In the survival studies of chronic diseases, the most frequently used methods ofdata analysis are the nonparametric estimations and comparisons of survival rates atdifferent times but there have been no matched methods for determining the required samplesize so far. Instead have been often used the parametric methods based on the assumptionof exponential distributions^[1～7] such that the relevant sample sizes maynot be satisfied with the prescribed power.
This paper reports a nonparametric procedure of the sample size determination forsurvival rate test.

Let X_i (i=1,…,n) be independent and identically distributed (i.i.d.)failure times of individuals with continuous distribution function F_x (t)=Pr(X_i ≤t).Its complement is survival function S_x (t)=1-F_x (t). Let C_i be i.i.d. censoring times with distribution function F_c (t)=Pr(C_i (t).The corresponding survival function is S_c (t)=1-F_c (t). Suppose C isindependent of X so that F_c is functionally independent of F_x .Suppose we observe a sequence of paires (T_i ,δ_i )(i=1,…,n) suchthat T_i =min(X_i ,C_i ) Where δ_i =1 if X_i ≤C_i and δ_i =0 if X_i >C_i . Permutation gives order statisticsT₍₁₎ <…(n) and the corresponding indicator δ₍₁₎ ,…,δ_(n) .
The Kaplan-Meier estimate^[8] for S_x (t) is

(1)

Its homogenetic effective sample size is given by

(2)

It declines with the censoring vector (1-δ_(i) )^[9] :

(3)

and reduces to the sample size when there is no censoring:
Let π be the expectation of at some interested time t₀ with π=π₀ under a hypothesis H₀ and π=π₁ under an alternativehypothesis H₁ , where π₁ ≠π₀ (two sided test) or π₁ >π₀ or π₁ <π₀ (one sided). By the asymptotic normality ofKaplan-Meier estimate^[10] , the statistic for festing H₀ is

(4)

where . Thus, we have Z～N(0,1) if H₀ is true. Bycontrast with its classical counterpart^[11] , the only change is that thesample size n is replaced by the observed homogenetic effective sample size in (4).

The determination of required homogenetic effective sample size can inherit theclassical asymptotic normal procedure^[12]

(5)

where Z_α and Z_β are the standard normal deviatesat levels α and β the essential design parameters respectively and D₀ =π₁ -π₀ is the selected maximum error, ie. the principle design parameter.
As for the required sample size, it can be realized by the inverse operation fromm(t₀ ) to n. Then it is useful to estimate the survival function of censoringtimes

(6)

It follows that ， which is the complement of the empirical distributionfunction. Writing it as and substituting it into (2) gives

Taking π₁ =S_x (t₀ ),Ψ=S_c (t₀ ),and b=1-δ_(i) , we obtain the expected homogenetic effective sample size

m(t₀ )=n(t₀ )Ψ+b/π₁ (7)

It still possesses the reducibility as Ψ=1,b=0, and m(t₀ )=n(t₀ )when there is no censoring. The inverse operation gives the required sample size

n(t₀ )=m(t₀ )/Ψ-b/(π₁ Ψ)

It contains the two parameters b and Ψ and both of them correspond to thecensoring rate. It is not possible to establish the relationship between b and Ψ bynonparametric approach. Thus we have to omit the term -b/(π₁ Ψ) and obtain ashorter equation

n(t₀ )=m(t₀ )/Ψ (8)

Fortunately, the term -b/(π₁ Ψ) has a very small value underusual clinical conditions and its influence upon the sample sizes determined is negligible(see the next section). thus we need a supplimentary design parameter the prescribed valueof the survival function of censoring times Ψ. Design parameters are usually specified inthe light of pilot or previous similar trials. Writing , we can find the prescribed value ofΨ from π₁ and the complement of empirical distribution function, where thelatter is especially easy to calculate.

We specify the values of design parameters on the basis of a previous clinicaltrial^[13] for planning a new clinical trial. In that trial, two dosageregimens of ⁶⁰ Co radiotherapy were compared between two groups of patientswith esophageal carcinoma, 34 each. The trial did not yield a significant difference ofefficacy betwenn two groups so that the pooled-sample one-year survival rate is calculatedbased on (1) as 0.485. Among the 68 patients, three were lost fo follow-up during 3-yearfollow-up period and 11 alive at the end of study. The survival function of censoringtimes is estimated by (6) as 0.915.
And now, consider a new modality of radiotherapy. It will be clinically acceptableas long as it can yield an increment of 0.2 of one-year survival rate. The question is howmany patients are needed for detecting the increment.
As a usual practice, we take α=0.05 (one-sided test) and β=0.1 and then Z_α =1.6448and Z_β =1.2816. The other design parameters are prespecified based on the data.We take π₀ =0.5 and this plus the increment D₀ =0.2 yields π₁ =0.7.The survival function of censoring times is taken as Ψ=0.9.
Using (5) and (8) yields the required homogenetic effective sample size m(t₀ )=49.682and the required sample size n(t₀ )=55.202 respectively. The trial needs 55patients.

By contrast with the existing methods of this kind, this procedure has suchcharacteristic as that it is thoroughly nonparametric. The term -b/(π₁ Ψ) isemitted at the second step of the procedure. It is shown in the simulations that thevariations in design results due to the omission are not greater than one sample unit.When the censoring level is nearer to 0.5 and π₁ is nearer to life tabletails, however, the omission leads to slightly conservative results. According to Simonand Wittes^[14] , not more than 15% of eligible patients should be consideredinevaluable for response due to various causes so that high censoring level is notpreferred. Life table tails are often the regions of greatest medical interest and thesource of most mistakes^[15] . It was shown by experience that thestatistical power is easily lost in the regions. A relatively conservative designprocedure in the regions is fortunately helpful for keeping power. The survival functionof censoring times is added as a supplimentary design parameter for this purpose.
The nonparametric procedure is matched with the rate test for censured data. It doesnot involve survival distributions. It reduces to its classical counterpart when there isno censoring. The observed power of the test coincides with the prescribed power in commonclinical situations. Thus it can be applied to planning survival studies of chronicdiseases.

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