Biost 536 HW2
October 12, 2014
1. Methods: The minimum and maximum values for observation time were calculated in
terms of days, stratified by whether the patient had died or not.
Results: As the table below shows, the minimum observation time for a patient in the
study who had not died was 1827 days (i.e. just over 5 years). Thus, the vital status of
everyone is known at 5 years.
Minimum observation time Maximum observation time
Died
68
2022
Not died
1827
2159
2a. Methods: Stratified analysis was used to compare the absolute difference in 5 year
survival between subjects with and without baseline atherosclerotic CVD (ASCVD,
defined by history of prior angina, myocardial infarction, transient ischemic attacks, or
stroke), adjusting for age and sex. Age (65-69, 70-74, 75-79, 80-84, 85+) was
categorized. The standardized population was the “unexposed” distribution, or those
without ASCVD. Two-sided p-value and 95% confidence interval were computed using
Wald statistics.
Results: 10.2% of subjects with ASCVD died within 5 years, while 31.3% of those
without ASCVD died within 5 years. The estimated probability of death within 5 years is
an absolute 18.1% lower for those with baseline ASCVD compared to those without.
Based on a 95% confidence interval, this observed difference would not be unusual if
the true difference in probabilities were anywhere from 11.5% to 24.6% higher in the
ASCVD group. A two-sided p-value of less than 0.05 suggests that we can with high
confidence reject the null hypothesis that the probability of 5-year mortality is not
associated with baseline ASCVD.
2b. Methods: The probabilities of dying within 5 years were compared between
subjects with and without ASCVD (as defined in 2a), adjusting for age (categorized as
2a) and sex using a linear regression model. Statistical inference in the difference in the
adjusted probabilities of death was based on the Wald statistic computed from the
regression slope parameters and its standard error as estimated using the Huber-White
sandwich estimator. Two-sided p value and 95% confidence interval were calculated
using approximate normal distribution for linear regression parameter estimates.
Results: 10.2% of subjects with ASCVD died within 5 years, while 31.3% of those
without ASCVD died within 5 years. We estimate from linear regression that death
within 5 years is an absolute 18.9% higher in those with ASCVD compared to those
without. Based on the 95% confidence interval, this observed difference would not be
judged unusual if the true difference in probabilities were anywhere from 12.2% to
25.6% higher in those with ASCVD compared to those without. A two sided p
value<0.001 suggests that we can with high confidence reject the null hypothesis that
the probability of 5 year mortality is not associated with baseline ASCVD.
2c. There was a difference of 0.8% in the adjusted difference in probability of 5 year
mortality between ASCVD groups using stratified analysis and linear regression. This is
due to the choice of standard population for the stratified analysis and the averaging
over strata for regression analysis.
3a. Methods: Using stratified analysis, the odds of dying within 5 years were compared
between those with and without ASCVD, adjusting for age and sex. Age (65-69, 70-74,
75-79, 80-84, 85+) was categorized. The standardized population was the “unexposed”
distribution, or those without ASCVD. Two-sided p-value and 95% confidence interval
were computed using Wald statistics.
Results: 10.2% of subjects with ASCVD died within 5 years, while 31.3% of those
without ASCVD died within 5 years. From stratified analysis, we estimate that the odds
of death within 5 years for those with ASCVD is 3.76 times the odds of death within 5
years for those without ASCVD. Based on a 95% confidence interval, this observed
odds ratio would not be judged unusual if the true odds ratio were anywhere between
2.62 and 6.12. A two-sided p value <0.05 suggests that we can with high confidence
reject the null hypothesis that the odds of 5 years mortality is not associated with
baseline ASCVD.
3b. Methods: The odds of dying within 5 years were compared between subjects with
and without ASCVD (as defined in 2a), adjusting for age (categorized as 2a) and sex
using a logistic regression model. Statistical inference on the adjusted ratio of odds of
death was based on the Wald statistic computed from the regression slope parameters
and its standard error as estimated using the Huber-White sandwich estimator. Twosided p value and 95% confidence interval were calculated using approximate normal
distribution for linear regression parameter estimates.
Results: 10.2% of subjects with ASCVD died within 5 years, while 31.3% of those
without ASCVD died within 5 years. From logistic regression, we estimate that the odds
of death within 5 years are 3.52 times higher for those with ASCVD compared to those
without. Based on the 95% confidence interval, this observed odds ratio would not be
judged unusual if the true survival odds ratio comparing those with and without ASCVD
was anywhere between 2.33 and 5.33. A two-sided p value<0.001 suggests that we can
with high confidence reject the null hypothesis that the odds of 5 year mortality is not
associated with baseline ASCVD.
3c. There was a difference in odds ratio between the stratified approach (OR=3.76) and
the logistic regression approach (OR=3.52). This is due to the choice of standard
population for the stratified analysis and the averaging over strata for regression
analysis.
4a. Methods: The ratios in risk of dying within 5 years were compared between
subjects with and without ASCVD (as defined in 2a), adjusting for age (categorized as
2a) and sex using stratified analysis. The standardized population was the “unexposed”
distribution, or those without ASCVD. Two-sided p-value and 95% confidence interval
were computed using Wald statistics.
Results: 10.2% of subjects with ASCVD died within 5 years, while 31.3% of those
without ASCVD died within 5 years. From stratified analysis, we estimate that the risk of
5 year mortality for those with ASCVD is 3.03 times that for those without ASCVD.
Based on a 95% confidence interval, this observed ratio would not be judged unusual if
the true ratio of survival were anywhere from 2.09 to 4.39. A two sided p value<0.05
suggests that we can with high confidence reject the null hypothesis that the probability
of 5 year mortality is not associated with ASCVD.
4b. Methods: The probabilities of dying within 5 years were compared between
subjects with and without ASCVD (as defined in 2a), adjusting for age (categorized as
2a) and sex using a Poisson regression model. Statistical inference on the adjusted
ratio of risk of death was based on the Wald statistic computed from the regression
slope parameters and its standard error as estimated using the Huber-White sandwich
estimator. Two-sided p value and 95% confidence interval were calculated using
approximate normal distribution for linear regression parameter estimates.
Results: 10.2% of subjects with ASCVD died within 5 years, while 31.3% of those
without ASCVD died within 5 years. From Poisson regression analysis, we estimate that
the probability of death within 5 years among those with ASCVD is 2.69 times the
probability of death within 5 years among those without ASCVD. Based on a 95%
confidence interval, this observed rate ratio would not be judged unusual if the true ratio
of survival probabilities were between 1.94 and 3.74. A two-sided p value<0.001
suggests that we can with high confidence reject the null hypothesis that the probability
of 5 year mortality is not associated with ASCVD.
4c. There was a difference in risk ratio between the stratified approach (OR=3.03) and
the logistic regression approach (OR=2.69). This is due to the choice of standard
population for the stratified analysis and the averaging over strata for regression
analysis.
5. All three measures of association (RD, RR, OR) showed an association between
baseline ASCVD and 5-year mortality. Unlike the other two, the risk difference shows an
absolute difference in survival probabilities, which is more applicable for understanding
public health impact. Thus the risk difference may be a better approach.
6a. Methods: Using stratified analysis, the incidence ratio was used to assess the
association between incidence of colorectal cancer (CRC) and birthplace (foreign vs.
US born) after adjustment by age (in 5 year categories) and sex. The standardized
population was the US population. Two-sided p-value and 95% confidence interval were
computed using Wald statistics.
Results: Among foreign born, there was an average of 34 CRC diagnoses per SEER
site. Among US born, there was an average of 193 CRC diagnoses per SEER site.
From stratified analysis, we estimate that the incidence of colorectal cancer among
those who were foreign born is 1.02 times that of those who were US born. From the
95% confidence interval, this observed ratio would not be judged unusual if the true
incidence ratio of were anywhere from 0.99 to 1.05. A two-sided p-value>0.05 suggests
that we cannot with high confidence reject the null hypothesis that the incidence of CRC
is not associated with birthplace.
6b. Methods: Using Poisson regression, the incidence of CRC was compared between
those who were foreign and US born, adjusting for age and sex. Statistical inference on
the adjusted ratio of CRC incidence was based on the Wald statistic computed from the
regression slope parameters and its standard error as estimated using the Huber-White
sandwich estimator. Two-sided p value and 95% confidence interval were calculated
using approximate normal distribution for linear regression parameter estimates.
Results: Among foreign born, there was an average of 34 CRC diagnoses per SEER
site. Among US born, there was an average of 193 CRC diagnoses per SEER site.
From Poisson regression, we estimate that the incidence of colorectal cancer among
those who were foreign born is 0.99 times that of those who were US born. From the
95% confidence interval, this observed ratio would not be judged unusual if the true
incidence ratio were anywhere from 0.90 and 1.10. A two-sided p-value of 0.90
suggests that we cannot with high confidence reject the null hypothesis that the
incidence of CRC is not associated with birthplace.
6c. In both cases, we were unable to reject the null hypothesis. The estimates from the
two methods differed slightly (1.02 vs. 0.99), due in part to the averaging over effect
modification in Poisson regression as opposed to the use of the directly standardized
rates in stratified analysis.