The SEQDESIGN Procedure

DESIGN Statement

<label:> DESIGN <options> ;

The DESIGN statement requests a new group sequential design. You can use multiple DESIGN statements, and each DESIGN statement corresponds to a separate group sequential design.

Table 83.2 summarizes the options available in the DESIGN statement.

Table 83.2: Design Statement Options

Option	Description
Design Parameters
ALPHA=	Specifies the Type I error probability level $\alpha$
ALT=	Specifies the type of alternative hypothesis
BETA=	Specifies the Type II error probability level $\beta$
BETAOVERLAP=	Checks for overlapping of the lower and upper $\beta$ boundaries
	in a two-sided design with error spending methods
BOUNDARYKEY=	Specifies the type of error probability to maintain
INFO=	Specifies the information levels
NSTAGES=	Specifies the number of stages
STOP=	Specifies the condition for early stopping
Boundary Methods
METHOD=	Specifies methods for boundary values

The required NSTAGES= option specifies the number of stages. The METHOD= option is required if the number of stages specified in the NSTAGES= option is greater than one. The following options can be used in the DESIGN statement. They are listed in alphabetical order.

ALPHA= $\alpha$ <( <LOWER= $\alpha _ l$ > <UPPER= $\alpha _ u$ > )>

specifies the Type I error probability $\alpha$ . The default is $\alpha = 0.05$ . The LOWER= and UPPER= options are applicable only for the two-sided design. The LOWER= option specifies the lower Type I error probability $\alpha _{l}$ , and the upper Type I error probability is computed as $\alpha _{u}= \alpha - \alpha _{l}$ . The UPPER= option specifies the upper Type I error probability $\alpha _{u}$ , and the lower Type I error probability is computed as $\alpha _{l}= \alpha - \alpha _{u}$ . If both LOWER= and UPPER= options are not specified, $\alpha _{l}= \alpha _{u}= \alpha /2$ .

If both the MAXINFO= and ALTREF= options are specified, then the Type I and Type II error probability levels cannot be met simultaneously. In this case, the ALPHA= option is applicable only with the BOUNDARYKEY=ALPHA option (which is the default), and the Type II error probability $\beta$ is derived.

ALT=LOWER | UPPER | TWOSIDED

specifies the type of alternative hypothesis in the design. For a test of $H_{0}: \theta = 0$ , the keywords LOWER, UPPER, and TWOSIDED correspond to the alternatives of $\theta < 0$ , $\theta > 0$ , and $\theta \ne 0$ , respectively. The default is ALT=TWOSIDED.

BETA= $\beta$ <( <LOWER= $\beta _ l$ > <UPPER= $\beta _ u$ > )>

specifies the Type II error probability level $\beta$ . The default is $\beta = 0.10$ . The LOWER= and UPPER= options are applicable only for the two-sided design. The LOWER= option specifies the lower Type II error probability level $\beta _{l}$ , and the UPPER= option specifies the upper Type II error probability level $\beta _{u}$ . If the LOWER= or UPPER= option is not specified, $\beta$ is used.

If both the MAXINFO= and ALTREF= options are specified, then the Type I and Type II error probability levels cannot be met simultaneously. In this case, the BETA= option is applicable only with the BOUNDARYKEY=BETA option, and the Type I error probability $\alpha$ is derived.

BETAOVERLAP=ADJUST | NOADJUST OVERLAP=ADJUST | NOADJUST

specifies whether to check for overlapping of the lower and upper $\beta$ boundaries for the two corresponding one-sided tests. This option applies to two-sided designs with STOP=ACCEPT or STOP=BOTH that are constructed with error spending methods, and this type of overlapping might result from a small $\beta$ spending at an interim stage. When you specify BETAOVERLAP=ADJUST, the procedure checks for this type of overlapping. If such overlapping is found, the $\beta$ boundaries for the two-sided design at that stage are set to missing, and the $\beta$ spending values at subsequent stages are adjusted, as described in the section Boundary Adjustments for Overlapping Lower and Upper $\bbeta$ Boundaries".

You can specify BETAOVERLAP=NOADJUST to request that no adjustment be made. The default is BETAOVERLAP=ADJUST.

BOUNDARYKEY=ALPHA | BETA | BOTH | NONE

specifies types of errors to be maintained in the resulting boundary. The default is BOUNDARYKEY=ALPHA if both ALTREF= and MAXINFO= options are specified. Otherwise, the default is BOUNDARYKEY=NONE for Whitehead methods with the STOP=BOTH option, and it is BOUNDARYKEY=BOTH for others.

See the section Applicable Boundary Keys for a detailed description of applicable boundary keys.

INFO=EQUAL INFO=CUM( numbers )

specifies relative information levels for all stages in the design. The INFO=EQUAL option specifies equally spaced information levels, and the INFO=CUM option specifies cumulative relative information levels. The default is INFO=EQUAL.

If the number of information levels specified in the INFO=CUM option is less than the number of stages specified in the NSTAGES= option, the last available information increment is used as the information increment for each subsequent stage.

METHOD=WHITEHEAD < ( TAU= $\tau$ < ( <LOWER= $\tau _ l$ > <UPPER= $\tau _ u$ > ) > ) > METHOD=method METHOD(boundary) = method

specifies the methods for the boundaries in the design, where $0 \le \tau < 0.5$ .

For a one-sided design, an $\alpha$ boundary is created with the STOP=REJECT or STOP=BOTH option, and a $\beta$ boundary is created with the STOP=ACCEPT or STOP=BOTH option. For a two-sided design, lower and upper $\alpha$ boundaries are created with the STOP=REJECT or STOP=BOTH option, and lower and upper $\beta$ boundaries are created with the STOP=ACCEPT or STOP=BOTH option.

There are three types of methods available in the SEQDESIGN procedure. The unified family methods and Haybittle-Peto methods derive boundary values with fixed boundary shape; the Whitehead methods derive boundary values by adjusting the boundary values generated from continuous monitoring; and the error spending methods derive the boundary values from the specified errors used at each stage. You can specify different methods for the same design, but all methods must be from the same group.

For a design with early stopping to reject or accept the null hypothesis, the METHOD=WHITEHEAD option uses Whitehead’s triangular design and double-triangular design for a one-sided design and two-sided design, respectively (Whitehead and Stratton, 1983; Whitehead, 1997, 2001). For a design with early stopping only to reject the null hypothesis or only to accept the null hypothesis, you can specify the slope of the boundary line in the score statistic scale with the TAU= $\tau$ option. The default is TAU=0.25. See the section Whitehead Methods for a detailed description of the Whitehead methods.

The following options specify available error spending methods for the boundary. Each of these methods can be specified with the METHOD= option for all boundaries, or with the METHOD(boundary) = option for an individual boundary. See the section Error Spending Methods for a detailed description of these error spending methods.

ERRFUNCGAMMA < ( GAMMA= $\gamma$ ) >: specifies a gamma cumulative error spending function for the boundary (Hwang, Shih, and DeCani, 1990). The GAMMA= option specifies the gamma parameter $\gamma$ in the function, where $\gamma \leq 3$ . The boundaries created with $\gamma =1$ are similar to the boundaries from the Pocock method, and the boundaries created with $\gamma =-4$ or $\gamma =-5$ are similar to the boundaries from the O’Brien-Fleming method. The default is GAMMA=–2, which is the average of $\gamma =1$ and $\gamma =-5$ .
ERRFUNCOBF: specifies the O’Brien-Fleming-type cumulative error spending function for the boundary (Lan and DeMets, 1983).
ERRFUNCPOC: specifies the Pocock-type cumulative error spending function for the boundary (Lan and DeMets, 1983).
ERRFUNCPOW < ( RHO= $\rho$ ) >: specifies a power cumulative error spending function for the boundary (Jennison and Turnbull, 2000, p. 148). The RHO= option specifies the power parameter $\rho$ in the function, where $\rho \geq 0.25$ . The boundaries created with $\rho =1$ are similar to the boundaries from the Pocock method, and the boundaries created with $\rho =3$ are similar to the boundaries from the O’Brien-Fleming method. The default is RHO=2, which is the average of $\rho =1$ and $\rho =3$ .
ERRSPEND ( numbers ): specifies the relative cumulative error spending at each stage.

With a fixed boundary shape, you can use the following available Haybittle-Peto methods and unified family methods to derive the boundary. You can specify each of these methods in the METHOD= option for all boundaries, or in the METHOD(boundary) = option for an individual boundary. See the section Haybittle-Peto Method for a detailed description of the Haybittle-Peto methods, and see the section Unified Family Methods for a detailed description of unified family methods.

HP | HAYBITTLE | PETO < ( Z= numbers | PVALUE= numbers ) >: specifies the Haybittle-Peto method (Haybittle, 1971; Peto et al., 1976). The values specified are used to create the boundary values. The boundary value at the final stage can be derived in the procedure to maintain the Type I and Type II error probability levels. The default is Z=3.
OBF | OBRIENFLEMING: specifies the O’Brien-Fleming method (O’Brien and Fleming, 1979). The O’Brien-Fleming method is equivalent to a power family method with RHO=0.5.
POC | POCOCK: specifies the Pocock method (Pocock, 1977). The Pocock method is equivalent to a power family method with RHO=0.
POW | POWER < ( RHO= $\rho$ ) >: specifies a power family method (Wang and Tsiatis, 1987; Emerson and Fleming, 1989; Pampallona and Tsiatis, 1994). The RHO= option specifies the power parameter $\rho$ in the power family method, where $\rho \geq -0.25$ . The power family method with $\rho =0$ corresponds to the Pocock method, and the power family method with $\rho =0.5$ corresponds to the O’Brien-Fleming method. The default is RHO=0.25, a value halfway between the Pocock and O’Brien-Fleming methods. A power family method is equivalent to a unified family method with RHO= $\rho$ and TAU=0.
TRI | TRIANGULAR < ( TAU= $\tau$ ) >: specifies a unified family triangular method (Kittelson and Emerson, 1999), where $0 \leq \tau \leq 1$ . The default is TAU=1.0. The triangular method is identical to the unified family method with RHO=0.5 and TAU= $\tau$ . Note that this unified family triangular method is different from Whitehead’s triangular method.
UNI | UNIFIED < ( <TAU= $\tau$ > <RHO= $\rho$ > ) >: specifies a unified family method (Kittelson and Emerson, 1999). The TAU= and RHO= options specify the $\tau$ and $\rho$ parameters in a unified family method, respectively, where $\rho \geq 0$ and $0 \leq \tau \leq 2\rho$ . The defaults are TAU=0 and RHO=0.25. See the section Unified Family Methods for a detailed description of the unified family methods.

The O’Brien-Fleming, Pocock, power family, and triangular methods are all special cases of the unified family methods. Table 83.3 summarizes the corresponding parameters in the unified family for these methods.

Table 83.3: Parameters in the Unified Family for Various Methods

		Unified Family
Method	Option	Rho	Tau
Pocock	POC	0	0
O’Brien-Fleming	OBF	0.5	0
Power family	POW (RHO= $\rho$ )	$\rho$	0
Triangular	TRI (TAU= $\tau$ )	0.5	$\tau$

Note that the power parameter $\rho = 1/2 - \Delta = \rho ^{*} - 1/2$ , where $\Delta$ is the power parameter used in Jennison and Turnbull (2000) and Wang and Tsiatis (1987) and $\rho ^{*}$ is the power parameter used in Kittelson and Emerson (1999).

If a method with specified parameters is used for all boundaries in the design, you can use the METHOD= option to specify the method. Otherwise, you can use the following METHOD(boundary)= options to specify different methods from the same group for the boundaries.

METHOD(ALPHA)=method METHOD(REJECT)=method

specifies the method for the $\alpha$ boundary of a one-sided design or the lower and upper $\alpha$ boundaries for a two-sided design.

METHOD(LOWERALPHA)=method METHOD(LOWERREJECT)=method

specifies the method for the lower $\alpha$ boundary of a two-sided design.

METHOD(UPPERALPHA)=method METHOD(UPPERREJECT)=method

specifies the method for the upper $\alpha$ boundary of a two-sided design.

METHOD(BETA)=method METHOD(ACCEPT)=method

specifies the method for the $\beta$ boundary of a one-sided design or the lower and upper $\beta$ boundaries for a two-sided design.

METHOD(LOWERBETA)=method METHOD(LOWERACCEPT)=method

specifies the method for the lower $\beta$ boundary of a two-sided design.

METHOD(UPPERBETA)=method METHOD(UPPERACCEPT)=method

specifies the method for the upper $\beta$ boundary of a two-sided design.

NSTAGES=number

specifies the number of stages for the design. This option is required in the DESIGN statement, and the maximum allowed number of stages is 25.

STOP=ACCEPT STOP=REJECT STOP=BOTH < ( BETABOUNDARY=BINDING | NONBINDING ) >

specifies the early stopping criterion for the design. The keywords ACCEPT, REJECT, and BOTH correspond to early stopping only to accept, only to reject, and either to accept or reject the null hypothesis $H_{0}$ , respectively. The default is STOP=REJECT. With STOP=BOTH, the BETABOUNDARY=BINDING suboption computes the Type I error probability with the acceptance boundary, and the BETABOUNDARY=NONBINDING suboption computes the Type I error probability without the acceptance boundary. The default is BETABOUNDARY=BINDING. The BETABOUNDARY=NONBINDING option is not applicable with METHOD=WHITEHEAD.