### Example 67.10 Analysis of Recurrent Events Data

Recurrent events data consist of times to a number of repeated events for each sample unit—for example, times of recurrent episodes of a disease in patients. Various ways of analyzing recurrent events data are described in the section Analysis of Multivariate Failure Time Data. The bladder cancer data listed in Wei, Lin, and Weissfeld (1989) are used here to illustrate these methods.

The data consist of 86 patients with superficial bladder tumors, which were removed when the patients entered the study. Of these patients, 48 were randomized into the placebo group, and 38 were randomized into the group receiving thiotepa. Many patients had multiple recurrences of tumors during the study, and new tumors were removed at each visit. The data set contains the first four recurrences of the tumor for each patient, and each recurrence time was measured from the patient’s entry time into the study.

The data consist of the following eight variables:

• Trt, treatment group (1=placebo and 2=thiotepa)

• Time, follow-up time

• Number, number of initial tumors

• Size, initial tumor size

• T1, T2, T3, and T4, times of the four potential recurrences of the bladder tumor. A patient with only two recurrences has missing values in T3 and T4.

In the data set Bladder, four observations are created for each patient, one for each of the four potential tumor recurrences. In addition to values of Trt, Number, and Size for the patient, each observation contains the following variables:

• ID, patient’s identification (which is the sequence number of the subject)

• Visit, visit number (with value k for the kth potential tumor recurrence)

• TStart, time of the (k – 1) recurrence for Visit=k, or the entry time 0 if VISIT=1, or the follow-up time if the (k – 1) recurrence does not occur

• TStop, time of the kth recurrence if Visit=k or follow-up time if the kth recurrence does not occur

• Status, event status of TStop (1=recurrence and 0=censored)

For instance, a patient with only one recurrence time at month 6 who was followed until month 10 will have values for Visit, TStart, TStop, and Status of (1,0,6,1), (2,6,10,0), (3,10,10,0), and (4,10,10,0), respectively. The last two observations are redundant for the intensity model and the proportional means model, but they are important for the analysis of the marginal Cox models. If the follow-up time is beyond the time of the fourth tumor recurrence, it is tempting to create a fifth observation with the time of the fourth tumor recurrence as the TStart value, the follow-up time as the TStop value, and a Status value of 0. However, Therneau and Grambsch (2000, Section 8.5) have warned against incorporating such observations into the analysis.

The following SAS statements create the data set Bladder:

data Bladder;
keep ID TStart TStop Status Trt Number Size Visit;
retain ID TStart 0;
array tt T1-T4;
infile datalines missover;
input Trt Time Number Size T1-T4;
ID + 1;
TStart=0;
do over tt;
Visit=_i_;
if tt = . then do;
TStop=Time;
Status=0;
end;
else do;
TStop=tt;
Status=1;
end;
output;
TStart=TStop;
end;
if (TStart < Time) then delete;
datalines;
1       0       1     1
1       1       1     3
1       4       2     1
1       7       1     1
1       10      5     1
1       10      4     1   6
1       14      1     1
1       18      1     1
1       18      1     3   5
1       18      1     1   12  16
1       23      3     3
1       23      1     3   10  15
1       23      1     1   3   16  23
1       23      3     1   3   9   21
1       24      2     3   7   10  16  24
1       25      1     1   3   15  25
1       26      1     2
1       26      8     1   1
1       26      1     4   2   26
1       28      1     2   25
1       29      1     4
1       29      1     2
1       29      4     1
1       30      1     6   28  30
1       30      1     5   2   17  22
1       30      2     1   3   6   8   12
1       31      1     3   12  15  24
1       32      1     2
1       34      2     1
1       36      2     1
1       36      3     1   29
1       37      1     2
1       40      4     1   9   17  22  24
1       40      5     1   16  19  23  29
1       41      1     2
1       43      1     1   3
1       43      2     6   6
1       44      2     1   3   6   9
1       45      1     1   9   11  20  26
1       48      1     1   18
1       49      1     3
1       51      3     1   35
1       53      1     7   17
1       53      3     1   3   15  46  51
1       59      1     1
1       61      3     2   2   15  24  30
1       64      1     3   5   14  19  27
1       64      2     3   2   8   12  13
2       1       1     3
2       1       1     1
2       5       8     1   5
2       9       1     2
2       10      1     1
2       13      1     1
2       14      2     6   3
2       17      5     3   1   3   5   7
2       18      5     1
2       18      1     3   17
2       19      5     1   2
2       21      1     1   17  19
2       22      1     1
2       25      1     3
2       25      1     5
2       25      1     1
2       26      1     1   6   12  13
2       27      1     1   6
2       29      2     1   2
2       36      8     3   26  35
2       38      1     1
2       39      1     1   22  23  27  32
2       39      6     1   4   16  23  27
2       40      3     1   24  26  29  40
2       41      3     2
2       41      1     1
2       43      1     1   1   27
2       44      1     1
2       44      6     1   2   20  23  27
2       45      1     2
2       46      1     4   2
2       46      1     4
2       49      3     3
2       50      1     1
2       50      4     1   4   24  47
2       54      3     4
2       54      2     1   38
2       59      1     3
;


First, consider fitting the intensity model (Andersen and Gill, 1982) and the proportional means model (Lin et al., 2000). The counting process style of input is used in the PROC PHREG specification. For the proportional means model, inference is based on the robust sandwich covariance estimate, which is requested by the COVS(AGGREGATE) option in the PROC PHREG statement. The COVM option is specified for the analysis of the intensity model to use the model-based covariance estimate. Note that some of the observations in the data set Bladder have a degenerated interval of risk. The presence of these observations does not affect the results of the analysis since none of these observations are included in any of the risk sets. However, the procedure will run more efficiently without these observations; consequently, in the following SAS statements, the WHERE clause is used to eliminate these redundant observations:

title 'Intensity Model and Proportional Means Model';
model (TStart, TStop) * Status(0) = Trt Number Size;
id id;
where TStart < TStop;
run;


Results of fitting the intensity model and the proportional means model are shown in Output 67.10.1 and Output 67.10.2, respectively. The robust sandwich standard error estimate for Trt is larger than its model-based counterpart, rendering the effect of thiotepa less significant in the proportional means model (p = 0.0747) than in the intensity model (p = 0.0215).

Output 67.10.1: Analysis of the Intensity Model

 Intensity Model and Proportional Means Model

The PHREG Procedure

Analysis of Maximum Likelihood Estimates
with Model-Based Variance Estimate
Parameter DF Parameter
Estimate
Standard
Error
Chi-Square Pr > ChiSq Hazard
Ratio
Trt 1 -0.45979 0.19996 5.2873 0.0215 0.631
Number 1 0.17165 0.04733 13.1541 0.0003 1.187
Size 1 -0.04256 0.06903 0.3801 0.5375 0.958

Output 67.10.2: Analysis of the Proportional Means Model

Analysis of Maximum Likelihood Estimates
with Sandwich Variance Estimate
Parameter DF Parameter
Estimate
Standard
Error
StdErr
Ratio
Chi-Square Pr > ChiSq Hazard
Ratio
Trt 1 -0.45979 0.25801 1.290 3.1757 0.0747 0.631
Number 1 0.17165 0.06131 1.296 7.8373 0.0051 1.187
Size 1 -0.04256 0.07555 1.094 0.3174 0.5732 0.958

Next, consider the conditional models of Prentice, Williams, and Peterson (1981). In the PWP models, the risk set for the (k + 1) recurrence is restricted to those patients who have experienced the first k recurrences. For example, a patient who experienced only one recurrence is an event observation for the first recurrence; this patient is a censored observation for the second recurrence and should not be included in the risk set for the third or fourth recurrence. The following DATA step eliminates those observations that should not be in the risk sets, forming a new input data set (named Bladder2) for fitting the PWP models. The variable Gaptime, representing the gap times between successive recurrences, is also created.

data Bladder2(drop=LastStatus);
retain LastStatus;
by ID;
if first.id then LastStatus=1;
if (Status=0 and LastStatus=0) then delete;
LastStatus=Status;
Gaptime=Tstop-Tstart;
run;


The following statements fit the PWP total time model. The variables Trt1, Trt2, Trt3, and Trt4 are visit-specific variables for Trt; the variables Number1, Number2, Numvber3, and Number4 are visit-specific variables for Number; and the variables Size1, Size2, Size3, and Size4 are visit-specific variables for Size.

title 'PWP Total Time Model with Noncommon Effects';
model (TStart,Tstop) * Status(0) = Trt1-Trt4 Number1-Number4
Size1-Size4;
Trt1= Trt * (Visit=1);
Trt2= Trt * (Visit=2);
Trt3= Trt * (Visit=3);
Trt4= Trt * (Visit=4);
Number1= Number * (Visit=1);
Number2= Number * (Visit=2);
Number3= Number * (Visit=3);
Number4= Number * (Visit=4);
Size1= Size * (Visit=1);
Size2= Size * (Visit=2);
Size3= Size * (Visit=3);
Size4= Size * (Visit=4);
strata Visit;
run;


Results of the analysis of the PWP total time model are shown in Output 67.10.3. There is no significant treatment effect on the total time in any of the four tumor recurrences.

Output 67.10.3: Analysis of the PWP Total Time Model with Noncommon Effects

 PWP Total Time Model with Noncommon Effects

The PHREG Procedure

Summary of the Number of Event and Censored Values
Stratum Visit Total Event Censored Percent
Censored
1 1 85 47 38 44.71
2 2 46 29 17 36.96
3 3 27 22 5 18.52
4 4 20 14 6 30.00
Total   178 112 66 37.08

Analysis of Maximum Likelihood Estimates
Parameter DF Parameter
Estimate
Standard
Error
Chi-Square Pr > ChiSq Hazard
Ratio
Trt1 1 -0.51757 0.31576 2.6868 0.1012 0.596
Trt2 1 -0.45967 0.40642 1.2792 0.2581 0.631
Trt3 1 0.11700 0.67183 0.0303 0.8617 1.124
Trt4 1 -0.04059 0.79251 0.0026 0.9592 0.960
Number1 1 0.23605 0.07607 9.6287 0.0019 1.266
Number2 1 -0.02044 0.09052 0.0510 0.8213 0.980
Number3 1 0.01219 0.18208 0.0045 0.9466 1.012
Number4 1 0.18915 0.24443 0.5989 0.4390 1.208
Size1 1 0.06790 0.10125 0.4498 0.5024 1.070
Size2 1 -0.15425 0.12300 1.5728 0.2098 0.857
Size3 1 0.14891 0.26299 0.3206 0.5713 1.161
Size4 1 0.0000732 0.34297 0.0000 0.9998 1.000

The following statements fit the PWP gap-time model:

title 'PWP Gap-Time Model with Noncommon Effects';
model Gaptime * Status(0) = Trt1-Trt4 Number1-Number4
Size1-Size4;
Trt1= Trt * (Visit=1);
Trt2= Trt * (Visit=2);
Trt3= Trt * (Visit=3);
Trt4= Trt * (Visit=4);
Number1= Number * (Visit=1);
Number2= Number * (Visit=2);
Number3= Number * (Visit=3);
Number4= Number * (Visit=4);
Size1= Size * (Visit=1);
Size2= Size * (Visit=2);
Size3= Size * (Visit=3);
Size4= Size * (Visit=4);
strata Visit;
run;


Results of the analysis of the PWP gap-time model are shown in Output 67.10.4. Note that the regression coefficients for the first tumor recurrence are the same as those of the total time model, since the total time and the gap time are the same for the first recurrence. There is no significant treatment effect on the gap times for any of the four tumor recurrences.

Output 67.10.4: Analysis of the PWP Gap-Time Model with Noncommon Effects

 PWP Gap-Time Model with Noncommon Effects

The PHREG Procedure

Analysis of Maximum Likelihood Estimates
Parameter DF Parameter
Estimate
Standard
Error
Chi-Square Pr > ChiSq Hazard
Ratio
Trt1 1 -0.51757 0.31576 2.6868 0.1012 0.596
Trt2 1 -0.25911 0.40511 0.4091 0.5224 0.772
Trt3 1 0.22105 0.54909 0.1621 0.6873 1.247
Trt4 1 -0.19498 0.64184 0.0923 0.7613 0.823
Number1 1 0.23605 0.07607 9.6287 0.0019 1.266
Number2 1 -0.00571 0.09667 0.0035 0.9529 0.994
Number3 1 0.12935 0.15970 0.6561 0.4180 1.138
Number4 1 0.42079 0.19816 4.5091 0.0337 1.523
Size1 1 0.06790 0.10125 0.4498 0.5024 1.070
Size2 1 -0.11636 0.11924 0.9524 0.3291 0.890
Size3 1 0.24995 0.23113 1.1695 0.2795 1.284
Size4 1 0.03557 0.29043 0.0150 0.9025 1.036

You can fit the PWP total time model with common effects by using the following SAS statements. However, the analysis is not shown here.

title2 'PWP Total Time Model with Common Effects';
model (tstart,tstop) * status(0) = Trt Number Size;
strata Visit;
run;


You can fit the PWP gap-time model with common effects by using the following statements. Again, the analysis is not shown here.

title2 'PWP Gap Time Model with Common Effects';
model Gaptime * Status(0) = Trt Number Size;
strata Visit;
run;


Recurrent events data are a special case of multiple events data in which the recurrence times are regarded as multivariate failure times and the marginal approach of Wei, Lin, and Weissfeld (1989) can be used. WLW fits a Cox model to each of the component times and makes statistical inference of the regression parameters based on a robust sandwich covariance matrix estimate. No specific correlation structure is imposed on the multivariate failure times. For the kth marginal model, let denote the row vector of regression parameters, let denote the maximum likelihood estimate of , let denote the covariance matrix obtained by inverting the observed information matrix, and let denote the matrix of score residuals. WLW showed that the joint distribution of can be approximated by a multivariate normal distribution with mean vector and robust covariance matrix

with the submatrix given by

In this example, there are four marginal proportional hazards models, one for each potential recurrence time. Instead of fitting one model at a time, you can fit all four marginal models in one analysis by using the STRATA statement and model-specific covariates as in the following statements. Using Visit as the STRATA variable on the input data set Bladder, PROC PHREG simultaneously fits all four marginal models, one for each Visit value. The COVS(AGGREGATE) option is specified to compute the robust sandwich variance estimate by summing up the score residuals for each distinct pattern of ID value. The TEST statement TREATMENT is used to perform the global test of no treatment effect for each tumor recurrence, the AVERAGE option is specified to estimate the parameter for the common treatment effect, and the E option displays the optimal weights for the common treatment effect.

title 'Wei-Lin-Weissfeld Model';
model TStop*Status(0)=Trt1-Trt4 Number1-Number4 Size1-Size4;
Trt1= Trt * (Visit=1);
Trt2= Trt * (Visit=2);
Trt3= Trt * (Visit=3);
Trt4= Trt * (Visit=4);
Number1= Number * (Visit=1);
Number2= Number * (Visit=2);
Number3= Number * (Visit=3);
Number4= Number * (Visit=4);
Size1= Size * (Visit=1);
Size2= Size * (Visit=2);
Size3= Size * (Visit=3);
Size4= Size * (Visit=4);
strata Visit;
id ID;
TREATMENT: test trt1,trt2,trt3,trt4/average e;
run;


Out of the 86 patients, 47 patients have only one tumor recurrence, 29 patients have two recurrences, 22 patients have three recurrences, and 14 patients have four recurrences (Output 67.10.5). Parameter estimates for the four marginal models are shown in Output 67.10.6. The 4 DF Wald test (Output 67.10.7) indicates a lack of evidence of a treatment effect in any of the four recurrences (p = 0.4105). The optimal weights for estimating the parameter of the common treatment effect are 0.67684, 0.25723, –0.07547, and 0.14140 for Trt1, Trt2, Trt3, and Trt4, respectively, which gives a parameter estimate of –0.5489 with a standard error estimate of 0.2853. A more sensitive test for a treatment effect is the 1 DF test based on this common parameter; however, there is still insufficient evidence for such effect at the 0.05 level (p = 0.0543).

Output 67.10.5: Summary of Bladder Tumor Recurrences in 86 Patients

 Wei-Lin-Weissfeld Model

The PHREG Procedure

Summary of the Number of Event and Censored Values
Stratum Visit Total Event Censored Percent
Censored
1 1 86 47 39 45.35
2 2 86 29 57 66.28
3 3 86 22 64 74.42
4 4 86 14 72 83.72
Total   344 112 232 67.44

Output 67.10.6: Analysis of Marginal Cox Models

Analysis of Maximum Likelihood Estimates
Parameter DF Parameter
Estimate
Standard
Error
StdErr
Ratio
Chi-Square Pr > ChiSq Hazard
Ratio
Trt1 1 -0.51762 0.30750 0.974 2.8336 0.0923 0.596
Trt2 1 -0.61944 0.36391 0.926 2.8975 0.0887 0.538
Trt3 1 -0.69988 0.41516 0.903 2.8419 0.0918 0.497
Trt4 1 -0.65079 0.48971 0.848 1.7661 0.1839 0.522
Number1 1 0.23599 0.07208 0.947 10.7204 0.0011 1.266
Number2 1 0.13756 0.08690 0.946 2.5059 0.1134 1.147
Number3 1 0.16984 0.10356 0.984 2.6896 0.1010 1.185
Number4 1 0.32880 0.11382 0.909 8.3453 0.0039 1.389
Size1 1 0.06789 0.08529 0.842 0.6336 0.4260 1.070
Size2 1 -0.07612 0.11812 0.881 0.4153 0.5193 0.927
Size3 1 -0.21131 0.17198 0.943 1.5097 0.2192 0.810
Size4 1 -0.20317 0.19106 0.830 1.1308 0.2876 0.816

Output 67.10.7: Tests of Treatment Effects

 Wei-Lin-Weissfeld Model

The PHREG Procedure

Linear Coefficients for Test TREATMENT
Parameter Row 1 Row 2 Row 3 Row 4 Average Effect
Trt1 1 0 0 0 0.67684
Trt2 0 1 0 0 0.25723
Trt3 0 0 1 0 -0.07547
Trt4 0 0 0 1 0.14140
Number1 0 0 0 0 0.00000
Number2 0 0 0 0 0.00000
Number3 0 0 0 0 0.00000
Number4 0 0 0 0 0.00000
Size1 0 0 0 0 0.00000
Size2 0 0 0 0 0.00000
Size3 0 0 0 0 0.00000
Size4 0 0 0 0 0.00000
CONSTANT 0 0 0 0 0.00000

Test TREATMENT Results
Wald
Chi-Square
DF Pr > ChiSq
3.9668 4 0.4105

Average Effect for Test TREATMENT
Estimate Standard Error z-Score Pr > |z|
-0.5489 0.2853 -1.9240 0.0543