樣本量估計(jì)及其在nQuery和SAS軟件上的實(shí)現(xiàn)*——均數(shù)比較(十一)

南方醫(yī)科大學(xué)生物統(tǒng)計(jì)學(xué)系(510515)

徐 瑩 高培純 徐笑寒 陳平雁△

1.2.2.11 基于比值高階交叉設(shè)計(jì)的非劣效性檢驗(yàn)

Chen(1997)[1]和Chow and Liu(2009)[2]給出的高階交叉設(shè)計(jì)的樣本量估計(jì)方法是建立在近似服從自由度為vk的t分布上，當(dāng)指標(biāo)為高優(yōu)指標(biāo)時(shí)，其檢驗(yàn)效能計(jì)算公式為：

(1-59)

當(dāng)指標(biāo)為低優(yōu)指標(biāo)時(shí)，對應(yīng)的檢驗(yàn)效能計(jì)算公式為：

(1-60)

在計(jì)算樣本量時(shí)，一般先設(shè)定樣本量初始值，然后迭代樣本量直到所得的檢驗(yàn)效能滿足條件為止。此時(shí)的樣本量，即研究所需的樣本量。

【例1-32】某公司欲驗(yàn)證一種治療風(fēng)濕病的仿制藥非劣于標(biāo)準(zhǔn)藥。擬采用2×4設(shè)計(jì)。研究者決定將非劣界值設(shè)置為0.2。根據(jù)以往研究，已知標(biāo)準(zhǔn)藥CV為0.4。假設(shè)仿制藥與標(biāo)準(zhǔn)藥真實(shí)比值為1。檢驗(yàn)水準(zhǔn)為0.05，試估計(jì)檢驗(yàn)效能為90%所需的樣本量。

nQuery Advanced 8.2實(shí)現(xiàn)：設(shè)定檢驗(yàn)水準(zhǔn)為α=0.05，檢驗(yàn)效能取90%。由題意知，μ2/μ1=1，NIM=0.2，CV=0.4。在nQuery Advanced 8.2主菜單選擇：

方法框中選擇：Higher-order Cross-Over Design for Two Means- Non-Inferiority-using Ratios

在彈出的樣本量計(jì)算窗口將各參數(shù)鍵入，如圖1-75所示，結(jié)果為N=29。

圖1-75 nQuery Advanced 8.2 關(guān)于例1-32樣本量估計(jì)的參數(shù)設(shè)置與計(jì)算結(jié)果

SAS9.4軟件實(shí)現(xiàn)：

proc IML;

start MTE91(Designtype,Better,a,Ratio,NIM,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(Better=1 | Better=0)then do;

error=0;end;

else do;

error=1;print"error" "Higher means better(1)/worse(0)";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<0) then do;

error=1;print"error" "True Ratio of means must be>0";end;

if(NIM<0) then do;

error=1;print"error" "Non-Inferiority margin must be>0";end;

if(Better=1) then do;

if(Ratio<= 1-NIM)then do;

error=1;print"error" "True Ratio should be>1 - Non-Inferiority margin if higher means better";end;

end;

if(Better=0) then do;

if(Ratio>= 1+NIM)then do;

error=1;print"error" "True Ratio should be<1+Non-Inferiority margin if higher means worse";end;

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;

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

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=(-log(Ratio)-log(1-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)-(log(Ratio)+log(1+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"]

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

NIM[label="Non-Inferiority Margin"]

CV[label="Coefficient of Variance，CV"]

power[label="Power(%)"]

N[label="N"];

end;

finish MTE91;

run MTE91(3,1,0.05,1,0.2,0.4,90);

quit;

SAS運(yùn)行結(jié)果：

圖1-76 SAS 9.4 關(guān)于例1-32樣本量估計(jì)的參數(shù)設(shè)置與計(jì)算結(jié)果

1.2.2.12 基于均差高階交叉設(shè)計(jì)的優(yōu)效性檢驗(yàn)

Chen(1997)[1]和Chow and Liu(2009)[2]給出的高階交叉設(shè)計(jì)的樣本量估計(jì)方法是建立在近似服從自由度為vk的t分布上，當(dāng)指標(biāo)為高優(yōu)指標(biāo)時(shí)，其檢驗(yàn)效能計(jì)算公式為：

(1-61)

當(dāng)指標(biāo)為低優(yōu)指標(biāo)時(shí)，對應(yīng)的檢驗(yàn)效能計(jì)算公式為：

(1-62)

在計(jì)算樣本量時(shí)，一般先設(shè)定樣本量初始值，然后迭代樣本量直到所得的檢驗(yàn)效能滿足條件為止。此時(shí)的樣本量，即研究所需的樣本量。

【例1-33】某公司欲驗(yàn)證一種治療風(fēng)濕病的仿制藥非劣于標(biāo)準(zhǔn)藥，擬采用2×3設(shè)計(jì)。研究者決定將優(yōu)效界值設(shè)置為5。根據(jù)以往類似研究，已知均方誤為100。假設(shè)仿制藥于標(biāo)準(zhǔn)藥真實(shí)差值為15。檢驗(yàn)水準(zhǔn)設(shè)置為0.05，試估計(jì)檢驗(yàn)效能為90%所需的樣本量。

nQuery Advanced 8.2 實(shí)現(xiàn)：設(shè)定檢驗(yàn)水準(zhǔn)為α=0.05，檢驗(yàn)效能取90%。由題意知，μ2-μ1=15，SM=5，=10。在nQuery Advanced 8.2主菜單選擇：

方法框中選擇：Higher-order Cross-Over Design for Two Means- Superiority by Margin-using Differences

在彈出的樣本量計(jì)算窗口將各參數(shù)鍵入，如圖1-77所示，結(jié)果為N=14。

圖1-77 nQuery Advisor 8.2 關(guān)于例1-33樣本量估計(jì)的參數(shù)設(shè)置與計(jì)算結(jié)果

SAS 9.4軟件實(shí)現(xiàn)：

proc IML;

start MTE71(Designtype,Better,a,Delta,SM,Sw,power);

error=0;

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

error=0;end;

elsedo;error=1;print"error" "Designtype must be 1 or 2 or 3 or 4";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(a>0.2 | a<0) then do;

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

if(Better=1) then do;

if(Delta<= abs(SM))then do;

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

end;

if(Better=0) then do;

if(Delta>= -abs(SM))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(SM))/(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(SM))/(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"]

SM[label="Superiority Margin"]

Sw[label="Within Standard Error"]

power[label="Power(%)"]

N[label="N"];

end;

finish MTE71;

run MTE71(2,1,0.05,15,5,10,90);

quit;

SAS運(yùn)行結(jié)果：

圖1-78 SAS 9.4 關(guān)于例1-33樣本量估計(jì)的參數(shù)設(shè)置與計(jì)算結(jié)果

1.2.2.13 基于比值高階交叉設(shè)計(jì)的優(yōu)效性檢驗(yàn)

Chen(1997)[1]和Chow(2009)[2]給出的高階交叉設(shè)計(jì)的樣本量估計(jì)方法是建立在近似服從自由度為vk的t分布上，當(dāng)指標(biāo)為高優(yōu)指標(biāo)時(shí)，其檢驗(yàn)效能計(jì)算公式為：

(1-63)

當(dāng)指標(biāo)為低優(yōu)指標(biāo)時(shí)對應(yīng)的檢驗(yàn)效能計(jì)算公式為：

(1-64)

在計(jì)算樣本量時(shí)，一般先設(shè)定樣本量初始值，然后迭代樣本量直到所得的檢驗(yàn)效能滿足條件為止。此時(shí)的樣本量，即研究所需的樣本量。

【例1-34】某公司欲驗(yàn)證一種治療風(fēng)濕病的仿制藥非劣于標(biāo)準(zhǔn)藥，擬采用2×4設(shè)計(jì)。研究者決定將優(yōu)效界值設(shè)置為0.2。根據(jù)以往研究，已知標(biāo)準(zhǔn)藥CV為0.4。假設(shè)仿制藥與標(biāo)準(zhǔn)藥真實(shí)比值為1.4。檢驗(yàn)水準(zhǔn)設(shè)置為0.05，試估計(jì)檢驗(yàn)效能為90%所需的樣本量。

nQuery Advanced 8.2 實(shí)現(xiàn)：設(shè)定檢驗(yàn)水準(zhǔn)為α=0.05，檢驗(yàn)效能取90%。由題意知，μ2/μ1=1.4，SM=0.2，CV=0.4。在nQuery Advanced 8.2主菜單選擇：

方法框中選擇：Higher-order Cross-Over Design for Two Means- Superiority by Margin- using Ratios

在彈出的樣本量計(jì)算窗口將各參數(shù)鍵入，如圖1-79所示，結(jié)果為N=60。

圖1-79 nQuery Advanced 8.2 關(guān)于例1-34樣本量估計(jì)的參數(shù)設(shè)置與計(jì)算結(jié)果

SAS 9.4軟件實(shí)現(xiàn)：

proc IML;

start MTE101(Designtype,Better,a,Ratio,SM,CV,power);

error=0;

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

error=0;end;

elsedo;error=1;print"error" "Designtype must be 1 or 2 or 3 or 4";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(a>0.2 | a<0) then do;

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

if(Ratio<= 0) then do;

error=1;print"error" "True Ratio of means must be>0";end;

if(SM<0) then do;

error=1;print"error" "Superiority margin must be>0";end;

if(Better=1) then do;

if(Ratio<= 1+SM)then do;

error=1;print"error" "True difference in means must>1+superiority margin if higher means better";end;

end;

if(Better=0) then do;

if(Ratio>= 1-SM)then do;

error=1;print"error" "True difference in means must<1 - superiority margin if higher means worse";end;

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;

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

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=(abs(log(Ratio))-log(1+SM))/(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=-(abs(log(Ratio))+log(1-SM))/(Sw*sqrt(b/n))+tinv(1-a,df);

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;

print

Design_Type

a[label="Test Significance Level"]

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

SM[label="Superiority margin"]

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

power[label="Power(%)"]

N[label="N"];

end;

finish MTE101;

run MTE101(3,1,0.05,1.4,0.2,0.4,90);

quit;

SAS運(yùn)行結(jié)果：

圖1-80 SAS 9.4 關(guān)于例1-34樣本量估計(jì)的參數(shù)設(shè)置與計(jì)算結(jié)果