1.
Methods
In order to assess the association between 5-year mortality and presence of
atherosclerotic disease (ASCVD) at baseline, groups with and without ASCVD at
baseline were compared using the odds ratio of mortality comparing the odds of 5-year
mortality in those with ASCVD to the odds of 5-year mortality in those without ASCVD
at baseline. Logistic regression methods were used with a primary predictor indicator
variable of ASCVD, based on history of stroke or coronary heart disease, defined as ‘yes’
for history of any one of the following: transient ischemic stroke, stroke, diagnosis or
angina or diagnosis of myocardial infarction (MI) before baseline MRI. A primary
outcome indicator variable for 5-year mortality was created, defined as ‘yes’ if the
participant had died during study follow-up and the total amount of observation time
contributed by the individual after their MRI was less than 5 years, and ‘no’ if they
survived greater than or equal to 5 years by study completion (Sept 16, 1997).
The odds of 5-year mortality was estimated using the sample proportion, and Wald-type
95% confidence intervals calculated based on the normal approximation maximum
likelihood estimates based on the binomial distribution. P-value for a test of the null
hypothesis of no difference in odds of mortality in groups defined by ASCVD status was
calculated based on a chi square test, and alpha level of 5% was pre-specified for
determination of statistical significance.
Results and Interpretation
Mortality within 5 years of MRI occurred in 68 of 217 (31.3%) participants with ASCVD
at baseline, and 53 of 518 (10.2%) those without ASCVD at baseline. We estimate that
those with ASCVD had 4.00-fold higher odds of 5-year mortality than did those without
ASCVD at baseline. Based on a 95% confidence interval, we would expect our observed
estimate to be consistent with a true relative increase in odds between 2.67-fold and 6.00fold, when comparing those with to those without ASCVD. Based on this data, we reject
the null hypothesis of no difference in likelihood of 5-year mortality by ASCVD status,
and (p-value <.0001 from 2-sided chi square test).
2A. Stratified analysis
Methods
The risk of 5-year mortality was estimated using the sample proportion. Stratified risk
difference analysis using internal standardization (standardized to the ‘exposed’) was
used determine the risk difference, or difference in proportions surviving up to 5 years in
groups differing by prevalence of ASCVD at baseline, adjusting for age at enrollment
(categorized as age <75, or >= 75 years) and sex. 95% confidence intervals for the
stratum-specific differences in proportions were calculated, as well as crude and age- and
sex-standardized differences in proportions (produced by internally-weighted stratumspecific estimates) of 5-year mortality in those with and without ASCVD. To test the null
hypothesis of no difference in proportions dying within 5 years by ASCVD status, after
standardization for age and sex, we assessed whether the 95% confidence intervals for the
standardized risk difference did or did not contain 0.
Results and interpretation
Mortality within 5 years of MRI occurred in 68 of 217 (31.3%) participants with ASCVD
at baseline, and 53 of 518 (10.2%) those without ASCVD at baseline. In an age-and-sexstandardized analysis, presence of ASCVD was associated with an absolute difference of
19.4% in the probability of complete remission (95% CI: 12.5% to 26.1%). Because our
95% confidence interval does not include 0, we reject the null hypothesis of no difference
in 5-year mortality between those with and without ASCVD at baseline, after adjustment
for age and sex.
Results and Interpretation
a. Regression
Groups differing in baseline prevalence of ASCVD were compared using the difference
in probability of survival in each arm: those with ASCVD minus those without ASCVD,
risk difference (RD= p1-p0, where p1 is sample proportion in exposed and p0 is sample
proportion in unexposed). The crude (unadjusted) probability of mortality in 5 years in
groups defined by ASCVD was calculated using the sample proportion. Point estimate
and 95% confidence intervals for and adjusted difference in proportions between ASCVD
exposed and unexposed, with adjustment for covariates age (continuous variable) and
sex, were calculated using a linear regression model with robust standard errors. The
binary predictor variable was presence of ASCVD at baseline (any ASCVD, no ASCVD)
and binary outcome variable was 5-year mortality. A continuous variable for age was
added to the model to attempt to avoid residual confounding due to inappropriate
categorization of age. A binary indicator variable of sex (male, not male) was included in
the model as an additional adjustment variables. The age and sex-adjusted estimate for
the difference in mortality probabilities was based on the regression parameter for
ASCVD status. A Wald-type 95% confidence interval and p-value testing the null
hypothesis of no difference in probabilities of 5-year mortality between ASCVD groups,
after adjustment for age and sex, were calculated assuming an approximately normal
distribution for the regression parameter estimates.
Results and Interpretation
Mortality within 5 years of MRI occurred in 68 of 217 (31.3%) participants with ASCVD
at baseline, and 53 of 518 (10.2%) those without ASCVD at baseline. Based on a linear
regression model adjusting for age and sex, we estimate that ASCVD at baseline was
associated with an 18.9% higher absolute probability of mortality within 5 years (95%
CI: 12.2% to 25.7%). Based on a 2-sided p-value of <.0001, we reject the null hypothesis
of no difference in risk of mortality in 5 years in those with and without ASCVD at
baseline.
2C. Small differences in the point estimates and p-values for the two analysis are due to
the different ways in which the data in each stratum of the covariates are weighted using
each approach, and the way in which variance is calculated in each method.
3a.
Methods
The association between 5-year mortality and ASCVD of baseline were compared using
the odds ratio comparing the odds of mortality in 5 years in those with ASCVD compared
to the odds of mortality in 5-years in those without ASCVD, adjusting for age (less than
75, >= 75 years old)and sex. Crude (unadjusted) estimates of the probability of inducing
a complete remission were estimated for each exposure using the sample proportions.
Mantel-Haenszel adjustment methods of an age and sex-adjusted odds ratio and 95%
confidence intervals, and P- value for the null hypothesis of no difference in mortality
probability in exposure groups, after adjustment for age and sex, were computed using
the M-H statistic assuming the chi square distribution.
Results and Interpretation
Mortality within 5 years of MRI occurred in 68 of 217 (31.3%) participants with ASCVD
at baseline, and 53 of 518 (10.2%) those without ASCVD at baseline. The odds of
mortality in 5 years among those with ASCVD at baseline was 3.44 times the odds of
mortality in 5 years among those without ASCVD at baseline, after adjusting for age and
sex. The M-H sex and age-adjusted 95% confidence interval was 2.26 to 5.24. Based on a
p-value of <.0001, we reject the null hypothesis of no difference in mortality probability
between groups with and without ASCVD, after adjusting for age and sex (p-value
compared to chi square distribution with 1df).
3b.
Methods
Groups differing in baseline prevalence of ASCVD were compared using the odds ratio
comparing in odds of survival in those with ASCVD relative to the odds of survival in
those without ASCVD. The crude, unadjusted probability of mortality in 5 years in
groups defined by ASCVD was calculated using the sample proportions. Point estimate
and 95% confidence intervals for an adjusted odds ratio comparing ASCVD exposed and
unexposed, with adjustment for covariates age (continuous variable) and sex, were
calculated using a logistic regression model. The binary predictor variable was presence
of ASCVD at baseline (any ASCVD, no ASCVD) and binary outcome variable was 5year mortality. A continuous variable for age was added to the model to attempt to avoid
residual confounding due to inappropriate categorization of age. A binary indicator
variable of sex (male, not male) was included in the model as an additional adjustment
variable. The age and sex-adjusted estimate for the odds ratio was based on the
regression parameter for ASCVD status. A Wald-type 95% confidence interval and pvalue testing the null hypothesis of no difference in probabilities of 5-year mortality
between ASCVD groups, after adjustment for age and sex, were calculated assuming an
approximately normal distribution for the regression parameter estimates.
Results and Interpretation
Mortality within 5 years of MRI occurred in 68 of 217 (31.3%) participants with ASCVD
at baseline, and 53 of 518 (10.2%) those without ASCVD at baseline. Based on a logistic
regression model, the sex and age-adjusted odds of mortality in 5 years among those with
ASCVD at baseline was 3.57 times the odds of mortality in 5 years among those without
ASCVD at baseline (95% confidence interval 2.36 to 5.39 higher odds), after adjusting
for age and sex. Based on a two-sided p-value of <.0001, we reject the null hypothesis of
no difference in 5-year mortality probability between groups with and without ASCVD,
after adjusting for age and sex.
3C. The stratified analysis in part A requires that adjustment variables be categorical, and
in the regression model in part B we used a continuous variable for age for adjustment,
which likely resulted in less residual confounding by age and more complete adjustment.
Differences in the point estimates for the two analysis are also due to the different ways
in which the data in each stratum of the covariates are weighted using each approach to
determine the overall OR using information from each stratum, and the way in which
variance is calculated in each method. The M-H estimates are closely related to the score
test, but we used the Wald test to test the hypothesis in the logistic regression model.
4A. Methods
To assess association between ASCVD and 5-year mortality, adjusting for the effects of
age and sex, a risk ratio comparing the probability of 5-year mortality in those with
ASCVD at baseline was compared to probability in those without ASCVD at baseline,
using Mantel-Haenszel adjustment methods. Crude (unadjusted) estimates of the
probability of 5-year mortality were estimated in each exposure group using the sample
proportions.
Point estimates and interval estimates for the ratio of 5-year mortality probabilities, after
adjustment for age and sex, were calculated using stratified analysis with MantelHaenszel adjustment methods. The primary outcome of interest was a binary indicator of
mortality and primary exposure of interest was presence or absence of ASCVD at
baseline. Stratum-specific estimates and 95% confidence intervals for the relative risk of
mortality were calculated in each strata of sex and age category (<75, > or = 75 years). A
M-H combined, or summary RR estimate, and 95% confidence interval for the summary
estimate were calculated by averaging the stratum-specific estimates using M-H weights
for estimates contributed by each strata, based on the amount of information in the strata.
No p-value for the statistical significant of the estimated parameter to test the null
hypothesis of no difference risk between exposure groups, after adjustment for sex and
age, is provided. Therefore, inference on the test of the null hypothesis was based on the
95% confidence intervals for the point estimate. Ideally, we would test the null
hypothesis with a M-H chi- square test of whether the M-H RR was 1 or not 1.
4A. Results and Interpretation
Mortality within 5 years of MRI occurred in 68 of 217 (31.3%) participants with ASCVD
at baseline, and 53 of 518 (10.2%) those without ASCVD at baseline. Based on a
stratified analysis using M-H adjustment methods to account for age and sex, we estimate
that the relative risk of (incidence rate ratio) of 5-year mortality in those with ASCVD
was 2.65 times that in those without ASCVD at baseline (95% CI 1.92, 3.65). Because
the 95% CI does not cross 1, we reject the null hypothesis of no difference in risk of 5year mortality between those with and without ASCVD, among those of similar age and
sex, at alpha level of 5%.
4B. Methods
To assess association between ASCVD and 5-year mortality, adjusting for the effects of
age and sex, a risk ratio comparing the probability of 5-year mortality in those with
ASCVD at baseline was compared to probability in those without ASCVD at baseline.
Crude (unadjusted) estimates of the probability of 5-year mortality were estimated in
each exposure group using the sample proportions. Point estimates and interval estimates
for the ratio of 5-year mortality probabilities were based on a poisson regression methods
using a binary indicator of mortality in a model that included an indicator of exposure
group (ASCVD vs. no ASCVD), an indicator of sex, and a continuous variable for age.
The age and sex-adjusted estimate of the risk associated with ASCVD was based on the
regression parameter for the indicator of ASCVD. A 95% confidence interval and p value
were computed using the approximate normal distribution for the regression parameter
estimates along with robust standard errors (Huber-White sandwich estimator) to account
for the mean-variance relationship of binary response variables in independent
observations.
4B. Results and interpretation
Mortality within 5 years of MRI occurred in 68 of 217 (31.3%) participants with ASCVD
at baseline, and 53 of 518 (10.2%) those without ASCVD at baseline. Based on a poisson
regression model with adjusting for age and sex, we estimate that the relative risk of
(incidence rate ratio) of 5-year mortality in those with ASCVD was 2.72 times that in
those without ASCVD at baseline (95% CI 1.96 to 3.76). Based on a 2-sided p-value of
<.0001, we reject the null hypothesis of no difference in risk of 5-year mortality between
those with and without ASCVD, among those of similar age and sex.
4C. The estimate for the RR in the stratified analysis comes from a pooling of the
weighted stratum-specific relative risks, from each stratum of combinations of
presence/absence of adjustment variables. The M-H weighting comes from the size of the
specific strata (how much data it contributes). The standard errors (and thus 95%
confidence intervals) for the estimates from M-H adjustment come from different
formula for estimating variance as do those from the regression model. The inference is
the same.
5. For this research question we are interested in association, not the public health impact,
so a relative measure of association is more appropriate. If the baseline prevalence of the
disease is high in the population/high in the unexposed, even with a small difference in
relative measure of effect (RR or OR) the risk difference will be very large, which
implies a strong association. I tend to prefer relative risk to assess association in a cohort
study (versus odds), because the relative risk is more interpretable to lay people than is
the odds ratio. When analyzing odds after adjustment for a precision variable, we may
observe a more extreme estimate because the odds using the average of the stratum
specific proportions is not the average of the stratum specific odds. There can be
attenuation due to the non-collapsibility of the OR.
6A. Methods
The association between incidence of colorectal cancer and birthplace, after adjustment
for age, sex and SEER site, was conducted by estimating the incidence rate ratio for
cancer comparing the rate in foreign born to non-foreign born. Crude rates were based on
event counts divided by person-time at risk in each exposure group. Point and interval
estimates for the ratio of cancer rates were calculated using standardization based on the
US population, to determine standardized rates of event counts (cases of cancer) per unit
of observation time (person-years of observation), in groups defined by foreign or nonforeign-born status. Standardization was used to adjust for differences in distributions of
exposure groups in SEER site, age, and sex. The sex, SEER site and age-standardized
IRR was based on the distribution of these three covariates in the US population census.
A 95% confidence interval and p-value were computed assuming the approximate normal
distribution for the standardized estimates. We tested the null hypothesis that the IRR =1
vs. didn’t equal 1.
Results and Interpretation
The crude incidence rate for colorectal cancer among those who were foreign born was
89.27 cases per 105 person-years of observation, and 32.59 cases per person-year of
observation among those who were not foreign born.
6B. Methods
The association between incidence of colorectal cancer and birthplace, after adjustment
for age, sex and SEER site, was conducted by estimating the incidence rate ratio for
cancer comparing the rate in foreign born to non-foreign born. Crude rates were based on
event counts divided by person-time at risk in each exposure group. Point and interval
estimates for the ratio of cancer rates were based on Poisson regression for a rate of event
counts (cases of cancer) per unit of observation time (person-years of observation), with
primary predictor and indicator of foreign born, in a model that included an indicator of
sex, SEER site (categorical, dummy variable), and age as a continuous variable. The sex,
SEER site and age-adjusted IRR was based on the regression parameter for the indicator
of foreign born. A 95% confidence interval and p-value were computed assuming the
approximate normal distribution for the regression parameter estimates, testing the null
hypothesis that the IRR =1 vs. didn’t equal 1.
Results and inference
The crude incidence rate for colorectal cancer among those who were foreign born was
89.27 cases per 105 person-years of observation, and 32.59 cases per 105 person-years of
observation among those who were not foreign born. In the Poisson regression model, the
incidence rate in foreign born was 0.95 (95% CI 0.93 to 0.97) that of the non-foreign
born than non foreign born, after adjustment for age, sex and SEER site. This indicates
that foreign-born males had approximately 5% lower rate than did non-foreign men, after
adjustment. Based on a two-sided p-value of <.0001, we reject the null hypothesis of no
relative difference (IRR=1) in cancer rates between foreign and non-foreign born.
6C. The difference in the statistical models is based on the way in which the subgroupspecific rates from each combination of the covariates of interest (stratum) are weighted
to create the overall averaged rates in the exposed and unexposed groups, as well as the
differences in the way in which the variance (and standard errors) are calculated. The
Poisson regression model uses a log link to estimate rates and then creates an average
difference on a log scale, and exponentiates the estimates to get the geometric mean in
each stratum (versus mean).