A study is being planned to investigate the association between systolic blood pressure (SBP) and independent variables age (AGE), smoking history (SMK = 0 if non-smoker, SMK = 1 if a current or previous smoker), and body size as measured by the quetelet index (QUET) described in Problem 2 of Chapter 5. SBP, AGE and QUET are continuous variables. The statistical analysis will involve multiple linear regression with SBP as the dependent variable.
a. Suppose that SMK is the only independent variable to be included in the regression model. Determine the approximate sample size necessary to detect a true slope with an absolute value of at least 1.5. From previous research, the population standard deviation of SBP is approximately 3 mm Hg.
b. Suppose that SMK, AGE, and QUET are all included as predictors in the regression analysis, with SMK being the main predictor of interest and AGE and QUET being included to control for confounding. Using the method described in Section 27.3.3, determine the approximate sample size necessary to detect a true slope for SMK that has an absolute value of at least 1.5. From previous research, the population standard deviation of SBP is about 3 mm Hg, and the population squared multiple correlation between SMK and the other two independent vari ables is approximately 0.10.
c. In part (b), recalculate the required sample size for population squared multiple correlations ranging from 0.10 to 0.90, in increments of 0.1. Comment on the relationship between the required sample size and the population squared multiple correlations.
d. Suppose that AGE is the only independent variable to be included in the regression model. Determine the sample size necessary to detect a true correlation between AGE and SBP of at least 0.40 in absolute value.
e. Suppose that AGE, SMK, and QUET are all included as predictors in the regression analysis, with AGE being the main predictor of interest. Using the method described in Section 27.3.3, determine the sample size necessary to detect a correlation between SBP and AGE of at least 0.40 in absolute value, assuming that the population squared multiple correlation between AGE and the other two independent variables is 0.10.