SIZE DETERMINATION FOR SURVIVAL RATE TEST

ZHAO Guolong

(Henan Medical University Zhengzhou 450052)

Heidi Holmes Morgan

(University of Kansas Medical Center Kansas 66160)

AbstractObjective This paper proposes a nonparametric procedure of the sample size determination for survival rate test. Methods Using the classical asymptotic normal procedure yields the required homogenetic effective sample size and using the inverse operation with the prespecified value of the survival function of censoring times yields the required sample size. Results It is matched with the rate test for censored data, does not involve survival distributions, and reduces to its classical counterpart when there is no censoring. The observed power of the test coincides with the prescribed power under usual clinical conditions. Conclusion It can be used for planning survival studies of chronic diseases.

Key worksCensorshipEffective sample sizeRate testSample size

中图分类号：O 213.9文献标识码：A

文章编号：1004-4337(2000)03-0273-05

In the survival studies of chronic diseases, the most frequently used methods of data analysis are the nonparametric estimations and comparisons of survival rates at different times but there have been no matched methods for determining the required sample size so far. Instead have been often used the parametric methods based on the assumption of exponential distributions^［1～7］ such that the relevant sample sizes may not be satisfied with the prescribed power.

This paper reports a nonparametric procedure of the sample size determination for survival rate test.

1Survival 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 is independent 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) such that 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 statistics T₍₁₎＜…＜T_(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 null hypothesis H₀ and π=π₁ under an alternative hypothesis H₁, where π₁≠π₀ (two sided test) or π₁＞π₀ or π₁＜π₀(one sided). By the asymptotic normality of Kaplan-Meier estimate^［10］, the statistic for festing H₀ is

(4)

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

2Determination of Required Sample Size

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

(5)

where Z_α and Z_β are the standard normal deviates at 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 from m(t₀) to n. Then it is useful to estimate the survival function of censoring times

(6)

It follows that ， which is the complement of the empirical distribution function. 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/(π₁Ψ)

It contains the two parameters b and Ψ and both of them correspond to the censoring rate. It is not possible to establish the relationship between b and Ψ by nonparametric approach. Thus we have to omit the term -b/(π₁Ψ) and obtain a shorter equation

n(t₀)=m(t₀)/Ψ(8)

Fortunately, the term -b/(π₁Ψ) has a very small value under usual 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 value of the survival function of censoring times Ψ. Design parameters are usually specified in the light of pilot or previous similar trials. Writing , we can find the prescribed value<