Mixed models: an essential tool for non-independent data analysis

Linear mixed models (LMM) are a method for analysing non-independent, multilevel/hierarchical, longitudinal or correlated data.

In their article in European Journal of Cardiothoracic Surgery [1], Wang et al. show an example of repeated measures analysis on the same individual with LMM. This utility of the LMMs had already been analysed in a more summarized way by Hickey et al. in a statistical primer published in Interactive Cardiovascular and Thoracic Surgery a few years ago [2] and other previous studies [3, 4]. Intuitively, one could think of the analysis of repeated measures with other tools, such as linear regressions, analysis of variance models (ANOVA) or repeated measures ANOVA (RM-ANOVA). The problem with linear models is that the observations on the same individual are not independent of each other, a necessary condition [5]. ANOVA or similar models, on the other hand, are not valid in the case of unbalanced measurements or are very sensitive to the presence of many missing values [6]. The great advantage of LMMs is that they support data dependency and are more robust against unbalanced or missing measurements.

LMMs are a modification of simple linear models that allow adjustment for both fixed and random effects and are very useful when there is no independence of the observations (as in the 2 statistical primers mentioned above) or when the data have a hierarchical structure [7]. In clinical research, hierarchical data analysis is even more common than repeated measurements on the same individual. A very common example is the analysis of series of patients operated on in different centers. When there are different ‘levels’ of data, such as patients treated in the same hospital, the variability can be conceived in 2 dimensions: that of individuals within the same hospital or that of different hospitals. Observations of patients within the same hospital are not independent, as patients tend to be more similar to each other than to patients attending another centre. However, research units at a higher level (in our case, hospitals) are independent.

There are different ways to deal with this problem: (i) ignore the random component of the hospital and assume that the variability of the observations is independent of the centers, (ii) aggregate the data by centre: this would give us an independent data sample but with many fewer observations and (iii) make as many regressions as there are centres, which would make it difficult to draw valid conclusions for all the centres and would also reduce the power of each point estimate.

LMMs can be considered an intermediate option between options 2 and 3. This is possible because LMMs can combine fixed and random effects. A fixed effect is one that does not vary, such as age, sex, hypertension and haemoglobin level, while, in the previous example, the hospital would be the grouping variable and responsible for the random effect. In the case of repeated within-subject measurements, the random effect would occur at the individual level [8]. These are the simplest examples of LMM utilities, in which we add a single random intercept to the equation of a linear regression model, but the model is flexible and allows the estimation of random coefficients (not only intercepts), or various nested levels.

LMMs are valid for the analysis of continuous dependent variables, but there are other analogous methods for the analysis of binomial variables (mixed effects logistic regression) or ordinals, as well as analogs of Poisson, ordinal logistic or negative binomial regressions. Some examples of LMM software for the most frequently used statistical packages are ‘mixed’ (Stata), lme/lmer (R), PROC MIXED (SAS) or ‘mixed models’ (STATA).

In conclusion, the statistical primers by Wang et al. [1] and Hickey et al. [2] are a great opportunity for readers of European Journal of Cardiothoracic Surgery and Interactive Cardiovascular and Thoracic Surgery to get familiar with a robust statistical tool, which can be of great help in dealing with very frequent analysis challenges in clinical research.

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