The SEQTEST Procedure

Example 88.6 Comparing Two Survival Distributions with a Log-Rank Test

This example requests a log-rank test that compares two survival distributions for the treatment effect (Jennison and Turnbull 2000, pp. 77–79; Whitehead 1997, pp. 36–39).

A clinic is studying the effect of a new cancer treatment. The study consists of mice exposed to a carcinogen and randomized to either the control group or the treatment group. The event of interest is the death from cancer induced by the carcinogen, and the response is the time from randomization to death.

Following the derivations in the section Test for Two Survival Distributions with a Log-Rank Test in the chapter The SEQDESIGN Procedure, the hypothesis $H_{0}: \theta = -\mr {log}(\lambda )= 0$ with an alternative hypothesis $H_{1}: \theta = \theta _{1} > 0$ can be used, where $\lambda $ is the hazard ratio between the treatment group and control group.

Suppose that from past experience, the median survival time for the control group is $t_{0}= 20$ weeks, and the study would like to detect a $t_{1}= 40$ weeks median survival time with a 80% power in the trial. Assuming exponential survival functions for the two groups, the hazard rates can be computed from

\[  S_{j}(t_{j}) = e^{-h_{j} t_{j}} = \frac{1}{2}  \]

where $j=0, 1$.

Thus, with $h_{0}=0.03466$ and $h_{1}=0.01733$, the hazard ratio $\lambda _{1}= h_{1} / h_{0}= 1/2$ and the alternative hypothesis is

\[  \theta _{1} = -\mr {log}(\lambda _{1})= -\mr {log} (\frac{1}{2}) = 0.69315  \]

The following statements invoke the SEQDESIGN procedure and request a four-stage group sequential design for normally distributed data. The design uses a one-sided alternative hypothesis with early stopping to reject and to accept the null hypothesis $H_{0}$. Whitehead’s triangular method is used to derive the boundaries.

ods graphics on;
proc seqdesign boundaryscale=score
               ;
   OneSidedWhitehead: design method=whitehead
                             nstages=4
                             boundarykey=alpha
                             alt=upper  stop=both
                             beta=0.20;
   samplesize model=twosamplesurvival
                    ( nullhazard=0.03466
                      hazard=0.01733
                      accrate=10);
run;
ods graphics off;

A Whitehead method creates boundaries that approximately satisfy the Type I and Type II error probability level specification. The BOUNDARYKEY=ALPHA option is used to adjust the boundary value at the last stage and to meet the specified Type I probability level.

The specified ACCRATE=10 option indicates that 10 mice will be accrued each week and the resulting minimum and maximum accrual times are displayed. With the BOUNDARYSCALE=SCORE option, the procedure displays the output boundaries with the score statistics.

The Design Information table in Output 88.6.1 displays design specifications and derived statistics.

Output 88.6.1: Design Information

The SEQDESIGN Procedure
Design: OneSidedWhitehead

Design Information
Statistic Distribution Normal
Boundary Scale Score
Alternative Hypothesis Upper
Early Stop Accept/Reject Null
Method Whitehead
Boundary Key Alpha
Alternative Reference 0.693147
Number of Stages 4
Alpha 0.05
Beta 0.20044
Power 0.79956
Max Information (Percent of Fixed Sample) 129.9894
Max Information 16.70638
Null Ref ASN (Percent of Fixed Sample) 62.6302
Alt Ref ASN (Percent of Fixed Sample) 74.00064


The Boundary Information table in Output 88.6.2 displays the information level, alternative reference, and boundary values at each stage.

Output 88.6.2: Boundary Information

Boundary Information (Score Scale)
Null Reference = 0
_Stage_   Alternative Boundary Values
Information Level Reference Upper
Proportion Actual Events Upper Beta Alpha
1 0.2500 4.176595 16.70638 2.89500 -0.95755 4.78775
2 0.5000 8.35319 33.41276 5.78999 1.91510 5.74529
3 0.7500 12.52979 50.11914 8.68499 4.78775 6.70284
4 1.0000 16.70638 66.82552 11.57998 7.81300 7.81300


With ODS Graphics enabled, a detailed boundary plot with the rejection and acceptance regions is displayed, as shown in Output 88.6.3.

Output 88.6.3: Boundary Plot


With the MODEL=TWOSAMPLESURVIVAL option in the SAMPLESIZE statement, the Sample Size Summary table in Output 88.6.4 displays the parameters for the sample size computation.

Output 88.6.4: Required Sample Size Summary

Sample Size Summary
Test Two-Sample Survival
Null Hazard Rate 0.03466
Hazard Rate (Group A) 0.01733
Hazard Rate (Group B) 0.03466
Hazard Ratio 0.5
log(Hazard Ratio) -0.69315
Reference Hazards Alt Ref
Accrual Uniform
Accrual Rate 10
Min Accrual Time 6.682552
Min Sample Size 66.82552
Max Accrual Time 25.40111
Max Sample Size 254.0111
Max Number of Events 66.82552


With a minimum accrual time of 6.6826 weeks and a maximum accrual time of 25.401 weeks, an accrual time of 20 weeks is used in the study.

The Numbers of Events table in Output 88.6.5 displays the required number of events for the group sequential clinical trial.

Output 88.6.5: Required Numbers of Events

Numbers of Events (D)
Two-Sample Log-Rank Test
_Stage_ D Information
1 16.71 4.1766
2 33.41 8.3532
3 50.12 12.5298
4 66.83 16.7064


The following statements invoke the SEQDESIGN procedure and provide more detailed sample size information:

proc seqdesign boundaryscale=score
               ;
   OneSidedWhitehead: design method=whitehead
                             nstages=4
                             boundarykey=alpha
                             alt=upper
                             stop=both
                             beta=0.20;
   samplesize model=twosamplesurvival
                    ( nullhazard=0.03466
                      hazard=0.01733
                      accrate=10 acctime=20);
   ods output Boundary=Bnd_Surv;
run;

The ODS OUTPUT statement with the BOUNDARY=BND_SURV option creates an output data set named BND_SURV which contains the resulting boundary information for the subsequent sequential tests.

With an accrual time of 20 weeks, the Sample Size Summary table in Output 88.6.6 displays the follow-up time for the trial.

Output 88.6.6: Required Sample Size Summary

The SEQDESIGN Procedure
Design: OneSidedWhitehead

Sample Size Summary
Test Two-Sample Survival
Null Hazard Rate 0.03466
Hazard Rate (Group A) 0.01733
Hazard Rate (Group B) 0.03466
Hazard Ratio 0.5
log(Hazard Ratio) -0.69315
Reference Hazards Alt Ref
Accrual Uniform
Accrual Rate 10
Accrual Time 20
Follow-up Time 6.474366
Total Time 26.47437
Max Number of Events 66.82552
Max Sample Size 200
Expected Sample Size (Null Ref) 161.5941
Expected Sample Size (Alt Ref) 172.4692
Follow-up Time (Ceiling Time) 7
Total Time (Ceiling Time) 27


The Numbers of Events and Sample Sizes table in Output 88.6.7 displays the required sample sizes for the group sequential clinical trial.

Output 88.6.7: Numbers of Events and Sample Sizes

Numbers of Events (D) and Sample Sizes (N)
Two-Sample Log-Rank Test
_Stage_ Fractional Time Ceiling Time
D D(Grp 1) D(Grp 2) Time N N(Grp 1) N(Grp 2) Information D D(Grp 1) D(Grp 2) Time N N(Grp 1) N(Grp 2) Information
1 16.71 5.82 10.89 11.9867 119.87 59.93 59.93 4.1766 16.74 5.83 10.91 12 120.00 60.00 60.00 4.1854
2 33.41 11.84 21.57 17.3585 173.58 86.79 86.79 8.3532 35.73 12.68 23.04 18 180.00 90.00 90.00 8.9322
3 50.12 18.01 32.11 21.7480 200.00 100.00 100.00 12.5298 51.07 18.37 32.70 22 200.00 100.00 100.00 12.7667
4 66.83 24.46 42.37 26.4744 200.00 100.00 100.00 16.7064 68.55 25.14 43.41 27 200.00 100.00 100.00 17.1378


Thus the study will perform three interim analyses after 12, 18, and 22 weeks and a final analysis after 27 weeks if the study does not stop at any of the interim analyses.

Note that the SEQDESIGN procedure does not compute numbers of events or sample sizes for all statistical models. If the number of events or sample size for a fixed-sample design is available, then the MODEL=INPUTNEVENTS or MODEL=INPUTNOBS option can be used to input fixed-sample information. For example, with a required fixed-sample number of events 51.41, the following SAMPLESIZE statement can be used to produce the same sample size results:

   samplesize model=inputnevents
                    ( d=51.41 sample=two
                      hazard=0.03466 0.01733
                      accrate=10 acctime=20);

Suppose that 120 mice are available after week 12 for the first interim analysis. Output 88.6.8 lists the 10 observations in the data set weeks_1.

Output 88.6.8: Clinical Trial Data

First 10 Obs in the Trial Data

Obs TrtGp Event Weeks
1 0 0 11
2 1 0 11
3 0 0 11
4 1 0 11
5 0 1 6
6 1 0 11
7 0 0 11
8 1 0 11
9 0 1 9
10 1 0 11


The TrtGp variable is a grouping variable with the value 0 for a mouse in the placebo control group and the value 1 for a mouse in the treatment group. The Weeks variable is the survival time variable measured in weeks and the Event variable is the censoring variable with the value 0 indicating censoring. That is, the values of Weeks are considered censored if the corresponding values of Event are 0; otherwise, they are considered as event times.

The following statements use the LIFETEST procedure to estimate the log-rank statistic at stage 1:

proc lifetest data=Surv_1;
   time Weeks*Event(0);
   test TrtGp;
   ods output logunichisq=Parms_Surv1;
run;

The following statements create and display (in Output 88.6.9) the data set for the log-rank statistic and its associated standard error:

data Parms_Surv1;
   set Parms_Surv1(rename=(Statistic=Estimate));
   if Variable='TrtGp';
   _Scale_='Score';
   _Stage_= 1;
   keep Variable _Scale_ _Stage_ StdErr Estimate;
run;

proc print data=Parms_Surv1;
   title 'Statistics Computed at Stage 1';
run;

Output 88.6.9: Statistics Computed at Stage 1

Statistics Computed at Stage 1

Obs Variable Estimate StdErr _Scale_ _Stage_
1 TrtGp 3.2004 1.9979 Score 1


The following statements invoke the SEQTEST procedure to test for early stopping at stage 1:

ods graphics on;
proc seqtest Boundary=Bnd_Surv
             Parms(Testvar=TrtGp)=Parms_Surv1
             infoadj=none
             boundaryscale=score
             ;
   ods output Test=Test_Surv1;
run;
ods graphics off;

The BOUNDARY= option specifies the input data set that provides the boundary information for the trial at stage 1, which was generated in the SEQDESIGN procedure. The PARMS=PARMS_SURV1 option specifies the input data set PARMS_SURV1 that contains the test statistic and its associated standard error at stage 1, and the TESTVAR=TRTGP option identifies the test variable TRTGP in the data set. The INFOADJ=NONE option maintains the information levels for future interim stages (2 and 3) at the values provided in the BOUNDARY= data set.

The ODS OUTPUT statement with the TEST=TEST_SURV1 option creates an output data set named TEST_SURV1 which contains the updated boundary information for the test at stage 1. The data set also provides the boundary information that is needed for the group sequential test at the next stage.

The Design Information table in Output 88.6.10 displays design specifications. By default (or equivalently if you specify BOUNDARYKEY=ALPHA), the maximum information and the Type I error level $\alpha $ are preserved. Since the computed information level at stage 1 is not the same as the value provided in the BOUNDARY= data set, the power has been modified.

Output 88.6.10: Design Information

The SEQTEST Procedure

Design Information
BOUNDARY Data Set WORK.BND_SURV
Data Set WORK.PARMS_SURV1
Statistic Distribution Normal
Boundary Scale Score
Alternative Hypothesis Upper
Early Stop Accept/Reject Null
Number of Stages 4
Alpha 0.05
Beta 0.20055
Power 0.79945
Max Information (Percent of Fixed Sample) 130.0335
Max Information 16.7063807
Null Ref ASN (Percent of Fixed Sample) 62.80859
Alt Ref ASN (Percent of Fixed Sample) 74.19158


The Test Information table in Output 88.6.11 displays the boundary values for the test statistic with the SCORE statistic scale. Since only the information level at stage 1 is specified in the DATA= data set, the information levels at subsequent stages are derived proportionally from the corresponding information levels provided in the BOUNDARY= data set. At stage 1, the score statistic 3.2004 is between the upper $\beta $ boundary value –1.0386 and the upper $\alpha $ boundary value 4.7142, so the trial continues to the next stage.

Output 88.6.11: Sequential Tests

Test Information (Score Scale)
Null Reference = 0
_Stage_   Alternative Boundary Values Test
Information Level Reference Upper TrtGp
Proportion Actual Upper Beta Alpha Estimate Action
1 0.2389 3.991698 2.76683 -1.03862 4.71423 3.20040 Continue
2 0.5000 8.35319 5.78999 1.91799 5.73973 .  
3 0.7500 12.52979 8.68499 4.78804 6.70287 .  
4 1.0000 16.70638 11.57998 7.81349 7.81349 .  


Note that the observed information level 3.9917 corresponds to a proportion of 0.2389 in information level. If the observed information level is much smaller than the target proportion of 0.25, then you need to increase the accrual rate, accrual time, or follow-up time to achieve the target maximum information level for the trial. Scharfstein and Tsiatis (1998) use the simulation and bootstrap methods to modify the trial at interim stages to achieve the target maximum information level. These modifications should be specified in the study protocol or study plan before the study starts.

With ODS Graphics enabled, a boundary plot with test statistics is displayed, as shown in Output 88.6.12. As expected, the test statistic is in the continuation region between the upper $\beta $ and $\alpha $ boundary values.

Output 88.6.12: Sequential Test Plot


Note that the input DATA= option can also be used for the test statistics. For example, the following statements create and display (in Output 88.6.13) the data set for the log-rank statistic and its associated standard error after the LIFETEST procedure. Since the log-rank statistic is a score statistic, the corresponding information level is the variance of the statistic.

proc lifetest data=Surv_1;
   time Weeks*Event(0);
   test TrtGp;
   ods output logunichisq=Parms_Surv1a;
run;
data Parms_Surv1a;
   set Parms_Surv1a(rename=(Statistic=TrtGp));
   keep _Scale_ _Stage_ _Info_ TrtGp;
   _Scale_='Score';
   _Stage_= 1;
   _Info_= StdErr * StdErr;
   if Variable='TrtGp';
run;
proc print data=Parms_Surv1a;
   title 'Statistics Computed at Stage 1';
run;

Output 88.6.13: Statistics Computed at Stage 1

Statistics Computed at Stage 1

Obs TrtGp _Scale_ _Stage_ _Info_
1 3.2004 Score 1 3.99170


The following statements can be used to invoke the SEQTEST procedure to test for early stopping at stage 1:

ods graphics on;
proc seqtest Boundary=Bnd_Surv
             Data(Testvar=TrtGp)=Parms_Surv1a
             infoadj=none
             boundaryscale=score
             ;
   ods output Test=Test_Surv1;
run;
ods graphics off;

The following statements use the LIFETEST procedure to compute the log-rank statistic and its associated standard error at stage 2:

proc lifetest data=Surv_2;
   time Weeks*Event(0);
   test TrtGp;
   ods output logunichisq=Parms_Surv2;
run;

The following statements create and display (in Output 88.6.14) the data set for the log-rank statistic and its associated standard error for each of the first two stages:

data Parms_Surv2;
   set Parms_Surv2 (rename=(Statistic=Estimate));
   if Variable='TrtGp';
   _Scale_='Score';
   _Stage_= 2;
   keep Variable _Scale_ _Stage_ StdErr Estimate;
run;

proc print data=Parms_Surv2;
   title 'Statistics Computed at Stage 2';
run;

Output 88.6.14: Statistics Computed at Stage 2

Statistics Computed at Stage 2

Obs Variable Estimate StdErr _Scale_ _Stage_
1 TrtGp 7.3136 2.9489 Score 2


The following statements invoke the SEQTEST procedure to test for early stopping at stage 2:

ods graphics on;
proc seqtest Boundary=Test_Surv1
             Parms(Testvar=TrtGp)=Parms_Surv2
             infoadj=none
             boundaryscale=score
             citype=lower
             ;
   ods output Test=Test_Surv2;
run;
ods graphics off;

The BOUNDARY= option specifies the input data set that provides the boundary information for the trial at stage 2, which was generated by the SEQTEST procedure at the previous stage. The PARMS= option specifies the input data set that contains the test statistic and its associated standard error at stage 2, and the TESTVAR= option identifies the test variable in the data set. The INFOADJ=NONE option maintains the information level for stage 3 at the value provided in the BOUNDARY= data set.

The ODS OUTPUT statement with the TEST=TEST_SURV2 option creates an output data set named TEST_SURV2 which contains the updated boundary information for the test at stage 2. The data set also provides the boundary information that is needed for the group sequential test at the next stage if the trial does not stop at the current stage.

The Test Information table in Output 88.6.15 displays the boundary values for the test statistic. The test statistic 7.31365 is larger than the corresponding upper $\alpha $ boundary 5.79334, so the study stops and rejects the null hypothesis. That is, there is evidence of reduction in hazard rate for the new treatment.

Output 88.6.15: Sequential Tests

The SEQTEST Procedure

Test Information (Score Scale)
Null Reference = 0
_Stage_   Alternative Boundary Values Test
Information Level Reference Upper TrtGp
Proportion Actual Upper Beta Alpha Estimate Action
1 0.2389 3.991698 2.76683 -1.03862 4.71423 3.20040 Continue
2 0.5205 8.696125 6.02769 2.17041 5.79334 7.31365 Reject Null
3 0.7500 12.52979 8.68499 4.76308 6.72917 .  
4 1.0000 16.70638 11.57998 7.81290 7.81290 .  


With ODS Graphics enabled, the Test Plot displays boundary values of the design and the test statistic at the first two stages, as shown in Output 88.6.16. It also shows that the test statistic is in the Rejection Region above the upper $\alpha $ boundary value at stage 2.

Output 88.6.16: Sequential Test Plot


After the stopping of a trial, the Parameter Estimates table in Output 88.6.17 displays the stopping stage, parameter estimate, unbiased median estimate, confidence limits, and p-value under the null hypothesis $H_{0}: \theta = 0$.

Output 88.6.17: Parameter Estimates

Parameter Estimates
Stagewise Ordering
Parameter Stopping
Stage
MLE p-Value for
H0:Parm=0
Median
Estimate
Lower 95% CL
TrtGp 2 0.841024 0.0139 0.810329 0.21615


As expected, the p-value 0.0139 is significant at the $\alpha = 0.05$ level and the lower 95% confidence limit is larger than $\theta _{0}= 0$. The p-value, unbiased median estimate, and lower confidence limit depend on the ordering of the sample space $(k, z)$, where k is the stage number and z is the standardized Z statistic. With the specified stagewise ordering, the p-value is $p_{1} + p_{2}$, where $p_{1}$ is the $\alpha $ spending at stage 1,

\[  p_{1} = P_{\theta =0} ( Z_{1} \sqrt {I_{1}} \geq 4.71423 ) = 0.00915  \]
\[  p_{2} = P_{\theta =0} ( Z_{2} \sqrt {I_{2}} \geq 7.31365 \; \; |\; \;  -1.03862 < Z_{1} \sqrt {I_{1}} < 4.71423 )  \]

where $Z_{k}$ is a standardized normal variate and $I_{k}$ is the information level at stage k for k = 1, 2.