Creating Fixed-Sample Designs
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The SEQDESIGN Procedure

Example 87.1 Creating Fixed-Sample Designs

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 $H_0: \theta = 0$ and the ALT=UPPER option specifies an upper alternative hypothesis $H_1: \theta = \theta _1 > 0$. The MEAN=0.25 option in the SAMPLESIZE statement specifies the upper alternative reference $\theta _1= 0.25$.

The options ALPHA=0.025 and BETA=0.10 specify the Type I error probability level $\alpha = 0.025$ and the Type II error probability level $\beta = 0.10$. That is, the design has a power $1-\beta = 0.90$ at $\theta _1= 0.25$.

The “Design Information” table in Output 87.1.1 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 87.1.1: One-Sided Fixed-Sample Design Information

The SEQDESIGN Procedure
Design: OneSidedFixedSample

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 87.1.2 displays the $\alpha $ and $\beta $ error levels. It also displays the derived drift parameter, which is the standardized reference improvement, $\theta _{1} \sqrt {I_{0}}$, where $\theta _{1}$ is the alternative reference and $I_{0}$ is the maximum information for the design. If either $\theta _{1}$ or $I_{0}$ is specified, the other statistic is derived in the SEQDESIGN procedure. For a fixed-sample design,

\[ \theta _{1} \sqrt {I_{0}} = \Phi ^{-1} (1-\alpha ) + \Phi ^{-1} (1-\beta ) = \Phi ^{-1} (0.975) + \Phi ^{-1} (0.90) = 3.2415 \]

Output 87.1.2: 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 87.1.3 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 87.1.3: 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 $\theta _1 \sqrt {I_1}$, where $I_1$ is the information level at stage 1. With a boundary value 1.96, the hypothesis of $\theta = 0$ is rejected if the standardized normal statistic $Z \geq 1.96$.

With ODS Graphics enabled, a detailed boundary plot with the rejection and acceptance regions is displayed, as shown in Output 87.1.4. The boundary values in the “Boundary Information” table in Output 87.1.3 are displayed in the plot.

Output 87.1.4: Boundary Plot


The “Sample Size Summary” table in Output 87.1.5 displays parameters for the sample size computation of the test for a normal mean.

Output 87.1.5: 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 87.1.6 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 87.1.6: 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 $H_{0}: \theta = 0$. The ALT=TWOSIDED option specifies a two-sided alternative hypothesis $H_1: \theta = \theta _1 \neq 0$. The ALTREF=1.2 option in the PROC SEQDESIGN statement specifies the alternative reference $\theta _1= \pm 1.2$.

The ALPHA=0.05 option (which is the default) specifies the two-sided Type I error probability level $\alpha =0.05$. That is, the lower and upper Type I error probabilities $\alpha _{l}=\alpha _{u}=0.025$. The BETA=0.10 option (which is the default) specifies the Type II error probability level $\beta = 0.10$, and the design has a power $1-\beta = 0.90$ at the alternative reference $\theta _1= \pm 1.2$.

The “Design Information” table in Output 87.1.7 displays design specifications and the derived power. With a specified alternative reference, the maximum information is derived.

Output 87.1.7: Two-Sided Fixed-Sample Design Information

The SEQDESIGN Procedure
Design: TwoSidedFixedSample

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 87.1.8 displays the $\alpha $ and $\beta $ errors, alternative references, and drift parameter. For a fixed-sample design, the derived drift parameter

\[ \theta _{1} \sqrt {I_{0}} = \Phi ^{-1}(1-\frac{\alpha }{2}) + \Phi ^{-1} (1-\beta ) = \Phi ^{-1} (0.975) + \Phi ^{-1} (0.90) = 3.2415 \]

Output 87.1.8: 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 $\theta _{1}= 1.2$, the maximum information

\[ I_{0} = \left( \frac{3.2415}{1.2} \right)^{2} = 7.2968 \]

The default “Boundary Information” table in Output 87.1.9 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 $\pm \theta _{1} \sqrt {I_{0}}$ are displayed.

Output 87.1.9: 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 $\theta = 0$ is rejected if the standardized normal statistic $Z \geq 1.96$ or $Z \leq -1.96$.

With ODS Graphics enabled, a detailed boundary plot with the rejection and acceptance regions is displayed, as shown in Output 87.1.10 . The boundary values in the “Boundary Information” table in Output 87.1.9 are displayed in the plot.

Output 87.1.10: Boundary Plot


The “Sample Size Summary” table in Output 87.1.11 displays parameters for the sample size computation of the test for a normal mean.

Output 87.1.11: 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 87.1.12 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 87.1.12: 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


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