BIOST 536: Assignment #3
Due October 30, 2014
Student Code: 3291
Question 1:
Methods: An association between participants’ ankle-arm index (AAI) and 4-year mortality can be
assessed using a logistic regression model with AAI as a continuous predictor of interest and four-year
mortality as a binary outcome variable. No participant observations need to be censored prior to four
years, since all participants who lived for the first four years of follow-up had data for at least four years;
the shortest follow-up time for participants who did not die was 1,480 days, which is 4.05 years. 121
observations were not included since these observations did not have baseline AAI measurements
recorded. Four-year mortality was estimated using sample proportions, with ratios of odds of mortality
compared using 95% confidence intervals using Wald-type maximum likelihood ratios. Two-sided pvalues were provided testing the null hypothesis of no difference in the odds of four-year mortality
across AAI measurements.
Inference: Participants with a higher baseline AAI measurement of one unit had 0.06 the odds of
participants with a baseline AAI measurement one unit lower (95% CI, 0.04, 0.10). Based on a two-sided
p value of < 0.001 we reject the null hypothesis that participants with AAI measurements differing by
one unit have the same odds of mortality.
Question 2
2a.
Methods: The mean, median, and range of data for the baseline AAI measurements were determined to
assess how the data are distributed. Since the outcome variable is binary (whether a participant died
within four years or not), its distribution was not evaluated. As above, the length of follow-up for
participants who did not die within four years was determined to identify any cases that should be
censored. A boxplot was also generated to assess the distribution of the data. Missing data was also
assessed.
Inference: The range of baseline AAI measurements is 0.28 to 2.4, the mean is 1.06, and the median is
1.08. The boxplot shows that the baseline AAI measurements are normally distributed. 121 observations
do not have baseline AAI data and will therefore not be included in analyses of the association with
baseline AAI and four-year mortality.
2b.
Methods: Baseline AAI measurements were modeled continuously and four-year mortality was included
as a binary outcome variable in a linear regression model. No participant’s observations were censored
prior to four years, so four-year mortality within each AAI group were estimated using sample
proportions, with differences in mortality rates compared using 95% confidence intervals computed
using Wald-type maximum likelihood methods. The two-sided p value testing the null hypothesis of no
difference in the probability of four year survival across AAI measurements was based on the standard
normal distribution.
Inference: The absolute risk difference of four year mortality between participants with AAI
measurements 1 unit higher was -0.28 when compared with participant’s with AAI measurements 1 unit
lower, which was statistically significant (95% CI, -0.23, -0.33; two-sided P<0.0001).
2c.
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BIOST536Homework_3_ID: 3291
Methods: Log baseline AAI measurements were modeled continuously and four-year mortality was
included as a binary outcome variable in a linear regression model. No participant’s observations were
censored prior to four years, so four-year mortality within each AAI group were estimated using sample
proportions, with differences in mortality rates compared using 95% confidence intervals computed
using Wald-type maximum likelihood methods. The two-sided p value testing the null hypothesis of no
difference in the probability of four year survival across AAI measurements was based on the standard
normal distribution.
Inference: The absolute risk difference of four year mortality between participants with log AAI
measurements 1 unit higher was -0.28 when compared with participant’s with log AAI measurements 1
unit lower, which was statistically significant (95% CI, -0.23, -0.32; two-sided P<0.0001).
2d.
Methods: Baseline AAI measurements were modeled continuously, as an untransformed and squared
term, and four-year mortality was included as a binary outcome variable in a linear regression model. No
participant’s observations were censored prior to four years, so four-year mortality within each AAI
group were estimated using sample proportions, with differences in mortality rates compared using 95%
confidence intervals computed using Wald-type maximum likelihood methods. The two-sided p value
testing the null hypothesis of no difference in the probability of four year survival across AAI
measurements was based on the standard normal distribution.
Inference: The absolute risk difference between participant’s with higher squared AAI measurements
and lower squared AAI measurements was -0.12, which was statistically significant (95% CI, -0.15, -0.10;
two-sided P<0.0001).
2e.
Methods: Baseline AAI measurements were modeled continuously with dummy variables representing
biologically relevant cut points (0.25, 0.55, 0.75, 0.95, 1.15, 1.35, 1.55, 2.4) and four-year mortality was
included as a binary outcome variable in a linear regression model. No participant’s observations were
censored prior to four years, so four-year mortality within each AAI group were estimated using sample
proportions, with differences in mortality rates compared using 95% confidence intervals computed
using Wald-type maximum likelihood methods. The two-sided p value testing the null hypothesis of no
difference in the probability of four year survival within AAI groups was based on the standard normal
distribution.
Inference: The absolute risk difference between adjacent groups of AAI, as you move into increasing AAI
measurements is -0.27; this finding was statistically significant (95% CI, -0.32, -0.23; two-sided P<0.001).
2f.
Methods: Baseline AAI measurements were modeled continuously with dummy variables for quintiles
were 0.70, 1.12, 1.54, 1.96 and four-year mortality was included as a binary outcome variable in a linear
regression model. No participant’s observations were censored prior to four years, so four-year
mortality within each AAI group were estimated using sample proportions, with differences in mortality
rates compared using 95% confidence intervals computed using Wald-type maximum likelihood
methods. The two-sided p value testing the null hypothesis of no difference in the probability of four
year survival within AAI groups was based on the standard normal distribution.
Inference: The absolute risk difference between adjacent groups of AAI, as you move into increasing AAI
measurements is -0.15; this finding was statistically significant (95% CI, -0.0.18, -0.12; two-sided
P<0.001).
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BIOST536Homework_3_ID: 3291
2g and 2h. I was unable to complete the answers to these problems by the due date/time, so they are
omitted.
Question 3
Methods: The mean, median, and range of data for the baseline AAI measurements were determined to
assess how the data are distributed. Since the outcome variable is binary (whether a participant died
within four years or not), its distribution was not evaluated. As above, the length of follow-up for
participants who did not die within four years was determined to identify any cases that should be
censored. A boxplot was also generated to assess the distribution of the data. Missing data was also
assessed.
Inference: The range of baseline AAI measurements if 0.28 to 2.48, the mean is 1.06, and the median is
1.08. The boxplot shows that the baseline AAI measurements are normally distributed. 121
observations do not have baseline AAI data and will therefore not be included in analyses of the
association with baseline AAI and four-year mortality.
3b.
Methods: Baseline AAI measurements were modeled continuously and four-year mortality was included
as a binary outcome variable in a logistic regression model. No participant’s observations were censored
prior to four years, so four-year mortality within each AAI group were estimated using sample
proportions, with ratios of odds of mortality compared using 95% confidence intervals computed using
Wald-type maximum likelihood methods. Two-sided p values testing the null hypothesis of no difference
in the odds of four year mortality for every increase in baseline AAI measurement were based on the
standard normal distribution.
Inference: For every one unit increase in baseline AAI measurements participants had 0.06 the odds of
dying within four years of study entry, which was statistically significant (95% CI, 0.39, 0.10; two-sided
P<0.0001).
3c.
Methods: Log baseline AAI measurements were modeled continuously and four-year mortality was
included as a binary outcome variable in a logistic regression model. No participant’s observations were
censored prior to four years, so four-year mortality within each AAI group were estimated using sample
proportions, with ratios of odds of mortality compared using 95% confidence intervals computed using
Wald-type maximum likelihood methods. Two-sided p values testing the null hypothesis of no difference
in the odds of four year mortality for every increase in log baseline AAI measurement were based on the
standard normal distribution.
Inference: For every one unit increase in log baseline AAI measurements participants had 0.09 the odds
of dying within four years of study entry, which was statistically significant (95% CI, 0.07, 0.15; two-sided
P<0.001).
3d.
Methods: Quadratic baseline AAI measurements were modeled continuously and four-year mortality
was included as a binary outcome variable in a logistic regression model. No participant’s observations
were censored prior to four years, so four-year mortality within each AAI group were estimated using
sample proportions, with ratios of odds of mortality compared using 95% confidence intervals computed
using Wald-type maximum likelihood methods. Two-sided p values testing the null hypothesis of no
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BIOST536Homework_3_ID: 3291
difference in the odds of four year mortality for every increase in squared baseline AAI measurement
were based on the standard normal distribution.
Inference: For every one unit increase in squared baseline AAI measurements participants had 6.5 the
odds of dying within four years of study entry, which was statistically significant (95% CI, 1.79, 23.76;
two-sided P=0.005).
3e.
Methods: Baseline AAI measurements were modeled continuously with dummy variables representing
biologically relevant cut points (0.25, 0.55, 0.75, 0.95, 1.15, 1.35, 1.55, 2.4) and four-year mortality was
included as a binary outcome variable in a logistic regression model. No participant’s observations were
censored prior to four years, so four-year mortality within each AAI group were estimated using sample
proportions, with ratios of odds of mortality compared using 95% confidence intervals computed using
Wald-type maximum likelihood methods. Two-sided p values testing the null hypothesis of no difference
in the odds of four year mortality across AAI cut-offs were based on the standard normal distribution.
Inference: As baseline AAI measurements increase between adjacent categories participants had 0.08
the odds of dying within four years of study entry, which was statistically significant (95% CI, 0.05, 0.12
two-sided P<0.001).
3f.
Methods: Baseline AAI measurements were modeled continuously with dummy variables for quintiles
and four-year mortality was included as a binary outcome variable in a logistic regression model. Since
the range of the AAI measurements was 0.278 to 2.38, the quintiles were 0.70, 1.12, 1.54, 1.96 and fouryear mortality was included as a binary outcome variable. No participant’s observations were censored
prior to four years, so four-year mortality within each AAI group were estimated using sample
proportions, with ratios of odds of mortality compared using 95% confidence intervals computed using
Wald-type maximum likelihood methods. Two-sided p values testing the null hypothesis of no difference
in the odds of four year mortality across AAI cut-offs were based on the standard normal distribution.
Inference: As baseline AAI measurements increase between adjacent categories participants had 0.18
the odds of dying within four years of study entry, which was statistically significant (95% CI, 0.12, 0.27
two-sided P<0.001).
3g and 3h. I was unable to complete the answers to these problems by the due date/time, so they are
omitted.
Question 4
4a.
Methods: The mean, median, and range of data for the baseline AAI measurements were determined to
assess how the data are distributed. Since the outcome variable is binary (whether a participant died
within four years or not), its distribution was not evaluated. As above, the length of follow-up for
participants who did not die within four years was determined to identify any cases that should be
censored. A boxplot was also generated to assess the distribution of the data. Missing data was also
assessed.
Inference: The range of baseline AAI measurements if 0.28 to 2.4, the mean is 1.06, and the median is
1.08. The boxplot shows that the baseline AAI measurements are normally distributed. 121
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BIOST536Homework_3_ID: 3291
observations do not have baseline AAI data and will therefore not be included in analyses of the
association with baseline AAI and four-year mortality.
4b.
Methods: Baseline AAI measurements were modeled continuously and four-year mortality was included
as a binary outcome variable in a Poisson regression model with robust standard errors. No participant’s
observations were censored prior to four years, so four-year mortality across AAI measurements was
estimated using sample proportions, with risk ratio in mortality rates compared using 95% confidence
intervals computed using Wald-type maximum likelihood methods. The two-sided p value testing the
null hypothesis of the probability of four year survival across AAI measurements was based on the
asymptotic normal distribution for the Poisson regression parameter estimates.
Inference: For every one unit increase in baseline AAI measurements participants were 2.34 times less
likely to die within four years of study entry, which was statistically significant (95% CI, 1.95, 2.73; twosided P<0.001).
4c.
Methods: Log baseline AAI measurements were modeled continuously and four-year mortality was
included as a binary outcome variable in a Poisson regression model with robust standard errors. No
participant’s observations were censored prior to four years, so four-year mortality across AAI
measurements was estimated using sample proportions, with risk ratio in mortality rates compared
using 95% confidence intervals computed using Wald-type maximum likelihood methods. The two-sided
p value testing the null hypothesis of the probability of four year survival across log AAI measurements
was based on the asymptotic normal distribution for the Poisson regression parameter estimates.
Inference: For every one unit increase in log baseline AAI measurements participants were 1.85 times
less likely to die within four years of study entry, which was statistically significant (95% CI, 1.57, 2.12;
two-sided P<0.001).
4d.
Methods: Quadratic baseline AAI measurements were modeled continuously and four-year mortality
was included as a binary outcome variable a Poisson regression model with robust standard errors. No
participant’s observations were censored prior to four years, so four-year mortality across AAI
measurements was estimated using sample proportions, with risk ratio in mortality rates compared
using 95% confidence intervals computed using Wald-type maximum likelihood methods. The two-sided
p value testing the null hypothesis of the probability of four year survival across AAI measurements was
based on the asymptotic normal distribution for the Poisson regression parameter estimates.
Inference: For every one unit increase in squared baseline AAI measurements participants were 1.12
times less likely to die within four years of study entry, which was statistically significant (95% CI, 1.53,
0.10; two-sided P<0.001).
4e.
Methods: Baseline AAI measurements were modeled continuously with dummy variables representing
biologically relevant cut points (0.25, 0.55, 0.75, 0.95, 1.15, 1.35, 1.55, 2.4) and four-year mortality was
included as a binary outcome variable a Poisson regression model with robust standard errors. No
participant’s observations were censored prior to four years, so four-year mortality across AAI
measurements was estimated using sample proportions, with risk ratio in mortality rates compared
using 95% confidence intervals computed using Wald-type maximum likelihood methods. The two-sided
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BIOST536Homework_3_ID: 3291
p value testing the null hypothesis of the probability of four year survival across AAI measurements was
based on the asymptotic normal distribution for the Poisson regression parameter estimates.
Inference: As baseline AAI measurements increase between adjacent categories participants were 2.15
times less likely to die within four years of study entry, which was statistically significant (95% CI, 1.78,
2.53; two-sided P<0.001).
4f.
Methods: Baseline AAI measurements were modeled continuously with dummy variables for quintiles
and four-year mortality was included as a binary outcome variable a Poisson regression model with
robust standard errors. Since the range of the AAI measurements was 0.278 to 2.38, the quintiles were
0.70, 1.12, 1.54, 1.96 and four-year mortality was included as a binary outcome variable. No
participant’s observations were censored prior to four years, so four-year mortality across AAI
measurements was estimated using sample proportions, with risk ratio in mortality rates compared
using 95% confidence intervals computed using Wald-type maximum likelihood methods. The two-sided
p value testing the null hypothesis of the probability of four year survival across AAI measurements was
based on the asymptotic normal distribution for the Poisson regression parameter estimates.
Inference: As baseline AAI measurements increase between adjacent categories participants were
1.5times less likely to die within four years of study entry, which was statistically significant (95% CI,
1.14, 1.86; two-sided P<0.001).
4g and 4h. I was unable to complete the answers to these problems by the due date/time, so they are
omitted.
Question 5.
Like the spline questions and comparison of graphed fitted values questions I was unable to answer this
question in by the due date/time.
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BIOST536Homework_3_ID: 3291