2.

Two-part model with independent random cluster effects for ventilator-free days in 2947 patients from the PART trial

Parameter	Estimate	Standard error	P-value
Part 1: multinomial logit model for odds of death without ROSC
Intercept	2.56	0.23	<0.001
Period 2	−0.23	0.20	0.286
Period 3	−0.02	0.21	0.918
Period 4	−0.52	0.28	0.083
Period 5	−0.21	0.81	0.797
Initial LT	−0.52	0.16	0.006
Part 1: multinomial logit model for odds of death after ROSC
Intercept	1.75	0.24	<0.001
Period 2	−0.32	0.21	0.156
Period 3	−0.05	0.22	0.825
Period 4	−0.28	0.29	0.355
Period 5	0.15	0.82	0.854
Initial LT	−0.35	0.17	0.060
Part 2: beta-binomial model for the number of ventilator-free days in survivors
Intercept	1.59	0.24	<0.001
Period 2	−0.21	0.20	0.325
Period 3	−0.04	0.20	0.862
Period 4	0.25	0.30	0.408
Period 5	−0.01	0.83	0.990
Initial LT	−0.03	0.16	0.852
	0.17	0.02
	0.40	0.12
	0.41	0.15
log-likelihood	−3082.5
AIC	6207.0
BIC	6500.1

A simulation study was conducted to investigate the performance of the two-part model with a shared random cluster effect in (7) and (8) on hypothesis testing in terms of type I error rates and power. Four cases were considered in detecting whether there is an overall intervention effect through testing the null hypothesis of : (1) No intervention effect in both Part 1 and Part 2 when ; (2) Intervention effect on death without or after ROSC in Part 1 only when , , and ; (3) Intervention effect on duration of mechanical ventilation in Part 2 only when and ; (4) Intervention effects in both Part 1 and Part 2 when , , and . In all four cases, we simulated data from the two-part model with a shared random cluster effect under a two-period, two-intervention, cluster randomized, cross-over design with , , and 50, 100, or 150. In each case, three different sample sizes, namely , 2000, or 3000 were considered, where . For each sample size, 1000 samples were generated from the two-part shared random cluster effect model with , , , , , , , , and .

Under each case, the two-part shared random cluster effect model was fitted to each of 1000 samples for each sample size. Table 3 displays simulated significance levels for rejecting the null hypothesis at based on 1000 samples drawn under the first case of no intervention effect in both Part 1 and Part 2 for each sample size. The results indicate that the probability of rejecting the null hypothesis given that it is true is close to the nominal level. Table 4 displays the observed rate of correct rejections of the null hypothesis at based on 1000 samples drawn under the second, third, or fourth case. As expected, power increases as the sample size increases in all three cases. The two-part shared random cluster effect model has better power to detect an overall intervention effect when intervention effects are present in both parts compared to intervention effects are present in only one part (either Part 1 or Part 2).

Table 3.

Simulated significance levels for rejecting the null hypothesis of based on 1000 replications fitted to the two-part shared random cluster effect model for each sample size when the null hypothesis is true (Case 1)

Nominal level	Sample size
	1000	2000	3000
0.05	0.055	0.056	0.062

Table 4.

Simulated power, in percentages, for rejecting the null hypothesis of based on 1000 replications fitted to the two-part shared random cluster effect model for each sample size when the null hypothesis is not true

Case	Sample size
	1000	2000	3000
Case 2: , , and	47.1	78.3	93.5
Case 3: and	27.4	49.7	72.8
Case 4: , , and	70.4	96.0	99.5

We also investigated type I error rates and power of the mis-specified two-part model with independent random cluster effects in testing the null hypothesis of when data were actually drawn from the two-part shared random cluster effect model under the four cases described above. Under each case, the mis-specified two-part model with independent random cluster effects was fitted to each of 1000 samples for each sample size. Table 5 displays simulated significance levels for rejecting the null hypothesis at based on 1000 samples drawn under the first case of no intervention effect in both Part 1 and Part 2 for each sample size. The results indicate that, under the null hypothesis, the mis-specified two-part model has a similar probability of rejecting the null as the two-part shared random cluster effect model, as shown in Table 3 and illustrated in Fig. 2, Case 1. Table 6 displays the observed rate of correct rejections of the null hypothesis at based on 1000 samples drawn under the second, third, or fourth case. Comparisons of simulated power in Tables 4 and 6 indicate no differences when Parts 1 and 2 are mis-specified as independent and modeled separately, as illustrated in Fig. 2, Cases 2–4.

Table 5.

Simulated significance levels for rejecting the null hypothesis of based on 1000 replications fitted to the mis-specified two-part model with independent random cluster effects for each sample size when the null hypothesis is true (Case 1)

Nominal level	Sample size
	1000	2000	3000
0.05	0.057	0.055	0.061

Fig. 2.

Comparison of simulated type I error rates and power between the two-part shared and mis-specified independent random cluster effect model across four scenarios

Table 6.

Simulated power, in percentages, for rejecting the null hypothesis of based on 1000 replications fitted to the mis-specified two-part model with independent random cluster effects for each sample size when the null hypothesis is not true

Case	Sample size
	1000	2000	3000
Case 2: , , and	47.0	78.3	93.4
Case 3: and	25.8	47.2	71.9
Case 4: , , and	69.8	95.9	99.5

Discussion

We have demonstrated the usefulness of the two-part model to determine intervention effects on the composite outcome measure, ventilator-free days. A two-part shared random cluster effect model and a two-part model with independent random cluster effects were fitted to data from the PART trial. We found that there were no substantial differences in parameter estimates in both models whereas the two-part shared random cluster effect model had higher log-likelihood, but lower AIC and BIC compared to the two-part model with independent random cluster effects. Initial LT reduced odds of death without ROSC and was marginally associated with decreased odds of death after ROSC compared with initial ETI in Part 1 whereas initial LT was not associated with duration of mechanical ventilation among patients alive in Part 2. From the two-part shared random cluster effect, the estimated dependency parameter was negative and significantly different from zero, suggesting that Part 1 and Part 2 were dependent, and the independence assumption on random cluster effects was not realistic. The negative dependency suggests that higher odds of death without or after ROSC was associated with lower odds of ventilator-free. This confirms the expectation that a patient who is less likely to achieve ROSC and survive is more likely to require prolonged mechanical ventilation if the patient indeed survives during hospitalization. While the association that sicker patients often require extended respiratory support may seem intuitive, it holds important clinical implications. Quantifying this relationship enables earlier identification of patient clusters characterized by both high mortality risk and prolonged ventilation needs. Recognizing such clusters early may guide proactive interventions in patients with out-of-hospital cardiac arrest (OHCA), including timely intensive care unit (ICU) transfers, early consideration of advanced therapies such as extracorporeal membrane oxygenation (ECMO), and intensified monitoring for complications such as ventilator-associated pneumonia, delirium, or ventilator-induced lung injury. Furthermore, this insight may support the timely initiation of prognostic discussions with patients and families to ensure that care aligns with individual goals and values.

The two-part shared random cluster effect model can be extended to allow an additional random cluster effect in Part 2 to account for extra-heterogeneity in odds of ventilator-free due to some cluster-specific (or EMS agency-specific) characteristics not shared with odds of death without or after ROSC in Part 1. Nonetheless, the PART data did not support including an additional random cluster effect in Part 2. Other types of dependency between odds of death without or after ROSC and odds of ventilator-free can be assumed. For example, we can consider three correlated cluster-specific random effects, one for odds of death without ROSC, one for odds of death after ROSC, and one for odds of ventilator-free under the assumption that the correlation between odds of death without or after ROSC and odds of ventilator-free is captured by the correlation among three normally distributed random cluster effects. When fitting a two-part model with three correlated random cluster effects to the PART data, we encountered convergence problems in the search for the maximum likelihood estimates for intervention effects, variances, and covariances due to high dimensional integration and sparse data in Part 2. Similar convergence problems were encountered when the data were fitted to a two-part model with two correlated random cluster effects, one for odds of death without or after ROSC in Part 1 and one for odds of ventilator-free in Part 2. To reduce integration dimensions and improve model convergence, periods have been included as fixed effects rather than random effects in our two-part modeling. We have assumed that the dispersion parameter is homogeneous and not associated with covariates of interest in Part 2. If investigators suspect the over-dispersion in number of ventilator-free days varying with different covariate combinations, an additional logit link function can be used to allow for heterogeneity of the dispersion parameter.

Our simulations found that the two-part shared random cluster effect model and mis-specified two-part model with independent random cluster effects have similar type I error rates and power for detecting an overall intervention effect under our parameter settings. The results seem to suggest that the two-part model with independent random cluster effects can be used with variance and covariance components including the dependency parameter being treated as nuisance parameters if the primary objectives of inference are the regression coefficients for intervention effects in Part 1 and Part 2. When researchers and clinicians are also interested in investigating whether the correlation structure between two parts is similar across groups or clusters, a two-part model with shared random effects should be used if they suspect that the two-parts may share some common random effects across these groups; otherwise, a two-part model with correlated random effects should be used for more efficient estimation of intervention effects. The two-part shared random effect model is a simplified version of the two-part model with correlated random effects, which can take several hours to fit, for example, using the SAS NLMIXED procedure, to reduce computational burden and improve model convergence. On the other hand, an independence assumption on random effects can be made for convenience and computational feasibility. Nevertheless, dependency between two parts cannot be assessed when Part 1 and Part 2 are modeled separately. Failing to account for dependency of Part 1 and Part 2 may lead to biased estimates or incorrect inferences. Su et al (2009) [13] showed that bias can be induced for regression coefficients and variance and covariance components when random effects are truly correlated but mis-specified as independent in a two-part mixed model.

In summary, choosing the correct correlation structure should be guided by the nature of data, research questions, and assumptions underlying models being used. We suggest fitting two-part models with shared random effects and two-part models with correlated random effects with initial values for the unknown parameters obtained from two-part models with independent random effects and then select the model with the best fit through hypothesis testing and model comparisons unless there is prior knowledge for choosing a certain model. As a final note, although the proposed two-part model was originally developed to assess treatment effects in a cluster randomized cross-over clinical trial, it is also applicable to traditional trials where the unit of randomization is the individual rather than the cluster. In trials without clustering, the model can be simplified by removing the cluster-specific random effects in both parts. This simplified version still allows separate modeling of the probability of mortality or survival (part 1) and ventilator-free days among survivors (part 2).

Conclusions

The proposed two-part model provides a more comprehensive assessment of intervention effects on ventilator-free days in critical care trials. Researchers and clinicians can obtain greater insights with this approach about the direction and magnitude of the intervention effects on mortality, ROSC, and duration of mechanical ventilation.

Abbreviations

AIC

Akaike information criterion

BIC

Bayesian information criterion

ECMO

Extracorporeal membrane oxygenation

EMS

Emergency medical services

ETI

Endotracheal intubation

ICU

Intensive care unit

LT

Laryngeal tube

OHCA

Out-of-hospital cardiopulmonary arrest

PART

Pragmatic Airway Resuscitation Trial

ROSC

Return to spontaneous circulation

VFD

Ventilator-free day

Authors’ contributions

Y.L., A.M., M.N.G., H.E.W., and M.K. contributed to the conception and design of this work. Y.L. developed the model. X.X. analyzed data and conducted simulations. Y.L. wrote the manuscript and M.K. edited it. All authors reviewed the final manuscript.

Funding

A.M. and M.K. are funded by the National Heart, Lung, and Blood Institute (R33HL162980) and the National Heart, Lung, and Blood Institute (UH2/UH3-HL125163).

M.K. and Y.L. are funded by the National Center for Advancing Translational Sciences Grant (UM1TR004400).

Data availability

Data are provided by Dr. Henry Wang at the Ohio State University Wexner Medical Center

Declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.