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The United States’ association between blood lead level

A cross-sectional study was undertaken in the United States to determine if there is an association between blood lead level (measured in micrograms per decilitre: pg/dL) and diastolic blood pressure (measured in millimetres of mercury: mmHg). It is hypothesised that as blood lead level increases, diastolic blood pressure increases.
The following data were recorded for each of 431 participants:
id: dbp: lead: age: sex: race: smoke: bmi: potassium: sodium: anti:
unique subject identifier diastolic blood pressure (mmHg) blood lead level (pg/dL) age (years) sex (1 = male, 2 = female) race (1= white, 2 = black, 3 = other) current smoker (1 = no, 2 = yes) body mass index (kg/m2) serum potassium (mmol/L) serum sodium (mmol/L) currently taking medication for high blood pressure (0 = no, 1 = yes)

(a) Based on the research question presented above, which is the outcome and which is the explanatory variable? [1 mark]

You are a part of a research team and you have been asked to give advice on the design of a case-control study to investigate the association between breast cancer and participation in mammography screening. The research team wants to be able to detect an odds ratio of at least 0.5 using a two-sided significance test at the 5% level. It is estimated that 50% of the control population will have participated in mammography. (a) If an equal number of cases and controls could be recruited in this study, what is the required sample size to achieve a study power of (i) 80% and (ii) 90%? [2 Marks]
(b) One of the researchers thinks that the prevalence of the exposure (mammography screening) in the population would be 30%, not 50%. i. if the researchers would like to achieve 90% power, how many cases and controls would be required to detect the same odds ratio as in (a)? (1 Mark]
ii. If the prevalence of exposure in the population was not known, would it be better to assume the prevalence of exposure in the population was 30% or 50%? Why? (2 Marks]