样本量估计及其在nQuery和SAS软件上的实现*——均数比较(十)

南方医科大学生物统计学系(510515)

高培纯 徐笑寒 徐 莹 陈平雁△

1.2.2.8 基于均差高阶交叉设计的等效性检验

Schuirmann’s(1987)[1]，Phillip(1990)[2]和Chen(1997)[3]给出的高阶交叉设计的样本量估计方法是建立在近似服从自由度为vk的t分布上，其检验效能的计算公式为：

(1-55)

在计算样本量时，一般先设定样本量初始值，然后迭代样本量直到所得的检验效能满足条件为止。此时的样本量，即研究所需的样本量。

①4×2设计(Balaam's design)，v1=4n-3，b1=2

②2×3设计(the two-sequence dual design)，v2=4n-4，b2=3/4

④4×4设计(the four-period with four sequences)，v4=12n-5，b4=1/4

【例1-29】欲比较两种降压药物对舒张压的降压效果。根据以往研究，患者使用对照药后平均舒张压为96mmHg，研究者认为使用新药后患者平均舒张压能降到92mmHg。已知均方误为324。新药均值与对照药均值差值置信区间在对照药均值±20%之内可认为两种药物等效，本研究等效性界值设定为-19.2和19.2。试验设计采用2×3设计，检验水准为0.05，欲估计检验效能为90%的样本量。

nQuery Advanced 8.2 实现：设定检验水准为α=0.05，检验效能取90%。由题意知，μ2-μ1=-4，LEL=-19.2，UEL=19.2，Sw=18。在nQuery Advanced 8.2主菜单选择：

Goal：Make conclusion Using：⊙Means

Number of Groups：⊙Two

Analysis Method：⊙Equivalence

方法框中选择：Higher-order Cross-Over Design for Two Means- Equivalence-using Differences

在弹出的样本量计算窗口将各参数键入，如图1-69所示，结果为N=20。

图1-69 nQuery Advanced 8.2 关于例1-29样本量估计的参数设置与计算结果

SAS 9.4软件实现：

proc IML;

start MTE51(Designtype,a,Delta,UEL,LEL,Sw,power);

error=0;

if(Designtype=1|Designtype=2|Designtype=3|Designtype=4)then do;

error=0;end;

else do;error=1;print"error" "Designtype must be 1 or 2 or 3 or 4";end;

if( a>0.2 | a<0) then do;

error=1;print"error" "Test significance level must be in 0-0.2";end;

if( Delta>UEL | Delta

error=1;print"error" "True difference in means set between upper and lower limits";end;

if(UEL<0 ) then do;

error=1;print"error" "Upper equivalence limit difference must be>0 ";end;

if(LEL>0 ) then do;

error=1;print"error" "Lower equivalence limit difference must be<0 ";end;

if(Sw<0) then do;

error=1;print"error" "Within standard error must be>=0"; end;

if(power>100 | power<1) then do;

error=1;print"error" "Power(%) must be in 1-100";end;

if(error=1) then stop;

if(error=0) then do;

if(Designtype=1) then do;n=1;b=2;end;

if(Designtype=2) then do;n=2;b=3/4;end;

if(Designtype=3) then do;n=1;b=11/20;end;

if(Designtype=4) then do;n=1;b=1/4;end;

do until(pw>=power/100);

if(Designtype=1) then do;df=4*n-3;end;

if(Designtype=2) then do;df=4*n-4;end;

if(Designtype=3) then do;df=6*n-5;end;

if(Designtype=4) then do;df=12*n-5;end;

tU=(-Delta+UEL)/(Sw*sqrt(b/n))-tinv(1-a,df);

tL=tinv(1-a,df)-(Delta-LEL)/(Sw*sqrt(b/n));

pw1=1-probt(tU,df);

pw2=1-probt(tL,df);

pw=pw2-pw1;n=n+0.01;

end;

if(Designtype=1) then do;N=ceil((n-0.01)*4);end;

if(Designtype=2) then do;N=ceil((n-0.01)*2);end;

if(Designtype=3) then do;N=ceil((n-0.01)*2);end;

if(Designtype=4) then do;N=ceil((n-0.01)*4);end;

if(Designtype=1) then do;Design_type="1. 4*2";end;

if(Designtype=2) then do;Design_type="2. 2*3";end;

if(Designtype=3) then do;Design_type="3. 2*4";end;

if(Designtype=4) then do;Design_type="4. 4*4";end;

print

Design_type

a[label="Test Significance Level"]

Delta[label="True Difference in Means.u2-u1"]

LEL[label="Lower Equivalence Limit"]

UEL[label="Upper Equivalence Limit"]

Sw[label="Within Standard Error(Sw)"]

power[label="Power(%)"]

N[label="N"];end;

finish MTE51;

run MTE51(2,0.05,-4,19.2,-19.2,18,90);

quit;

SAS 9.4运行结果：

图1-70 SAS 9.4关于例1-29样本量估计的参数设置与计算结果

1.2.2.9 基于比值高阶交叉设计的等效性检验

Schuirmann’s (1987)[1]，Phillip(1990)[2]和Chen(1997)[3]给出的高阶交叉设计的样本量估计方法是建立在近似服从自由度为vk的t分布上，其检验效能的计算公式为：

(1-56)

在计算样本量时，一般先设定样本量初始值，然后迭代样本量直到所得的检验效能满足条件为止。此时的样本量，即研究所需的样本量。

【例1-30】某公司欲验证新开工厂生产的药物与旧工厂生产的药物是否等效。根据以往研究数据，旧工厂生产药物CV为0.4。假定新旧工厂生产药物均值比值为0.96。根据指导原则，新工厂生产的药物与旧工厂生产的药物的均值比不低于0.8且不高于1.25可认为等效。若采用2×3设计，试估计检验效能为90%的样本量。

nQuery Advanced 8.2实现：设定检验水准为α=0.05，检验效能取90%。由题意知，μ2/μ1=0.96，LER=0.8，UER=1.25，CV=0.4。在nQuery Advanced 8.2主菜单选择：

方法框中选择：Higher-order Cross-Over Design for Two Means- Equivalence-using Ratio

在弹出的样本量计算窗口将各参数键入，如图1-71所示，结果为N=60。

图1-71 nQuery Advanced 8.2 关于例1-30样本量估计的参数设置与计算结果

SAS9.4软件实现：

proc IML;

start MTE81(Designtype,a,Ratio,UER,LER,CV,power);

error=0;

if(Designtype=1|Designtype=2|Designtype=3|Designtype=4)then do;

error=0;end;

else do;

error=1;print"error" "Designtype must be 1 or 2 or 3 or 4";end;

if( a>0.2 | a<0) then do;

error=1;print"error" "Test significance level must be in 0-0.2";end;

if( Ratio>UER | Ratio

error=1;print"error" "True ratio of means set between upper and lower equivalence ratio limits";end;

if( UER<1 ) then do;

error=1;print"error" "Upper equivalence limit ratio must be>= 1";end;

if( LER>= 1 | LER<= 0) then do;

error=1;print"error" "Lower equivalence limit ratio must be in 0-1";end;

if( Ratio>UER | Ratio

error=1;print"error" "Ture ratio must be set between Upper and Lower equivalence ratio limit";end;

if( CV<0) then do;

error=1;print"error" "Coefficient of variance must be>= 0";end;

if(power>100 | power<1) then do;

error=1;print"error" "Power(%) must be in 1 - 100";end;

if(error=1) then stop;

if(error=0) then do;

if(Designtype=1) then do;n=1;b=2;end;

if(Designtype=2) then do;n=2;b=3/4;end;

if(Designtype=3) then do;n=1;b=11/20;end;

if(Designtype=4) then do;n=1;b=1/4;end;

CVm=sqrt(log(CV**2+1));

do until(pw>=power/100);

if(Designtype=1) then do;df=4*n-3;end;

if(Designtype=2) then do;df=4*n-4;end;

if(Designtype=3) then do;df=6*n-5;end;

if(Designtype=4) then do;df=12*n-5;end;

tU=(-abs(log(Ratio))+log(UER))/(CVm*sqrt(b/n))-tinv(1-a,df);

tL=tinv(1-a,df)-(abs(log(Ratio))-log(LER))/(CVm*sqrt(b/n));

pw1=1-probt(tU,df);

pw2=1-probt(tL,df);

pw=pw2-pw1;n=n+0.01;

end;

if(Designtype=1) then do;N=ceil((n-0.01)*4);end;

if(Designtype=2) then do;N=ceil((n-0.01)*2);end;

if(Designtype=3) then do;N=ceil((n-0.01)*2);end;

if(Designtype=4) then do;N=ceil((n-0.01)*4);end;

if(Designtype=1) then do;Design_Type="1.4*2";end;

if(Designtype=2) then do;Design_Type="2.2*3";end;

if(Designtype=3) then do;Design_Type="3.2*4";end;

if(Designtype=4) then do;Design_Type="4.4*4";end;

print

Design_Type

a[label="Test Significance Level"]

Ratio[label="True Ratio of Means,u2/u1"]

LER[label="Lower Equivalence Ratio"]

UER[label="Upper Equivalence Ratio"]

CV[label="Coefficient of Variance(non-logarithmic),CV"]

power[label="Power(%)"]

N[label="N"];end;

finish MTE81;

run MTE81(2,0.05,0.96,1.25,0.8,0.4,90);

quit;

SAS运行结果；

图1-72 SAS 9.4 关于例1-30样本量估计的参数设置与计算结果

1.2.2.10 基于均差高阶交叉设计的非劣效性检验

Chen(1997)[3]和Chow and Liu (2009)[4]给出的高阶交叉设计的样本量估计方法是建立在近似服从自由度为vk的t分布上，当指标为高优指标时，其检验效能计算公式为：

(1-57)

当指标为低优指标时，对应的检验效能计算公式为：

(1-58)

在计算样本量时，一般先设定样本量初始值，然后迭代样本量直到所得的检验效能满足条件为止。此时的样本量，即研究所需的样本量。

【例1-31】某公司欲验证一种治疗风湿病的仿制药非劣于标准药，拟采用2×3设计。研究者决定将非劣界值设置为-5。根据以往类似研究，已知均方误为100。假设仿制药与标准药真实差值为0。检验水准设置为0.05，试估计检验效能为90%所需的样本量。

nQuery Advanced 8.2 实现：设定检验水准为α=0.05，检验效能取90%。由题意知，μ2-μ1=0，NIM=-5，Sw=10。在nQuery Advanced 8.2主菜单选择：

方法框中选择：Higher-order Cross-Over Design for Two Means- Non-Inferiority-using Differences

在弹出的样本量计算窗口将各参数键入，如图1-73所示，结果为N=53。

图1-73 nQuery Advanced 8.2 关于例1-31样本量估计的参数设置与计算结果

SAS 9.4软件实现：

proc IML;

start MTE61(Designtype,Better,a,Delta,NIM,Sw,power);

error=0;

if(Designtype=1|Designtype=2|Designtype=3|Designtype=4)then do;

error=0;end;

else do;

error=1;print"error" "Designtype must be 1 or 2 or 3 or 4";end;

if( a>0.2 | a<0) then do;

error=1;print"error" "Test significance level must be in 0-0.2";end;

if( Better=1 | Better=0)then do;

error=0;end;

else do;

error=1;print"error" "Higher means better(1)/worse(0)";end;

if( Better=1 ) then do;

if( Delta<= -abs(NIM))then do;

error=1;print"error" "True difference in means must>Non-Inferiority margin if higher means better";end;

end;

if( Better=0 ) then do;

if( Delta>= abs(NIM) )then do;

error=1;print"error" "True difference in means must

end;

if(Sw<0) then do;

error=1;print"error" "Within standard error must be>=0";end;

if(power>100 | power<1) then do;

error=1;print"error" "Power(%) must be in 1-100";end;

if(error=1) then stop;

if(error=0) then do;

if(Designtype=1) then do;n=1;b=2;end;

if(Designtype=2) then do;n=2;b=3/4;end;

if(Designtype=3) then do;n=1;b=11/20;end;

if(Designtype=4) then do;n=1;b=1/4;end;

if(Better=1) then do;

do until(pw>=power/100);

if(Designtype=1) then do;df=4*n-3;end;

if(Designtype=2) then do;df=4*n-4;end;

if(Designtype=3) then do;df=6*n-5;end;

if(Designtype=4) then do;df=12*n-5;end;

t=(Delta+abs(NIM))/(Sw*sqrt(b/n))-tinv(1-a,df);

pw=probt(t,df);n=n+0.01;

end;

end;

if(Better=0) then do;

do until(pw>=power/100);

if(Designtype=1) then do;df=4*n-3;end;

if(Designtype=2) then do;df=4*n-4;end;

if(Designtype=3) then do;df=6*n-5;end;

if(Designtype=4) then do;df=12*n-5;end;

t=tinv(1-a,df)-(-Delta+abs(NIM))/(Sw*sqrt(b/n));

pw=probt(t,df);n=n+0.01;

end;

end;

if(Designtype=1) then do;N=ceil((n-0.01)*4);end;

if(Designtype=2) then do;N=ceil((n-0.01)*2);end;

if(Designtype=3) then do;N=ceil((n-0.01)*2);end;

if(Designtype=4) then do;N=ceil((n-0.01)*4);end;

if(Designtype=1) then do;Design_type="1. 4*2";end;

if(Designtype=2) then do;Design_type="2. 2*3";end;

if(Designtype=3) then do;Design_type="3. 2*4";end;

if(Designtype=4) then do;Design_type="4. 4*4";end;

if better=1 then Better_="better";else Better_="worse";

print

Design_type

Better_[label="Higher Mean Values are"]

a[label="Test Significance Level"]

Delta[label="True Difference in Means.u2-u1"]

NIM[label="Non-Inferiority Margin"]

Sw[label="Within Standard Error"]

power[label="Power(%)"]

N[label="N"];

end;

finish MTE61;

run MTE61(2,1,0.05,0,-5,10,90);

quit;

SAS运行结果：

图1-74 SAS 9.4 关于例1-31样本量估计的参数设置与计算结果