This supplementary material includes three parts: some preliminary results, four examples, an experiment, three new algorithms, and all proofs of the results in the paper [4].CONTENTS

We examined the causal direction between gut microbiota–dependent metabolite trimethylamine N-oxide (TMAO) or its predecessors and cardiometabolic diseases, such as risk of type 2 diabetes mellitus (T2DM), coronary artery disease (CAD), myocardial infarction (MI), stroke, atrial fibrillation (AF), and chronic kidney disease (CKD). We used genetic variants as instruments to test the causal associations. Genetically predicted higher TMAO and carnitine were not associated with higher odds of T2DM, AF, CAD, MI, stroke, and CKD after Bonferroni correction (P ≤ 0.0005). However, we observed that genetically increased choline showed a suggestive association with higher risk of T2DM (odds ratio 1.84 [95% CI 1.00–3.42] per 10 units, P = 0.05). In contrast, genetically predicted higher betaine (0.68 [0.48–0.95] per 10 units, P = 0.023) was suggestively associated with a lower risk of T2DM. We observed a suggestive association of genetically increased choline with a lower level of body fat percentage (β ± SE −0.28 ± 0.11, P = 0.013) but a higher estimated glomerular filtration rate (0.10 ± 0.05, P = 0.034). We further found that T2DM (0.130 ± 0.036, P < 0.0001) and CKD (0.483 ± 0.168, P = 0.004) were causally associated with higher TMAO levels. Our Mendelian randomization findings support that T2DM and kidney disease increase TMAO levels and that observational evidence for cardiovascular diseases may be due to confounding or reverse causality.

A surrogate end point is often used to evaluate the effects of treatments or exposures on the true end point in medical researches. Various criteria for the statistical surrogate, principal surrogate and strong surrogate have been proposed. We first illustrate that, with a surrogate end point that is defined by these criteria, it is possible that a treatment has a positive effect on the surrogate, which in turn has a positive effect on the true end point, but the treatment has a negative effect on the true end point. We define such a phenomenon as a surrogate paradox. The surrogate paradox also means that the sign of the treatment effect on the true end point is unpredictable by the effect signs of both the treatment on the surrogate and the surrogate on the true end point. Then we propose two notions for a consistent surrogate and a strictly consistent surrogate to avoid the surrogate paradox. With the causal network that was presented by Lauritzen, we discuss the conditions for a strong surrogate to be a consistent surrogate and a strictly consistent surrogate. Copyright 2007 Royal Statistical Society.

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