table 5.3b operating characteristics of
6.5. appendix e. derivation of

Let f θ )(., be the density function for random variable X. According to Neyman and Pearson, the most powerful test of H o : =θθ 0 versus 1 1 H : = θθ decides in favor of H1 or H0 if α cLn > or α cLn < , respectively, where = ∏
n i n i i *xfxfL* );(/);( 1 θθ 0 is the likelihood ratio, and α c is determined to have the size α . When the sample size is not fixed in advance, further improvement is possible by using Wald's Sequential Probability Ratio Test (SPRT). The SPRT continues to sample as long as n << ALB for some constant 1 << AB , stops sampling and decides in favor of H1 as soon as n > AL , and stops sampling and decides in favor of H0 as soon as n < BL .  
The usual measures of performance of such a procedure are the error probabilities αand β of rejecting H0 when = θθ 0 , and of accepting H0 when =θθ 1 , respectively, and the expected sample size *NENE* )()|( ≡ jj θ . Wald and Wolfowitz showed that among all tests, sequential or not, for which Pr0 (reject H0 ) ≤α and Pr1 (reject H0 ) ≤ β , and for which NE )(j are finite, j=0,1, the SPRT with error probabilities α and β minimizes )( 0 NE and )( 1 NE . If, in addition, the xx ,...2,1 are independent and identically distributed (i.i.d.) with density function xf θ ),( , with monotone likelihood ratio inτ x)( , then any SPRT for testing 0 θ against )( 01 > θθ has nondecreasing power function.  
For the SPRT with error probabilities αand β , the SPRT boundaries are given approximately by A −= /)1( αβ and B /( −= αβ )1 . The operating characteristics of the SPRT are given by
)1(),,,,( /( ) )( )()(
10θ θ θ θθβαθ h h h O A −= − BA where h θ )( is the non-trivial solution to the equation
∫ =1);()),(/);(( )(
1 2 *dxxfxfxf* h θθ θ θ .  
The formula NE θ −= θ + θ *zEBOAO* θ );(/]log)(log)](1[[();( provides the average sample number for an arbitrary θ . The sample size distribution is very highly skewed, 2 ≈ *NENVar* )]([)( . Thus we will consider a truncated test with maximum sample size of N0 and simulate to obtain the operating characteristics of the test.
table 5.3b operating characteristics of
derivation of the sprt for

Suppose that we wish to construct a sequential test for the composite null hypothesis that the rate of overall mortality at an early time point t is less than or equal to p0 versus the alternative hypothesis that it is greater than or equal to p0. Let us assume that the survival times, TTT n ,...,, 21 ,
are i.i.d. with exponential density function T eTf θ θθ − ),( = . Although an exponential model may not fit well for overall mortality, it usually provides a reasonable model over a short time frame for modeling toxicity, so in all discussion below we assume that exponential survival times are censored at time point t. In the exponential parameterization, a t-day survival rate of p0 translates into a mean survival of µ0=-t/ln(1-p0) (rate parameter θ 0 = -ln(1-p0)/t). .  
The SPRT is derived with reference to a simple null and alternative hypothesis for the rate parameter, in this case, 0 : H o θ θ = versus 1 1 H : = θθ . The log-likelihood ratio for the exponential  
in the presence of censoring is  1 0 1 0 10 log ( ; ) log ( , ) (log( ) log( )) ( ) n n n i i i i i i ∏ ∏ fx fx d θ − θ θ θ θθ = − −− ∑T , where d is the number of
events. The SPRT can be represented graphically when plotting the number of deaths (d) on the y axis against the total time on study ∑
n i Ti on the x axis. The continuation region in terms of d is  
bounded by two parallel lines given by  $$\left[\frac{\log(B)}{(\log\theta_{1}-\log\theta_{0})}\right]+\left[\frac{(\theta_{1}-\theta_{0})}{(\log\theta_{1}-\log\theta_{0})}\right]\sum_{i}^{\wedge}T_{i}<d<\left[\frac{\log(A)}{(\log\theta_{1}-\log\theta_{0})}\right]+\left[\frac{(\theta_{1}-\theta_{0})}{(\log\theta_{1}-\log\theta_{0})}\right]\sum_{i}^{\wedge}T_{i}$$  with common slope $(\theta_{1}-\theta_{0})/(\log\theta_{1}-\log\theta_{0})$, and intercepts $\log A/(\ln\theta_{1}-\ln\theta_{0})$ and $\log\theta_{1}-\log\theta_{0}$.  
1 0 log / (ln ln ) B θ θ − , for the upper and lower bounds, respectively. For monitoring purposes, at an interim analysis calendar time point s, suppose that d(s) events have occurred and that the total time on study is ( )
n i i
∑T s .The cumulative number of events d(s) is plotted on the y axis against the  
total time on study, ( )
$\text{by,}\sum T_i(s)\,.$ Wl
i
i
∑T s . When this graph crosses the upper boundary, the null hypothesis is rejected. In practice, monitoring will be scheduled monthly after the start of enrollment to the study. A truncated version of the SPRT can be obtained by specifying a maximum sample size. We truncate the SPRT by declaring that if the test has failed to terminate after the maximum sample size, that the null hypothesis will be accepted. Since the probability that the untruncated SPRT would reject the null at the maximum sample size is negligible, it makes little difference how the final boundary value is selected, and this rule is chosen for simplicity. The operating characteristics of this proposed truncated SPRT for censored exponential data can be estimated by simulation.
table 5.3b operating characteristics of
6.6. appendix f. references

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