Table 2 Missing data generation process for the simulation study by varying the proportion of missingness in the daily assessments, using different missing data mechanisms and varying the strength of association between the auxiliary variable and missingness

1. Proportion of missingness, p: 1%, 5%, 20%, 35% of participant-days missing exposure value

2. Types of missingness:

a. Missing Completely at Random (MCAR): p% of the participant-days/assessments in each of the 1000 simulated datasets were removed completely at random.

b. Missing at Random (MAR): MAR mechanism was simulated under a logistic regression model as a function of an auxiliary variable, the SOFA score, with varying correlations with missingness:

\( logit\ \left({missing}_{ij}\right)=\alpha +{\beta}_{SOFA_{ij}}\ast {SOFA}_{ij} \),

where \( {\beta}_{SOFA_{ij}} \) = 0.01, 0.1 and 0.2 representing weak, moderate and strong relationships with missingness. Here α was manipulated for different combinations of \( {\beta}_{SOFA_{ij}} \) to generate the required proportion of missingness p.

c. Missing Not at Random (MNAR): MNAR mechanism was generated under a logistic regression model with the probability of missingness having a weak, moderate, and strong association with the daily delirium status.

logit(missing_ij) = α + β_{del _ miss} ∗ delirium_ij, where β_{del _ miss} took on values 0.1, 0.5 and 1.0 representing weak, moderate and strong relationships with missingness. Here α was manipulated for different combinations of β_{del _ miss} to generate the required proportion of missingness p.