BIOST 536
Homework 2
1. The data was checked to determine whether there were any cases where 5
year mortality was censored. All study participants who were followed up for
less than five years (defined as 5 times 365.25 days) died during the course
of the study. Therefore, there was no censoring of five year mortality in the
dataset.
Groups defined by the risk factor of having ASCVD were compared using the
odds ratio comparing the odds of dying within five years among those who
had ASCVD compared to those who did not have ASCVD. Point and interval
estimates for the risk factor odds ratio were based on a logistic regression of
the binary indicator of dying within five years on a model that included a
binary indicator of having ASCVD. The estimate of treatment effect was based
on the regression parameter for the indicator of having ASCVD. A 95%
confidence interval and p value were computed assuming the approximate
normal distribution for the regression parameter estimate.
Based on the aforementioned logistic regression model, having ASCVD is
estimated to be associated with 4.00-fold higher odds of dying within five
years (95% CI: 2.67-fold to 6.00-fold higher odds). Based on a two-sided p
value of < 0.01, we reject the null hypothesis that the having ASCVD is not
associated with five year mortality.
2.
a. A stratified analysis was performed, with a binary indicator for five
year mortality as the outcome, and a binary indicator of having ASCVD
in the model. The analysis stratified by sex and a binary indicator of
being over 74 years in age. A standardized risk difference was
calculated, using weights with respect to the exposed distribution. A
95% confidence interval and p value were computed assuming the
approximate normal distribution for the regression parameter
estimate.
Based on the analysis described above, those with ASCVD are
estimated to have 0.19 higher risk of death within five years than
those without ASCVD (95% CI: .13 to .26 higher risk), after adjusting
for age and sex. Based on a two-sided p value of < 0.05, we reject the
null hypothesis that ASCVD is not associated with five year mortality,
after adjusting for age and sex.
b. Groups defined by the risk factor of having ASCVD were compared
using the risk difference comparing the risk of dying within five years
among those who had ASCVD compared to those who did not have
ASCVD. Point and interval estimates for the risk factor risk difference
were based on a linear regression of the binary indicator of dying
within five years on a model that included a binary indicator of having
ASCVD, as well as a binary indicator for sex, and a binary indicator of
BIOST 536
Homework 2
being over 74 years of age. The estimate of treatment effect was based
on the regression parameter for the indicator of having ASCVD. A 95%
confidence interval and p value were computed assuming the
approximate normal distribution for the regression parameter
estimate, using robust standard errors.
Based on the analysis described above, those with ASCVD are
estimated to have 0.19 higher risk of death within five years than
those without ASCVD (95% CI: .12 to .26 higher risk), after adjusting
for age and sex. Based on a two-sided p value of < 0.01, we reject the
null hypothesis that ASCVD is not associated with five year mortality,
after adjusting for age and sex.
c. The results in the two models were very similar. If the results
displayed had not been rounded to two digits, the differences would
have been more readily noticeable. The small differences are likely
due to the fact that in the first analysis, a certain standardization was
chosen (using weights with respect to the exposed distribution),
whereas in the second analysis, no such standardization was done.
3.
a. A stratified analysis was performed, with a binary indicator for five
year mortality as the outcome, and a binary indicator of having ASCVD
in the model. The analysis stratified by sex and a binary indicator of
being over 74 years in age. A standardized odds ratio was calculated,
using the default weights in STATA (leading to the Mantel–Haenszel
standardization). A 95% confidence interval and p value were
computed assuming the approximate normal distribution for the
regression parameter estimate.
Based on the analysis described above, those with ASCVD are
estimated to have 3.44-fold odds of death within five years, compared
to those without ASCVD (95% CI: 2.28 to 5.18-fold higher odds), after
adjusting for age and sex. Based on a two-sided p value of < 0.05, we
reject the null hypothesis that ASCVD is not associated with five year
mortality, after adjusting for age and sex.
b. Groups defined by the risk factor of having ASCVD were compared
using the odds ratio comparing the odds of dying within five years to
those who had ASCVD compared to those who did not have ASCVD.
Point and interval estimates for the risk factor odds ratio were based
on a logistic regression of the binary indicator of dying within five
years on a model that included a binary indicator of having ASCVD, as
well as a binary indicator for sex, and a binary indicator of being over
74 years of age. The estimate of treatment effect was based on the
regression parameter for the indicator of having ASCVD. A 95%
BIOST 536
Homework 2
confidence interval and p value were computed assuming the
approximate normal distribution for the regression parameter
estimate.
Based on the analysis described above, those with ASCVD are
estimated to have 3.55-fold odds of death within five years, compared
to those without ASCVD (95% CI: 2.35 to 5.36-fold higher odds), after
adjusting for age and sex. Based on a two-sided p value of < 0.01, we
reject the null hypothesis that ASCVD is not associated with five year
mortality, after adjusting for age and sex.
c. Note that in the first analysis for this problem, the Mantel-Haenszel
statistic is closely related to score test from the logistic regression in
the second analysis, but the test in the second analysis is based off of a
Wald test. For this reason, the conclusions from the analyses pretty
much agree, even though the numbers are not exactly the same.
4.
a. A stratified analysis was performed, with a binary indicator for five
year mortality as the outcome, and a binary indicator of having ASCVD
in the model. The analysis stratified by sex and a binary indicator of
being over 74 years in age. A standardized risk ratio was calculated,
using the default weights in STATA (leading to the Mantel–Haenszel
standardization). A 95% confidence interval and p value were
computed assuming the approximate normal distribution for the
regression parameter estimate.
Based on the analysis described above, those with ASCVD are
estimated to have 2.65-fold risk of death within five years, compared
to those without ASCVD (95% CI: 1.92 to 3.65-fold higher risk), after
adjusting for age and sex. Based on a two-sided p value of < 0.05, we
reject the null hypothesis that ASCVD is not associated with five year
mortality, after adjusting for age and sex.
b. Groups defined by the risk factor of having ASCVD were compared
using the risk ratio comparing the risk of dying within five years to
those who had ASCVD compared to those who did not have ASCVD.
Point and interval estimates for the risk factor risk ratio were based
on a poisson regression of the binary indicator of dying within five
years on a model that included a binary indicator of having ASCVD, as
well as a binary indicator for sex, and a binary indicator of being over
74 years of age. The estimate of treatment effect was based on the
regression parameter for the indicator of having ASCVD. A 95%
confidence interval and p value were computed assuming the
approximate normal distribution for the regression parameter
estimate.
BIOST 536
Homework 2
Based on the analysis described above, those with ASCVD are
estimated to have 2. 73-fold risk of death within five years, compared
to those without ASCVD (95% CI: 1.90 to 3.94-fold higher risk), after
adjusting for age and sex. Based on a two-sided p value of < 0.01, we
reject the null hypothesis that ASCVD is not associated with five year
mortality, after adjusting for age and sex.
c. Note that in the first analysis for this problem, the Mantel-Haenszel
statistic is closely related to score test from the poisson regression in
the second analysis, but the test in the second analysis is based off of a
Wald test. For this reason, the conclusions from the analyses pretty
much agree, even though the numbers are not exactly the same.
5. The three approaches are similar in that an association between ASCVD and
some transformation of the risk of five year mortality is being investigated,
after adjusting for age and sex. However, in the first approach the risk is not
transformed, while the second and third approaches, logit and log
transformations are used, respectively.
Because for most people, odds are harder to interpret than risk, the first and
third approaches may be more desirable than the second approach. Whether
the first or third approach is preferred depends on whether relative
differences or absolute differences are more relevant in a scientific sense. If
comparisons on a relative scale are most important, then the third approach
is preferred. Conversely, if comparisons on an absolute scale are most
important, then the first approach is preferred.
6.
a. A stratified analysis was performed, with a binary indicator of getting
colorectal cancer as the outcome, and a binary indicator of being born
outside of the U.S. in the model. Person-years observied are adjusted
for in the model. The analysis stratified by age and SEER site. A
standardized risk ratio was calculated, using weights such that the
standardization is to the U.S. population. A 95% confidence interval
and p value were computed assuming the approximate normal
distribution for the regression parameter estimate.
Based on the analysis described above, those born outside the U.S. are
estimated to have 1.02-fold risk of colorectal cancer, compared to
those born in the U.S. (95% CI: 0.99 to 1.05-fold higher risk), after
adjusting for person-years observed, age, and SEER site. Based on a
two-sided p value of > 0.05, we fail to reject the null hypothesis that
ASCVD is not associated with five year mortality, after adjusting for
age and sex.
BIOST 536
Homework 2
b. Groups defined by the risk factor of place of birth were compared
using the risk ratio comparing the risk of colorectal cancer among
those born outside the U.S. compared to those born in the U.S. Point
and interval estimates for the risk factor risk ratio were based on a
poisson regression of the binary indicator of getting colorectal cancer
on a model that included a binary indicator of being born outside the
U.S., as well as number of person-years observed, SEER site, and age.
The estimate of treatment effect was based on the regression
parameter for the indicator of being born outside of the U.S. A 95%
confidence interval and p value were computed assuming the
approximate normal distribution for the regression parameter
estimate.
Based on the analysis described above, those born outside the U.S. are
estimated to have 0.116-fold risk of colorectal cancer, compared to
those born in the U.S. (95% CI: 0.114 to 0.119-fold higher risk), after
adjusting for number of person-years observed, SEER site, and age.
Based on a two-sided p value of < 0.01, we reject the null hypothesis
that location of birth is not associated with colorectal cancer, after
adjusting for number of person-years observed, SEER site, and age.
c. In the first analysis, standardization to the U.S. population is done.
This is not done in the second analysis. For this reason, we see wildly
different answers in the two analyses.