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Our group is working on a dataset of approximately 1000 patients with 10 complete variables at the time of an acute disease, who subsequently completed a questionnaire of 5 Likert items (questions with possible ordinal answers 1 to 5, each exploring one dimension of quality of life) at three regularly spaced time points over 1 year (times 1, 2, and 3). So, each individual would have, in addition to the baseline variables, 5 Likert items at 3 time points (=15 items). We need this dataset as cases for a matched cohort study in which the Likert items are the outcome. The external control population has no missing data.

There was loss to follow-up over time, with a missingness rate of ~9% for the 5 questions at time 1, ~23% at time 2, and ~37% at time 3.

We decided to multiply impute this repeated measure.

Using package mice in R, which we have so far used often but only for cross-sectional data, we decided to perform the imputation in wide format (one column per time per item) rather than using a multilevel imputation model in long format for these two reasons:

  1. In prof. Stef van Buuren's book "Flexible imputation of missing data", Chapter 9 ("Longitudinal data"), page 222, explicitly states that multiple imputation can be done in the wide format (and later suggests this may be especially practical if individuals are not seen at different times - e.g., if the time intervals are regular, as in this case):

Multiple imputation of longitudinal data is conveniently done when data are in the wide format. Apart from the fact that the columns are ordered in time, there is nothing special about the imputation problem. We may thus apply the techniques from the earlier chapters to longitudinal data.

  1. the proportional odds logistic regression is an appropriate imputation method for ordered categorical variables (such as a Likert scale), but it is not implemented in the multilevel imputation methods of mice or, to the best of our knowledge, other software packages. However, it is possible if ordinary, non-multilevel methods are used.

Therefore, the head of the dataset looks like this, with 5*3 = 15 Likert columns (there are 10 baseline variable, of which here age and sex are visible):

enter image description here

However, we suspected and confirmed before the imputation that these questionnaire items are highly correlated with each other (after all, they explore dimensions of quality of life): both the 5 items at the same time point and the same items at 3 time points produce Spearman's rhos (and, if considered linear, Pearson's rs) of ~1.0!

Most baseline variables are also correlated with these quality of life measures, but these correlations are 1) biologically plausible, 2) expected, 3) the basis of the rationale for inclusion in the imputation model, and 4) not as strong as the correlations between the 5 Likert items at the same timepoint. Relative to them, the MAR assumption is tenable.

Indeed, when we generaetd 40 imputed datasets and first 5, then 20 iterations, there is no convergence: the lines are parallel instead of intermingled, and show a trend instead of coalescing toward one value. This is especially evident in time 2 and 3, which have higher missingness rates. Example of the traceplot for the 3 columns corresponding to the first Likert item at each of the 3 timepoints:

enter image description here

How should we interpret this diagnostic result, and what action would you recommend? Our hypotheses:

  1. In most guides on multiple imputation, the typical example leading to this situation is the presence of variables describing the same quantity, e.g. including height, weight and BMI in the imputation model and allowing them to impute each other with no passive imputation. Is this our situation? Should we adapt the imputation model by not allowing each item to impute the other items at the same time point, but, possibly, only at different time points; or even by not allowing at all these items to impute each other (and only using baseline variables to impute them)?
  2. As a sign that the Likert variables, being strongly correlated with each other, should not be used to impute each other at all in a wide format and we should perform multilevel imputation at the cost of not using polr (but then, under what condition is prof. van Buuren's recommendation to impute repeated measures in wide format valid)?
  3. As a sign of some other possible problem?

We are aware that additional diagnostics may be appropriate as well as a priori knowledge of the relationship between variables and welcome any suggestions.

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