Generalized additive models to analyze non-linear trends in biomedical longitudinal data using R: Beyond repeated measures ANOVA and Linear Mixed Models

Abstract

In biomedical research, the outcome of longitudinal studies has been traditionally analyzed using the repeated measures analysis of variance (rm-ANOVA) or more recently, linear mixed models (LMEMs). Although LMEMs are less restrictive than rm-ANOVA in terms of correlation and missing observations, both methodologies share an assumption of linearity in the measured response, which results in biased estimates and unreliable inference when they are used to analyze data where the trends are non-linear, which is a common occurrence in biomedical research. In contrast, generalized additive models (GAMs) relax the linearity assumption, and allow the data to determine the fit of the model while permitting missing observations and different correlation structures. Therefore, GAMs present an excellent choice to analyze non-linear longitudinal data in the context of biomedical research. This paper summarizes the limitations of rm-ANOVA and LMEMs and uses simulated data to visually show how both methods produce biased estimates when used on non-linear data. We also present the basic theory of GAMs, and using trends of oxygen saturation in tumors reported in the biomedical literature, we simulate example longitudinal data (2 treatment groups, 10 subjects per group, 6 repeated measures for each group) to demonstrate how these models can be computationally implemented. We show that GAMs are able to produce estimates that are consistent with the trends of biomedical non-linear data even in the case when missing observations exist (with 40% of the observations missing), allowing reliable inference from the data. To make this work reproducible, the code and data used in this paper are available at: https://github.com/aimundo/GAMs-biomedical-research.

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“…Therefore we fitted generalized additive models (GAMs) with an interaction between time and treatment in each case. A GAM is a model that allows to fit non-linear trends in longitudinal data by using basis functions, thus overcoming the limitation of linear models (such as repeated measures ANOVA or a linear mixed model) which give biased estimates when used in data with non-linear trends 48 . Because GAMs allow the data to dictate the fit of the model, they are advantageous to perform comparisons between the different trends in the data.…”

“…Because GAMs allow the data to dictate the fit of the model, they are advantageous to perform comparisons between the different trends in the data. Pairwise comparisons between the different groups were obtained by calculating the difference in the fitted smooths in each case (the trend over time), and constructing a 95% simultaneous confidence interval (CI) around that difference 48,49 . In this way, the time intervals where a significant difference exists between the fitted trends will correspond to those intervals of time where the simultaneous CI does not cover zero 48,50 .…”

“…Pairwise comparisons between the different groups were obtained by calculating the difference in the fitted smooths in each case (the trend over time), and constructing a 95% simultaneous confidence interval (CI) around that difference 48,49 . In this way, the time intervals where a significant difference exists between the fitted trends will correspond to those intervals of time where the simultaneous CI does not cover zero 48,50 . Because the simultaneous CIs contain the whole function at a nominal level (95%), they are robust against false positives 47 .…”