Fifty people participated in a medical study. Each participant contributed two
eye measurements (labeled y ). Thus the study had a total of 100 measurements
on y from the 50 participants. It is suspected that the two eye measurements
from the same person are correlated. Without sufficient knowledge of the study
background, a data analyst performed a multiple linear regression analysis using a
model with three regressors, x_(1),x_(2),x_(3), and assuming that all 100 measurements
are statistically independent (i.e., no correlation between participants and within
each participant). The model can be expressed as E(y|x_(1),x_(2),x_(3))=eta _(0)+eta _(1)x_(1)+
eta _(2)x_(2)+eta _(3)x_(3). Thus the data analyst performed OLS (i.e., ordinary least-squares)
estimation for all regression coefficients.
a) The data analyst assumed that all 100 measurements are statistically
independent and used the OLS estimator to generate t test for testing
significance of x_(1) at the target significance level alpha . Is the type I error
probability of this t test no larger than the targeted widetilde(alpha ) level? Provide
mathematical proof for your answer. [10 points]
b) Later the data analyst was told that the statistical independence assumption
did not hold. The data analyst was given the variance-covariance matrix V=
sigma ^(2)([1,0.20],[0.20,1]) for each pair of eye measurements of each participant,
where sigma ^(2) is unknown and needs to be estimated.
(b1) Should the analyst adjust estimation and statistical testing for the
regression coefficient vector B=(eta _(0),eta _(1),eta _(2),eta _(3))^(') ? Provide mathematical
proof for
(b2) Construct an unbiased estimator for sigma ^(2) and show unbiasedness.
(b3) Construct the best linear unbiased estimator for eta _(1) [10 points] (b4) Construct a statistically valid test for testing H0: eta _(1) = 0 . [10 points] your answer. [10 points]
1. Fifty people participated in a medical study. Each participant contributed two
eye measurements (labeled y ). Thus the study had a total of 100 measurements
on y from the 50 participants. It is suspected that the two eye measurements
from the same person are correlated. Without sufficient knowledge of the study background, a data analyst performed a multiple linear regression analysis using a model with three regressors, x1, x2, 3, and assuming that all 100 measurements are statistically independent (i.e., no correlation between participants and within each participant). The model can be expressed as E(y (x1,x2,x) = o + x1 + B,x2 + P3x3 . Thus the data analyst performed OLS (i.e., ordinary least-squares) estimation for all regression coefficients.
a) The data analyst assumed that all 100 measurements are statistically
independent and used the OLS estimator to generate t test for testing significance of x, at the target significance level a. Is the type I error probability of this t test no larger than the targeted a level? Provide mathematical proof for your answer. [10 points] b) Later the data analyst was told that the statistical independence assumption
did not hold. The data analyst was given the variance-covariance matrix V = 0.20 for each pair of eye measurements of each participant, 0.20 1 where o2 is unknown and needs to be estimated.
(b1) Should the analyst adjust estimation and statistical testing for the
regression coefficient vector B = (Po, P1, P2 P3)? Provide mathematical proof for your answer. [10 points]