Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
11/26/07 - 1/20
BIOST/STAT 579 – Data Analysis
Longitudinal Data Analysis
Longitudinal data present special opportunities and challenges that don’t exist with cross-sectional data
analysis. First, there are questions that can be addressed with longitudinal data that cannot even be
considered with cross-sectional data. For example, with longitudinal data one can distinguish between crosssectional and longitudinal associations. A second feature of longitudinal data is that there are three distinct
types of models that can be considered: 1) marginal models, 2) random-effects models, and 3) transition
models. These models address different scientific questions so the choice of model type is extremely
important and needs to be made based on the scientific question of interest rather than on statistical
considerations. The third issue to be considered is modeling the correlation structure. With longitudinal data
there is a wide variety of correlation structures to choose from.
Outline
1. Longitudinal versus cross-sectional associations
2. Contrasts between three approaches to modeling longitudinal data
3. Modeling longitudinal correlation structures
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Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
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1. Longitudinal versus cross-sectional associations
Example I (Potthoff & Roy growth data):
Distances between two facial landmarks for 27 children (11 females, 16 males) at ages 8, 10,
12, and 14 years. Spaghetti plots of the data are shown below. (Note: age is centered at 11.)
-3
-2
-1
0
1
2
3
25
20
Distance (mm)
25
20
Distance (mm)
30
Distance vs. Age (Males)
30
Distance vs. Age (Females)
-3
-2
-1
Age (yrs)
0
1
2
3
Age (yrs)
Recall that a linear model with gender, age, and gender-age interaction gave the same
coefficients as separate analyses of subject means and individual regression slopes on age.
Linear models: (Estimate  SE)
Variable
Intercept
Male
Age
Male*Age
Linear model on
raw data
(Estimate 
Robust SE)
22.6  0.61
2.32  0.75
0.48  0.06
0.30  0.12
Linear model on
means (Estimate
 SE)
Linear model on
slopes (Estimate
 SE)
22.6  0.59
2.32  0.76
0.48  0.10
0.30  0.13
For these data the overall linear model gives the same results as the analyses of subjectspecific means and slopes. This means that the age coefficient in the overall model represents
the change in mean corresponding to aging of an individual child. This property results from
the fact that the data are balanced. Let’s try a different data set to see what happens if the
data are not balanced.
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Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
Example II (Tooth loss in periodontitis patients):
11/26/07 - 3/20
A cohort study was done on the effectiveness of treatment of periodontitis. The study enrolled
approximately 800 patients with periodontitis and followed them for up to 10 years. The
outcome variable is the number of teeth extracted in each follow-up year.
Results of the three analyses as above:
Variable
Linear model on
raw data
(Estimate  robust
SE)
Linear model on
means (Estimate
 SE)
Linear model on
slopes (Estimate 
SE)
Intercept
0.181  0.019
0.187  0.022
Male
0.022  0.030
0.012  0.030
Age
-0.001  0.002
0.000  0.003*
-0.060  0.016
Male X Age
0.001  0.004
0.002  0.004*
0.029  0.022
*Note that we can fit age and male X age terms to the subject means because the average value of
age varies across subjects (unlike the Potthoff-Roy data).
The results show there is a longitudinal association (within subjects) between extractions and
age but no cross-sectional association (between subjects). (The overall analysis gives results
similar to the between subjects analysis in this case.) The analysis of slopes is not very precise
because each slope is given equal weight regardless of its variability. A better analysis, which
allows simultaneous estimation of cross-sectional and longitudinal associations is between and
within-cluster regression, based on fitting the following model:
E(Y) = Intercept + Male * { Ave. Age + (Age – Ave. Age) }
Variable
Linear model
(Estimate 
Robust SE)
Intercept
0.180  0.019
Male
0.022  0.030
Ave. Age
0.002  0.003
Age Diff.
-0.025  0.009
Male X Ave. Age
0.000  0.004
Male X Age Diff.
0.014  0.012
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Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
11/26/07 - 4/20
Note that the overall estimates for Age and Male X Age (from previous table) are
intermediate between the cross-sectional and longitudinal estimates, but are closer to
the cross-sectional estimates.
Alternative parametrization:
E(Y) = Intercept + Male*{ Baseline Age + Year }
(Note: Year = Age – Baseline Age)
Variable
Estimate  Robust
SE
Intercept
0.072  0.151
Male
0.056  0.214
Baseline Age
0.002  0.003
Year
-0.022  0.008
Male X BL Age
-0.001  0.004
Male X Year
0.015  0.011
Between and within-regression for a log-linear model:
log(E(Y)) = Intercept + Male * { Baseline Age + Year }
Estimates and robust SEs:
Variable
Estimate  Robust
SE
Intercept
-2.362  0.847
Male
0.396  1.128
Baseline Age
0.012  0.016
Year
-0.130  0.049
Male X Baseline Age
-0.005  0.022
Male X Year
0.092  0.063
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11/26/07 - 5/20
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Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
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2. Contrasts between three approaches to modeling longitudinal
data
Three modeling approaches:

Marginal Model (Direct Approach)

Random Effects Model

Transition Model
Example I (Potthoff & Roy growth data):
Marginal Model (“direct” approach):
E(Y) = Intercept + Male + Age + Male X Age
Random Effects Model:
E(Y|U) = Intercept + Male + Age + Male X Age + U
U = random effect at subject level (iid) ~ N(0, V)
Transition Model:
E(Y|YPrevious) = Intercept + Male + Age + Male X Age + YPrevious
YPrevious = value of Y at previous time point
Results of three model fits for the Potthoff-Roy data:
Marginal Model*
Estimate  SE
Random- Effects
Model
Transition Model*
Intercept
22.6  0.61
22.6  0.59
9.57  3.67
Male
2.32  0.75
2.32  0.76
1.03  0.51
Age
0.48  0.06
0.48  0.09
0.18  0.12
Male X Age
0.30  0.12
0.30  0.12
0.31  0.13
NA
NA
0.60  0.17
Variable
YPrevious
* SEs are robust SEs.
Note: The marginal and RE models give the same results here because the data are
balanced. The transition model gives very different results from the other two models
because it includes adjustment for a prior outcome.
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Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
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Example II (Periodontitis study):
Variable
Marginal Model
Random- Effects
Model
Transition Model
Intercept
0.181  0.019
0.180  0.021
0.133  0.018
Male
0.022  0.030
-0.002  0.003
0.040  0.028
Age
-0.001  0.002
0.022  0.030
0.001  0.002
Male X Age
0.001  0.004
0.002  0.004
0.001  0.003
NA
NA

YPrevious
Marginal Model
Estimate  SE
Random- Effects
Model
Transition Model
Intercept
0.072  0.151
0.079  0.158
0.078  0.136
Male
0.056  0.214
0.043  0.216
0.022  0.186
BL-Age
0.002  0.003
0.002  0.003
0.001  0.003
Year
-0.022  0.008
-0.023  0.007
0.004  0.006
Male X BL-Age
-0.001  0.004
0.000  0.004
0.011  0.007
Male X Year
0.015  0.011
0.015  0.010
0.000  0.004
NA
NA
0.006  0.010
Variable
YPrevious
Note: The marginal and RE models give similar results, which is not surprising (for
linear models). The transition model does not differ too much from the other two
models, in this case, because the prior outcome is not strongly associated with the
outcome.
Example III (Logistic regression for a cross-over trial):
Ref: Diggle et al., p.180
Marginal Model:
logit(E(Y)) = Intercept + Treatment + Period
Random Effects Model:
logit(E(Y|U)) = Intercept + Treatment + Period + U
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Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
U = random effect at subject level (iid) ~ N(0, V)
11/26/07 - 8/20
Results:
Marginal Model*
Random- Effects
Model
Intercept
0.67  0.29
2.2  1.0
Treatment
0.57  0.23
1.8  0.93
Period
-0.30  0.23
-1.0  0.84
Variable
* SEs are robust SEs.
Notes:
The estimated random effects SD is very large (5.0), which contributes to the large
differences between model estimates.
Differences between coefficient estimates are due to different interpretations of the two
models (as opposed to different properties of the parameter estimators).
Ratios of coefficient estimates to their SEs are similar in the two models, which is often
observed.
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Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
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3. Modeling longitudinal correlation structures
Example (Potthoff-Roy data):
Variable
Intercept
Male
Age
Male*Age
Est.
22.6
2.32
0.48
0.30
Independent
Naive
Robust
SE
SE
0.34
0.61
0.44
0.75
0.15
0.06
0.20
0.12
Est.
22.6
2.32
0.48
0.30
Exchangeable
Naïve
Robust
SE
SE
0.57
0.61
0.74
0.75
0.10
0.06
0.12
0.12
Est.
22.6
2.42
0.48
0.29
Autoregressive
Naïve
Robust
SE
SE
0.52
0.61
0.67
0.75
0.14
0.06
0.18
0.12
Note: Independent and Exchangeable analyses give same estimates and robust SEs
(but not naïve SEs) because the data are balanced. The AR-1 analysis gives similar
estimates and robust SEs (which are equal to 2 decimal places). Naïve SEs in AR-1
analysis are noticeably different than robust SEs.
The observed correlation structure is closer to exchangeable than AR-1:
Age
Age
Age
Age
8
10
12
14
Age 8
1
.63
.71
.60
Age 10
.63
1
.63
.76
Age 12
.71
.63
1
.79
Age 14
.60
.76
.79
1
The correlation structure is similar, but with very different values of the correlations,
for males and females:
Males:
8
10
12
14
Age 8
1
.44
.56
.32
Age 10
.44
1
.39
.63
Age 12
.56
.39
1
.59
Age 14
.32
.63
.59
1
Age 8
Age 10
Age 12
Age 8
1
.83
.86
Age 10
.83
1
.90
Age 12
.86
.90
1
Age 14
.84
.88
.95
Age
Age
Age
Age
Females:
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Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
Age 14
.84
.88
11/26/07 - 10/20
.95
1
Note: these differences can be seen in the spaghetti plots.
Note: such variations in correlation structure are often ignored but can be capitalized on to
increase estimation precision (Stoner and Leroux, 2002)
References
Diggle, Heagerty, Liang and Zeger (2002). Longitudinal Data Analysis.
Potthoff and Roy (1964) Biometrika (facial measurement data)
Stoner and Leroux (2002) Biometrika (improving precision compared to GEE)
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Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
11/26/07 - 11/20
Survival Analysis
Example (NHANES cardiovascular hospitalization or death):
What the data look like:
chdhosp: indicator of whether or not subject had the event of interest (cardiovascular
hospitalization or death)
time: time of occurrence of the event or time of censoring (end of follow-up) whichever came
first
Consider subset of subjects > 70 years of age at baseline.
n
mean min
max
chdhosp 1110 0.5153
0 1.0000
time
1110 9.1047
0 22.0616
table(round(chdsurv$time[chdsurv$age>70],0),chdsurv$chdhosp[chdsurv$age>70])
0 1
0 24 23
1 21 60
2 20 40
3 27 41
4 30 33
5 20 47
6 28 33
7 15 22
8 27 33
9 40 28
10 29 25
11 19 42
12 22 21
13 23 31
14 27 21
15 19 22
16 23 10
17 12 16
18 12 12
19 32 10
20 38 2
21 28 0
22 2 0
Note: the above give a picture of the data but are not descriptives you would report.
Descriptive statistics to report: Kaplan-Meier estimate of the survival curve, which is
the probability of “surviving” (being event-free) to a given time.
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Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
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0.6
0.4
Female
Male
0.0
0.2
Survival Probability
0.8
1.0
Kaplan-Meier Estimates (Age > 70) by Gender
with 95% Confidence Limits
0
5
10
15
20
Year
How Kaplan-Meier is calculated (first 10 values, females, n=597):
Time Risk Event Survival
0.0000 597
0 1.0000
0.0274 590
0 1.0000
0.0548 589
1 0.9983
0.1150 588
1 0.9966
0.1506 587
1 0.9949
0.1725 586
0 0.9949
0.1862 585
1 0.9932
0.1916 584
0 0.9932
0.2108 583
1 0.9915
0.4408 582
0 0.9915
SE
0.0000
0.0000
0.0017
0.0024
0.0029
0.0029
0.0034
0.0034
0.0038
0.0038
Regression analysis
The Cox proportional hazards model is the most popular model (like logistic regression
for binary data there are alternatives which don’t seem to be used much). The Cox
model is a linear model for the log of the hazard rate (instantaneous probability of
failure given survival to a given time).
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Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
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Example: Comparison of hazards for males and females among subjects greater than
70 years of age at baseline.
coxph(formula = Surv(time, chdhosp) ~ gender, data = chdsurv, subset =
(age > 70))
n= 1110
coef exp(coef) se(coef)
z
p
gendermale 0.309
1.36
0.0843 3.67 0.00024
gendermale
exp(coef) exp(-coef) lower .95 upper .95
1.36
0.734
1.16
1.61
The hazard for cardiovascular hospitalization or death is 1.36 times as high for males
as for females. This hazard ratio is highly statistically significant (significantly different
from 1).
Example: Apply Cox regression to the HW problem on comparing male and female risk
of CHD.
summary(coxph( Surv(time,chdhosp) ~ gender, data=chdsurv))
Call:
coxph(formula = Surv(time, chdhosp) ~ gender, data = chdsurv)
n= 11313
coef exp(coef) se(coef)
z p
gendermale 0.708
2.03
0.0388 18.3 0
gendermale
exp(coef) exp(-coef) lower .95 upper .95
2.03
0.493
1.88
2.19
Rsquare= 0.029
(max possible= 0.986 )
Likelihood ratio test= 331 on 1 df,
p=0
Wald test
= 334 on 1 df,
p=0
Score (logrank) test = 348 on 1 df,
p=0
summary(coxph( Surv(time,chdhosp) ~ gender+age, data=chdsurv))
Call:
coxph(formula = Surv(time, chdhosp) ~ gender + age, data = chdsurv)
n= 11313
coef exp(coef) se(coef)
z p
gendermale 0.5291
1.70 0.03887 13.6 0
age
0.0765
1.08 0.00169 45.2 0
exp(coef) exp(-coef) lower .95 upper .95
gendermale
1.70
0.589
1.57
1.83
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Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
age
1.08
0.926
Rsquare= 0.238
(max possible=
Likelihood ratio test= 3074 on
Wald test
= 2289 on
Score (logrank) test = 2965 on
1.08
11/26/07 - 14/20
1.08
0.986 )
2 df,
p=0
2 df,
p=0
2 df,
p=0
summary(coxph( Surv(time,chdhosp) ~ gender + age + povindex + physact,
data=chdsurv))
Call:
coxph(formula = Surv(time, chdhosp) ~ gender + age + povindex +
physact, data = chdsurv)
n=10873 (440 observations deleted due
coef exp(coef) se(coef)
gendermale 0.582057
1.790 0.040131
age
0.073910
1.077 0.001734
povindex
-0.000685
0.999 0.000119
physactYes -0.214802
0.807 0.041118
gendermale
age
povindex
physactYes
to missingness)
z
p
14.50 0.0e+00
42.62 0.0e+00
-5.74 9.5e-09
-5.22 1.8e-07
exp(coef) exp(-coef) lower .95 upper .95
1.790
0.559
1.654
1.936
1.077
0.929
1.073
1.080
0.999
1.001
0.999
1.000
0.807
1.240
0.744
0.874
Rsquare= 0.244
(max possible=
Likelihood ratio test= 3035 on
Wald test
= 2313 on
Score (logrank) test = 2981 on
0.986 )
4 df,
p=0
4 df,
p=0
4 df,
p=0
summary(coxph( Surv(time,chdhosp) ~ gender+age+povindex+physact+priorhd*diabetes,
data=chdsurv))
Call:
coxph(formula = Surv(time, chdhosp) ~ gender + age + povindex +
physact + priorhd * diabetes, data = chdsurv)
n=10873 (440 observations deleted due
coef exp(coef)
gendermale
0.575933
1.779
age
0.069264
1.072
povindex
-0.000549
0.999
physactYes
-0.149642
0.861
priorhdYes
1.074203
2.928
diabetesYes
0.792959
2.210
priorhdYes:diabetesYes -0.202058 0.817
gendermale
to missingness)
se(coef)
z
p
0.040236 14.31 0.0e+00
0.001769 39.16 0.0e+00
0.000118 -4.66 3.2e-06
0.041467 -3.61 3.1e-04
0.055510 19.35 0.0e+00
0.076011 10.43 0.0e+00
0.144600 -1.40 1.6e-01
exp(coef) exp(-coef) lower .95 upper .95
1.779
0.562
1.644
1.925
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Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
age
povindex
physactYes
priorhdYes
diabetesYes
priorhdYes:diabetesYes
1.072
0.999
0.861
2.928
2.210
0.817
Rsquare= 0.275
(max possible=
Likelihood ratio test= 3499 on
Wald test
= 2968 on
Score (logrank) test = 4175 on
0.933
1.001
1.161
0.342
0.453
1.224
1.068
0.999
0.794
2.626
1.904
0.615
11/26/07 - 15/20
1.075
1.000
0.934
3.264
2.565
1.085
0.986 )
7 df,
p=0
7 df,
p=0
7 df,
p=0
summary(coxph( Surv(time,chdhosp) ~ gender + age + povindex + physact +
priorhd*diabetes + bpdia + I(bpdia^2) + weight + I(weight^2) + height,
data=chdsurv))
Call:
coxph(formula = Surv(time, chdhosp) ~ gender + age + povindex +
physact + priorhd * diabetes + bpdia + I(bpdia^2) + weight
+ I(weight^2) + height, data = chdsurv)
n=10815 (498 observations deleted due
coef exp(coef)
gendermale
6.11e-01
1.842
age
6.93e-02
1.072
povindex
-4.59e-04
1.000
physactYes
-1.25e-01
0.883
priorhdYes
1.04e+00
2.823
diabetesYes
7.68e-01
2.156
bpdia
5.80e-03
1.006
I(bpdia^2)
1.57e-05
1.000
weight
-1.28e-02
0.987
I(weight^2)
1.27e-04
1.000
height
-6.86e-01
0.504
priorhdYes:diabetesYes -1.90e-01 0.827
to missingness)
se(coef)
z
5.66e-02 10.790
1.87e-03 37.031
1.18e-04 -3.888
4.17e-02 -2.998
5.58e-02 18.586
7.62e-02 10.089
1.17e-02 0.495
6.31e-05 0.249
7.49e-03 -1.705
4.45e-05 2.846
3.39e-01 -2.025
1.45e-01 -1.313
p
0.0000
0.0000
0.0001
0.0027
0.0000
0.0000
0.6200
0.8000
0.0880
0.0044
0.0430
0.1900
exp(coef) exp(-coef) lower .95 upper .95
gendermale
1.842
0.543
1.649
2.059
age
1.072
0.933
1.068
1.076
povindex
1.000
1.000
0.999
1.000
physactYes
0.883
1.133
0.813
0.958
priorhdYes
2.823
0.354
2.530
3.149
diabetesYes
2.156
0.464
1.857
2.503
bpdia
1.006
0.994
0.983
1.029
I(bpdia^2)
1.000
1.000
1.000
1.000
weight
0.987
1.013
0.973
1.002
I(weight^2)
1.000
1.000
1.000
1.000
height
0.504
1.985
0.259
0.978
priorhdYes:diabetesYes
0.827
1.210
0.622
1.098
Rsquare= 0.281
(max possible= 0.986 )
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Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
Likelihood ratio test= 3566
Wald test
= 2951
Score (logrank) test = 4191
on 12 df,
on 12 df,
on 12 df,
11/26/07 - 16/20
p=0
p=0
p=0
summary(coxph( Surv(time,chdhosp) ~ gender + age + povindex + physact +
priorhd*diabetes + hitched + breakdwn, data=chdsurv))
Call:
coxph(formula = Surv(time, chdhosp) ~ gender + age + povindex +
physact + priorhd * diabetes + hitched + breakdwn, data =
chdsurv)
n=10873 (440 observations deleted due
coef exp(coef)
gendermale
0.581903
1.789
age
0.069304
1.072
povindex
-0.000532
0.999
physactYes
-0.149064
0.862
priorhdYes
1.069120
2.913
diabetesYes
0.788180
2.199
hitchedYes
0.080791
1.084
breakdwnYes
0.208847
1.232
priorhdYes:diabetesYes -0.222180 0.801
gendermale
age
povindex
physactYes
priorhdYes
diabetesYes
hitchedYes
breakdwnYes
priorhdYes:diabetesYes
to missingness)
se(coef)
z
p
0.040357 14.42 0.0e+00
0.001771 39.14 0.0e+00
0.000118 -4.51 6.5e-06
0.041507 -3.59 3.3e-04
0.055556 19.24 0.0e+00
0.076041 10.37 0.0e+00
0.089802 0.90 3.7e-01
0.096550 2.16 3.1e-02
0.144980 -1.53 1.3e-01
exp(coef) exp(-coef) lower .95 upper .95
1.789
0.559
1.653
1.937
1.072
0.933
1.068
1.075
0.999
1.001
0.999
1.000
0.862
1.161
0.794
0.935
2.913
0.343
2.612
3.248
2.199
0.455
1.895
2.553
1.084
0.922
0.909
1.293
1.232
0.812
1.020
1.489
0.801
1.249
0.603
1.064
Rsquare= 0.276
(max possible=
Likelihood ratio test= 3505 on
Wald test
= 2966 on
Score (logrank) test = 4176 on
0.986 )
9 df,
p=0
9 df,
p=0
9 df,
p=0
16
Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
For comparison, results from logistic and linear regression:
11/26/07 - 17/20
Logistic regression models:
round(summary(glm(chd~gender,family=binomial,data=chd))$coef,3)
(Intercept)
gendermale
Estimate Std. Error z value Pr(>|z|)
-3.545
0.075 -47.276
0
0.826
0.098
8.416
0
round(summary(glm(chd~gender+age,family=binomial,data=chd))$coef,3)
Estimate Std. Error z value Pr(>|z|)
(Intercept)
-8.745
0.353 -24.778
0
gendermale
0.575
0.101
5.724
0
age
0.090
0.005 16.943
0
round(summary(glm(chd~gender+age+povindex+physact,family=binomial,data=chd))$coef,3)
(Intercept)
gendermale
age
povindex
physactYes
Estimate Std. Error z value Pr(>|z|)
-8.066
0.378 -21.364
0.000
0.653
0.103
6.310
0.000
0.085
0.005 15.546
0.000
-0.001
0.000 -2.238
0.025
-0.507
0.113 -4.494
0.000
round(summary(glm(chd~gender+age+povindex+physact+priorhd*diabetes,family=binomial,d
ata=chd))$coef,3)
Estimate Std. Error z value Pr(>|z|)
(Intercept)
-7.889
0.383 -20.576
0.000
gendermale
0.601
0.106
5.652
0.000
age
0.074
0.006 13.197
0.000
povindex
-0.001
0.000 -1.665
0.096
physactYes
-0.347
0.116 -2.977
0.003
priorhdYes
1.520
0.122 12.491
0.000
diabetesYes
0.724
0.197
3.680
0.000
priorhdYes:diabetesYes
-0.163
0.306 -0.531
0.595
round(summary(glm(chd~gender+age+povindex+physact+priorhd*diabetes+bpdia+I(bpdia^2)+
weight+I(weight^2)+height,family=binomial,data=chd))$coef,3)
(Intercept)
gendermale
age
povindex
physactYes
priorhdYes
diabetesYes
Estimate Std. Error z value Pr(>|z|)
-4.662
1.904 -2.449
0.014
0.726
0.149
4.883
0.000
0.074
0.006 12.776
0.000
0.000
0.000 -1.429
0.153
-0.331
0.117 -2.828
0.005
1.510
0.122 12.327
0.000
0.729
0.198
3.689
0.000
17
Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
11/26/07 - 18/20
bpdia
-0.024
0.027
-0.863
0.388
I(bpdia^2)
weight
I(weight^2)
height
priorhdYes:diabetesYes
0.000
-0.026
0.000
-0.763
-0.167
0.000
0.020
0.000
0.880
0.307
0.872
-1.336
1.355
-0.868
-0.544
0.383
0.181
0.175
0.386
0.586
round(summary(glm(chd~gender+age+povindex+physact+priorhd*diabetes+hitched+breakdwn,
family=binomial,data=chd))$coef,3)
(Intercept)
gendermale
age
povindex
physactYes
priorhdYes
diabetesYes
hitchedYes
breakdwnYes
priorhdYes:diabetesYes
Estimate Std. Error z value Pr(>|z|)
-8.025
0.444 -18.090
0.000
0.609
0.107
5.701
0.000
0.074
0.006 13.190
0.000
-0.001
0.000 -1.626
0.104
-0.348
0.117 -2.984
0.003
1.514
0.122 12.416
0.000
0.719
0.197
3.653
0.000
0.127
0.236
0.537
0.592
0.180
0.247
0.730
0.466
-0.167
0.307 -0.543
0.587
Linear models:
round(summary(glm(chd~gender, family=quasi(link=identity, variance="mu(1-mu)"),
data=chd))$coef, 4)
(Intercept)
gendermale
Estimate Std. Error t value Pr(>|t|)
0.0281
0.0020 13.7204
0
0.0338
0.0042 8.0374
0
round(summary(glm(chd~gender+I(age-50), start=c(.03,.02,.001),
family=quasi(link=identity,variance="mu(1-mu)"), data=chd))$coef, 3)
Estimate Std. Error t value Pr(>|t|)
(Intercept)
0.035
0.002 20.185
0
gendermale
0.014
0.003
4.815
0
I(age - 50)
0.001
0.000 20.185
0
There were 27 warnings (use warnings() to see them)
round(summary(glm(chd~gender+I(age-50) + povindex + physact,
start=c(.03,.02,0,0,0),family=quasi(link=identity, variance="mu(1-mu)"),
data=chd))$coef, 3)
(Intercept)
gendermale
I(age - 50)
povindex
physactYes
Estimate Std. Error t value Pr(>|t|)
0.037
0.003 14.526
0.000
0.021
0.003
6.181
0.000
0.001
0.000
8.562
0.000
0.000
0.000 -3.284
0.001
-0.006
0.003 -2.163
0.031
18
Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
11/26/07 - 19/20
There were 29 warnings (use warnings() to see them)
round(summary(glm(chd~gender+I(age-50) + povindex + physact + priorhd*diabetes,
start=c(.03,.02,0,0,0,0,0,0), family = quasi(link=identity,variance="mu(1mu)"),data=chd))$coef,3)
Estimate Std. Error t value Pr(>|t|)
(Intercept)
0.031
0.002 13.355
0.000
gendermale
0.019
0.003
6.269
0.000
I(age - 50)
0.001
0.000
7.864
0.000
povindex
0.000
0.000 -3.141
0.002
physactYes
-0.005
0.002 -1.816
0.069
priorhdYes
0.070
0.010
6.938
0.000
diabetesYes
0.018
0.009
1.915
0.056
priorhdYes:diabetesYes
0.033
0.032
1.020
0.308
There were 29 warnings (use warnings() to see them)
round(summary(glm(chd~gender+I(age-50) + povindex + physact + priorhd*diabetes +
bpdia + I(bpdia^2) + weight + I(weight^2) + height, start =
c(.03,.02,0,0,0,0,0,0,0,0,0,0,0), family = quasi(link=identity,variance="mu(1mu)"),data=chd))$coef,3)
Estimate Std. Error t value Pr(>|t|)
(Intercept)
0.096
0.046
2.071
0.038
gendermale
0.022
0.004
5.676
0.000
I(age - 50)
0.001
0.000
7.427
0.000
povindex
0.000
0.000 -1.847
0.065
physactYes
-0.004
0.003 -1.691
0.091
priorhdYes
0.066
0.010
6.574
0.000
diabetesYes
0.017
0.009
1.845
0.065
bpdia
-0.001
0.001 -0.687
0.492
I(bpdia^2)
0.000
0.000
0.601
0.548
weight
0.000
0.000 -0.894
0.372
I(weight^2)
0.000
0.000
0.770
0.441
height
-0.013
0.021 -0.648
0.517
priorhdYes:diabetesYes
0.030
0.031
0.966
0.334
There were 27 warnings (use warnings() to see them)
round(summary(glm(chd~gender+I(age-50) + povindex + physact + priorhd*diabetes +
hitched + breakdwn, start = c(.03,.02,0,0,0,0,0,0,0,0), family =
quasi(link=identity,variance="mu(1-mu)"),data=chd))$coef,3)
(Intercept)
gendermale
I(age - 50)
povindex
physactYes
priorhdYes
diabetesYes
Estimate Std. Error t value Pr(>|t|)
0.030
0.005
6.444
0.000
0.019
0.003
6.279
0.000
0.001
0.000
7.830
0.000
0.000
0.000 -3.079
0.002
-0.005
0.002 -1.816
0.069
0.070
0.010
6.917
0.000
0.018
0.009
1.903
0.057
19
Biost/Stat 579 – Autumn 2008 – Longitudinal Data Analysis & Survival Analysis
hitchedYes
0.000
0.005
0.078
breakdwnYes
0.002
0.008
0.287
priorhdYes:diabetesYes
0.032
0.032
1.019
There were 29 warnings (use warnings() to see them)
11/26/07 - 20/20
0.938
0.774
0.308
20