The SEQDESIGN Procedure

Sample Size Computation

Subsections:

Input Sample Size for Fixed-Sample Design
Input Number of Events for Fixed-Sample Design
Uniform Accrual without Losses to Follow Up
Uniform Accrual with Losses to Follow Up
Truncated Exponential Accrual without Losses to Follow Up
Truncated Exponential Accrual with Losses to Follow Up

The SEQDESIGN procedure assumes that the data are from a multivariate normal distribution and the sequence of the standardized test statistics $\{ Z_{1}, Z_{2}, \ldots , Z_{K}\}$ has the following canonical joint distribution:

$(Z_{1}, Z_{2}, \ldots , Z_{K})$ is multivariate normal
$Z_{k} \sim N \left( \, \theta \sqrt {I_{k}}, \, 1 \right)$
$\mr{Cov}( Z_{k_1}, Z_{k_2})= \sqrt {I_{k_1}/I_{k_2}}$ , $1 \leq k_1 \leq k_2 \leq K$

where K is the total number of stages and $I_{k}$ is the information available at stage k.

If the test statistic is computed from the data that are not from a normal distribution, such as a binomial distribution, then it is assumed that the test statistic is computed from a large sample such that the statistic has an approximately normal distribution.

In a typical clinical trial, the sample size required depends on the Type I error probability level $\alpha$ , alternative reference $\theta _1$ , power $1-\beta$ , and variance of the response variable. Given a one-sided null hypothesis $H_{0}: \theta = 0$ with an upper alternative hypothesis $H_{1}: \theta = \theta _1$ , the information required for a fixed-sample test is given by

$I_{0} = \frac{ ( \Phi ^{-1}(1-\alpha ) + \Phi ^{-1}(1-\beta ) )^{2} }{ {\theta }^{2}_{1} }$

The parameter $\theta$ and the subsequent alternative reference $\theta _1$ depend on the test specified in the clinical trial. For example, suppose you are comparing two binomial populations $p_{a} = p_{b}$ ; then $\theta = p_{a} - p_{b}$ is the difference between two proportions if the proportion difference statistic is used, and $\theta = \mr{log} \left( \frac{p_{a} (1-p_{b})}{p_{b} (1-p_{a})} \right)$ , the log odds ratio for the two proportions if the log odds ratio statistic is used.

If the maximum likelihood estimate $\hat{\theta }$ from the likelihood function can be derived, then the asymptotic variance for $\hat{\theta }$ is $\mr{Var}( \hat{\theta })= 1/I$ , where I is Fisher information for $\theta$ . The resulting statistic $\hat{\theta }$ corresponds to the MLE statistic scale as specified in the BOUNDARYSCALE=MLE option in the PROC SEQDESIGN statement, $\hat{\theta } \sqrt {I}$ corresponds to the standardized Z scale (BOUNDARYSCALE=STDZ), and $\hat{\theta } \, I$ corresponds to the score statistic scale (BOUNDARYSCALE=SCORE).

Alternatively, if the score statistic S is derived in a statistical procedure, it can be used as the test statistic and its asymptotic variance is given by Fisher information, I. In this case, $S/ \sqrt {I}$ corresponds to the standardized Z scale and $S / I$ corresponds to the MLE statistic scale.

For a group sequential trial, the maximum information $I_{X}$ is derived in the SEQDESIGN procedure with the specified $\alpha$ , $\beta$ , and $\theta _1$ . With the maximum information

$I_{X} = \frac{1}{\mr{Var}( \hat{\theta })}$

the sample size required for a specified test statistic in the trial can be evaluated or estimated from the known or estimated variance of the response variable. Note that different designs might produce different maximum information levels for the same hypothesis, and this in turn might require a different number of observations for the trial.

If each observation in the data set provides one unit of information in a hypothesis testing, such as a one-sample test for the mean, the required sample size for the sequential design can be derived from the maximum information. However, for a survival analysis, an individual in the survival time data might provide only partial information because of censoring. In this case, the required number of events can be derived from the maximum information. With addition accrual information, the sample size can also be computed.

The SEQDESIGN procedure provides sample size computation for some one-sample and two-sample tests in the SAMPLESIZE statement. It also provides sample size computation for tests of a parameter in regression models such as normal regression, logistic regression, and proportional hazards regression. In addition, the procedure can also compute the required sample size or number of events from the corresponding number in the fixed-sample design.

Table 101.11 lists the options available in the SAMPLESIZE statement.

Table 101.11: SAMPLESIZE Statement Options

Option	Description
Fixed-Sample Models
INPUTNOBS	Specifies sample size for fixed-sample design
INPUTNEVENTS	Specifies number of events for fixed-sample design
One-Sample Models
ONESAMPLEMEAN	Specifies one-sample Z test for mean
ONESAMPLEFREQ	Specifies one-sample test for binomial proportion
Two-Sample Models
TWOSAMPLEMEAN	Specifies two-sample Z test for mean difference
TWOSAMPLEFREQ	Specifies two-sample test for binomial proportions
TWOSAMPLESURVIVAL	Specifies log-rank test for two survival distributions
Regression Models
REG	Specifies test for a regression parameter
LOGISTIC	Specifies test for a logistic regression parameter
PHREG	Specifies test for a proportional hazards regression parameter

The MODEL=INPUTNOBS and MODEL=INPUTNEVENTS options are described next, and the remaining options are described in the next three sections.

Input Sample Size for Fixed-Sample Design

The MODEL=INPUTNOBS option derives the sample size required for a group sequential trial from the sample size $n_{0}$ for the corresponding fixed-sample design. With the N= $n_{0}$ option specifying the sample size $n_{0}$ for a fixed-sample design, the sample size required for a group sequential trial is then computed as

$N_{X} = \frac{I_{X}}{I_{0}} \, \, n_{0}$

where $I_{X}$ is the maximum information for the group sequential design and $I_{0}$ is the information for the corresponding fixed-sample design. The information ratio between $I_{X}$ and $I_{0}$ is derived in the SEQDESIGN procedure.

The SAMPLE=ONE option specifies a one-sample test, and the SAMPLE=TWO option specifies a two-sample test. For a two-sample test, the WEIGHT= option specifies the sample size allocation weights for the two groups.

Input Number of Events for Fixed-Sample Design

The MODEL=INPUTNEVENTS option derives the number of events required for a group sequential trial from the number of events $d_{0}$ for the corresponding fixed-sample design. With the D= $d_{0}$ option specifies the number of events $d_{0}$ for a fixed-sample survival analysis, the number of events required for a group sequential trial is then computed as

$d_{X} = \frac{I_{X}}{I_{0}} \, \, d_{0}$

The ACCRUAL= option specifies the method for individual accrual. The ACCRUAL=UNIFORM option (which is the default) specifies that the individual accrual is uniform in the accrual time $T_ a$ with a constant accrual rate $r_ a$ , and the ACCRUAL=EXP(PARM= $\gamma _0$ ) option specifies that the individual accrual is truncated exponential with a scaled power parameter $\gamma _0$ , where $\gamma _0 \geq -10$ and $\gamma _0 \neq 0$ . With a scaled parameter $\gamma _0$ , the power parameter for the truncated exponential with the accrual time $T_ a$ is given by $\gamma = \gamma _0 / T_ a$ .

The LOSS= option specifies the individual loss to follow up in the sample size computation. The LOSS=NONE option (which is the default) specifies no loss to follow up, and the EXP(POWER= $\tau$ ) option specifies exponential loss function with a power parameter $\tau$ .

With the computed number of events $d_{X}$ for a group sequential survival design, the required total sample size and sample size at each stage can be derived from specifications of hazard rates, accrual information, and losses to follow-up information. For each study group, the hazard rate h is constant (which corresponds to an exponential survival distribution) in the sample size computation.

The next four subsections describe required sample sizes for uniform accrual (with and without losses to follow up) and for truncated exponential accrual (with and without losses to follow up).

Uniform Accrual without Losses to Follow Up

For a study group with a constant hazard rate h, if the individual accrual is uniform in the accrual time $T_ a$ with a constant accrual rate $r_ a$ , Kim and Tsiatis (1990, pp. 83–84) show that the expected number of events by time t is given by

$D_{h}(t) = \left\{ \begin{array}{cl} r_{a} \left( t - \frac{1 - e^{-ht}}{h} \right) & \mbox{if} \, \, t \leq T_{a} \\ r_{a} \left( T_{a} - \frac{e^{-ht}}{h} (e^{h T_{a}} -1) \right) & \mbox{if} \, \, t > T_{a} \end{array} \right.$

For a one-sample design (such as a proportional hazards regression), the expected number of events by time t is $E(t) = D_{h}(t)$ , where h is the hazard rate for the group. For a two-sample design (such as a log-rank test for two survival distributions), the expected number of events by time t is

$E(t) = \frac{R}{R+1} D_{h_{a}}(t) + \frac{1}{R+1} D_{h_{b}}(t)$

where $h_ a$ and $h_ b$ are hazard rates in groups A and B, respectively, and R is the ratio of the sample size allocation weights $w_ a/w_ b$ .

If the accrual rate $r_ a$ is specified with one of the three time parameters—the accrual time, follow-up time, and total study time—then PROC SEQDESIGN derives the other two time parameters by solving the equation for the expected number of events. Similarly, if the accrual rate $r_ a$ is not specified but two of the three time parameters are specified, then PROC SEQDESIGN derives the accrual rate.

If the accrual rate $r_ a$ is specified without the accrual time $T_ a$ , follow-up time $T_ f$ , and total study time $T= T_ a + T_ f$ , the minimum and maximum accrual times can be computed from the following equation, as described in Kim and Tsiatis (1990, p. 85):

$\frac{d_{X}}{r_ a} \leq T_ a \leq E^{-1}(d_ X)$

With the accrual rate $r_ a$ and the accrual time $T_ a$ , the total sample size is

$N_{X} = r_ a \; T_ a$

At each stage k, the number of events is given by

$d_ k = \frac{I_ k}{I_ X} \, \, d_ X$

The corresponding time $T_ k$ can be derived from the equation for the expected number of events, $E(t)= d_ k$ , and the resulting sample size is computed as

$N_ k= \left\{ \begin{array}{cl} r_{a} \; T_ k & \mbox{if} \, \, T_ k \leq T_ a \\ r_{a} \; T_ a & \mbox{if} \, \, T_ k > T_ a \end{array} \right.$

Uniform Accrual with Losses to Follow Up

With the LOSS=EXP(HAZARD= $\tau$ ) option, the individual loss to follow up has an exponential loss distribution function

$H(t)= 1 - e^{-\tau t}$

where $\tau \geq 0$ is the loss hazard rate. The loss hazard rate can also be specified implicitly with the MEDTIME= $t_{\tau }$ suboption through the median loss time $t_{\tau }$ .

For a study group with a constant hazard rate h, if the individual accrual is uniform in the accrual time $T_ a$ with a constant accrual rate $r_ a$ and the individual loss to follow up has an exponential loss distribution function $H(t)$ , Lachin and Foulkes (1986, p. 511) derive the expected number of events by time t (where $t > T_ a$ ) as

$D_{h}(t, T_ a) = r_{a} \, T_{a} \; \frac{h}{h+\tau } \left( 1 - \frac{1}{(h+\tau ) T_ a} e^{-(h+\tau ) t} (e^{h+\tau ) T_{a}} -1) \right)$

For $t \leq T_ a$ , the SEQDESIGN procedure estimates the expected number of events by time t as

$D_{h}(t, T_ a) = r_{a} \, t \; \frac{h}{h+\tau } \left( 1 - \frac{1}{(h+\tau ) \, t} (1 - e^{-(h+\tau ) t}) \right)$

For a one-sample design (such as a proportional hazards regression), the expected number of events by time t is $E(t, T_ a) = D_{h}(t, T_ a)$ , where h is the hazard rate for the group. For a two-sample design (such as a log-rank test for two survival distributions), the expected number of events by time t is

$E(t, T_ a) = \frac{R}{R+1} D_{h_ a}(t, T_ a) + \frac{1}{R+1} D_{h_ b}(t, T_ a)$

where $h_ a$ and $h_ b$ are hazard rates in groups A and B, respectively, and R is the ratio of the sample size allocation weights $w_{a}/w_{b}$ .

If the accrual rate $r_{a}$ is specified with one of the three time parameters—the accrual time, follow-up time, and total study time—then PROC SEQDESIGN derives the other two time parameters by solving the equation for the expected number of events. Similarly, if the accrual rate $r_{a}$ is not specified, but two of the three time parameters are specified, then PROC SEQDESIGN derives the accrual rate.

If the accrual rate $r_ a$ is specified without the accrual time $T_ a$ , follow-up time $T_ f$ , and total study time $T= T_ a + T_ f$ , the SEQDESIGN procedure computes the minimum accrual time $T_ a$ by solving the equation

$E(\infty , T_ a) = d_ X$

A closed-form solution is then given by

$\frac{d_{X}}{r_{a}} \; \left( \frac{R}{R+1} \; \frac{h_ a}{h_ a+\tau } \, + \, \frac{1}{R+1} \; \frac{h_ b}{h_ b+\tau } \right)^{-1}$

Similarly, the SEQDESIGN procedure derives the maximum accrual time $T_ a$ by solving the equation

$E(T_ a, T_ a) = d_ X$

The maximum accrual time $T_ a$ is then obtained by an iterative process.

With the accrual rate $r_{a}$ and the accrual time $T_{a}$ , the total sample size is

$N_{X} = r_{a} \; T_{a}$

At each stage k, the number of events is given by

$d_{k} = \frac{I_{k}}{I_{X}} \, \, d_{X}$

The corresponding time $T_ k$ can be derived from the equation for the expected number of events, $E(t)= d_{k}$ , and the resulting sample size is computed as

$N_ k= \left\{ \begin{array}{cl} r_{a} \; T_ k & \mbox{if} \, \, T_ k \leq T_{a} \\ r_{a} \; T_ a & \mbox{if} \, \, T_ k > T_{a} \end{array} \right.$

Truncated Exponential Accrual without Losses to Follow Up

For a study group with a constant hazard rate h, if the individual accrual is truncated exponential with parameter $\gamma$ over the accrual period from 0 to $T_{a}$ with distribution

$F(t)= \frac{1 - e^{-\gamma t}}{1 - e^{-\gamma T_ a}} \; \; \; \; \mbox{for} \, \, 0 \leq t \leq T_{a}$

Lachin and Foulkes (1986, p. 510) derive the expected number of events by time t (where $t > T_ a$ ) as

$D_{h}(t, N) = N \left( 1 + \frac{\gamma }{h-\gamma } \; \, \frac{e^{-ht} - e^{-h (t-T_ a)} e^{-\gamma T_ a}}{1 - e^{-\gamma T_ a}} \right)$

where N is the total sample size.

For $t \leq T_ a$ , the SEQDESIGN procedure estimates the expected number of events by time t as

$D_{h}(t, N) = N \, F(t) \left( 1 + \frac{\gamma }{h-\gamma } \; \frac{e^{-h t} - e^{-\gamma t}}{1 - e^{-\gamma t}} \right)$

For the truncated exponential accrual function with a parameter $\gamma$ over the accrual period from 0 to $T_{a}$ , you specify the scaled parameter $\gamma _0= \gamma T_ a$ in the ACCRUAL=EXP(PARM= $\gamma _0$ ) option, where $\gamma _0 \geq -10$ and $\gamma _0 \neq 0$ .

For a one-sample design (such as a proportional hazards regression), the expected number of events by time t is $E(t, N) = D_{h}(t, N)$ , where h is the hazard rate for the group. For a two-sample design (such as a log-rank test for two survival distributions), the expected number of events by time t is

$E(t, N) = \frac{R}{R+1} D_{h_{a}}(t, N) + \frac{1}{R+1} D_{h_{b}}(t, N)$

where $h_{a}$ and $h_{b}$ are hazard rates in groups A and B, respectively, and R is the ratio of the sample size allocation weights $w_{a}/w_{b}$ .

If the total sample size N is specified, then at least one of the three time parameters—the accrual time, follow-up time, and total study time—must be specified, and then PROC SEQDESIGN derives the other two time parameters by solving the equation for the expected number of events. Similarly, if the total sample size N is not specified, then at least two of the three time parameters must be specified, and PROC SEQDESIGN derives the sample size.

If the accrual sample size N is not specified, the SEQDESIGN procedure computes the minimum accrual sample size by solving the equation

$E(\infty , N) = d_ X$

That is, $N= d_ X$ .

Similarly, if the total sample size N is not specified but the accrual time $T_ a$ is specified, the SEQDESIGN procedure derives the maximum accrual sample size N by solving the equation

$E(T_ a, N) = d_ X$

At each stage k, the number of events is given by

$d_{k} = \frac{I_{k}}{I_{X}} \, \, d_{X}$

The corresponding time $T_ k$ can be derived from the equation for the expected number of events, $E(t)= d_{k}$ , and the resulting sample size is computed as

$N_ k= \left\{ \begin{array}{cl} N \; F(T_ k) & \mbox{if} \, \, T_ k \leq T_{a} \\ N & \mbox{if} \, \, T_ k > T_{a} \end{array} \right.$

Truncated Exponential Accrual with Losses to Follow Up

With the LOSS=EXP(HAZARD= $\tau$ ) option, the individual loss to follow up has an exponential loss distribution function

$H(t)= 1 - e^{-\tau t}$

where $\tau \geq 0$ is the loss hazard rate. The loss hazard rate can also be specified implicitly with the MEDTIME= $t_{\tau }$ suboption through the median loss time $t_{\tau }$ .

For a study group with a constant hazard rate h, if the individual accrual is truncated exponential with parameter $\gamma$ over the accrual period from 0 to $T_{a}$ with distribution

$F(t)= \frac{1 - e^{-\gamma t}}{1 - e^{-\gamma T_ a}} \; \; \; \; \mbox{for} \, \, 0 \leq t \leq T_{a}$

and the individual loss to follow up has an exponential loss distribution function $H(t)$ , Lachin and Foulkes (1986, p. 513) derive the expected number of events by time t (where $t > T_ a$ ) as

$D_{h}(t, N) = N \, \frac{h}{h+\tau } \, \left( 1 + \frac{\gamma }{h+\tau -\gamma } \; \, \frac{e^{-(h+\tau ) t} - e^{-(h+\tau )(t-T_ a)} e^{-\gamma T_ a}}{1 - e^{-\gamma T_ a}} \right)$

For $t \leq T_ a$ , the SEQDESIGN procedure estimates the expected number of events by time t as

$D_{h}(t, N) = N \, F(t) \, \frac{h}{h+\tau } \, \left( 1 + \frac{\gamma }{h+\tau -\gamma } \; \, \frac{e^{-(h+\tau ) t} - e^{-\gamma t}}{1 - e^{-\gamma t}} \right)$

$E(t, N) = \frac{R}{R+1} D_{h_{a}}(t, N) + \frac{1}{R+1} D_{h_{b}}(t, N)$

where $h_{a}$ and $h_{b}$ are hazard rates in groups A and B, respectively, and R is the ratio of the sample size allocation weights $w_{a}/w_{b}$ .

If the accrual sample size is not specified, the SEQDESIGN procedure computes the minimum sample size N by solving the equation

$E(\infty , N) = d_ X$

A closed-form solution is then given by

$d_ X \; \left( \frac{R}{R+1} \; \frac{h_ a}{h_ a+\tau } \, + \, \frac{1}{R+1} \; \frac{h_ b}{h_ b+\tau } \right)^{-1}$

Similarly, if the accrual sample size N is not specified but the accrual time $T_ a$ is specified, the SEQDESIGN procedure derives the maximum accrual sample size N by solving the equation

$E(T_ a, N) = d_ X$

At each stage k, the number of events is given by

$d_{k} = \frac{I_{k}}{I_{X}} \, \, d_{X}$

The corresponding time $T_ k$ can be derived from the equation for the expected number of events, $E(t)= d_{k}$ , and the resulting sample size is computed as

$N_ k= \left\{ \begin{array}{cl} N \; F(T_ k) & \mbox{if} \, \, T_ k \leq T_{a} \\ N & \mbox{if} \, \, T_ k > T_{a} \end{array} \right.$

The following three sections describe examples of test statistics with their resulting information levels, which can then be used to derive the required sample size. The maximum likelihood estimators are used for all tests except to compare two survival distributions with a log-rank test, where a score statistic is used.