1. Risk of bias assessment ‐ detailed assessment methods.

Risk of bias element	Criteria for low risk of bias	Criteria for unclear	Criteria for high risk of bias
Selection bias: random sequence generation	The trial authors needed to have described the method used to generate the allocation sequence in sufficient detail to allow an assessment of whether it should produce comparable groups. For example “the randomisation sequence was computer‐generated”. We allowed that a good method of randomisation was strongly implied if the trial authors discussed stratification and/or blocking. Therefore, if they were not explicit about their randomisation method but did describe stratification or blocking we assessed this as low risk.	The trial authors have not described their method in sufficient detail for the assessment of whether it would produce comparable groups. For example, the trial authors state “the trial was randomised” and provide no further information.	The randomisation method was assessed as not truly random, and may not produce comparable groups.
Selection bias: allocation concealment	The trial authors needed to have described the method used to conceal allocation sequence in sufficient detail to determine whether the allocations could have been foreseen in advance of, or during, enrolment. Good methods included putting allocation codes in opaque, sealed envelopes (ideally prepared by someone outside the treatment or assessment teams and sequentially numbered), using a telephone allocation system after the participants had consented to participate or providing a random number that links to a specific set of capsules prepared and distributed centrally or by an arms‐length pharmacist.	The authors gave insufficient detail as to method.	The allocation was known in advance of participants consenting to take part in the trial.
Performance bias: blinding of participants and personnel	The trial authors needed to have described all measures used, if any, to blind trial participants and personnel from knowledge of which intervention a participant received. Ideally, they should also have provided information relating to whether the intended blinding was effective. For example, the authors could say “both the intervention and placebo capsules looked and tasted the same.” However if the trial authors did not provide information on whether the blinding was effective, but sufficient detail was given on a good method of blinding, then it was assumed that the blinding was effective and the risk of bias was low.	Insufficient methodological details were provided e.g. “the trial was blinded.”	The trial was unblinded or where blinding was broken, e.g. “the capsules were visually identical but the participants reported a strong fishy flavour in the intervention group only.”
Detection bias: blinding of outcome assessment	Trial authors needed to have described measures used, if any, to blind outcome assessors from knowledge of which intervention a participant received. Ideally, they should also have provided information relating to whether the intended blinding was effective. For example, the authors could say “the outcome assessors had no knowledge of the group allocation, and both the intervention and placebo capsules looked and tasted the same so the self‐assessment scales were also blinded.” However if the trial authors did not provide information on whether the blinding was effective, but sufficient detail was given on a good method of blinding of the assessors, then it was assumed that the blinding was effective and the risk of bias is low. All biochemical assessment (lipids, glucose, CRP, insulin, PSA etc.) were considered at low risk of detection bias if outcome assessor blinding or double blinding was stated.	Insufficient methodological details were provided e.g. “the trial was blinded.”	The trial was unblinded or blinding was broken, e.g. for a self‐assessment measure “the capsules were visually identical but the participants reported a strong fishy flavour in the intervention group only.” (Because the level of blinding could vary by outcome assessment of risk of bias was based on blinding of the review's primary outcome(s). Where primary outcomes had different assessments we opted for the higher risk of bias but noted that risk of bias was lower for other outcomes.
Attrition bias: incomplete outcome data	The trial authors needed to describe the completeness of outcome data for each main outcome, including attrition and exclusions from the analysis. They needed to report the number of attrition/exclusions, the numbers in each group at each time point, reasons for attrition/exclusion and any re‐inclusions in analyses. Ideally, they would report how they imputed any missing data e.g. last observation carried forward. There needed to be a reasonable balance of attrition/exclusions between trial arms and ≤ 20% of the sample should be lost over a year.	The trial authors didn't state reasons for attrition/exclusion, or were unclear about the numbers lost to attrition/exclusion in each trial arm.	The trial authors demonstrated a substantial difference in the rates of attrition/ exclusions between the trial arms and/or > 20% of the baseline sample was lost over a year (> 10% over 6 months).
Reporting bias: selective outcome reporting	The trial authors needed to have published their trial protocol or trials registry entry before the end of the trial’s recruitment period i.e. prospectively. They needed to have reported on all of the primary and secondary outcomes listed in the protocol/registry entry. Reporting additional secondary outcomes in the results paper(s), although not ideal, was deemed to still be low risk.	No trial protocol or trials registry entry was found, it was registered retrospectively, or the dates of registration and participant recruitment were unclear.	The trial authors did not report at least one primary or secondary outcome listed in the protocol/registry entry OR the results paper(s) reported a primary outcome that was not listed at all in the protocol or not listed as a primary outcome in the protocol.
Other sources of bias: Attention bias	The trial authors needed to have reported that participants in all trial arms received the same amount of attention and time from researchers and clinical teams. For example, “All participants attended the clinic for a baseline assessment which took 2 hours. They were then followed with monthly telephone calls, and finally attended for a 6 month assessment at the clinic which took 1 hour.” If the trial only differed by the content of the capsules, and the assessment schedule was not stated to differ between the two arms, it was assumed to be at low risk.	The trial authors did not state the attention each arm received.	Participants in different arms received different amounts of attention. For example, “The intervention group only attended for additional assessments at months 2, 4, and 6” or “the rates of relapse differed substantially between the groups which led to differing amounts of treatment time and attention,” or “the intervention group received a 40 minute dietary education session.”
Other sources of bias: limited compliance	The higher PUFA arm had to demonstrate an increase in PUFA fats over control in a body biomarker (total PUFA had to be assessed by at least LA plus one or more further components of PUFA), or greater reduction in TC in the higher PUFA arm.	Biomarker data not reported or not in a way that could be interpreted. Where lipid biomarker and TC contradicted each other we chose unclear.	Measures of compliance were reported but did not suggest higher total PUFA in the appropriate arm.
Other sources of bias: other	In the absence of any additional issues this item was coded "low risk of bias"		If fraud concerns had been raised and the paper had been withdrawn, or the trial author had been found guilty of fraud by a legal or medical entity the paper was excluded from the review. However if fraud concerns were raised, but the journal had not withdrawn the paper, and the trial author had not been formally sanctioned; then the trial was included in the review, but concerns were raised here, and the risk of bias for this item was high.

LA: linoleic acid; PUFA: polyunsaturated fatty acids; TC: total cholesterol