The SEQDESIGN Procedure

Sample Size Computation

The SEQDESIGN procedure assumes that the data are from a multivariate normal distribution and the sequence of the standardized test statistics StartSet upper Z 1 comma upper Z 2 comma ellipsis comma upper Z Subscript upper K Baseline EndSet has the following canonical joint distribution:

  • left-parenthesis upper Z 1 comma upper Z 2 comma ellipsis comma upper Z Subscript upper K Baseline right-parenthesis is multivariate normal

  • upper Z Subscript k Baseline tilde upper N left-parenthesis theta StartRoot upper I Subscript k Baseline EndRoot comma 1 right-parenthesis

  • normal upper C normal o normal v left-parenthesis upper Z Subscript k 1 Baseline comma upper Z Subscript k 2 Baseline right-parenthesis equals StartRoot upper I Subscript k 1 Baseline slash upper I Subscript k 2 Baseline EndRoot   ,   1 less-than-or-equal-to k 1 less-than-or-equal-to k 2 less-than-or-equal-to upper K

where K is the total number of stages and upper I Subscript 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 minus beta, and variance of the response variable. Given a one-sided null hypothesis upper H 0 colon theta equals 0 with an upper alternative hypothesis upper H 1 colon theta equals theta 1, the information required for a fixed-sample test is given by

upper I 0 equals StartFraction left-parenthesis normal upper Phi Superscript negative 1 Baseline left-parenthesis 1 minus alpha right-parenthesis plus normal upper Phi Superscript negative 1 Baseline left-parenthesis 1 minus beta right-parenthesis right-parenthesis squared Over theta 1 squared EndFraction

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 Subscript a Baseline equals p Subscript b; then theta equals p Subscript a Baseline minus p Subscript b is the difference between two proportions if the proportion difference statistic is used, and theta equals normal l normal o normal g left-parenthesis StartFraction p Subscript a Baseline left-parenthesis 1 minus p Subscript b Baseline right-parenthesis Over p Subscript b Baseline left-parenthesis 1 minus p Subscript a Baseline right-parenthesis EndFraction right-parenthesis, the log odds ratio for the two proportions if the log odds ratio statistic is used.

If the maximum likelihood estimate ModifyingAbove theta With caret from the likelihood function can be derived, then the asymptotic variance for ModifyingAbove theta With caret is normal upper V normal a normal r left-parenthesis ModifyingAbove theta With caret right-parenthesis equals 1 slash upper I, where I is Fisher information for theta. The resulting statistic ModifyingAbove theta With caret corresponds to the MLE statistic scale as specified in the BOUNDARYSCALE=MLE option in the PROC SEQDESIGN statement, ModifyingAbove theta With caret StartRoot upper I EndRoot corresponds to the standardized Z scale (BOUNDARYSCALE=STDZ), and ModifyingAbove theta With caret upper 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, upper S slash StartRoot upper I EndRoot corresponds to the standardized Z scale and upper S slash upper I corresponds to the MLE statistic scale.

For a group sequential trial, the maximum information upper I Subscript upper X is derived in the SEQDESIGN procedure with the specified alpha, beta, and theta 1. With the maximum information

upper I Subscript upper X Baseline equals StartFraction 1 Over normal upper V normal a normal r left-parenthesis ModifyingAbove theta With caret right-parenthesis EndFraction

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 11 lists the options available in the SAMPLESIZE statement.

Table 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

upper N Subscript upper X Baseline equals StartFraction upper I Subscript upper X Baseline Over upper I 0 EndFraction n 0

where upper I Subscript upper X is the maximum information for the group sequential design and upper I 0 is the information for the corresponding fixed-sample design. The information ratio between upper I Subscript upper X and upper 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 Subscript upper X Baseline equals StartFraction upper I Subscript upper X Baseline Over upper I 0 EndFraction d 0

where upper I Subscript upper X is the maximum information for the group sequential design and upper I 0 is the information for the corresponding fixed-sample design. The information ratio between upper I Subscript upper X and upper 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.

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 upper T Subscript a with a constant accrual rate r Subscript 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 greater-than-or-equal-to negative 10 and gamma 0 not-equals 0. With a scaled parameter gamma 0, the power parameter for the truncated exponential with the accrual time upper T Subscript a is given by gamma equals gamma 0 slash upper T Subscript 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 Subscript upper 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 upper T Subscript a with a constant accrual rate r Subscript a, Kim and Tsiatis (1990, pp. 83–84) show that the expected number of events by time t is given by

upper D Subscript h Baseline left-parenthesis t right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column r Subscript a Baseline left-parenthesis t minus StartFraction 1 minus e Superscript minus h t Baseline Over h EndFraction right-parenthesis 2nd Column if t less-than-or-equal-to upper T Subscript a Baseline 2nd Row 1st Column r Subscript a Baseline left-parenthesis upper T Subscript a Baseline minus StartFraction e Superscript minus h t Baseline Over h EndFraction left-parenthesis e Superscript h upper T Super Subscript a Superscript Baseline minus 1 right-parenthesis right-parenthesis 2nd Column if t greater-than upper T Subscript a Baseline EndLayout

For a one-sample design (such as a proportional hazards regression), the expected number of events by time t is upper E left-parenthesis t right-parenthesis equals upper D Subscript h Baseline left-parenthesis t right-parenthesis, 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

upper E left-parenthesis t right-parenthesis equals StartFraction upper R Over upper R plus 1 EndFraction upper D Subscript h Sub Subscript a Baseline left-parenthesis t right-parenthesis plus StartFraction 1 Over upper R plus 1 EndFraction upper D Subscript h Sub Subscript b Baseline left-parenthesis t right-parenthesis

where h Subscript a and h Subscript b are hazard rates in groups A and B, respectively, and R is the ratio of the sample size allocation weights w Subscript a Baseline slash w Subscript b.

If the accrual rate r Subscript 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 Subscript 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 Subscript a is specified without the accrual time upper T Subscript a, follow-up time upper T Subscript f, and total study time upper T equals upper T Subscript a Baseline plus upper T Subscript f, the minimum and maximum accrual times can be computed from the following equation, as described in Kim and Tsiatis (1990, p. 85):

StartFraction d Subscript upper X Baseline Over r Subscript a Baseline EndFraction less-than-or-equal-to upper T Subscript a Baseline less-than-or-equal-to upper E Superscript negative 1 Baseline left-parenthesis d Subscript upper X Baseline right-parenthesis

With the accrual rate r Subscript a and the accrual time upper T Subscript a, the total sample size is

upper N Subscript upper X Baseline equals r Subscript a Baseline upper T Subscript a

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

d Subscript k Baseline equals StartFraction upper I Subscript k Baseline Over upper I Subscript upper X Baseline EndFraction d Subscript upper X

The corresponding time upper T Subscript k can be derived from the equation for the expected number of events, upper E left-parenthesis t right-parenthesis equals d Subscript k, and the resulting sample size is computed as

upper N Subscript k Baseline equals StartLayout Enlarged left-brace 1st Row 1st Column r Subscript a Baseline upper T Subscript k Baseline 2nd Column if upper T Subscript k Baseline less-than-or-equal-to upper T Subscript a Baseline 2nd Row 1st Column r Subscript a Baseline upper T Subscript a Baseline 2nd Column if upper T Subscript k Baseline greater-than upper T Subscript a Baseline EndLayout

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

upper H left-parenthesis t right-parenthesis equals 1 minus e Superscript minus tau t

where tau greater-than-or-equal-to 0 is the loss hazard rate. The loss hazard rate can also be specified implicitly with the MEDTIME=t Subscript tau suboption through the median loss time t Subscript tau.

For a study group with a constant hazard rate h, if the individual accrual is uniform in the accrual time upper T Subscript a with a constant accrual rate r Subscript a and the individual loss to follow up has an exponential loss distribution function upper H left-parenthesis t right-parenthesis, Lachin and Foulkes (1986, p. 511) derive the expected number of events by time t (where t greater-than upper T Subscript a) as

upper D Subscript h Baseline left-parenthesis t comma upper T Subscript a Baseline right-parenthesis equals r Subscript a Baseline upper T Subscript a Baseline StartFraction h Over h plus tau EndFraction left-parenthesis 1 minus StartFraction 1 Over left-parenthesis h plus tau right-parenthesis upper T Subscript a Baseline EndFraction e Superscript minus left-parenthesis h plus tau right-parenthesis t Baseline left-parenthesis e Superscript h plus tau right-parenthesis upper T Super Subscript a Superscript Baseline minus 1 right-parenthesis right-parenthesis

For t less-than-or-equal-to upper T Subscript a, the SEQDESIGN procedure estimates the expected number of events by time t as

upper D Subscript h Baseline left-parenthesis t comma upper T Subscript a Baseline right-parenthesis equals r Subscript a Baseline t StartFraction h Over h plus tau EndFraction left-parenthesis 1 minus StartFraction 1 Over left-parenthesis h plus tau right-parenthesis t EndFraction left-parenthesis 1 minus e Superscript minus left-parenthesis h plus tau right-parenthesis t Baseline right-parenthesis right-parenthesis

For a one-sample design (such as a proportional hazards regression), the expected number of events by time t is upper E left-parenthesis t comma upper T Subscript a Baseline right-parenthesis equals upper D Subscript h Baseline left-parenthesis t comma upper T Subscript a Baseline right-parenthesis, 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

upper E left-parenthesis t comma upper T Subscript a Baseline right-parenthesis equals StartFraction upper R Over upper R plus 1 EndFraction upper D Subscript h Sub Subscript a Baseline left-parenthesis t comma upper T Subscript a Baseline right-parenthesis plus StartFraction 1 Over upper R plus 1 EndFraction upper D Subscript h Sub Subscript b Baseline left-parenthesis t comma upper T Subscript a Baseline right-parenthesis

where h Subscript a and h Subscript b are hazard rates in groups A and B, respectively, and R is the ratio of the sample size allocation weights w Subscript a Baseline slash w Subscript b.

If the accrual rate r Subscript 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 Subscript 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 Subscript a is specified without the accrual time upper T Subscript a, follow-up time upper T Subscript f, and total study time upper T equals upper T Subscript a Baseline plus upper T Subscript f, the SEQDESIGN procedure computes the minimum accrual time upper T Subscript a by solving the equation

upper E left-parenthesis normal infinity comma upper T Subscript a Baseline right-parenthesis equals d Subscript upper X

A closed-form solution is then given by

StartFraction d Subscript upper X Baseline Over r Subscript a Baseline EndFraction left-parenthesis StartFraction upper R Over upper R plus 1 EndFraction StartFraction h Subscript a Baseline Over h Subscript a Baseline plus tau EndFraction plus StartFraction 1 Over upper R plus 1 EndFraction StartFraction h Subscript b Baseline Over h Subscript b Baseline plus tau EndFraction right-parenthesis Superscript negative 1

Similarly, the SEQDESIGN procedure derives the maximum accrual time upper T Subscript a by solving the equation

upper E left-parenthesis upper T Subscript a Baseline comma upper T Subscript a Baseline right-parenthesis equals d Subscript upper X

The maximum accrual time upper T Subscript a is then obtained by an iterative process.

With the accrual rate r Subscript a and the accrual time upper T Subscript a, the total sample size is

upper N Subscript upper X Baseline equals r Subscript a Baseline upper T Subscript a

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

d Subscript k Baseline equals StartFraction upper I Subscript k Baseline Over upper I Subscript upper X Baseline EndFraction d Subscript upper X

The corresponding time upper T Subscript k can be derived from the equation for the expected number of events, upper E left-parenthesis t right-parenthesis equals d Subscript k, and the resulting sample size is computed as

upper N Subscript k Baseline equals StartLayout Enlarged left-brace 1st Row 1st Column r Subscript a Baseline upper T Subscript k Baseline 2nd Column if upper T Subscript k Baseline less-than-or-equal-to upper T Subscript a Baseline 2nd Row 1st Column r Subscript a Baseline upper T Subscript a Baseline 2nd Column if upper T Subscript k Baseline greater-than upper T Subscript a Baseline EndLayout

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 upper T Subscript a with distribution

upper F left-parenthesis t right-parenthesis equals StartFraction 1 minus e Superscript minus gamma t Baseline Over 1 minus e Superscript minus gamma upper T Super Subscript a Superscript Baseline EndFraction for 0 less-than-or-equal-to t less-than-or-equal-to upper T Subscript a Baseline

Lachin and Foulkes (1986, p. 510) derive the expected number of events by time t (where t greater-than upper T Subscript a) as

upper D Subscript h Baseline left-parenthesis t comma upper N right-parenthesis equals upper N left-parenthesis 1 plus StartFraction gamma Over h minus gamma EndFraction StartFraction e Superscript minus h t Baseline minus e Superscript minus h left-parenthesis t minus upper T Super Subscript a Superscript right-parenthesis Baseline e Superscript minus gamma upper T Super Subscript a Superscript Baseline Over 1 minus e Superscript minus gamma upper T Super Subscript a Superscript Baseline EndFraction right-parenthesis

where N is the total sample size.

For t less-than-or-equal-to upper T Subscript a, the SEQDESIGN procedure estimates the expected number of events by time t as

upper D Subscript h Baseline left-parenthesis t comma upper N right-parenthesis equals upper N upper F left-parenthesis t right-parenthesis left-parenthesis 1 plus StartFraction gamma Over h minus gamma EndFraction StartFraction e Superscript minus h t Baseline minus e Superscript minus gamma t Baseline Over 1 minus e Superscript minus gamma t Baseline EndFraction right-parenthesis

For the truncated exponential accrual function with a parameter gamma over the accrual period from 0 to upper T Subscript a, you specify the scaled parameter gamma 0 equals gamma upper T Subscript a in the ACCRUAL=EXP(PARM=gamma 0) option, where gamma 0 greater-than-or-equal-to negative 10 and gamma 0 not-equals 0.

For a one-sample design (such as a proportional hazards regression), the expected number of events by time t is upper E left-parenthesis t comma upper N right-parenthesis equals upper D Subscript h Baseline left-parenthesis t comma upper N right-parenthesis, 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

upper E left-parenthesis t comma upper N right-parenthesis equals StartFraction upper R Over upper R plus 1 EndFraction upper D Subscript h Sub Subscript a Baseline left-parenthesis t comma upper N right-parenthesis plus StartFraction 1 Over upper R plus 1 EndFraction upper D Subscript h Sub Subscript b Baseline left-parenthesis t comma upper N right-parenthesis

where h Subscript a and h Subscript b are hazard rates in groups A and B, respectively, and R is the ratio of the sample size allocation weights w Subscript a Baseline slash w Subscript 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

upper E left-parenthesis normal infinity comma upper N right-parenthesis equals d Subscript upper X

That is, upper N equals d Subscript upper X.

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

upper E left-parenthesis upper T Subscript a Baseline comma upper N right-parenthesis equals d Subscript upper X

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

d Subscript k Baseline equals StartFraction upper I Subscript k Baseline Over upper I Subscript upper X Baseline EndFraction d Subscript upper X

The corresponding time upper T Subscript k can be derived from the equation for the expected number of events, upper E left-parenthesis t right-parenthesis equals d Subscript k, and the resulting sample size is computed as

upper N Subscript k Baseline equals StartLayout Enlarged left-brace 1st Row 1st Column upper N upper F left-parenthesis upper T Subscript k Baseline right-parenthesis 2nd Column if upper T Subscript k Baseline less-than-or-equal-to upper T Subscript a Baseline 2nd Row 1st Column upper N 2nd Column if upper T Subscript k Baseline greater-than upper T Subscript a Baseline EndLayout

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

upper H left-parenthesis t right-parenthesis equals 1 minus e Superscript minus tau t

where tau greater-than-or-equal-to 0 is the loss hazard rate. The loss hazard rate can also be specified implicitly with the MEDTIME=t Subscript tau suboption through the median loss time t Subscript 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 upper T Subscript a with distribution

upper F left-parenthesis t right-parenthesis equals StartFraction 1 minus e Superscript minus gamma t Baseline Over 1 minus e Superscript minus gamma upper T Super Subscript a Superscript Baseline EndFraction for 0 less-than-or-equal-to t less-than-or-equal-to upper T Subscript a Baseline

and the individual loss to follow up has an exponential loss distribution function upper H left-parenthesis t right-parenthesis, Lachin and Foulkes (1986, p. 513) derive the expected number of events by time t (where t greater-than upper T Subscript a) as

upper D Subscript h Baseline left-parenthesis t comma upper N right-parenthesis equals upper N StartFraction h Over h plus tau EndFraction left-parenthesis 1 plus StartFraction gamma Over h plus tau minus gamma EndFraction StartFraction e Superscript minus left-parenthesis h plus tau right-parenthesis t Baseline minus e Superscript minus left-parenthesis h plus tau right-parenthesis left-parenthesis t minus upper T Super Subscript a Superscript right-parenthesis Baseline e Superscript minus gamma upper T Super Subscript a Superscript Baseline Over 1 minus e Superscript minus gamma upper T Super Subscript a Superscript Baseline EndFraction right-parenthesis

For t less-than-or-equal-to upper T Subscript a, the SEQDESIGN procedure estimates the expected number of events by time t as

upper D Subscript h Baseline left-parenthesis t comma upper N right-parenthesis equals upper N upper F left-parenthesis t right-parenthesis StartFraction h Over h plus tau EndFraction left-parenthesis 1 plus StartFraction gamma Over h plus tau minus gamma EndFraction StartFraction e Superscript minus left-parenthesis h plus tau right-parenthesis t Baseline minus e Superscript minus gamma t Baseline Over 1 minus e Superscript minus gamma t Baseline EndFraction right-parenthesis

For the truncated exponential accrual function with a parameter gamma over the accrual period from 0 to upper T Subscript a, you specify the scaled parameter gamma 0 equals gamma upper T Subscript a in the ACCRUAL=EXP(PARM=gamma 0) option, where gamma 0 greater-than-or-equal-to negative 10 and gamma 0 not-equals 0.

For a one-sample design (such as a proportional hazards regression), the expected number of events by time t is upper E left-parenthesis t comma upper N right-parenthesis equals upper D Subscript h Baseline left-parenthesis t comma upper N right-parenthesis, 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

upper E left-parenthesis t comma upper N right-parenthesis equals StartFraction upper R Over upper R plus 1 EndFraction upper D Subscript h Sub Subscript a Baseline left-parenthesis t comma upper N right-parenthesis plus StartFraction 1 Over upper R plus 1 EndFraction upper D Subscript h Sub Subscript b Baseline left-parenthesis t comma upper N right-parenthesis

where h Subscript a and h Subscript b are hazard rates in groups A and B, respectively, and R is the ratio of the sample size allocation weights w Subscript a Baseline slash w Subscript 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 is not specified, the SEQDESIGN procedure computes the minimum sample size N by solving the equation

upper E left-parenthesis normal infinity comma upper N right-parenthesis equals d Subscript upper X

A closed-form solution is then given by

d Subscript upper X Baseline left-parenthesis StartFraction upper R Over upper R plus 1 EndFraction StartFraction h Subscript a Baseline Over h Subscript a Baseline plus tau EndFraction plus StartFraction 1 Over upper R plus 1 EndFraction StartFraction h Subscript b Baseline Over h Subscript b Baseline plus tau EndFraction right-parenthesis Superscript negative 1

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

upper E left-parenthesis upper T Subscript a Baseline comma upper N right-parenthesis equals d Subscript upper X

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

d Subscript k Baseline equals StartFraction upper I Subscript k Baseline Over upper I Subscript upper X Baseline EndFraction d Subscript upper X

The corresponding time upper T Subscript k can be derived from the equation for the expected number of events, upper E left-parenthesis t right-parenthesis equals d Subscript k, and the resulting sample size is computed as

upper N Subscript k Baseline equals StartLayout Enlarged left-brace 1st Row 1st Column upper N upper F left-parenthesis upper T Subscript k Baseline right-parenthesis 2nd Column if upper T Subscript k Baseline less-than-or-equal-to upper T Subscript a Baseline 2nd Row 1st Column upper N 2nd Column if upper T Subscript k Baseline greater-than upper T Subscript a Baseline EndLayout

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.

Last updated: December 09, 2022