Repeated determinations can be made during the course of a study of variables thought to be related to survival. Consider an experiment to study the dosing effect of a tumorpromoting agent. Fortyfive rodents initially exposed to a carcinogen were randomly assigned to three dose groups. After the first death of an animal, the rodents were examined every week for the number of papillomas. Investigators were interested in determining the effects of dose on the carcinoma incidence after adjusting for the number of papillomas.
The input data set TUMOR consists of the following 19 variables:
ID
(subject identification)
Time
(survival time of the subject)
Dead
(censoring status where 1=dead and 0=censored)
Dose
(dose of the tumorpromoting agent)
P1
–P15
(number of papillomas at the 15 times that animals died. These 15 death times are weeks 27, 34, 37, 41, 43, 45, 46, 47, 49,
50, 51, 53, 65, 67, and 71. For instance, subject 1 died at week 47; it had no papilloma at week 27, five papillomas at week
34, six at week 37, eight at week 41, and 10 at weeks 43, 45, 46, and 47. For an animal that died before week 71, the number
of papillomas is missing for those times beyond its death.)
The following SAS statements create the data set TUMOR:
data Tumor; infile datalines missover; input ID Time Dead Dose P1P15; label ID='Subject ID'; datalines; 1 47 1 1.0 0 5 6 8 10 10 10 10 2 71 1 1.0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 3 81 0 1.0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 81 0 1.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 81 0 1.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 65 1 1.0 0 0 0 1 1 1 1 1 1 1 1 1 1 7 71 0 1.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 69 0 1.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 67 1 1.0 0 0 1 1 2 2 2 2 3 3 3 3 3 3 10 81 0 1.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 37 1 1.0 9 9 9 12 81 0 1.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 77 0 1.0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 14 81 0 1.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 81 0 1.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 54 0 2.5 0 1 1 1 2 2 2 2 2 2 2 2 17 53 0 2.5 0 0 0 0 0 0 0 0 0 0 0 0 18 38 0 2.5 5 13 14 19 54 0 2.5 2 6 6 6 6 6 6 6 6 6 6 6 20 51 1 2.5 15 15 15 16 16 17 17 17 17 17 17 21 47 1 2.5 13 20 20 20 20 20 20 20 22 27 1 2.5 22 23 41 1 2.5 6 13 13 13 24 49 1 2.5 0 3 3 3 3 3 3 3 3 25 53 0 2.5 0 0 1 1 1 1 1 1 1 1 1 1 26 50 1 2.5 0 0 2 3 4 6 6 6 6 6 27 37 1 2.5 3 15 15 28 49 1 2.5 2 3 3 3 3 4 4 4 4 29 46 1 2.5 4 6 7 9 9 9 9 30 48 0 2.5 15 26 26 26 26 26 26 26 31 54 0 10.0 12 14 15 15 15 15 15 15 15 15 15 15 32 37 1 10.0 12 16 17 33 53 1 10.0 3 6 6 6 6 6 6 6 6 6 6 6 34 45 1 10.0 4 12 15 20 20 20 35 53 0 10.0 6 10 13 13 13 15 15 15 15 15 15 20 36 49 1 10.0 0 2 2 2 2 2 2 2 2 37 39 0 10.0 7 8 8 38 27 1 10.0 17 39 49 1 10.0 0 6 9 14 14 14 14 14 14 40 43 1 10.0 14 18 20 20 20 41 28 0 10.0 8 42 34 1 10.0 11 18 43 45 1 10.0 10 12 16 16 16 16 44 37 1 10.0 0 1 1 45 43 1 10.0 9 19 19 19 19 ;
The number of papillomas (NPap
) for each animal in the study was measured repeatedly over time. One way of handling timedependent repeated measurements
in the PHREG procedure is to use programming statements to capture the appropriate covariate values of the subjects in each
risk set. In this example, NPap
is a timedependent explanatory variable with values that are calculated by means of the programming statements shown in
the following SAS statements:
proc phreg data=Tumor; model Time*Dead(0)=Dose NPap; array pp{*} P1P14; array tt{*} t1t15; t1=27; t2=34; t3=37; t4=41; t5=43; t6=45; t7=46; t8=47; t9=49; t10=50; t11=51; t12=53; t13=65; t14=67; t15=71; if Time < tt[1] then NPap=0; else if time >= tt[15] then NPap=P15; else do i=1 to dim(pp); if tt[i] <= Time < tt[i+1] then NPap= pp[i]; end; run;
At each death time, the NPap
value of each subject in the risk set is recalculated to reflect the actual number of papillomas at the given death time.
For instance, subject one in the data set Tumor
was in the risk sets at weeks 27 and 34; at week 27, the animal had no papilloma, while at week 34, it had five papillomas.
Results of the analysis are shown in Output 67.7.1. After the number of papillomas is adjusted for, the dose effect of the tumorpromoting agent is not statistically significant.
Output 67.7.1: Cox Regression Analysis on the Survival of Rodents
Model Information  

Data Set  WORK.TUMOR 
Dependent Variable  Time 
Censoring Variable  Dead 
Censoring Value(s)  0 
Ties Handling  BRESLOW 



Summary of the Number of Event and Censored Values 


Total  Event  Censored  Percent Censored 
45  25  20  44.44 
Convergence Status 

Convergence criterion (GCONV=1E8) satisfied. 
Model Fit Statistics  

Criterion  Without Covariates 
With Covariates 
2 LOG L  166.793  143.269 
AIC  166.793  147.269 
SBC  166.793  149.707 
Testing Global Null Hypothesis: BETA=0  

Test  ChiSquare  DF  Pr > ChiSq 
Likelihood Ratio  23.5243  2  <.0001 
Score  28.0498  2  <.0001 
Wald  21.1646  2  <.0001 
Analysis of Maximum Likelihood Estimates  

Parameter  DF  Parameter Estimate 
Standard Error 
ChiSquare  Pr > ChiSq  Hazard Ratio 
Dose  1  0.06885  0.05620  1.5010  0.2205  1.071 
NPap  1  0.11714  0.02998  15.2705  <.0001  1.124 
Another way to handle timedependent repeated measurements in the PHREG procedure is to use the counting process style of
input. Multiple records are created for each subject, one record for each distinct pattern of the timedependent measurements.
Each record contains a T1
value and a T2
value representing the time interval (T1
,T2
] during which the values of the explanatory variables remain unchanged. Each record also contains the censoring status at
T2
.
One advantage of using the counting process formulation is that you can easily obtain various residuals and influence statistics that are not available when programming statements are used to compute the values of the timedependent variables. On the other hand, creating multiple records for the counting process formulation requires extra effort in data manipulation.
Consider a counting process style of input data set named Tumor1
. It contains multiple observations for each subject in the data set Tumor
. In addition to variables ID
, Time
, Dead
, and Dose
, four new variables are generated:
T1
(left endpoint of the risk interval)
T2
(right endpoint of the risk interval)
NPap
(number of papillomas in the time interval (T1
,T2
])
Status
(censoring status at T2
)
For example, five observations are generated for the rodent that died at week 47 and that had no papilloma at week 27, five
papillomas at week 34, six at week 37, eight at week 41, and 10 at weeks 43, 45, 46, and 47. The values of T1
, T2
, NPap
, and Status
for these five observations are (0,27,0,0), (27,34,5,0), (34,37,6,0), (37,41,8,0), and (41,47,10,1). Note that the variables
ID
, Time
, and Dead
are not needed for the estimation of the regression parameters, but they are useful for plotting the residuals.
The following SAS statements create the data set Tumor1
:
data Tumor1(keep=ID Time Dead Dose T1 T2 NPap Status); array pp{*} P1P14; array qq{*} P2P15; array tt{1:15} _temporary_ (27 34 37 41 43 45 46 47 49 50 51 53 65 67 71); set Tumor; T1 = 0; T2 = 0; Status = 0; if ( Time = tt[1] ) then do; T2 = tt[1]; NPap = p1; Status = Dead; output; end; else do _i_=1 to dim(pp); if ( tt[_i_] = Time ) then do; T2= Time; NPap = pp[_i_]; Status = Dead; output; end; else if (tt[_i_] < Time ) then do; if (pp[_i_] ^= qq[_i_] ) then do; if qq[_i_] = . then T2= Time; else T2= tt[_i_]; NPap= pp[_i_]; Status= 0; output; T1 = T2; end; end; end; if ( Time >= tt[15] ) then do; T2 = Time; NPap = P15; Status = Dead; output; end; run;
In the following SAS statements, the counting process MODEL specification is used. The DFBETA statistics are output to a SAS
data set named Out1
. Note that Out1
contains multiple observations for each subject—that is, one observation for each risk interval (T1
,T2
].
proc phreg data=Tumor1; model (T1,T2)*Status(0)=Dose NPap; output out=Out1 resmart=Mart dfbeta=db1db2; id ID Time Dead; run;
The output from PROC PHREG (not shown) is identical to Output 67.7.1 except for the “Summary of the Number of Event and Censored Values” table. The number of event observations remains unchanged between the two specifications of PROC PHREG, but the number of censored observations differs due to the splitting of each subject’s data into multiple observations for the counting process style of input.
Next, the MEANS procedure sums up the component statistics for each subject and outputs the results to a SAS data set named
Out2
:
proc means data=Out1 noprint; by ID Time Dead; var Mart db1db2; output out=Out2 sum=Mart db_Dose db_NPap; run;
Finally, DFBETA statistics are plotted against subject ID for easy identification of influential points:
title 'DfBetas for Dose'; proc sgplot data=Out2; yaxis label="DfBeta" grid; refline 0 / axis=y; scatter y=db_Dose x=ID; run; title 'DfBetas for NPap'; proc sgplot data=Out2; yaxis label="DfBeta" grid; refline 0 / axis=y; scatter y=db_NPap x=ID; run;
The plots of the DFBETA statistics are shown in Output 67.7.2 and Output 67.7.3. Subject 30 appears to have a large influence on both the Dose
and NPap
coefficients. Subjects 31 and 35 have considerable influences on the DOSE coefficient, while subjects 22 and 44 have rather
large influences on the NPap
coefficient.