Supplemental Data 3
Generalized linear mixed-effects model with random intercept
For a proper analysis of the repeated measures data (i.e., several measurements along time during MV for each subject), we built a generalized linear mixed-effects model (GLMM) with random intercept.
Briefly, the idea is to explain variations in the desired response, Y (i.e. DC), by a set of variables X = {Xj} that represent the fixed effects of the model, taking into account the degree to which responses vary across subjects (the random effects). Explanatory variables were derived from the breathing pattern and respiratory mechanisms on a breath-by-breath basis during PCV, VCV and VCVDF modes.
The GLMM model assumed a negative binomial distribution (NB) for the dependent variable. The reason for this probability distribution was that our response variable is discrete (number of occurrences of DC), limited to non-negative values, and positively skewed with most of the observations having a value close to zero (Supplemental Fig.
3, Supplemental Digital Content 7, http://links.lww.com/CCM/D714). NB distribution has often been used in regression models with count data, as here (S5). Considering we are interested in modeling the number of events as a rate, we incorporated an exposure variable (the total number of respiratory cycles per hour) in the model, which indicates the number of times the DC events could have happened.
We did not estimate the sample size needed to achieve a specific power. No general and well-established sample size methodology exists for mixed models in longitudinal studies with specific correlations among observations from the same individual.
Simulation methods are recommended, but this is out of the scope of our study. We have analyzed an observational study with all the available data (9,694,573 breaths and 9251 hours of mechanical ventilation).
Mathematical formulation of the model
The following is an explanation of the model from which we derived the statistics reported in Supplemental Figure 4 (Supplemental Digital Content 8, http://links.lww.com/CCM/D715) and Table 2 (Results section in the main text).
If Y is a response variable consisting of counts with values restricted to Y∈[0,+∞] , then the expected value of Y can be modeled as a negative binomial distribution:
Y∨X NB(λ , α),
where λ is the expected value for the counts, λ=E(Y∨X) , and α is the overdispersion parameter of the distribution.
To model the expected rate of the occurrence of events rather than the counts, the log- linear regression model with negative binomial distribution is:
log(λ/N)=β0+β1X1+…+βpXp log(λ)−log(N)=β0+β1X1+…+βpXp
log(λ)=β0+β1X1+…+βpXp+log(N) , where λ/N is the expected value of the rate Y/N
N is the number of respiratory cycles per hour, the unit of analysis in our study
log(N) is an "offset", a term with fixed coefficient equal to 1 that is included to account for the number of respiratory cycles.
β are the coefficients of the p explanatory variables Xs, plus the intercept (β0).
Merging the above within a generalized linear mixed-effects model and rewriting to fit our case,
λ b N
(¿¿0k i∗vmodeki)+log(¿¿ji)+εji, +¿
log(¿ ¿ji)=β0+β1I(VCV)+β2I(VCVDF)¿
¿
where: λji is the expected jth value for level i of the grouping variable "Patient"; here, the expected value of DC events.
βm (m = 1, ..., p) are the fixed-effects coefficients plus the intercept (β0).
I(VCV) and I(VCVDF) are dummy variables to indicate the ventilatory mode; its value is 1 for the present mode and 0 otherwise. The level with no dummy variable (PCV) is the reference category.
b0ki are the intercept random effects for each level i of the grouping variable
"Patient". Here, the intercept was allowed to vary by modes within patient to account for the possible heterogeneity the ventilatory mode impacts on the response; vmode represents the categorical variable for the ventilatory modes (with k levels). This interaction random term (b0ki*vmodeki) significantly improved model fits versus the more simplistic random intercept model (i.e. that where the influence of patient i is explained by a single intercept value), as it was suggested by a likelihood ratio test.
log(Nji) is the offset term, i.e., the logarithm of the total number of breaths per hour for each patient.
εji is the vector of random errors.
We also investigated the effect of the time (i.e. hours) on DC by using a random intercept and slope model to account for trends over time within patients, but the variable time was not significant, thus the model described above was chosen for simplicity. Statistics derived from that model is reported in Supplemental Figure 4 (Supplemental Digital Content 8, http://links.lww.com/CCM/D715) and in the Results
section in the main text.
Additionally, to investigate factors affecting the development of DC, we accommodated physiological variables thought to affect DC (i.e., insp. time, peak flow, respiratory rate, etc.) in the above regression equation accordingly. We used a multivariate approach allowing separate fits per modes for the population (vmode:InspTime + vmode:PeakFlow and so on, where vmode is the categorical variable for the ventilatory modes). Statistics are reported in Table 2 (main text).
We used the lme4 package (S6) and the lsmeans package (S7) to fit and analyze the GLMM.
Supplemental References
XS5. Cameron AC, Trivedi PK. Regression Analysis of Count Data: Cambridge University Press; 2013.
S6. Bates D, Maechler M, Bolker B, et al. Fitting Linear Mixed-Effects Models Using lme4. Journal of Statistical Software, 2015;67(1):1-48.
S7. Lenth RV. Least-Squares Means: The R Package lsmeans. Journal of Statistical Software 2016;69(1):1-33.