Survival Models and TimeDependent Variables

1
Survival Models and Time-Dependent Variables

• Larry J. Brant, Ph.D.
• Gerontology Research Center
• National Institute on Aging
• Baltimore, MD USA

April 26, 2006
2
Follow-up Studies
A follow-up study is a longitudinal study in
which the outcome of interest is the time until
the occurrence of a particular endpoint, such as
death or disease.
x
x
x
x
(beginning)
follow-up
(end)
In these studies, a group(s) of individuals is
usually followed over a long period of time
(times of observation marked by x), and the times
at which the outcome of interest occurs are
recorded for individuals in the study. At the
observation times before the studys end,
information on factors believed to be related to
the event of interest is collected and these
factors are often termed risk factors, explanatory
variables, or covariates.
3
Presentation of Results
In follow-up studies, investigators are often
interested in describing survival patterns and
comparing these patterns between different study
groups. Also, of interest is to examine the
relationship of the risk factors to the length of
time to the occurrence of the event.
4
Example
Systolic blood pressure measurements on 932 male
participants of the BLSA are used to examine the
risk of blood pressure on the time to the
occurrence of the event coronary heart
disease (CHD). CHD is defined as the occurrence
of a coronary death, the diagnosis of a
myocardial infarction by history or
pathologic Q-waves, or angina pectoris.
5
Data Summary
Of the 932 participants, 185 experienced the
event CHD. A maximum of 5 repeated examinations
or visits were chosen per person so as to be as
uniformly spaced as possible over the study
period. On average, 3.1 visits were considered
per person with an average of 3.0 systolic blood
pressure readings per visit.
6
Descriptive Statistics of BLSA Data
Systolic Blood Censored
CHD Pressure (n 747)
(n 185)
Age at baseline 45.24 15.77 56.34
13.21 Follow-up time 18.48 7.87
13.38 7.15
Individuals are referred to as censored when
their survival times are unknown either because
of lost to follow-up during the study or because
the study has ended without the event occurring
to the individual.
7
Methods of Survival Analysis

• Life-table method
• Product-limit or Kaplan-Meier method
• Proportional hazards or Cox model
• Proportional hazards model with time-dependent
  covariates

8
Product-Limit Method
This method estimates the proportion of a sample
of individuals who survive a given length of time
under the same circumstances without the event of
study occurring from a set of observed survival
times (including censored times) for the given
sample.
9
Tests for Comparing Survival Curves Generated
from the Kaplan-Meier Method
SAS provides three tests for comparing the
survival curves between two groups or
strata Log-rank test Wilcoxon
test Likelihood-ratio test
Note The log-rank test places more weight on
the larger survival times, the
Wilcoxon test places more weight on the early
part of the survival curve, and the
likelihood-ratio test is a test of the assumption
that the survival distribution is
exponential. Since the likelihood-ratio (lr)
test is computed from -2log(lr), the
appropriateness of the exponential distribution
assumption can be assessed by
examining the plot of log(S(t)) vs. time.
10
SAS Example
To compare the survival patterns between 850
hypertensive (SBP 140) and 82 non-hypertensive
(SBP lt 140) BLSA men, consider the SAS code
title1 'luigisuv01.sas performs Kaplan-Meier
method' title2 'CHD survival for Hypertensive vs
Non-Hypertensive' data a infile
luigi030106 input XRAY 1-4 fage 6-9 .1 survtime
11-14 .1 chd 16 sbpmean 18-20 psmkc 22 sbpcat
0 if sbpmean gt 140 then sbpcat 1 proc
lifetest method km plots (c,s,ls,lls)notable
time survtimechd(0) strata sbpcat run
11
Comments on SAS Code

• method km denotes Kaplan-Meir method
• It is also possible to use pl for
  product-limit method.
• 2. plots (c,s,ls,lls) requests plots of the
  censoring patterns
• for the two strata hypertensive
  and non-hypertensive
• and plots for each strata of the
  estimated survival
• functions as well as the negative
  log and log of the
• negative log of the estimated
  survival functions
• against time.

12
Comments on Plots
Plot c is useful in examining for possible
informative censoring, while plots ls and lls
give an empirical check of the appropriateness of
the exponential and Weibull models for the
survival curves If the ls plot is linear, then
the survival curve can be modeled using an
exponential model, and if the lls plot is linear,
then the survival curve can be modeled using a
Weibull model.
13
Output for Hypertension Example

luigisuv01.sas performs Kaplan-Meier Method
March 9, 2006 1 CHD survival for Hypertensive
vs Non-Hypertensive The LIFETEST Procedure
Summary of the Number of Censored and
Uncensored Values
Percent Stratum
sbpcat Total Failed Censored
Censored 1 0 850
144 706 83.06 2
1 82 41 41 50.00
--------------------------------------------------
----------------- Total
932 185 747 80.15
14
luigisuv01.sas performs Kaplan-Meier method
March 9, 2006 2 CHD survival for Hypertensive
vs Non-Hypertensive Testing Homogeneity of
Survival Curves for survtime over Strata
Rank Statistics sbpcat Log-Rank
Wilcoxon 0 -31.600 -22240 1
31.600 22240 Covariance Matrix
for the Log-Rank Statistics sbpcat
0 1 0 8.81365
-8.81365 1 -8.81365 8.81365
Covariance Matrix for the Wilcoxon Statistics
sbpcat 0 1 0
4708021 -4708021 1 -4708021
4708021
15
luigisuv01.sas performs Kaplan-Meier method
March 9, 2006 3 CHD survival for Hypertensive
vs Non-Hypertensive Test of Equality over
Strata Pr
gt Test Chi-Square DF Chi-Square
Log-Rank 113.2971 1 lt.0001
Wilcoxon 105.0585 1 lt.0001
-2Log(LR) 53.7211 1 lt.0001
16
luigisuv01.sas performs Kaplan-Meier method
March 9, 2006 4 CHD survival for Hypertensive
vs Non-Hypertensive
Censored Observations
17
luigisuv01.sas performs Kaplan-Meier method
March 9, 2006 5 CHD survival for Hypertensive
vs Non-Hypertensive Survival
Function Estimates
18
luigisuv01.sas performs Kaplan-Meier method
March 9, 2006 6 CHD survival for Hypertensive
vs Non-Hypertensive
-log(Survival Function) Estimates
19
luigisuv01.sas performs Kaplan-Meier method
March 9, 2006 7 CHD survival for Hypertensive
vs Non-Hypertensive
log(-log(Survival Function)) Estimates
20
Testing Survival Curves Across Groups or Strata
Often you might wish to estimate and compare the
survival curves across different strata
represented by different diagnostic or treatment
groups, genders, age groups, or some other
distinguishing characteristics. To do this,
first Kaplan-Meier estimates of the survival
curve for each strata can be calculated and then
compared to each other using a test of
homogeneity over strata. Perhaps the most widely
used such test is the log-rank or Mantel- Cox
test.
21
Log-Rank Test of Homogeneity Over Strata
The log-rank statistic can be interpreted as the
difference between the observed number of
failures and the expected number of failures
under the null hypothesis of an
identical survival function for all strata. The
log-rank statistic is considered to have a
chi-square distribution with c-1 degrees of
freedom (c number of strata).
22
Smoking Example
Using a similar SAS code as the hypertension
example, we now test the CHD survival patterns
between 172 current smokers and 760 non-current
smokers.
23
Output for Smoking Example
luigisuv02.sas performs Kaplan-Meier method
March 9, 2006 1 CHD survival for Current
Smokers vs Non-Current Smokers The LIFETEST
Procedure Summary of the Number of
Censored and Uncensored Values

Percent Stratum psmkc Total
Failed Censored Censored 1
0 760 143 617
81.18 2 1 172
42 130 75.58 ----------------------
---------------------------------------------
Total 932 185
747 80.15
24
luigisuv02.sas performs Kaplan-Meier method
March 9, 2006 2 CHD survival for Current
Smokers vs Non-Current Smokers Testing
Homogeneity of Survival Curves for survtime over
Strata Rank Statistics psmkc
Log-Rank Wilcoxon 0 -9.2231
-7272.0 1 9.2231 7272.0
Covariance Matrix for the Log-Rank Statistics
psmkc 0 1 0
26.9371 -26.9371 1 -26.9371
26.9371 Covariance Matrix for the Wilcoxon
Statistics psmkc 0 1
0 11535576 -1.154E7 1
-1.154E7 11535576
25
luigisuv02.sas performs Kaplan-Meier method
March 9, 2006 3 CHD survival for Current
Smokers vs Non-Current Smokers Test of
Equality over Strata
Pr gt Test Chi-Square
DF Chi-Square Log-Rank 3.1580
1 0.0756 Wilcoxon 4.5843 1
0.0323 -2Log(LR) 2.7841 1
0.0952
26
Luigisuv02.sas performs Kaplan-Meier method
March 9, 2006 4 CHD survival for Current
Smokers vs Non-Current Smokers
Censored Observations
27
luigisuv02.sas performs Kaplan-Meier method
March 9, 2006 5 CHD survival for Current
Smokers vs Non-Current Smokers
Survival Function Estimates
28
luigisuv02.sas performs Kaplan-Meier method
March 9, 2006 6 CHD survival for Current
Smokers vs Non-Current Smokers
-log(Survival Function) Estimates
29
luigisuv02.sas performs Kaplan-Meier method
March 9, 2006 7 CHD survival for Current
Smokers vs Non-Current Smokers
log(-log(Survival Function)) Estimates
30
Proportional Hazards Model
The proportional hazards model (sometimes called
the Cox Model) is a well-recognized statistical
tool for exploring the relationship between the
survival of an individual and several explanatory
or risk factor values. Note Further, the
proportional hazards model, like the
product-limit method for calculating a survival
curve, uses the exact times at which events
occur, but generalizes the the usual methods of
survival analysis (such as the product- limit
method) by providing estimates of survival as a
function of specific risk factor values.
31
Proportional Hazards Model Formulation
The proportional hazards model may be written
as h(t) h0(t) exp(ß1x1 ß2x2
ßnxn) and is the product of the baseline hazards
function h0(t) and exp(ß1x1 ß2x2 ßnxn)
the function of the risk factors under study.
32
Relative Risk
The relative risk of individual 1 with
covariates x1 (x11,x21, ,xn1) relative to
individual 0 with covariates x0 (x10,x20,
,xn0) is given by h(tx1)/h(tx0)
exp(ß1(x11-x10)ßn(xn1-xn0)).
33
Comments

• The hazard function h(t) is a conditional
  probability that
• the event under study occurs at
  approximately time t
• given that the event has not occurred before
  t. If time is
• age, then h(t) represents the age-specific
  event rates.
• The survival function S(t) is the probability
  that the event
• under study does not occur before time t,
  that is, the
• probability that an individual survives to
  or longer than
• time t. Note that h(t) - d/dt(ln(S(t)).
  Thus, as we saw
• earlier, if the plot of ln(S(t)) vs. t is
  linear, then S(t) is
• exponential and h(t) is a constant.

34
SAS Example
To use the proportional hazards model to examine
the relationship between systolic blood pressure
measured at baseline and CHD in the 932 BLSA male
participants mentioned earlier, consider the SAS
code
program luigisuv03.sas models proportional
hazards data a infile luigi030106 input XRAY
1-4 fage 6-9 .1 survtime 11-14 .1 chd 16 sbpmean
18-20 psmkc 22 proc phreg model survtimechd(0)
sbpmean title1 'luigisuv03.sas performs
Proportional Hazards Model' title2 'Proportional
Hazards model for SBP, CHD Survival' run
35
Output for SBP Proportional Hazards Model
luigisuv03.sas performs Proportional Hazards
Model April 6, 2006 1 Proportional Hazards
model for SBP, CHD Survival The PHREG
Procedure Model Information Data
Set WORK.A Dependent Variable
survtime Censoring Variable chd
Censoring Value(s) 0 Ties Handling
BRESLOW Summary of the Number of Event and
Censored Values
Percent Total Event Censored
Censored 932 185 747
80.15
36
luigisuv03.sas performs Proportional Hazards
Model April 6, 2006 2 Proportional Hazards
model for SBP, CHD Survival Convergence Status
Convergence criterion (GCONV1E-8)
satisfied. Model Fit Statistics
Without With Criterion
Covariates Covariates -2 LOG L
2326.734 2235.457 AIC 2326.734
2237.457 SBC 2326.734
2240.678
37
luigisuv03.sas performs Proportional Hazards
Model April 6, 2006 3 Proportional Hazards
model for SBP, CHD Survival Testing Global Null
Hypothesis BETA0 Test
Chi-Square DF Pr gt ChiSq Likelihood
Ratio 91.2772 1 lt.0001
Score 121.2392 1
lt.0001 Wald 120.7367 1
lt.0001 Analysis
of Maximum Likelihood Estimates
Parameter Standard
Hazard Variable DF Estimate Error
Chi-Square Pr gt ChiSq Ratio sbpmean
1 0.04360 0.00397 120.7367
lt.0001 1.045
38
Relative Risk of SBP for CHD
The relative risk for a 10 mm Hg increase in
Systolic Blood Pressure (SBP) is given
by exp(0.0436 x 10) 1.55. Thus, there is 55
greater probability to develop CHD for every 10
mm HG increase in SBP.
39
SBP and Smoking Example
Using similar SAS code, we now compute the
relative risk of SBP and current smoking with
regard to CHD. The SAS code is
program luigisuv04.sas models proportional
hazards data a infile luigi030106 input XRAY
1-4 fage 6-9 .1 survtime 11-14 .1 chd 16 sbpmean
18-20 psmkc 22 proc phreg model survtimechd(0)
sbpmean psmkc title1 'luigisuv04.sas performs
Proportional Hazards Model' title2 'Proportional
Hazards model for SBP Smoking, CHD
Survival' run
40
Output for Smoking and SBP Proportional Hazards
Model
luigisuv04.sas performs Proportional Hazards
Model April 6, 2006 1 Proportional Hazards
model for SBP Smoking, CHD Survival
The PHREG Procedure Model
Information Data Set WORK.A
Dependent Variable survtime Censoring
Variable chd Censoring Value(s) 0
Ties Handling BRESLOW Summary of
the Number of Event and Censored Values
Percent Total
Event Censored Censored 932
185 747 80.15
41
luigisuv03.sas performs Proportional Hazards
Model April 6, 2006 2 Proportional Hazards
model for SBP Smoking, CHD Survival Convergence
Status Convergence criterion
(GCONV1E-8) satisfied. Model Fit
Statistics Without
With Criterion Covariates Covariates
-2 LOG L 2326.734 2228.672 AIC
2326.734 2232.672 SBC
2326.734 2239.113
42
luigisuv03.sas performs Proportional Hazards
Model April 6, 2006 3 Proportional Hazards
model for SBP Smoking, CHD Survival Testing
Global Null Hypothesis BETA0 Test
Chi-Square DF Pr gt ChiSq
Likelihood Ratio 98.0624 2
lt.0001 Score 128.9858 2
lt.0001 Wald 127.2952
2 lt.0001
Analysis of Maximum Likelihood Estimates
Parameter Standard
Hazard Variable DF Estimate
Error Chi-Square Pr gt ChiSq Ratio
sbpmean 1 0.04476 0.00400
125.4882 lt.0001 1.046 psmkc 1
0.48178 0.17693 7.4148 0.0065
1.619
43
Relative Risk of SBP and Smoking for CHD
The relative risk for a 10 mm HG increase in
Systolic Blood Pressure (SBP) and being a current
cigarette smoker is given by exp(0.0448 x 10
0.482) 2.53. Thus there is a 153 greater
probability to develop CHD for current cigarette
smokers with a 10 mm Hg increase in SBP.
44
Time-Dependent Covariates
A time-dependent covariate is one whose value for
each individual can change over time during the
course of the study. For example, it is possible
to use time-dependent covariates to examine the
effect of changing blood pressure values or other
measures of blood chemistry during the course of
a study. Or it is possible to use a
time-dependent variable to study the effect of
individuals changing treatment groups.
45
Remark on Time-Dependent Covariates
The estimation procedure for the time-dependent
covariates proportional hazards model requires
that each individual in the data set have a
time-dependent covariate value for every time at
which an event occurs in the data set.
46
Mixed-Effects Models and Time-Dependent Covariates
Mixed-effects models with both fixed- and
random-effects are useful tools in generating
estimated values of a time-dependent covariate at
any time during the follow-up course of a
given study. For many time-dependent covariate
measures there is an inability of getting an
observed value that reflects the true value of
the covariate. This has been shown to lead to a
so-called measurement error bias which in turn
can lead to regression dilution
bias. Mixed-effects models provide an additional
advantage in that they have been shown to be a
useful tool in dealing with measurement error
bias.
47
Measurement-Error Bias and Regression Dilution
Bias
The graph shows the relationship between the
value of a risk factor (covariate) and the
probability of a disease based on the true value
of the risk factor and those predicted from
observed values. Failure to account for
regression toward the mean can result in an
underestimate of the true strength of the
relationship between a covariate and the
probability of the disease.
48
Blood Pressure and Measurement-Error Bias
One clinical measure well known to be associated
with measurement- error bias is blood pressure.
The graph below indicates how the use of
mixed-effects models have proven useful in
reducing this bias and the subsequent resulting
regression dilution bias.
mse is 0.549 mse is 0.512 mse is 0.430
From a simulation study based on 500 simulations,
the mean-squared error (mse) for the beta
estimate of SBP using a single, mean and mixed
model predicted covariate values for SBP are
shown above where for reference the mse for the
true value of SBP from the assumed distribution
is 0.396.
49
Predicting SBP Values for Time-Dependent
Covariates
The mixed-effects model for predicting SBP values
at all times where CHD events occur is
sbppred 124.1 b0 (7.01 b1)suvt
(-0.449)suvt2 (-1.046)fage
0.0243fage2 (-0.00012)fage3
(-0.500)suvt?fage
0.0111suvt?fage2 (-0.0000737)suvt?fage3
0.0325suvt2?fage
(-0.000712)suvt2fage2
0.0000049suvt2fage3 where suvt represents the
individuals survival time from the beginning of
the study and fage represents the individuals
age at the beginning of the follow-up period.
50
Example of the Predicted SBP Values
51
SAS Example
To use the time-dependent covariate proportional
hazards model to examine the relationship between
systolic blood pressure measured longitudinally
over the course of the study and CHD in the 932
BLSA males mentioned earlier, consider the SAS
code
/ Modified coxlmet.sas in chm .coxlme
Use lme models to compute updated covariate
sbp-value and fit time dependent Cox
model / DATA A INFILE CHMBP797
INPUT XRAY 1-4 VISIT 6-7 SBP 9-11 FAGE 13-17 .1
FOLLUP 19-22 .1 data a1 set a by xray if
first.xray DATA B INFILE CHMCHD
INPUT XRAY 1-4 SURVTIME 6-9 .1 EVENT 11 data
small merge a1 b by xray keep xray survtime
event fage
52
SAS Example (continued)
data bsa inFILE Bsa inPUT XRAY b0 b1 data
singall merge small bsa f2fagefage
f3fagef2 by xray proc phreg model
survtimeevent(0) sbppred sbpt title1
'luigisuv06.sas performs time-dependent
proportional hazards model' title2 'Updated
Covariates LME of single observations for all
visits' title3 'Time-dependent SBP and CHD
survival' sbppred 124.08063412 b0
survtime(7.00991990 b1)
survtimesurvtime(-0.44900400)
FAGE(-1.04585782) F20.02431169
F3(-0.00011965) survtimefage(-0.50047459)
survtimef2(0.01111585)
survtimef3(-0.00007367)
survtimesurvtimefage(0.03254626)
survtimesurvtimeF2(-0.00071242)
survtimesurvtimeF3(0.00000491) sbpt
sbppredlog(survtime) proportionality_test test
sbpt run
53
Output for SBP Time-Dependent Proportional
Hazards Model
luigisuv06.sas performs time-dependent
proportional hazards model 1 Updated Covariates
LME of single observations for all
visits Time-dependent SBP and CHD survival The
PHREG Procedure Model Information
Data Set WORK.SINGALL Dependent
Variable SURVTIME Censoring Variable
EVENT Censoring Value(s) 0 Ties Handling
BRESLOW Summary of the Number of
Event and Censored Values
Percent Total Event
Censored Censored 932 184
748 80.26
54
Output for SBP Time-Dependent Model (continued)
luigisuv06.sas performs time-dependent
proportional hazards model 2 Updated Covariates
LME of single observations for all
visits Time-dependent SBP and CHD survival
Convergence Status
Convergence criterion (GCONV1E-8) satisfied.
Model Fit Statistics
Without With Criterion Covariates
Covariates -2 LOG L 2313.060
2181.237 AIC 2313.060
2185.237 SBC 2313.060 2191.666
55
Output for SBP Time-Dependent Model (continued)
luigisuv06.sas performs time-dependent
proportional hazards model 3 Updated Covariates
LME of single observations for all
visits Time-dependent SBP and CHD survival
Testing Global Null Hypothesis BETA0
Test Chi-Square DF Pr gt
ChiSq Likelihood Ratio 131.8232 2
lt.0001 Score 174.9177
2 lt.0001 Wald
161.3938 2 lt.0001
Analysis of Maximum Likelihood Estimates
Parameter Standard
Hazard Variable DF Estimate
Error Chi-Square Pr gt ChiSq Ratio
sbppred 1 0.09046 0.01508
35.9937 lt.0001 1.095 sbpt 1
-0.01774 0.00599 8.7681 0.0031
0.982
56
Output for SBP Time-Dependent Model (continued)
luigisuv06.sas performs time-dependent
proportional hazards model 4 Updated Covariates
LME of single observations for all
visits Time-dependent SBP and CHD survival The
PHREG Procedure Linear Hypotheses
Testing Results
Wald Label Chi-Square DF
Pr gt ChiSq proportionality_test
8.7681 1 0.0031
57
Proportionality of the Proportional Hazards Model
Recall that the proportional hazards model is
written as h(t) h0(t) exp(ß1x1 ß2x2
ßnxn) where all the terms in the model are
multiplicative, in particular, the covariate
terms. Thus, when modeling a proportional
hazards model a key assumption is the
proportionality or proportional hazards with
regard to the covariates.
58
Tests of Proportionality for the Proportional
Hazards Model
As we have just seen, if time-dependent
covariates are being used, then tests for
proportionality can be carried out by
considering interaction terms of the
time-dependent covariates and the log of survival
time. Interaction terms that are significant
indicate that the corresponding covariate is not
proportional. Also, using the test statement in
SAS proc phreg, one can test all the
time- dependent covariates together for overall
proportionality. For the fixed or non-time
dependent covariate case, one can do a similar
test, or stratify the covariate into groups and
examine graphs of the survival function versus
survival time or the graph of the log(-log(surviva
l function)) versus log(survival time) for
each stratified group.
59
Interpretation of the Stratified Curves
If the covariate satisfies the proportionality
assumption, then the graph for each strata of the
survival function versus the survival time should
result in parallel curves, while similarly the
graphs of the log(-log(survival function) versus
the log(survival time) should result in parallel
lines.
60
Testing Proportionality for SBP and Smoking
program luigisuv08.sas models proportional
hazards data a infile luigi030106 input XRAY
1-4 fage 6-9 .1 survtime 11-14 .1 chd 16 sbpmean
18-20 psmkc 22 proc phreg model survtimechd(0)
sbpmean sbpt psmkct sbpt sbpmeanlog(survtime
) psmkct psmkclog(survtime) proportionality_t
est test sbpt, psmkct title1 'luigisuv08.sas
performs Proportional Hazards Model-Fixed
Covariates' title2 'Proportional Hazards model
for SBP and Present Cig Smk, CHD
Survival' title3 ' Tests for Proportionality
Using Interactions' run
61
Testing Proportionality for SBP and Smoking
(continued)
luigisuv08.sas performs Proportional Hazards
Model-Fixed Covariates 1 Proportional Hazards
model for SBP and Present Cig Smk, CHD
Survival Tests for Proportionality Using
Interactions The PHREG Procedure
Model Information Data Set
WORK.A Dependent Variable survtime
Censoring Variable chd Censoring Value(s)
0 Ties Handling BRESLOW
Summary of the Number of Event and Censored
Values
Percent Total Event Censored
Censored 932 184 748
80.26
62
Testing Proportionality for SBP and Smoking
(continued)
luigisuv08.sas performs Proportional Hazards
Model-Fixed Covariates 2 Proportional Hazards
model for SBP and Present Cig Smk, CHD
Survival Tests for Proportionality Using
Interactions
Convergence Status Convergence
criterion (GCONV1E-8) satisfied.
Model Fit Statistics Without
With Criterion Covariates
Covariates -2 LOG L 2313.060
2222.188 AIC 2313.060
2228.188 SBC 2313.060
2237.833 Testing Global Null
Hypothesis BETA0 Test
Chi-Square DF Pr gt ChiSq Likelihood
Ratio 90.8716 3 lt.0001
Score 121.7070 3
lt.0001 Wald 118.7662 3
lt.0001
63
Testing Proportionality for SBP and Smoking
(continued)
luigisuv08.sas performs Proportional Hazards
Model-Fixed Covariates 3 Proportional Hazards
model for SBP and Present Cig Smk, CHD
Survival Tests for Proportionality Using
Interactions Analysis of
Maximum Likelihood Estimates
Parameter Standard
Hazard Variable DF Estimate Error
Chi-Square Pr gt ChiSq Ratio sbpmean
1 0.05269 0.01302 16.3752
lt.0001 1.054 sbpt 1 -0.00415
0.00571 0.5298 0.4667 0.996
psmkct 1 0.14367 0.07216
3.9642 0.0465 1.154
Linear Hypotheses Testing Results
Wald Label
Chi-Square DF Pr gt ChiSq
proportionality_test 4.6323 2
0.0987
64
Testing Proportionality for SBP - The Graphical
Approach
A lt130 mm Hg B 130-160 mm Hg C gt160 mm Hg
65
Testing Proportionality for SBP - The Graphical
Approach
A lt130 mm Hg B 130-160 mm Hg C gt160 mm Hg
66
Testing Proportionality for Smoking - Graphical
Approach
A no smoker B current smoker
67
Testing Proportionality for Smoking - Graphical
Approach
A no smoker B current smoker
68
My next seminar will be Wednesday, September 27
entitled Predicting Change over Time of
Continuous Variables.