This example demonstrates a twosided group sequential test that uses an error spending design with early stopping to accept the null hypothesis . The example is similar to Example 84.2 but with early stopping to accept .
A study is conducted to examine the effects of Age
(years), Weight
(kg), RunTime
(time in minutes to run 1.5 miles), RunPulse
(heart rate while running), and MaxPulse
(maximum heart rate recorded while running) on Oxygen
(oxygen intake rate, ml per kg body weight per minute). The primary interest is whether oxygen intake rate is associated
with weight.
The hypothesis is tested using the following linear model:

The null hypothesis is , where is the regression parameter for the variable Weight
. Suppose that is the reference improvement that should be detected at a 0.90 level. Then the maximum information can be derived in the SEQDESIGN procedure.
Following the derivations in the section “Test for a Parameter in the Regression Model” in the chapter “The SEQDESIGN Procedure,” the required sample size can be derived from

where is the variance of the response variable in the regression model, is the proportion of variance of Weight
explained by other covariates, and is the variance of Weight
.
Further suppose that from past experience, , , and . Then the required sample size can be derived using the SAMPLESIZE statement in the SEQDESIGN procedure.
The following statements invoke the SEQDESIGN procedure and request a threestage group sequential design for normally distributed data to test the null hypothesis of a regression parameter against the alternative :
ods graphics on; proc seqdesign altref=0.10; OBFErrorFunction: design method=errfuncgamma stop=accept nstages=3 info=cum(2 3 4); samplesize model=reg( variance=5 xvariance=64 xrsquare=0.10); ods output Boundary=Bnd_Fit; run; ods graphics off;
By default (or equivalently if you specify ALPHA=0.05 and BETA=0.10), the procedure uses a Type I error probability 0.05 and a Type II error probability 0.10. The ALTREF=0.10 option specifies a power of at the alternative hypothesis . The INFO=CUM(2 3 4) option specifies that the study perform the first interim analysis with information proportion —that is, after half of the total observations are collected.
The ODS OUTPUT statement with the BOUNDARY=BND_FIT option creates an output data set named BND_FIT
which contains the resulting boundary information for the subsequent sequential tests.
The “Design Information” table in Output 84.3.1 displays design specifications and derived statistics. Since the alternative reference is specified, the maximum information is derived.
Output 84.3.1: Error Spending Design Information
Design Information  

Statistic Distribution  Normal 
Boundary Scale  Standardized Z 
Alternative Hypothesis  TwoSided 
Early Stop  Accept Null 
Method  Error Spending 
Boundary Key  Both 
Alternative Reference  0.1 
Number of Stages  3 
Alpha  0.05 
Beta  0.1 
Power  0.9 
Max Information (Percent of Fixed Sample)  103.9245 
Max Information  1091.972 
Null Ref ASN (Percent of Fixed Sample)  75.00521 
Alt Ref ASN (Percent of Fixed Sample)  101.8099 
The “Boundary Information” table in Output 84.3.2 displays information level, alternative reference, and boundary values at each stage.
Output 84.3.2: Boundary Information
Boundary Information (Standardized Z Scale) Null Reference = 0 


_Stage_  Alternative  Boundary Values  
Information Level  Reference  Lower  Upper  
Proportion  Actual  N  Lower  Upper  Beta  Beta  
1  0.5000  545.9862  47.39463  2.33663  2.33663  0.44937  0.44937 
2  0.7500  818.9792  71.09195  2.86178  2.86178  1.13583  1.13583 
3  1.0000  1091.972  94.78926  3.30450  3.30450  1.91428  1.91428 
With ODS Graphics enabled, a detailed boundary plot with the rejection and acceptance regions is displayed, as shown in Output 84.3.3. The boundary plot also displays the information level and critical value for the corresponding fixedsample design.
Output 84.3.3: Boundary Plot
With the MODEL=REG option in the SAMPLESIZE statement, the “Sample Size Summary” table in Output 84.3.4 displays the parameters for the sample size computation.
Output 84.3.4: Required Sample Size Summary
Sample Size Summary  

Test  Reg Parameter 
Parameter  0.1 
Variance  5 
X Variance  64 
R Square (X)  0.1 
Max Sample Size  94.78926 
Expected Sample Size (Null Ref)  68.41207 
Expected Sample Size (Alt Ref)  92.86057 
The “Sample Sizes” table in Output 84.3.5 displays the required sample sizes for the group sequential clinical trial.
Output 84.3.5: Required Sample Sizes
Sample Sizes (N) Z Test for Regression Parameter 


_Stage_  Fractional N  Ceiling N  
N  Information  N  Information  
1  47.39  546.0  48  553.0 
2  71.09  819.0  72  829.4 
3  94.79  1092.0  95  1094.4 
Thus, 48, 72, and 95 individuals are needed in stages 1, 2, and 3, respectively. Since the sample sizes are derived from estimated values of , , and , the actual information levels might not achieve the target information levels. Thus, instead of specifying sample sizes in the protocol, you can specify the maximum information levels. Then if an actual information level is much less than the target level, you can increase the sample sizes for the remaining stages to achieve the desired information levels and power.
Suppose that 48 individuals are available at stage 1. Output 84.3.6 lists the first 10 observations of the trial data.
Output 84.3.6: Clinical Trial Data
First 10 Obs in the Trial Data 
Obs  Oxygen  Age  Weight  RunTime  RunPulse  MaxPulse 

1  54.5521  44  87.7676  11.6949  178.435  181.607 
2  52.2821  40  75.4853  9.8872  184.433  183.667 
3  62.1871  44  89.0638  8.7950  155.540  167.108 
4  65.3269  42  67.7310  8.4577  162.926  173.877 
5  59.9809  37  93.1902  9.3228  179.033  180.144 
6  52.5588  47  75.9044  12.0385  177.753  175.033 
7  51.7838  40  73.5422  11.6607  175.838  178.140 
8  57.0024  43  81.2861  11.2219  160.963  171.770 
9  48.0775  44  85.2290  13.1789  173.722  176.548 
10  68.3357  38  80.2490  8.5066  171.824  184.011 
The following statements use the REG procedure to estimate the slope and its associated standard error at stage 1:
proc reg data=Fit_1; model Oxygen=Age Weight RunTime RunPulse MaxPulse; ods output ParameterEstimates=Parms_Fit1; run;
The following statements create and display (in Output 84.3.7) the input data set that contains slope and its associated standard error for the SEQTEST procedure:
data Parms_Fit1; set Parms_Fit1; if Variable='Weight'; _Scale_='MLE'; _Stage_= 1; keep _Scale_ _Stage_ Variable Estimate StdErr; run; proc print data=Parms_Fit1; title 'Statistics Computed at Stage 1'; run;
Output 84.3.7: Statistics Computed at Stage 1
Statistics Computed at Stage 1 
Obs  Variable  Estimate  StdErr  _Scale_  _Stage_ 

1  Weight  0.04660  0.04308  MLE  1 
The following statements invoke the SEQTEST procedure to test for early stopping at stage 1:
ods graphics on; proc seqtest Boundary=Bnd_Fit Parms(testvar=Weight)=Parms_Fit1 infoadj=none errspendadj=errfuncgamma stopprob order=lr ; ods output Test=Test_Fit1; 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_FIT1 option specifies the input data set PARMS_FIT1
that contains the test statistic and its associated standard error at stage 1, and the TESTVAR=WEIGHT option identifies the
test variable WEIGHT
in the data set. The INFOADJ=NONE option maintains the information level for stage 2 at the value provided in the BOUNDARY=
data set.
The ORDER=LR option uses the LR ordering to derive the pvalue, the unbiased median estimate, and the confidence limits for the regression slope estimate. The ERRSPENDADJ=ERRFUNCGAMMA option adjusts the boundaries with the updated error spending values generated from a gamma cumulative error spending function.
The ODS OUTPUT statement with the TEST=TEST_FIT1 option creates an output data set named TEST_FIT1
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 84.3.8 displays the design specifications. By default (or equivalently if you specify BOUNDARYKEY=ALPHA), the boundary values are modified for the new information levels to maintain the Type I level. The maximum information remains the same as in the BOUNDARY= data set, but the derived Type II error probability and power are different because of the new information level.
Output 84.3.8: Design Information
Design Information  

BOUNDARY Data Set  WORK.BND_FIT 
Data Set  WORK.PARMS_FIT1 
Statistic Distribution  Normal 
Boundary Scale  Standardized Z 
Alternative Hypothesis  TwoSided 
Early Stop  Accept Null 
Number of Stages  3 
Alpha  0.05 
Beta  0.10007 
Power  0.89993 
Max Information (Percent of Fixed Sample)  103.9498 
Max Information  1091.97232 
Null Ref ASN (Percent of Fixed Sample)  75.15846 
Alt Ref ASN (Percent of Fixed Sample)  101.8296 
With the STOPPROB option, the “Expected Cumulative Stopping Probabilities” table in Output 84.3.9 displays the expected stopping stage and the cumulative stopping probability of accepting the null hypothesis at each stage under various hypothetical references , where is the alternative reference and by default. You can specify other values for with the CREF= option.
Output 84.3.9: Stopping Probabilities
Expected Cumulative Stopping Probabilities Reference = CRef * (Alt Reference) 


CRef  Expected Stopping Stage 
Source  Stopping Probabilities  
Stage_1  Stage_2  Stage_3  
0.0000  1.895  Accept Null  0.33304  0.76607  0.95000 
0.5000  2.409  Accept Null  0.17680  0.40947  0.62828 
1.0000  2.918  Accept Null  0.02636  0.05453  0.10007 
1.5000  2.997  Accept Null  0.00109  0.00166  0.00242 
The “Test Information” table in Output 84.3.10 displays the boundary values for the test statistic. By default (or equivalently if you specify BOUNDARYSCALE=STDZ), these statistics are displayed with the standardized Z scale. The information level at stage 1 is derived from the standard error in the PARMS= data set,

Output 84.3.10: Sequential Tests
Test Information (Standardized Z Scale) Null Reference = 0 


_Stage_  Alternative  Boundary Values  Test  
Information Level  Reference  Lower  Upper  Weight  
Proportion  Actual  Lower  Upper  Beta  Beta  Estimate  Action  
1  0.4934  538.7887  2.32118  2.32118  0.43033  0.43033  1.08174  Continue 
2  0.7500  818.9792  2.86178  2.86178  1.13623  1.13623  .  
3  1.0000  1091.972  3.30450  3.30450  1.91431  1.91431  . 
At stage 1, the standardized Z statistic 1.08174 is greater than the upper boundary 0.43033, so the trial continues to the next stage.
With ODS Graphics enabled, a boundary plot with test statistics is displayed, as shown in Output 84.3.11. As expected, the test statistic is in the continuation region.
Output 84.3.11: Sequential Test Plot
The following statements use the REG procedure to estimate the slope and its associated standard error at stage 2:
proc reg data=Fit_2; model Oxygen=Age Weight RunTime RunPulse MaxPulse; ods output ParameterEstimates=Parms_Fit2; run;
Note that the data set Fit_2
contains both the data from stage 1 and the data from stage 2,
The following statements create and display (in Output 84.3.12) the input data set that contains slope and its associated standard error at stage 2 for the SEQTEST procedure:
data Parms_Fit2; set Parms_Fit2; if Variable='Weight'; _Scale_='MLE'; _Stage_= 2; keep _Scale_ _Stage_ Variable Estimate StdErr; run; proc print data=Parms_Fit2; title 'Statistics Computed at Stage 2'; run;
Output 84.3.12: Statistics Computed at Stage 2
Statistics Computed at Stage 2 
Obs  Variable  Estimate  StdErr  _Scale_  _Stage_ 

1  Weight  0.02925  0.03490  MLE  2 
The following statements invoke the SEQTEST procedure to test for early stopping at stage 2:
ods graphics on; proc seqtest Boundary=Test_Fit1 Parms(testvar=Weight)=Parms_Fit2 errspendadj=errfuncgamma order=lr pss plots=(asn power) ; ods output Test=Test_Fit2; 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.
Since the data set PARMS_FIT2
does not contain the test information at stage 1, the information level at stage 1 in the TEST_FIT1
data set is used to generate boundary values for the test.
The ORDER=LR option uses the LR ordering to derive the pvalue, unbiased median estimate, and confidence limits for the regression slope estimate.
The ODS OUTPUT statement with the TEST=TEST_FIT2 option creates an output data set named TEST_FIT2
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.
The “Design Information” table in Output 84.3.13 displays design specifications. By default (or equivalently if you specify BOUNDARYKEY=ALPHA), the boundary values are modified for the new information levels to maintain the Type I level.
Output 84.3.13: Design Information
Design Information  

BOUNDARY Data Set  WORK.TEST_FIT1 
Data Set  WORK.PARMS_FIT2 
Statistic Distribution  Normal 
Boundary Scale  Standardized Z 
Alternative Hypothesis  TwoSided 
Early Stop  Accept Null 
Number of Stages  3 
Alpha  0.05 
Beta  0.10009 
Power  0.89991 
Max Information (Percent of Fixed Sample)  103.9566 
Max Information  1091.97232 
Null Ref ASN (Percent of Fixed Sample)  75.18254 
Alt Ref ASN (Percent of Fixed Sample)  101.8349 
The derived Type II error probability and power are different because of the new information levels.
With the PSS option, the “Power and Expected Sample Sizes” table in Output 84.3.14 displays powers and expected mean sample sizes under various hypothetical references , where is the alternative reference and are the default values in the CREF= option.
Output 84.3.14: Power and Expected Sample Size Information
Powers and Expected Sample Sizes Reference = CRef * (Alt Reference) 


CRef  Power  Sample Size 
Percent FixedSample 

0.0000  0.02500  75.1825 
0.5000  0.37154  88.5975 
1.0000  0.89991  101.8349 
1.5000  0.99758  103.8843 
With the PLOTS=ASN option, the procedure displays a plot of expected sample sizes under various hypothetical references, as shown in Output 84.3.15. By default, expected sample sizes under the hypotheses , , are displayed, where is the alternative reference.
Output 84.3.15: ASN Plot
With the PLOTS=POWER option, the procedure displays a plot of the power curves under various hypothetical references for all designs simultaneously, as shown in Output 84.3.16. By default, powers under hypothetical references are displayed, where by default. You can specify values with the CREF= option. The values are displayed on the horizontal axis.
Output 84.3.16: Power Plot
Under the null hypothesis, , the power is 0.025, which is the upper Type I error probability. Under the alternative hypothesis, , the power is 0.89991, which is one minus the Type II error probability, as displayed in the “Design Information” table in Output 84.3.13.
The “Test Information” table in Output 84.3.17 displays the boundary values for the test statistic with the default standardized Z scale. At stage 2, the standardized slope estimate 0.83805 is between the lower and upper boundary values. The trial stops to accept the null hypothesis that the variable Weight
has no effect on the oxygen intake rate after adjusting for other covariates.
Output 84.3.17: Sequential Tests
Test Information (Standardized Z Scale) Null Reference = 0 


_Stage_  Alternative  Boundary Values  Test  
Information Level  Reference  Lower  Upper  Weight  
Proportion  Actual  Lower  Upper  Beta  Beta  Estimate  Action  
1  0.4934  538.7887  2.32118  2.32118  0.43033  0.43033  1.08174  Continue 
2  0.7517  820.8509  2.86505  2.86505  1.14239  1.14239  0.83805  Accept Null 
3  1.0000  1091.972  3.30450  3.30450  1.91408  1.91408  . 
Since the data set PARMS_FIT2
contains the test information only at stage 2, the information level at stage 1 in the TEST_FIT1
data set is used to generate boundary values for the test.
With ODS Graphics enabled, a boundary plot with test statistics is displayed, as shown in Output 84.3.18. As expected, the test statistic is in the acceptance region between the lower and upper boundaries at the final stage.
Output 84.3.18: Sequential Test Plot
After a trial is stopped, the “Parameter Estimates” table in Output 84.3.19 displays the stopping stage, parameter estimate, unbiased median estimate, confidence limits, and the pvalue under the null hypothesis . As expected, the pvalue 0.3056 is not significant at the level, and the confidence interval does contain the value zero. The pvalue, unbiased median estimate, and confidence limits depend on the ordering of the sample space , where k is the stage number and z is the standardized Z statistic. With the specified LR ordering, the pvalues are computed with the ordering if . See the section Available Sample Space Orderings in a Sequential Test for a detailed description of the LR ordering.
Output 84.3.19: Parameter Estimates
Parameter Estimates LR Ordering 


Parameter  Stopping Stage 
MLE  pValue for H0:Parm=0 
Median Estimate 
95% Confidence Limits  
Weight  2  0.029251  0.3056  0.037080  0.03368  0.10532 