Epidemiological and statistical considerations

We used two types of observational study designs in the present study: Cohort (paper I and paper III) and cross-sectional study design (paper II).

Cohort study design: We followed patients from baseline ten weeks after renal transplantation to the onset of an event (death or graft loss) or to the end of the observational period (1^st of February 2014) for patients who did not suffer an event (censoring). We were particularly interested in associations between the primary exposure variable plasma marine n-3 PUFA levels (the sum of plasma EPA, DHA and docosapentaenoic acid [DPA] levels measured at ten weeks post-transplant) and risk of certain events, which we assessed by Cox proportional hazard regression analysis (156). Cox regression estimates the relative hazard of an event for explanatory variables included in the Cox model. In this study, we may interpret relative hazard as relative risk. Consequently, hazard ratio (HR) = 1.0 indicate a neutral risk, HR > 1.0 indicate an increased risk and HR < 1.0 a decreased risk of the event in question. We assumed that plasma marine n-3 PUFA levels had a linear association with events and that associations were stable throughout the observational period (proportional hazard). These assumptions were tested as they had to be met in order to accept the Cox regression model (156).

In addition to plasma marine n-3 PUFA levels, other explanatory variables were added to the Cox models (covariates). A variable that influence the association between the primary exposure variable and outcome (usually an event) or dependent variable (e.g. cardiovascular risk markers) is called a confounder and might either hide, exaggerate or distort the true estimates of the association between the primary exposure variable and outcome / dependent variable (e.g. between plasma marine n-3 PUFAs and cardiovascular mortality). As an example, current smoking was negatively associated with plasma marine n-3 PUFA levels.

Without adjustment for smoking status in the Cox regression analysis, associations between plasma marine n-3 PUFA levels and cardiovascular mortality would be influenced by the association between smoking and cardiovascular mortality. We used pre-defined traditional and transplant-specific risk factors for death and graft loss as covariates in the Cox models.

Competing risk in survival analysis occurs when an event (e.g. death due to cardiovascular disease) modifies the chance for another event to occur (e.g. death due to cancer). Some risk factors (e.g. current smoking) increase the risk of both events, but one event may be more likely to occur before the other (e.g. death due to cardiovascular disease occured before the

patient would have died of cancer). Under these circumstances standard Cox regression will not be optimal for assessment of associations between exposure and certain events (in this example smoking status and death due to cancer, but theoretically a similar scenario could be the case for marine n-3 PUFAs) (157). Renal graft dysfunction increases the risk of both graft failure and death (22), and therefore death competes with the risk of death censored graft loss (158). In contrast, overall graft loss comprises both grafts lost to recipient death and death censored graft loss and would therefore not be influenced by competing risk due to recipient death. In the setting of possible competing risk, standard Cox regression could be used when the research question investigates the etiological relationship between risk factors and a given outcome (e.g. plasma marine n-3 PUFA levels and cardiovascular mortality) (157). This approach requires the assumption that competing risks are independent (which would not be the case for smoking and death due to cancer, whereas it is more likely to be the case with plasma marine n-3 PUFA levels). Subdistribution hazard method is an alternative approach which was investigated for cause-specific mortality in this study and would especially be appropriate if we were interested in predicting the probability of a given outcome at a given time for a patient population with certain characteristics (e.g. current smokers and risk of death due to cancer) (157). Some argue that results obtained by both approaches should be reported when competing risks are suspected, and statistical packages have been updated in recent years to perform these analyses (159). Since Kaplan-Meier patient and graft survival curves for events susceptible to competing risk might be misleading (160), we chose to use the Kaplan-Meier method only for all-cause mortality and overall graft loss (161).

Cross-sectional study design: We collected data on explanatory variables (including plasma marine n-3 PUFA levels) and parameters of interest (various cardiovascular risk markers) at ten weeks post-transplant. Linear regression was used to study correlations between plasma marine n-3 PUFA levels and cardiovascular risk markers, with and without adjustment for other explanatory variables (162). Unstandardized (Unstd.) regression coefficients (β-coeff.) indicate the slope of regression lines, whereas standardized regression coefficients (Std. β-coeff.) scale the explanatory variable to its standard deviation. Std. β-coeff. make it possible to compare relative associations of different explanatory variables with the same dependent variable. Linear regression may also be used to assess how much of the variance (R²) in the dependent variable (e.g. fasting plasma glucose) is explained by the explanatory variables included in the final regression models. In age and gender adjusted models, recipient age and gender, as well as plasma marine n-3 PUFA levels were forced into the final model. In

multivariable linear regression we selected the variables that were included in the final model in addition to plasma marine n-3 PUFA levels in a stepwise backward manner: We started with all candidate variables and then deleted one variable per step from the model until all variables satisfied the model criterion of p<0.10 and no further improvement was possible.

In the present study, there were several potential sources of error. Random error might occur when measurements are unprecise or when wrong data are entered into a database.

Systematical error (bias) can have many sources. Information bias may arise from

measurement error, but also from misclassification (e.g. death due to MI classified as SCD).

Selection bias may occur when patients are not randomly selected for inclusion in a study and if patients not included in the study differ from the remaining population, the results would not be representative of the population intended to be analyzed. Since the observational time started at ten weeks post-transplant, events that occurred before this time-point were not included in the analyses and this lead to immortal time bias. Immortal time refers to a time period during which the outcome cannot occur. Confounding, as discussed earlier, is probably the most important source of bias in the present study. Regression analysis (adjusting for covariates) reduces the impact of confounders and was chosen as the general statistical approach in the current study (156, 157, 162).

Other statistical methods used in this study, graphing methods, use of restrictions, handling of interactions, missing data or explanatory variables with a skewed distribution, as well as some other sources of bias are discussed in later sections. The primary exposure variable in the main analysis was plasma marine n-3 PUFA levels. In addition, we performed subgroup analyses with either plasma EPA levels or plasma DHA levels. We also performed subgroup analyses with n-6 PUFA to n-3 PUFA ratio (the ratio of plasma n-6 PUFA levels [the sum of arachidonic, linoleic, gammalinoleic, dihomogammalinolenic, adrenic and eicosadienoic acid levels] to plasma n-3 PUFA levels [the sum of marine fatty acids EPA, DHA, DPA and alpha-linolenic acid derived from plants]) as the primary exposure variable. Since plasma marine n-3 PUFA levels predominantly consisted of EPA and DHA and most of plasma n-6 PUFA levels consisted of AA and linoleic acid, the ratio of EPA + DHA to AA + linoleic acid produced nearly identical results as n-6 PUFA to n-3 PUFA ratio, and neither of these, nor EPA to AA ratio showed any benefit compared to plasma marine n-3 PUFA, EPA and DHA levels. Results obtained with n-6 PUFA to n-3 PUFA ratio are not mention in the results section in this thesis, but are described in papers I-III.