Percentage of people per age group in ORCADES who holiday outside the UK at least once a year.

<p>People over 50 take significantly more holidays than those under 50.</p

Likelihood-based deviance information criterion (DIC) scores for conventional causal (M4) and conventional reverse causal (M5) models, both (i) assume absence of pleiotropic effects of instruments on biomarkers and outcomes, (ii) explicitly exclude unmeasured confounders from modelling and (iii) account for the noise in the measurement; and for the model where the association between the biomarker and outcome is modelled <i>entirely</i> by unmeasured confounders (M6); these models have been compared in Experiment 2.

<p>Digits after decimal point have been omitted from the table.</p><p>Setting 1: <i>precxt</i> = 200, <i>precx</i> = 200, <i>precy</i> = 100; Setting 2: <i>precxt</i> = 1000, <i>precx</i> = 1000, <i>precy</i> = 0.1; Setting 3: <i>precxt</i> = 100, <i>precx</i> = 100, <i>precy</i> = 100. Sparsity parameter gamma is set to 0.025 in all models. In model with confounders (M6) <i>precz</i> = 1.</p>**<p>indicates the best model for each setting; preferred modelling hypotheses are characterized by lower DICs.</p

Characteristics of ORCADES Study participants (n = 1972).

<p>Characteristics of ORCADES Study participants (n = 1972).</p

Study cohort.

<p>Mean (standard deviation) or median (interquartile range) is shown for continuous variables and number (percent) is shown for categorical variables.</p><p>Physical activity is estimated from the reported hours of cycling and other sports activities (4 categories) and Carstairs Deprivation Index was used to describe socio-economic status.</p

Likelihood of causal association between low 25-OHD and colorectal cancer is compared with the reverse causal hypothesis (proposing CRC leads to lower 25-OHD), on the complete dataset and for a range of parameter settings.

<p>Deviance information criterion (DIC) score differences between models are shown; positive values indicate that causal association is more likely. Mean DIC is calculated as the average DIC for all causal and reverse causal models considered for any given parameter setting (smaller values indicate better models). Large positive DIC differences provide overwhelming evidence for a direct causal relation between low 25-OHD and colorectal cancer. Details on DIC components are in <b><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0063475#pone.0063475.s003" target="_blank">Table S2</a>.</b> Digits after decimal point have been omitted from the table.</p>*<p>Settings: S1: precx = 1000, precxt = 1000, precy = 0.1; S2: precx = 100, precxt = 100, precy = 100; S3: precx = 1000, precxt = 1000, precy = 10; S4: precx = 100,precxt = 100, precy = 200; S5: precx = 20, precxt = 20, precy = 200.</p

Likelihood of causal association between low 25-OHD and colorectal cancer is compared with the reverse causal hypothesis, (proposing CRC leads to lower 25-OHD), in a subset of data comprising a random sample of 500 cases and 500 controls.

<p>DIC score differences arising from the comparison of the full causal and reverse causal models, for a range of parameter settings are shown. Positive values indicate preference for the causal model. Mean DIC (black line) represents the average DIC for all causal and reverse causal models considered (lower mean DIC scores suggest better models), for any given setting of sparsity <i>gam1</i> parameter (higher <i>gam1</i> favours sparser models - links between nodes are increasingly more likely to be pruned). We consider independent gamma priors on the associations concerning confounding effects (<i>gam2</i>) in order to attenuate the strong effect of confounder and to artificially boost the importance of the link between 25-OHD and colorectal cancer. Overall, optimal models are the denser ones (characterised by smaller values of <i>gam1</i> parameter, most links remain in the model), and large positive DIC differences provide overwhelming evidence for a direct causal relation between low 25-OHD and colorectal cancer.</p

Graphical representation of the basic model: <i>y</i> – outcome (colorectal cancer, CRC); <i>x</i> – true concentration of the biomarker, 25-OHD; <i>xt</i> – measured concentration of the biomarker, 25-OHD; <i>g</i> – a vector of predictor variables: age, sex, smoking, BMI, physical activity, family history, NSAIDs intake, socioeconomic status, total caloric intake, alcohol intake, consumption of red meat, dietary vitamin D intake and SNPs associated with CRC or 25-OHD; <i>z</i> – unmeasured, hidden confounders.

<p>Link <i>u</i> represents the effect of predictor variables on 25-OHD, <i>w</i> is the effect of 25-OHD on CRC, <i>wg</i> is the effect of predictor variables on the CRC, <i>v</i> is the effect of unmeasured confounders on the 25-OHD and <i>wz</i> is the effect of unmeasured confounders on colorectal cancer.</p

Likelihood of causal association between low 25-OHD and colorectal cancer is compared with the reverse causal hypothesis, (proposing CRC leads to lower 25-OHD), on the complete dataset and for a range of parameter settings.

<p>DIC score differences between models are shown; positive values indicate that causal association is more likely. Mean DIC (red line) is calculated as the average DIC for all causal and reverse causal models considered for any given parameter setting (smaller values indicate better models). Large positive DIC differences provide overwhelming evidence for a direct causal relation between low 25-OHD and colorectal cancer. * Settings: S1: precx = 1000, precxt = 1000, precy = 0.1; S2: precx = 100, precxt = 100, precy = 100; S3: precx = 1000, precxt = 1000, precy = 10; S4: precx = 100, precxt = 100, precy = 200; S5: precx = 20, precxt = 20, precy = 200.</p

Results of linear regression for complete cases and imputed data using May-adjusted vitamin D as the outcome.

<p>Results of linear regression for complete cases and imputed data using May-adjusted vitamin D as the outcome.</p

Comparison of farmers (n = 265) and non-farmers (n = 1649) on variables of interest in Orkney.

<p>Farmers are anyone who identified their primary profession as farmer. Unpaired t-tests applied to continuous data; chi-square tests applied to categorical data.</p