BIOST 536
1
Homework 4
1.
Coefficient 𝑖
𝛽𝑖
𝑒 𝛽𝑖
0: Intercept
-4.15 0.02
1: Age
0.04 1.05
2: Diagnosis of Angina
0.25 1.28
3: Diagnosis of Myocardial Infarction 0.57 1.77
𝑒 𝛽0 is an estimate of the odds of having regions identified on the MRI scan
which were suggestive of infarcts, among those of age zero and no diagnosis
of coronary heart disease prior to the MRI.
When comparing two groups differing in age by one year, but with the same
prior coronary heart disease diagnosis status, 𝑒 𝛽1 is an estimate of the ratio
of odds of having regions identified on the MRI scan which were suggestive
of infarcts, comparing the older group to the younger group.
When looking at study participants of the same age, 𝑒 𝛽2 is an estimate of the
ratio of odds of having regions identified on the MRI scan which were
suggestive of infarcts, comparing those with prior diagnosis of angina to
those with no prior diagnosis of coronary heart disease.
When looking at study participants of the same age, 𝑒 𝛽3 is an estimate of the
ratio of odds of having regions identified on the MRI scan which were
suggestive of infarcts, comparing those with prior diagnosis of myocardial
infarction to those with no prior diagnosis of coronary heart disease.
2.
Coefficient 𝑖
0: Intercept
1: Age
2: Diagnosis of Angina
3: Diagnosis of Myocardial Infarction
4: Age × Diagnosis of Angina
5: Age × Diagnosis of Myocardial Infarction
𝛽𝑖
-4.35
0.05
0.41
2.31
0.00
-0.02
𝑒 𝛽𝑖
0.01
1.05
1.51
10.07
1.00
0.98
𝑒 𝛽0 is an estimate of the odds of having regions identified on the MRI scan
which were suggestive of infarcts, among those of age zero and no diagnosis
of coronary heart disease prior to the MRI.
When comparing two groups differing in age by one year, but with no prior
coronary heart disease diagnosis status, 𝑒 𝛽1 is an estimate of the ratio of
odds of having regions identified on the MRI scan which were suggestive of
infarcts, comparing the older group to the younger group.
BIOST 536
Homework 4
2
Among those of age zero, 𝑒 𝛽2 is an estimate of the ratio of odds of having
regions identified on the MRI scan which were suggestive of infarcts,
comparing those with prior diagnosis of angina to those with no prior
diagnosis of coronary heart disease.
Among those of age zero, 𝑒 𝛽3 is an estimate of the ratio of odds of having
regions identified on the MRI scan which were suggestive of infarcts,
comparing those with prior diagnosis of myocardial infarction to those with
no prior diagnosis of coronary heart disease.
When comparing two groups differing in age by one year, with prior
diagnosis of angina, 𝑒 𝛽4 is an estimate of the ratio of odds of having regions
identified on the MRI scan which were suggestive of infarcts, comparing the
older group to the younger group.
When comparing two groups differing in age by one year, with prior
diagnosis of myocardial infarction, 𝑒 𝛽5 is an estimate of the ratio of odds of
having regions identified on the MRI scan which were suggestive of infarcts,
comparing the older group to the younger group.
3. Methods: The binary indicator of having regions identified on the MRI scan
which were suggestive of infarcts was analyzed using logistic regression on a
binary indicator of having been diagnosed with angina prior to the MRI, a
binary indicator of having been diagnosed with myocardial infarction prior
to the MIR, and linear splines modeling a categorization of age into 7
intervals: age less than 65, 65 < age < 70, 70 < age < 75, 75 < age < 80, 80 <
age < 85, 85 < age < 90, and 90 < age. The linearity in the association between
age and the log odds of having such regions identified, after holding constant
the status of prior diagnosis coronary heart disease, was then evaluated by
simultaneously testing that all regression coefficients for the linear spline
variables were equal to each other, at the 𝛼 = 0.05 level. The model-based
estimate of the standard error of the regression parameters was used to
compute a two-sided p value from a Wald test of association.
Results: Logistic regression analysis of having regions identified on the MRI
scan which were suggestive of infarcts, across age groups using a linear
spline model fit to the categorization of age, after adjusting for the status of
prior diagnosis coronary heart disease, finds statistically significant evidence
that the association between age and having such regions identified is not
linear (two-sided p < 0.0005).
4. Methods: The binary indicator of having regions identified on the MRI scan
which were suggestive of infarcts was analyzed using logistic regression on
age in years modeled continuously, ldl levels in mg/dl modeled continuously,
and ldl levels squared modeled continuously. The U-shaped association
BIOST 536
Homework 4
3
between ldl and the log odds of having such regions identified, after holding
age constant, was then evaluated by testing that the regression coefficient for
ldl squared was zero, at the 𝛼 = 0.05 level. The model-based estimate of the
standard error of the regression parameter was used to compute a two-sided
p value from a Wald test of association.
Results: Logistic regression analysis of having regions identified on the MRI
scan which were suggestive of infarcts, on ldl, ldl squared, and age, finds
statistically significant evidence that the association between ldl and having
such regions identified is U-shaped (two-sided p = 0.02).