## Example 51.1 Product-Limit Estimates and Tests of Association

The data presented in Appendix I of Kalbfleisch and Prentice (1980) are coded in the following DATA step. The response variable, SurvTime, is the survival time in days of a lung cancer patient. Negative values of SurvTime are censored values. The covariates are Cell (type of cancer cell), Therapy (type of therapy: standard or test), Prior (prior therapy: 0=no, 10=yes), Age (age in years), DiagTime (time in months from diagnosis to entry into the trial), and Kps (performance status). A censoring indicator variable Censor is created from the data, with the value 1 indicating a censored time and the value 0 indicating an event time. Since there are only two types of therapy, an indicator variable, Treatment, is constructed for therapy type, with value 0 for standard therapy and value 1 for test therapy.

```data VALung;
drop check m;
retain Therapy Cell;
infile cards column=column;
length Check \$ 1;
label SurvTime='failure or censoring time'
Kps='karnofsky index'
DiagTime='months till randomization'
Age='age in years'
Prior='prior treatment?'
Cell='cell type'
Therapy='type of treatment'
Treatment='treatment indicator';
M=Column;
input Check \$ @@;
if M>Column then M=1;
if Check='s'|Check='t' then input @M Therapy \$ Cell \$ ;
else input @M SurvTime Kps DiagTime Age Prior @@;
if SurvTime > .;
censor=(SurvTime<0);
SurvTime=abs(SurvTime);
Treatment=(Therapy='test');
datalines;
standard squamous
72 60  7 69  0   411 70  5 64 10   228 60  3 38  0   126 60  9 63 10
118 70 11 65 10    10 20  5 49  0    82 40 10 69 10   110 80 29 68  0
314 50 18 43  0  -100 70  6 70  0    42 60  4 81  0     8 40 58 63 10
144 30  4 63  0   -25 80  9 52 10    11 70 11 48 10
standard small
30 60  3 61  0   384 60  9 42  0     4 40  2 35  0    54 80  4 63 10
13 60  4 56  0  -123 40  3 55  0   -97 60  5 67  0   153 60 14 63 10
59 30  2 65  0   117 80  3 46  0    16 30  4 53 10   151 50 12 69  0
22 60  4 68  0    56 80 12 43 10    21 40  2 55 10    18 20 15 42  0
139 80  2 64  0    20 30  5 65  0    31 75  3 65  0    52 70  2 55  0
287 60 25 66 10    18 30  4 60  0    51 60  1 67  0   122 80 28 53  0
27 60  8 62  0    54 70  1 67  0     7 50  7 72  0    63 50 11 48  0
392 40  4 68  0    10 40 23 67 10
8 20 19 61 10    92 70 10 60  0    35 40  6 62  0   117 80  2 38  0
132 80  5 50  0    12 50  4 63 10   162 80  5 64  0     3 30  3 43  0
95 80  4 34  0
standard large
177 50 16 66 10   162 80  5 62  0   216 50 15 52  0   553 70  2 47  0
278 60 12 63  0    12 40 12 68 10   260 80  5 45  0   200 80 12 41 10
156 70  2 66  0  -182 90  2 62  0   143 90  8 60  0   105 80 11 66  0
103 80  5 38  0   250 70  8 53 10   100 60 13 37 10
test squamous
999 90 12 54 10   112 80  6 60  0   -87 80  3 48  0  -231 50  8 52 10
242 50  1 70  0   991 70  7 50 10   111 70  3 62  0     1 20 21 65 10
587 60  3 58  0   389 90  2 62  0    33 30  6 64  0    25 20 36 63  0
357 70 13 58  0   467 90  2 64  0   201 80 28 52 10     1 50  7 35  0
30 70 11 63  0    44 60 13 70 10   283 90  2 51  0    15 50 13 40 10
test small
25 30  2 69  0  -103 70 22 36 10    21 20  4 71  0    13 30  2 62  0
87 60  2 60  0     2 40 36 44 10    20 30  9 54 10     7 20 11 66  0
24 60  8 49  0    99 70  3 72  0     8 80  2 68  0    99 85  4 62  0
61 70  2 71  0    25 70  2 70  0    95 70  1 61  0    80 50 17 71  0
51 30 87 59 10    29 40  8 67  0
24 40  2 60  0    18 40  5 69 10   -83 99  3 57  0    31 80  3 39  0
51 60  5 62  0    90 60 22 50 10    52 60  3 43  0    73 60  3 70  0
8 50  5 66  0    36 70  8 61  0    48 10  4 81  0     7 40  4 58  0
140 70  3 63  0   186 90  3 60  0    84 80  4 62 10    19 50 10 42  0
45 40  3 69  0    80 40  4 63  0
test large
52 60  4 45  0   164 70 15 68 10    19 30  4 39 10    53 60 12 66  0
15 30  5 63  0    43 60 11 49 10   340 80 10 64 10   133 75  1 65  0
111 60  5 64  0   231 70 18 67 10   378 80  4 65  0    49 30  3 37  0
;
```

In the following statements, PROC LIFETEST is invoked to compute the product-limit estimate of the survivor function for each type of cancer cell and to analyze the effects of the variables Age, Prior, DiagTime, Kps, and Treatment on the survival of the patients. These prognostic factors are specified in the TEST statement, and the variable Cell is specified in the STRATA statement. ODS Graphics must be enabled before producing graphs. Graphical displays of the product-limit estimates (S), the negative log estimates (LS), and the log of negative log estimates (LLS) are requested through the PLOTS= option in the PROC LIFETEST statement. Because of a few large survival times, a MAXTIME of 600 is used to set the scale of the time axis; that is, the time scale extends from 0 to a maximum of 600 days in the plots. The variable Therapy is specified in the ID statement to identify the type of therapy for each observation in the product-limit estimates. The OUTTEST option specifies the creation of an output data set named Test to contain the rank test matrices for the covariates.

```ods graphics on;
proc lifetest data=VALung plots=(s,ls,lls) outtest=Test maxtime=600;
time SurvTime*Censor(1);
id Therapy;
strata Cell;
test Age Prior DiagTime Kps Treatment;
run;
ods graphics off;
```

Output 51.1.1 through Output 51.1.4 display the product-limit estimates of the survivor functions for the four cell types. Summary statistics of the survival times are also shown. The median survival times are 51 days, 156 days, 51 days, and 118 days for patients with adeno cells, large cells, small cells, and squamous cells, respectively.

Output 51.1.1 Estimation Results for Cell=adeno
The LIFETEST Procedure

Product-Limit Survival Estimates
SurvTime   Survival Failure Survival Standard
Error
Number
Failed
Number
Left
Therapy
0.000   1.0000 0 0 0 27
3.000   0.9630 0.0370 0.0363 1 26 standard
7.000   0.9259 0.0741 0.0504 2 25 test
8.000   . . . 3 24 standard
8.000   0.8519 0.1481 0.0684 4 23 test
12.000   0.8148 0.1852 0.0748 5 22 standard
18.000   0.7778 0.2222 0.0800 6 21 test
19.000   0.7407 0.2593 0.0843 7 20 test
24.000   0.7037 0.2963 0.0879 8 19 test
31.000   0.6667 0.3333 0.0907 9 18 test
35.000   0.6296 0.3704 0.0929 10 17 standard
36.000   0.5926 0.4074 0.0946 11 16 test
45.000   0.5556 0.4444 0.0956 12 15 test
48.000   0.5185 0.4815 0.0962 13 14 test
51.000   0.4815 0.5185 0.0962 14 13 test
52.000   0.4444 0.5556 0.0956 15 12 test
73.000   0.4074 0.5926 0.0946 16 11 test
80.000   0.3704 0.6296 0.0929 17 10 test
83.000 * . . . 17 9 test
84.000   0.3292 0.6708 0.0913 18 8 test
90.000   0.2881 0.7119 0.0887 19 7 test
92.000   0.2469 0.7531 0.0850 20 6 standard
95.000   0.2058 0.7942 0.0802 21 5 standard
117.000   0.1646 0.8354 0.0740 22 4 standard
132.000   0.1235 0.8765 0.0659 23 3 standard
140.000   0.0823 0.9177 0.0553 24 2 test
162.000   0.0412 0.9588 0.0401 25 1 standard
186.000   0 1.0000 . 26 0 test

 Note: The marked survival times are censored observations.

Output 51.1.2 Estimation Results for Cell=large
The LIFETEST Procedure

Stratum 2: Cell = large

Product-Limit Survival Estimates
SurvTime   Survival Failure Survival Standard
Error
Number
Failed
Number
Left
Therapy
0.000   1.0000 0 0 0 27
12.000   0.9630 0.0370 0.0363 1 26 standard
15.000   0.9259 0.0741 0.0504 2 25 test
19.000   0.8889 0.1111 0.0605 3 24 test
43.000   0.8519 0.1481 0.0684 4 23 test
49.000   0.8148 0.1852 0.0748 5 22 test
52.000   0.7778 0.2222 0.0800 6 21 test
53.000   0.7407 0.2593 0.0843 7 20 test
100.000   0.7037 0.2963 0.0879 8 19 standard
103.000   0.6667 0.3333 0.0907 9 18 standard
105.000   0.6296 0.3704 0.0929 10 17 standard
111.000   0.5926 0.4074 0.0946 11 16 test
133.000   0.5556 0.4444 0.0956 12 15 test
143.000   0.5185 0.4815 0.0962 13 14 standard
156.000   0.4815 0.5185 0.0962 14 13 standard
162.000   0.4444 0.5556 0.0956 15 12 standard
164.000   0.4074 0.5926 0.0946 16 11 test
177.000   0.3704 0.6296 0.0929 17 10 standard
182.000 * . . . 17 9 standard
200.000   0.3292 0.6708 0.0913 18 8 standard
216.000   0.2881 0.7119 0.0887 19 7 standard
231.000   0.2469 0.7531 0.0850 20 6 test
250.000   0.2058 0.7942 0.0802 21 5 standard
260.000   0.1646 0.8354 0.0740 22 4 standard
278.000   0.1235 0.8765 0.0659 23 3 standard
340.000   0.0823 0.9177 0.0553 24 2 test
378.000   0.0412 0.9588 0.0401 25 1 test
553.000   0 1.0000 . 26 0 standard

 Note: The marked survival times are censored observations.

Output 51.1.3 Estimation Results for Cell=small
The LIFETEST Procedure

Stratum 3: Cell = small

Product-Limit Survival Estimates
SurvTime   Survival Failure Survival Standard
Error
Number
Failed
Number
Left
Therapy
0.000   1.0000 0 0 0 48
2.000   0.9792 0.0208 0.0206 1 47 test
4.000   0.9583 0.0417 0.0288 2 46 standard
7.000   . . . 3 45 standard
7.000   0.9167 0.0833 0.0399 4 44 test
8.000   0.8958 0.1042 0.0441 5 43 test
10.000   0.8750 0.1250 0.0477 6 42 standard
13.000   . . . 7 41 standard
13.000   0.8333 0.1667 0.0538 8 40 test
16.000   0.8125 0.1875 0.0563 9 39 standard
18.000   . . . 10 38 standard
18.000   0.7708 0.2292 0.0607 11 37 standard
20.000   . . . 12 36 standard
20.000   0.7292 0.2708 0.0641 13 35 test
21.000   . . . 14 34 standard
21.000   0.6875 0.3125 0.0669 15 33 test
22.000   0.6667 0.3333 0.0680 16 32 standard
24.000   0.6458 0.3542 0.0690 17 31 test
25.000   . . . 18 30 test
25.000   0.6042 0.3958 0.0706 19 29 test
27.000   0.5833 0.4167 0.0712 20 28 standard
29.000   0.5625 0.4375 0.0716 21 27 test
30.000   0.5417 0.4583 0.0719 22 26 standard
31.000   0.5208 0.4792 0.0721 23 25 standard
51.000   . . . 24 24 standard
51.000   0.4792 0.5208 0.0721 25 23 test
52.000   0.4583 0.5417 0.0719 26 22 standard
54.000   . . . 27 21 standard
54.000   0.4167 0.5833 0.0712 28 20 standard
56.000   0.3958 0.6042 0.0706 29 19 standard
59.000   0.3750 0.6250 0.0699 30 18 standard
61.000   0.3542 0.6458 0.0690 31 17 test
63.000   0.3333 0.6667 0.0680 32 16 standard
80.000   0.3125 0.6875 0.0669 33 15 test
87.000   0.2917 0.7083 0.0656 34 14 test
95.000   0.2708 0.7292 0.0641 35 13 test
97.000 * . . . 35 12 standard
99.000   . . . 36 11 test
99.000   0.2257 0.7743 0.0609 37 10 test
103.000 * . . . 37 9 test
117.000   0.2006 0.7994 0.0591 38 8 standard
122.000   0.1755 0.8245 0.0567 39 7 standard
123.000 * . . . 39 6 standard
139.000   0.1463 0.8537 0.0543 40 5 standard
151.000   0.1170 0.8830 0.0507 41 4 standard
153.000   0.0878 0.9122 0.0457 42 3 standard
287.000   0.0585 0.9415 0.0387 43 2 standard
384.000   0.0293 0.9707 0.0283 44 1 standard
392.000   0 1.0000 . 45 0 standard

 Note: The marked survival times are censored observations.

Output 51.1.4 Estimation Results for Cell=squamous
The LIFETEST Procedure

Stratum 4: Cell = squamous

Product-Limit Survival Estimates
SurvTime   Survival Failure Survival Standard
Error
Number
Failed
Number
Left
Therapy
0.000   1.0000 0 0 0 35
1.000   . . . 1 34 test
1.000   0.9429 0.0571 0.0392 2 33 test
8.000   0.9143 0.0857 0.0473 3 32 standard
10.000   0.8857 0.1143 0.0538 4 31 standard
11.000   0.8571 0.1429 0.0591 5 30 standard
15.000   0.8286 0.1714 0.0637 6 29 test
25.000   0.8000 0.2000 0.0676 7 28 test
25.000 * . . . 7 27 standard
30.000   0.7704 0.2296 0.0713 8 26 test
33.000   0.7407 0.2593 0.0745 9 25 test
42.000   0.7111 0.2889 0.0772 10 24 standard
44.000   0.6815 0.3185 0.0794 11 23 test
72.000   0.6519 0.3481 0.0813 12 22 standard
82.000   0.6222 0.3778 0.0828 13 21 standard
87.000 * . . . 13 20 test
100.000 * . . . 13 19 standard
110.000   0.5895 0.4105 0.0847 14 18 standard
111.000   0.5567 0.4433 0.0861 15 17 test
112.000   0.5240 0.4760 0.0870 16 16 test
118.000   0.4912 0.5088 0.0875 17 15 standard
126.000   0.4585 0.5415 0.0876 18 14 standard
144.000   0.4257 0.5743 0.0873 19 13 standard
201.000   0.3930 0.6070 0.0865 20 12 test
228.000   0.3602 0.6398 0.0852 21 11 standard
231.000 * . . . 21 10 test
242.000   0.3242 0.6758 0.0840 22 9 test
283.000   0.2882 0.7118 0.0820 23 8 test
314.000   0.2522 0.7478 0.0793 24 7 standard
357.000   0.2161 0.7839 0.0757 25 6 test
389.000   0.1801 0.8199 0.0711 26 5 test
411.000   0.1441 0.8559 0.0654 27 4 standard
467.000   0.1081 0.8919 0.0581 28 3 test
587.000   0.0720 0.9280 0.0487 29 2 test
991.000   0.0360 0.9640 0.0352 30 1 test
999.000   0 1.0000 . 31 0 test

 Note: The marked survival times are censored observations.

The distribution of event and censored observations among the four cell types is summarized in Output 51.1.5.

Output 51.1.5 Summary of Censored and Uncensored Values
Summary of the Number of Censored and Uncensored Values
Stratum Cell Total Failed Censored Percent
Censored
1 adeno 27 26 1 3.70
2 large 27 26 1 3.70
3 small 48 45 3 6.25
4 squamous 35 31 4 11.43
Total   137 128 9 6.57

The graph of the estimated survivor functions is shown in Output 51.1.6. The adeno cell curve and the small cell curve are much closer to each other than they are to the large cell curve or the squamous cell curve. The survival rates of the adeno cell patients and the small cell patients decrease rapidly to approximately 29% in 90 days. Shapes of the large cell curve and the squamous cell curve are quite different, although both decrease less rapidly than those of the adeno and small cells. The squamous cell curve decreases more rapidly initially than the large cell curve, but the role is reversed in the later period.

Output 51.1.6 Graph of the Estimated Survivor Functions

The graph of the negative log of the estimated survivor functions is displayed in Output 51.1.7. Output 51.1.8 displays the log of the negative log of the estimated survivor functions against the log of time.

Output 51.1.7 Graph of Negative Log of the Estimated Survivor Functions

Output 51.1.8 Graph of Log of the Negative Log of the Estimated Survivor Functions

Results of the homogeneity tests across cell types are given in Output 51.1.9. The log-rank and Wilcoxon statistics and their corresponding covariance matrices are displayed. Also given is a table that consists of the approximate chi-square statistics, degrees of freedom, and p-values for the log-rank, Wilcoxon, and likelihood ratio tests. All three tests indicate strong evidence of a significant difference among the survival curves for the four types of cancer cells (p<0.0001).

Output 51.1.9 Homogeneity Tests across Cell Types
Rank Statistics
Cell Log-Rank Wilcoxon
large -8.549 -1085.0
small 14.898 1278.0
squamous -16.655 -890.0

Covariance Matrix for the Log-Rank Statistics
large -4.0701 24.1990 -7.8117 -12.3172
small -4.4087 -7.8117 21.7543 -9.5339
squamous -4.4873 -12.3172 -9.5339 26.3384

Covariance Matrix for the Wilcoxon Statistics
large -34718 151241 -59948 -56576
small -46639 -59948 175590 -69002
squamous -39831 -56576 -69002 165410

Test of Equality over Strata
Test Chi-Square DF Pr >
Chi-Square
Log-Rank 25.4037 3 <.0001
Wilcoxon 19.4331 3 0.0002
-2Log(LR) 33.9343 3 <.0001

Results of the log-rank test of the prognostic variables are shown in Output 51.1.10. The univariate test results correspond to testing each prognostic factor marginally. The joint covariance matrix of these univariate test statistics is also displayed. In computing the overall chi-square statistic, the partial chi-square statistics following a forward stepwise entry approach are tabulated.

Consider the log-rank test in Output 51.1.10. Since the univariate test for Kps has the largest chi-square (43.4747) among all the covariates, Kps is entered first. At this stage, the partial chi-square and the chi-square increment for Kps are the same as the univariate chi-square. Among all the covariates not in the model (Age, Prior, DiagTime, Treatment), Treatment has the largest approximate chi-square increment (1.7261) and is entered next. The approximate chi-square for the model that contains Kps and Treatment is 43.4747+1.7261=45.2008 with 2 degrees of freedom. The third covariate entered is Age. The fourth is Prior, and the fifth is DiagTime. The overall chi-square statistic in the last line of the output is the partial chi-square for including all the covariates. It has a value of 46.4200 with 5 degrees of freedom, which is highly significant (p<0.0001).

Output 51.1.10 Log-Rank Test of the Prognostic Factors
Univariate Chi-Squares for the Log-Rank Test
Variable Test
Statistic
Standard
Error
Chi-Square Pr >
Chi-Square
Label
Age -40.7383 105.7 0.1485 0.7000 age in years
Prior -19.9435 46.9836 0.1802 0.6712 prior treatment?
DiagTime -115.9 97.8708 1.4013 0.2365 months till randomization
Kps 1123.1 170.3 43.4747 <.0001 karnofsky index
Treatment -4.2076 5.0407 0.6967 0.4039 treatment indicator

Covariance Matrix for the Log-Rank Statistics
Variable Age Prior DiagTime Kps Treatment
Age 11175.4 -301.2 -892.2 -2948.4 119.3
Prior -301.2 2207.5 2010.9 78.6 13.9
DiagTime -892.2 2010.9 9578.7 -2295.3 21.9
Kps -2948.4 78.6 -2295.3 29015.6 61.9
Treatment 119.3 13.9 21.9 61.9 25.4

Forward Stepwise Sequence of Chi-Squares for the Log-Rank Test
Variable DF Chi-Square Pr >
Chi-Square
Chi-Square
Increment
Pr >
Increment
Label
Kps 1 43.4747 <.0001 43.4747 <.0001 karnofsky index
Treatment 2 45.2008 <.0001 1.7261 0.1889 treatment indicator
Age 3 46.3012 <.0001 1.1004 0.2942 age in years
Prior 4 46.4134 <.0001 0.1122 0.7377 prior treatment?
DiagTime 5 46.4200 <.0001 0.00665 0.9350 months till randomization

You can establish this forward stepwise entry of prognostic factors by passing the matrix corresponding to the log-rank test to the RSQUARE method in the REG procedure, as follows. PROC REG finds the sets of variables that yield the largest chi-square statistics.

```data RSq;
set Test;
if _type_='LOG RANK';
_type_='cov';
proc print data=RSq;
run;
proc reg data=RSq(type=COV);
model SurvTime=Age Prior DiagTime Kps Treatment
/ selection=rsquare;
title 'All Possible Subsets of Covariates for the log-rank Test';
run;
```

Output 51.1.11 displays the univariate statistics and their covariance matrix for the log-rank test.

Output 51.1.11 Log-Rank Statistics and Covariance Matrix
Obs _TYPE_ _NAME_ SurvTime Age Prior DiagTime Kps Treatment
1 cov SurvTime 46.42 -40.74 -19.94 -115.86 1123.14 -4.208
2 cov Age -40.74 11175.44 -301.23 -892.24 -2948.45 119.297
3 cov Prior -19.94 -301.23 2207.46 2010.85 78.64 13.875
4 cov DiagTime -115.86 -892.24 2010.85 9578.69 -2295.32 21.859
5 cov Kps 1123.14 -2948.45 78.64 -2295.32 29015.62 61.945
6 cov Treatment -4.21 119.30 13.87 21.86 61.95 25.409

Results of the best subset regression are shown in Output 51.1.12. The variable Kps generates the largest univariate test statistic among all the covariates, the pair Kps and Age generate the largest test statistic among any other pairs of covariates, and so on. The entry order of covariates is identical to that of PROC LIFETEST.

Output 51.1.12 Best Subset Regression from the REG Procedure
 All Possible Subsets of Covariates for the log-rank Test

The REG Procedure
Model: MODEL1
Dependent Variable: SurvTime

R-Square Selection Method

Number in
Model
R-Square Variables in Model
1 0.9366 Kps
1 0.0302 DiagTime
1 0.0150 Treatment
1 0.0039 Prior
1 0.0032 Age
2 0.9737 Kps Treatment
2 0.9472 Age Kps
2 0.9417 Prior Kps
2 0.9382 DiagTime Kps
2 0.0434 DiagTime Treatment
2 0.0353 Age DiagTime
2 0.0304 Prior DiagTime
2 0.0181 Prior Treatment
2 0.0159 Age Treatment
2 0.0075 Age Prior
3 0.9974 Age Kps Treatment
3 0.9774 Prior Kps Treatment
3 0.9747 DiagTime Kps Treatment
3 0.9515 Age Prior Kps
3 0.9481 Age DiagTime Kps
3 0.9418 Prior DiagTime Kps
3 0.0456 Age DiagTime Treatment
3 0.0438 Prior DiagTime Treatment
3 0.0355 Age Prior DiagTime
3 0.0192 Age Prior Treatment
4 0.9999 Age Prior Kps Treatment
4 0.9976 Age DiagTime Kps Treatment
4 0.9774 Prior DiagTime Kps Treatment
4 0.9515 Age Prior DiagTime Kps
4 0.0459 Age Prior DiagTime Treatment
5 1.0000 Age Prior DiagTime Kps Treatment