This example demonstrates a two-sided 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 three-stage 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 | Two-Sided |
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 fixed-sample design.
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 p-value, 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 | Two-Sided |
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.
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 p-value, 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 | Two-Sided |
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 Fixed-Sample |
||
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.
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.
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.
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 p-value under the null hypothesis . As expected, the p-value 0.3056 is not significant at the level, and the confidence interval does contain the value zero. The p-value, 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 p-values 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 | p-Value for H0:Parm=0 |
Median Estimate |
95% Confidence Limits | |
Weight | 2 | 0.029251 | 0.3056 | 0.037080 | -0.03368 | 0.10532 |