BIOST 536 HW #3
Due: October 28, 2014, 21:30pm
Code: 3663
1. Methods: The minimum of the observation time among patients still alive was compared to 4 years.
Results: Te minimal time of follow up for the patients alive is 4.05 years meaning that they have at least 4 years of
follow. So we can analyze this data using a binary outcome (death) and apply logistic regression.
2. a) Methods: univariate statistical techniques are employed to describe the binary variable death (proportion) and the
continuous positive variable ankle-arm index (mean, standard deviation, minimum, maximum and variation coefficient).
Results: There were 1121 (22.4%) deaths within the first 4 years of follow up as shown on table 1; the ankle-arm index
(aai) ranged from 0.28 to 2.38 with mean 1.06 and small variation (16.5%).
Table 1 – Descriptives of ankle-arm index
Deaths with 4 year follow up
No
Yes
3879 (77.6%)
Total, N (%)
1121 (22.4%)
Ankle-arm index
Min - Max
Mean (SD)
Missings, N (%)
0.28 - 2.38
1.08 (0.157)
81 (2.1%)
0.30 - 1.89
0.99 (0.211)
40 (3.6%)
Total
5000 (100%)
0.28 - 2.38
1.06 (0.175)
121 (2.4%)
2.b) Methods: the association between death within the first 4 years and the baseline ankle-arm index (aai) is assed
through a linear regression with robust standard error. Only 4879 records are used due to misingness on aai. P-values
from Wald test are reported.
Results: As shown on table 2 there is half reduction in mortality for 1 unity increase on ankle-arm index i.e for each 1/10
increase of aai the mortality decreases in 5% (95%CI: 4.2 - 5.8%).
Table 2 – Risk difference modeled through linear regression
Variable
Ankle-arm index
Constant
Linear Coefficient (95% CI)
p-value
-0.50 (-0.58 – -0.42)
0.76 (0.67 – 0.84)
< 0.001
< 0.001
2.c) Methods: the association between death within the first 4 years and the natural logarithm of the baseline anklearm index is assed through a linear regression with robust standard error. Only 4879 records are used due to
missingness on aai. P-values from Wald test are reported.
Results: As shown on table 3 there is almost half (0.47) reduction in mortality for 1 unity increase on natural logarithm of
ankle-arm index.
Table 3 – Risk difference modeled through linear regression
Variable
Log ankle-arm index
Constant
Linear Coefficient (95% CI)
p-value
-0.47 (-0.55 – -0.40)
0.24 (0.23 – 0.26)
< 0.001
< 0.001
2.d) Methods: the association between death within the first 4 years and the baseline ankle-arm index is assed through
a linear regression with robust standard error. A quadratic term of baseline ankle-arm index is included on the model.
Only 4879 records are used due to missingness on aai. P-values from Wald test and from Fisher test are reported.
Results: As shown on table 4 the quadratic aai is statistically significant indicating an U shaped relation between aai and
risk difference. Because of this the risk difference has not the same slope on different segments of aai; the slope is linear
-1.45 + 0.47aai. This indicates a reduction of risk mortality not more than 14.5% for each 1/10 increase on aai up to an
aai of 1.54 from where there is an increased risk of death.
Table 4 – Risk difference modeled through linear regression
Variable
Ankle-arm index
Squared ankle-arm index
Constant
Linear Coefficient (95% CI)
p-value
-1.45 (-1.87 – -1.02)
0.47 (0.27 – 0.67)
1.21 (0.99 – 1.44)
< 0.001
< 0.001
< 0.001
Overall p < 0.0001 through Fisher Test.
2.e) Methods: the association between death within the first 4 years and the baseline ankle-arm index is assed through
a linear regression with robust standard error. The baseline ankle-arm index is transformed into an ordinal variable using
these cut-points 0.25, 0.55, 0.75, 0.95, 1.15, 1.35, 1.55, 2.4. Only 4879 records are used due to misingness on aai. Pvalues from Wald test and from Fisher test are reported. also simultaneous testing of all parameters equal to 0
Results: As shown on table 5 there is a trend on risk mortality difference at different levels of baseline aai, varying from
3% increase up to 35% decrease in cumulative mortality on values of aai with 1.15 to 1.35.
Table 5 – Risk difference modeled through linear regression using an ordinal categorized aai
Variable
Ordinal ankle-arm index
< 0.55
0.55 – 0.75
0.75 – 0.95
0.95 – 1.15
1.15 – 1.35
1.35 – 1.55
≥1.55
Constant
Linear Coefficient (95% CI)
p-value
(reference)
0.03 (-0.10 – 0.16)
-0.13 (-0.25 – -0.01)
-0.29 (-0.41 - -0.18)
-0.35 (-0.46 - -0.23)
-0.26 (-0.40 - -0.12)
-0.13 (-0.39 – 0.12)
0.620
0.037
< 0.001
< 0.001
< 0.001
0.303
0.49 (0.37 – 0.60)
< 0.001
Overall p < 0.0001 through Fisher Test.
2.f) Methods: the association between death within the first 4 years and the baseline ankle-arm index is assed through
a linear regression with robust standard error. The baseline ankle-arm index is transformed into an ordinal variable using
these cut-points based on 7 equidistant quantiles (septile) of the cumulative distribution of ankle-arm index . Only 4879
records are used due to misingness on aai. P-values from Wald test and from Fisher test are reported. Testing of all
parameters equal to 0…
Results: As shown on table 6 there is a trend on risk mortality difference at different septile levels of baseline aai,
varying from 21% up to 32% decrease in cumulative mortality. All reductions are statistically significant.
Table 6 – Risk difference modeled through linear regression using an ordinal categorized aai
Variable
Ordinal ankle-arm index
1st septile
2nd septile
3rd septile
4th septile
5th septile
6th septile
7th septile
Constant
Linear Coefficient (95% CI)
p-value
(reference)
-0.21 (-0.26 – -0.16)
-0.25 (-0.30 – -0.21)
-0.27 (-0.32 - -0.22)
-0.28 (-0.33 - -0.24)
-0.32 (-0.36 - -0.28)
-0.29 (-0.33 – -0.24)
< 0.001
< 0.001
< 0.001
< 0.001
< 0.001
< 0.001
0.45 (0.42 – 0.49)
< 0.001
Overall p < 0.0001 through Fisher Test.
2.g) Methods: the association between death within the first 4 years and the baseline ankle-arm index is assed through
a linear regression with robust standard error. The baseline ankle-arm index is transformed into an ordinal variable using
these cut-points based on 7 equidistant quantiles (septile) of the cumulative distribution of ankle-arm index . The
resulting variable is incorporate on the regression as a continuous variable. Only 4879 records are used due to
misingness on aai. P-values from Wald test are reported.
Results: As shown on table 7 for each septile of AAI there is a statistically significant reduction in cumulative mortality of
4%.
Table 7 – Risk difference modeled trough linear regression using septiles of AAI
Variable
Ankle-arm index
Constant
Linear Coefficient (95% CI)
p-value
-0.04 (-0.05 – -0.03)
0.38 (0.35 – 0.41)
< 0.001
< 0.001
2.h) Methods: line plots are used to show different linear regressions fitting the proportion of death within first 4 years
of follow up on untransformed AAI, logarithmesed AAI, untransformed AAI and quadratic term, AAI clinical relevant
cutpoints and septiles of AAI.
Results: All predicted values from the different risk difference regressions are shown on figure 1. The simple linear
model without transformation suggests a constant reduction of the risk difference to a point where the proportion
below 0 (which is inadmissible). But other models indicate an U or at least a reduction on the impact of AAI on risk
difference between 1 and 1.5. It is to note that on models with categorical or ordered predictors the more levels or
using of continuous small interval increment details on the nature of the relationship between AAI and risk-difference.
Figure 1 – Different linear regression fitting of proportion of deaths over AAI
3.a) Methods: To assess the association between AAI and death within first 4 years of follow up odds-ratio (OR) is
calculated for 1 unity increase on AAI. OR is shown also for 1/10 increase on AAI which is more clinical usefull. A logistic
regression of death on AAI is used to obtain the OR. P-values from Wald test are reported.
Results: The odd of dying in the first 4 years of follow up is 0.06 times per 1 unity increase of AAI (table 8); this is
equivalent to an increase of the odds-ratio in 6.77x10-13 (95% CI: 1.45x10-14 - 3.17x10-11) per 1/10 increase of AAI. Should
be noted this OR is close to zero (below 1) so it has the effect of protection on increase of AAI.
Table 8 – Odds-ratio modeled trough logistic regression
Variable
Ankle-arm index
Odds-Ratio (95% CI)
p-value
0.06 (0.04 – 0.09)
< 0.001
3.b) The answer to this question is the same as on 3a.
3.c) Methods: the association between death within the first 4 years and the natural logarithm of the baseline anklearm index is assed through a logistic regression. Only 4879 records are used due to missingness on aai. P-values from
Wald test are reported.
Results: As shown on table 9 the odds of dying within first 4 years is 0.087 for each 1 unity increase in the logarithm of
the AAI.
Table 9 – Odds-ratio modeled trough logistic regression on AAI logarithm
Variable
Log ankle-arm index
Odds-Ratio (95% CI)
p-value
0.087 (0.060 – 0.123)
< 0.001
3.d) Methods: the association between death within the first 4 years and the baseline ankle-arm index is assed through
a logistic regression. A quadratic term of baseline ankle-arm index is included on the model. Only 4879 records are used
due to missingness on AAI. P-values from Wald test and from likelihood-ratio chi-squared test are reported.
Results: As shown on table 10 the quadratic AAI is statistically significant indicating an U shaped relation between AAI
and Odds-Ratio. It is to note that the Squared ankle-arm index has a smaller influence on the OR compared to the
untransformed term (1/0.0026 = 384.6) suggesting predominance of the influence of non-transformed component of
AAI within the range of the AAI on this dataset. Meaning that the odd of dying on first 4 years of follow up is reduced at
maximum 385 times per one unity increase of AAI. There is an association b/w death and aai…
Table 10 – Odds-Ratios modeled trough logistic regression
Variable
Ankle-arm index
Squared ankle-arm index
Odds-Ratio (95% CI)
p-value
0.0026 (0.0003 – 0.0217)
5.3196 (1.7875 – 15.8308)
< 0.001
< 0.001
Overall p < 0.0001 through likelihood-ratio chi-squared test.
3.e) Methods: the association between death within the first 4 years and the baseline ankle-arm index is assed through
a logistic regression. The baseline ankle-arm index is transformed into an ordinal variable using these cut-points 0.25,
0.55, 0.75, 0.95, 1.15, 1.35, 1.55, 2.4. Only 4879 records are used due to misingness on aai. P-values from Wald test and
from Fisher test are reported. Testing of all parameters…
Results: As shown on table 11 there is a decreasing trend on the odds to dye within first 4 years at different levels of
baseline AAI, decreasing from 1.14 to 0.17 on AAI within 1.15 to 1.35 and then ascending again. Is there an association?
Table 11 – Odds-Ratios modeled through logistic regression using an ordinal categorized AAI
Variable
Ordinal ankle-arm index
< 0.55
0.55 – 0.75
0.75 – 0.95
0.95 – 1.15
1.15 – 1.35
1.35 – 1.55
≥1.55
Odds-Ratio (95% CI)
p-value
(reference)
1.14 (0.68 – 1.93)
0.59 (0.36 – 0.96)
0.26 (0.16 – 0.41)
0.17 (0.11 – 0.28)
0.31 (0.16 – 0.60)
0.58 (0.19 – 1.72)
0.620
0.033
< 0.001
< 0.001
< 0.001
0.323
Overall p < 0.0001 through likelihood-ratio chi-squared test.
3.f) Methods: the association between death within the first 4 years and the baseline ankle-arm index is assed through
a logistic regression. The baseline ankle-arm index is transformed into an ordinal variable using these cut-points based
on 7 equidistant quantiles (septile) of the cumulative distribution of ankle-arm index . Only 4879 records are used due to
misingness on aai. P-values from Wald test and from likelihood-ratio chi-squared test are reported.
Results: As shown on table 12 there is a reducing trend on odds to die within first 4 years risk at increasing septile levels
of baseline AAI, varying from 0.39 up to 0.19 and increasing on last septile to 0.24.
Table 12 – Odds-Ratios modeled through logistic regression using an ordinal categorized AAI
Variable
Ordinal ankle-arm index
1st septile
2nd septile
3rd septile
4th septile
5th septile
6th septile
7th septile
Odds-Ratio (95% CI)
p-value
(reference)
0.39 (0.31 – 0.49)
0.31 (0.24 – 0.38)
0.27 (0.21 – 0.35)
0.25 (0.19 – 0.32)
0.19 (0.14 – 0.24)
0.24 (0.19 – 0.31)
< 0.001
< 0.001
< 0.001
< 0.001
< 0.001
< 0.001
Overall p < 0.0001 through likelihood-ratio chi-squared test.
3.g) Methods: the association between death within the first 4 years and the baseline ankle-arm index is assed through
a logistic regression. The baseline ankle-arm index is transformed into an ordinal variable using these cut-points based
on 7 equidistant quantiles (septile) of the cumulative distribution of ankle-arm index . The resulting variable is
incorporate on the regression as a continuous variable. Only 4879 records are used due to misingness on aai. P-values
from Wald test are reported.
Results: As shown on table 13 for each septile of AAI there is a statistically significant reduction in odds to die within
first 4 years by 1.27 (1/0.79) times.
Table 13 – Odds-Ratios modeled through logistic regression using septiles of AAI
Variable
Ankle-arm index
Risk-Ratio (95% CI)
p-value
0.79 (0.76 – 0.82)
< 0.001
3.h) Methods: line plots are used to show different logistic regressions fitting the proportion of death within first 4 years
of follow up on untransformed AAI, logarithmesed AAI, untransformed AAI and quadratic term, AAI clinical relevant
cutpoints and septiles of AAI.
Results: All predicted values from the different logistic regressions are shown on figure 2. All models agree that the risk
of death has a decreasing trend before somewhere 1.2 to 1.4 on AAI. Then different models show marked pattern. It is
for note that the quadratic and the ordered clinical relevant cut points show increase in risk, while the continuous
septiles show an unchangeable risk. All models show plausible proportion i.e between 0 and 1.
Choice of model for confounder vs poi vs precision?
Figure 2 – Different logistic regression fitting of proportion of deaths over AAI
4.a) Methods: To assess the association between AAI and death within first 4 years of follow up incidence rate ratios (or
risk ratio, RR) is calculated for 1 unity increase on AAI. RR is shown also for 1/10 increase on AAI which is more clinical
useful. A poisson regression of death on AAI is used to obtain the RR. P-values from Wald test are reported.
Results: The risk in the first 4 years of follow up is 0.09 times per 1 unity increase of AAI (table 14); this is equivalent to
an increase of the risk-relative in 5.08x10-11 (95% CI: 2.79x10-12 – 9.23x10-10) per 1/10 increase of AAI. Should be noted
this OR is close to zero (below 1) so it has the effect of protection on increase of AAI.
Table 14 – Risk-ratio modeled trough poisson regression
Variable
Ankle-arm index
Risk-Ratio (95% CI)
p-value
0.09 (0.07 – 0.12)
< 0.001
4.b) The answer to this question is the same as on 4a.
4.c) Methods: the association between death within the first 4 years and the natural logarithm of the baseline anklearm index is assed through a poisson regression. Only 4879 records are used due to missingness on aai. P-values from
Wald test are reported.
Results: As shown on table 15 the risk of dying within first 4 years is 0.156 for each 1 unity increase in the logarithm of
the AAI.
Table 15 – Risk-ratio modeled trough poisson regression on AAI logarithm
Variable
Log ankle-arm index
Risk-Ratio (95% CI)
p-value
0.156 (0.125 – 0.195)
< 0.001
4.d) Methods: the association between death within the first 4 years and the baseline ankle-arm index is assed through
a poisson regression. A quadratic term of baseline ankle-arm index is included on the model. Only 4879 records are used
due to missingness on AAI. P-values from Wald test and from likelihood-ratio chi-squared test are reported.
Results: As shown on table 16 the quadratic AAI is not statistically significant though the model is. There is not much
change on the log-likelihood (less than 1 from the simple model to the quadratic term). Anyway it is to note that the
Squared ankle-arm index has a smaller influence on the RR compared to the untransformed term (1/0.093448 = 10.7)
suggesting predominance of the influence of non-transformed component of AAI within the range of the AAI on this
dataset. Meaning that the risk of dying within first 4 years of follow up is reduced at maximum 11 times per one unity
increase of AAI.
Table 16 – Risk-Ratios modeled trough poisson regression
Variable
Ankle-arm index
Squared ankle-arm index
Risk-Ratio (95% CI)
p-value
0.0345 (0.0063 – 0.1895)
1.7526 (0.6805 – 4.5135)
< 0.001
0.245
Overall p < 0.0001 through likelihood-ratio chi-squared test.
4.e) Methods: the association between death within the first 4 years and the baseline ankle-arm index is assed through
a poisson regression. The baseline ankle-arm index is transformed into an ordinal variable using these cut-points 0.25,
0.55, 0.75, 0.95, 1.15, 1.35, 1.55, 2.4. Only 4879 records are used due to misingness on aai. P-values from Wald test and
from Fisher test are reported. Testing all parameters=0
Results: As shown on table 17 there is a decreasing trend on the risk to dye within first 4 years at different levels of
baseline AAI, decreasing from 0.95 to 0.20 on AAI within 1.15 to 1.35 and then ascending again. Is there an association?
Table 17 – Risk-Ratios modeled trough poisson regression using an ordinal categorized AAI
Variable
Ordinal ankle-arm index
< 0.55
0.55 – 0.75
0.75 – 0.95
0.95 – 1.15
1.15 – 1.35
1.35 – 1.55
≥1.55
Risk-Ratio (95% CI)
p-value
(reference)
0.95 (0.66 – 1.38)
0.60 (0.42 – 0.85)
0.29 (0.21 – 0.40)
0.20 (0.14 – 0.29)
0.34 (0.20 – 0.58)
0.52 (0.22 – 1.24)
0.801
0.004
< 0.001
< 0.001
< 0.001
0.141
Overall p < 0.0001 through likelihood-ratio chi-squared test.
4.f) Methods: the association between death within the first 4 years and the baseline ankle-arm index is assed through
a poisson regression. The baseline ankle-arm index is transformed into an ordinal variable using these cut-points based
on 7 equidistant quantiles (septile) of the cumulative distribution of ankle-arm index . Only 4879 records are used due to
misingness on aai. P-values from Wald test and from likelihood-ratio chi-squared test are reported.
Results: As shown on table 16 there is a reducing trend on risk to die within first 4 years risk at increasing septile levels
of baseline AAI, varying from 0.45 up to 0.23 and increasing on last septile to 0.29.
Table 18 – Risk-Ratios modeled through poisson regression using an ordinal categorized AAI
Variable
Ordinal ankle-arm index
1st septile
2nd septile
3rd septile
4th septile
5th septile
6th septile
7th septile
Risk-Ratio (95% CI)
p-value
(reference)
0.45 (0.38 – 0.55)
0.36 (0.30 – 0.44)
0.33 (0.27 – 0.41)
0.30 (0.25 – 0.38)
0.23 (0.19 – 0.30)
0.29 (0.24 – 0.37)
< 0.001
< 0.001
< 0.001
< 0.001
< 0.001
< 0.001
Overall p < 0.0001 through likelihood-ratio chi-squared test.
4.g) Methods: the association between death within the first 4 years and the baseline ankle-arm index is assed through
a poisson regression. The baseline ankle-arm index is transformed into an ordinal variable using these cut-points based
on 7 equidistant quantiles (septile) of the cumulative distribution of ankle-arm index . The resulting variable is
incorporate on the regression as a continuous variable. Only 4879 records are used due to misingness on aai. P-values
from Wald test are reported.
Results: As shown on table 19 for each septile of AAI there is a statistically significant reduction in risk to die within first
4 years by 1.25 (1/0.80) times.
Table 19 – Risk-Ratios modeled through poisson regression using septiles of AAI
Variable
Ankle-arm index
Risk-Ratio (95% CI)
p-value
0.80 (0.78 – 0.83)
< 0.001
4.h) Methods: line plots are used to show different poisson regressions fitting the incidence of death within first 4 years
(per year) of follow up on untransformed AAI, logarithmesed AAI, untransformed AAI and quadratic term, AAI clinical
relevant cutpoints and septiles of AAI. All plots have been smoothed trough the use of lowess.
Results: All predicted values from the different poisson regressions are shown on figure 3. All models generally show
decrease on the risk of death except the ordered and relevant cutpoints, the dummy septiles and continuous septiles
which suggest a U like curve. Dummy variables should not look connected as below
Figure 3 – Different poisson regression fitting of incidence of deaths over AAI
5. Methods: Different risk associations (risk difference, odds-ratio and risk-ratios) are compared. Diverse tables and plots
produced on previous questions are reused.
Results: generally all analysis suggests a decreasing in risk of death before when AAI moves from 0 to 1. The linear
regression untransformed and logarithmzed potentially produce implausible proportions (negative). Other models show
some concavity (with different degrees on each model) on the interval 1.2 – 1.4 and ascending risk for larger AAI values.
Anyway the extreme values of AAI (below 0.2 or above 2) have few people and that affects estimates, specially those
from categorized (non continuous predictors) producing very different fitting.
-dummy variables are saturate
linear splines better than linear/log/quadratic fits at more extreme values