This example demonstrates a one-sided fixed-sample design and a two-sided fixed-sample design. The following statements request a fixed-sample design with an upper alternative:
ods graphics on; proc seqdesign pss ; OneSidedFixedSample: design nstages=1 alt=upper alpha=0.025 beta=0.10 ; samplesize model=onesamplemean(mean=0.25); run; ods graphics off;
In the DESIGN statement, the label OneSidedFixedSample
identifies the design in the output tables. The NSTAGES=1 option specifies that the design has only one stage; this corresponds
to a fixed-sample design. In the SEQDESIGN procedure, the null hypothesis for the design is and the ALT=UPPER option specifies an upper alternative hypothesis . The MEAN=0.25 option in the SAMPLESIZE statement specifies the upper alternative reference .
The options ALPHA=0.025 and BETA=0.10 specify the Type I error probability level and the Type II error probability level . That is, the design has a power at .
The “Design Information” table in Output 83.1.1 displays design specifications and the derived statistics such as power. As expected, the derived statistics such as maximum information and average sample number (in percentage of its corresponding fixed-sample information) are 100 for the fixed-sample design (NSTAGES=1). Also, for a fixed-sample design, the STOP= and METHOD= options in the DESIGN statement are not applicable.
Output 83.1.1: One-Sided Fixed-Sample Design Information
Design Information | |
---|---|
Statistic Distribution | Normal |
Boundary Scale | Standardized Z |
Alternative Hypothesis | Upper |
Alternative Reference | 0.25 |
Number of Stages | 1 |
Alpha | 0.025 |
Beta | 0.1 |
Power | 0.9 |
Max Information (Percent of Fixed Sample) | 100 |
Max Information | 168.1188 |
Null Ref ASN (Percent of Fixed Sample) | 100 |
Alt Ref ASN (Percent of Fixed Sample) | 100 |
The “Method Information” table in Output 83.1.2 displays the and error levels. It also displays the derived drift parameter, which is the standardized reference improvement, , where is the alternative reference and is the maximum information for the design. If either or is specified, the other statistic is derived in the SEQDESIGN procedure. For a fixed-sample design,
Output 83.1.2: Method Information
Method Information | ||||
---|---|---|---|---|
Boundary | Alpha | Beta | Alternative Reference |
Drift |
Upper Alpha | 0.02500 | 0.10000 | 0.25 | 3.241516 |
The “Boundary Information” table in Output 83.1.3 displays information level, alternative reference, and boundary value at each stage. The information proportion indicates the proportion of maximum information available at the stage. With only one stage for a fixed-sample design, the proportion is 1. With the SAMPLESIZE statement, the required sample size N is also displayed under the heading “Information Level.”
Output 83.1.3: Boundary Information
Boundary Information (Standardized Z Scale) Null Reference = 0 |
|||||
---|---|---|---|---|---|
_Stage_ | Alternative | Boundary Values | |||
Information Level | Reference | Upper | |||
Proportion | Actual | N | Upper | Alpha | |
1 | 1.0000 | 168.1188 | 168.1188 | 3.24152 | 1.95996 |
By default (or equivalently if you specify BOUNDARYSCALE=STDZ), output alternative references and boundaries are displayed with the standardized normal Z scale. The alternative reference on the standardized Z scale at stage 1 is given by , where is the information level at stage 1. With a boundary value 1.96, the hypothesis of is rejected if the standardized normal statistic .
With ODS Graphics enabled, a detailed boundary plot with the rejection and acceptance regions is displayed, as shown in Output 83.1.4. The boundary values in the “Boundary Information” table in Output 83.1.3 are displayed in the plot.
The “Sample Size Summary” table in Output 83.1.5 displays parameters for the sample size computation of the test for a normal mean.
Output 83.1.5: Sample Size Summary
Sample Size Summary | |
---|---|
Test | One-Sample Mean |
Mean | 0.25 |
Standard Deviation | 1 |
Max Sample Size | 168.1188 |
Expected Sample Size (Null Ref) | 168.1188 |
Expected Sample Size (Alt Ref) | 168.1188 |
The “Sample Sizes (N)” table in Output 83.1.6 displays the derived sample sizes, in both fractional and integer numbers. With the resulting integer sample sizes, the corresponding
information level is slightly larger than the level from the design. This can increase the power slightly if the integer sample
size is used in the trial.
Output 83.1.6: Derived Sample Sizes
Sample Sizes (N) One-Sample Z Test for Mean |
||||
---|---|---|---|---|
_Stage_ | Fractional N | Ceiling N | ||
N | Information | N | Information | |
1 | 168.12 | 168.1 | 169 | 169.0 |
The following statements request a two-sided fixed-sample design with a specified alternative reference:
ods graphics on; proc seqdesign altref=1.2 pss ; TwoSidedFixedSample: design nstages=1 alt=twosided alpha=0.05 beta=0.10 ; samplesize model=twosamplemean(stddev=2 weight=2); run; ods graphics off;
In the SEQDESIGN procedure, the null hypothesis for the design is . The ALT=TWOSIDED option specifies a two-sided alternative hypothesis . The ALTREF=1.2 option in the PROC SEQDESIGN statement specifies the alternative reference .
The ALPHA=0.05 option (which is the default) specifies the two-sided Type I error probability level . That is, the lower and upper Type I error probabilities . The BETA=0.10 option (which is the default) specifies the Type II error probability level , and the design has a power at the alternative reference .
The “Design Information” table in Output 83.1.7 displays design specifications and the derived power. With a specified alternative reference, the maximum information is derived.
Output 83.1.7: Two-Sided Fixed-Sample Design Information
Design Information | |
---|---|
Statistic Distribution | Normal |
Boundary Scale | Standardized Z |
Alternative Hypothesis | Two-Sided |
Alternative Reference | 1.2 |
Number of Stages | 1 |
Alpha | 0.05 |
Beta | 0.1 |
Power | 0.9 |
Max Information (Percent of Fixed Sample) | 100 |
Max Information | 7.296822 |
Null Ref ASN (Percent of Fixed Sample) | 100 |
Alt Ref ASN (Percent of Fixed Sample) | 100 |
The “Method Information” table in Output 83.1.8 displays the and errors, alternative references, and drift parameter. For a fixed-sample design, the derived drift parameter
Output 83.1.8: Method Information
Method Information | ||||
---|---|---|---|---|
Boundary | Alpha | Beta | Alternative Reference |
Drift |
Upper Alpha | 0.02500 | 0.10000 | 1.2 | 3.241516 |
Lower Alpha | 0.02500 | 0.10000 | -1.2 | -3.24152 |
With a specified alternative reference , the maximum information
The default “Boundary Information” table in Output 83.1.9 displays information level, alternative reference, and boundary values. By default (or equivalently if you specify BOUNDARYSCALE=STDZ), the alternative reference and boundary values are displayed with the standardized normal Z scale. Thus, the standardized alternative references are displayed.
Output 83.1.9: Boundary Information
Boundary Information (Standardized Z Scale) Null Reference = 0 |
|||||||
---|---|---|---|---|---|---|---|
_Stage_ | Alternative | Boundary Values | |||||
Information Level | Reference | Lower | Upper | ||||
Proportion | Actual | N | Lower | Upper | Alpha | Alpha | |
1 | 1.0000 | 7.296822 | 131.3428 | -3.24152 | 3.24152 | -1.95996 | 1.95996 |
With boundary values of –1.96 and 1.96, the hypothesis of is rejected if the standardized normal statistic or .
With ODS Graphics enabled, a detailed boundary plot with the rejection and acceptance regions is displayed, as shown in Output 83.1.10 . The boundary values in the “Boundary Information” table in Output 83.1.9 are displayed in the plot.
The “Sample Size Summary” table in Output 83.1.11 displays parameters for the sample size computation of the test for a normal mean.
Output 83.1.11: Sample Size Summary
Sample Size Summary | |
---|---|
Test | Two-Sample Means |
Mean Difference | 1.2 |
Standard Deviation | 2 |
Max Sample Size | 131.3428 |
Expected Sample Size (Null Ref) | 131.3428 |
Expected Sample Size (Alt Ref) | 131.3428 |
Weight (Group A) | 2 |
Weight (Group B) | 1 |
The “Sample Sizes (N)” table in Output 83.1.12 displays the derived sample sizes, in both fractional and integer numbers. With the WEIGHT=2 option, the allocation ratio is 2 for the first group and 1 for the second group. With the resulting integer sample sizes, the corresponding information level is slightly larger than the level from the design. This can increase the power slightly if the integer sample size is used in the trial.
Output 83.1.12: Derived Sample Sizes
Sample Sizes (N) Two-Sample Z Test for Mean Difference |
||||||||
---|---|---|---|---|---|---|---|---|
_Stage_ | Fractional N | Ceiling N | ||||||
N | N(Grp 1) | N(Grp 2) | Information | N | N(Grp 1) | N(Grp 2) | Information | |
1 | 131.34 | 87.56 | 43.78 | 7.2968 | 132 | 88 | 44 | 7.3333 |