Several methods for meta-analysis of diagnostic tests have been proposed. However, when several diagnostic tests are evaluated, the situation is problematic since the within-studies correlation needs to be taken into account. Niki L. Dimou, Maria Adam and Pantelis G. Bagos A Multivariate Method for Meta-Analysis and Comparison of Diagnostic Tests. 2015, Statistics in Medicine (in press).We present an extension of the bivariate random effects meta-analysis for the log-transformed sensitivity and specificity that can be applied for two or more diagnostic tests. The advantage of this method is that a closed-form expression is derived for the calculation of the within-studies covariances. The method allows the direct calculation of sensitivity and specificity, as well as, the diagnostic odds ratio (DOR), the area under curve (AUC) and the parameters of the Summary Receiver Operator's Characteristic (SROC) curve, along with the means for a formal comparison of these quantities for different tests. There is no need for individual patient data or the simultaneous evaluation of both diagnostic tests in all studies. The method is simple and fast, it can be extended for several diagnostic tests and can be fitted in nearly all statistical packages. clearset more offinput study tp1 fp1 fn1 tn1 tp2 fp2 fn2 tn2 n111 n110 n101 n100 n001 n000 n010 n011***Replace zero values***gen zero=0 replace zero=1 if tp1==0|fp1==0|fn1==0|tn1==0|tp2==0|fp2==0|fn2==0|tn2==0 replace tp1=tp1+.5 if zero==1 replace fp1=fp1+.5 if zero==1 replace fn1=fn1+.5 if zero==1 replace tn1=tn1+.5 if zero==1replace tp2=tp2+.5 if zero==1 replace fp2=fp2+.5 if zero==1 replace fn2=fn2+.5 if zero==1 replace tn2=tn2+.5 if zero==1***Calculation of logit Sensitivity-(1-Specificity)***gen b1=log(tp1/fn1)gen b2=log(fp1/tn1)gen b3=log(tp2/fn2)gen b4=log(fp2/tn2)***Calculation of TPR-FPR***gen tpr1=exp(b1)/(1+exp(b1))gen fpr1=exp(b2)/(1+exp(b2))gen tpr2=exp(b3)/(1+exp(b3))gen fpr2=exp(b4)/(1+exp(b4))***Calculaton of the Variance***gen V11=1/tp1+1/1/fn1gen V22=1/tn1+1/fp1gen V33=1/tp2+1/1/fn2gen V44=1/tn2+1/fp2***Extrapolation of the data***gen total1=tp1+fp1+fn1+tn1 gen total2=tp2+fp2+fn2+tn2 gen total_cases1=tp1+fn1gen total_cases2=tp2+fn2 gen total_controls1=fp1+tn1 gen total_controls2=fp2+tn2 gen both=1 if total1!=. & total2!=.gen not_equal1=1 if total1<total2 & both==1gen not_equal2=1 if total1>total2 & both==1gen extrapolated_tp1=total_cases2*tp1/total_cases1 if not_equal1==1 gen extrapolated_fn1=total_cases2*fn1/total_cases1 if not_equal1==1 gen extrapolated_fp1=total_controls2*fp1/total_controls1 if not_equal1==1 gen extrapolated_tn1=total_controls2*tn1/total_controls1 if not_equal1==1 gen extrapolated_tp2=total_cases1*tp2/total_cases2 if not_equal2==1gen extrapolated_fn2=total_cases1*fn2/total_cases2 if not_equal2==1gen extrapolated_fp2=total_controls1*fp2/total_controls2 if not_equal2==1gen extrapolated_tn2=total_controls1*tn2/total_controls2 if not_equal2==1replace tp1=extrapolated_tp1 if not_equal1==1 replace fn1=extrapolated_fn1 if not_equal1==1 replace fp1=extrapolated_fp1 if not_equal1==1 replace tn1=extrapolated_tn1 if not_equal1==1 replace tp2=extrapolated_tp2 if not_equal2==1replace fn2=extrapolated_fn2 if not_equal2==1replace fp2=extrapolated_fp2 if not_equal2==1replace tn2=extrapolated_tn2 if not_equal2==1replace total1=total2 if not_equal1==1 replace total_cases1=total_cases2 if not_equal1==1 replace total_controls1=total_controls2 if not_equal1==1 replace total2=total1 if not_equal2==1replace total_cases2=total_cases1 if not_equal2==1replace total_controls2=total_controls1 if not_equal2==1***Fitting missing cells for Yi=1******Calculation of common OR given Yi=1***gen logOR_Y_1=log((n111*n100)/(n110*n101))replace logOR_Y_1=log(((0.5+n111)*(0.5+n100))/((0.5+n110)*(0.5+n101))) if n111==0|n110==0|n101==0|n100==0gen std_logOR_Y_1=sqrt(1/n111+1/n110+1/n101+1/n100)replace std_logOR_Y_1=sqrt(1/(n111+0.5)+1/(n110+0.5)+1/(n101+0.5)+1/(n100+0.5)) if n111==0|n110==0|n101==0|n100==0metan logOR_Y_1 std_logOR_Y_1, eform random nograph rename tp1 arename fn1 brename tp2 crename fn2 dgen e=.replace e=r(ES)***Generating values from equations (B11-14)***cap drop x1-x4gen x1=(-sqrt((-a*e-b-c*e+c)^2-4*a*c*(e-1)*e)+a*e+b+c*e-c)/(2*(e-1))gen x2=(sqrt((-a*e-b-c*e+c)^2-4*a*c*(e-1)*e)+a*e-2*a-b-c*e+c)/(2*(e-1))gen x3=(sqrt((-a*e-b-c*e+c)^2-4*a*c*(e-1)*e)-a*e-b+c*e-c)/(2*(e-1))gen x4=(-sqrt((-a*e-b-c*e+c)^2-4*a*c*(e-1)*e)+a*e+2*b*e-b-c*e+c)/(2*(e-1))replace n111=x1 if n111==.replace n110=x2 if n110==.replace n101=x3 if n101==.replace n100=x4 if n100==.***Fitting missing cells for Yi=0******Calculation of common OR given Yi=0***gen logOR_Y_0=log((n011*n000)/(n010*n001))replace logOR_Y_0=log(((0.5+n011)*(0.5+n000))/((0.5+n010)*(0.5+n001))) if n011==0|n010==0|n001==0|n000==0gen std_logOR_Y_0=sqrt(1/n011+1/n010+1/n001+1/n000)replace std_logOR_Y_0=sqrt(1/(n011+0.5)+1/(n010+0.5)+1/(n001+0.5)+1/(n000+0.5)) if n011==0|n010==0|n001==0|n000==0metan logOR_Y_0 std_logOR_Y_0, eform random nograph replace e=r(ES)rename a tp1rename b fn1rename c tp2rename d fn2rename fp1 arename tn1 brename fp2 crename tn2 d***Generating values from equations (B15-18)***cap drop x1-x4 gen x1=(-sqrt((-a*e-b-c*e+c)^2-4*a*c*(e-1)*e)+a*e+b+c*e-c)/(2*(e-1))gen x2=(sqrt((-a*e-b-c*e+c)^2-4*a*c*(e-1)*e)+a*e-2*a-b-c*e+c)/(2*(e-1))gen x3=(sqrt((-a*e-b-c*e+c)^2-4*a*c*(e-1)*e)-a*e-b+c*e-c)/(2*(e-1))gen x4=(-sqrt((-a*e-b-c*e+c)^2-4*a*c*(e-1)*e)+a*e+2*b*e-b-c*e+c)/(2*(e-1))replace n011=x1 if n011==.replace n010=x2 if n010==.replace n001=x3 if n001==.replace n000=x4 if n000==.rename a fp1rename b tn1rename c fp2rename d tn2***Calculation of the Covariance***drop zerogen zero=1 if n111==0|n110==0|n101==0|n100==0|n011==0|n010==0|n001==0|n000==0replace n111=n111+0.5 if zero==1replace n110=n110+0.5 if zero==1replace n101=n101+0.5 if zero==1replace n100=n100+0.5 if zero==1replace n011=n011+0.5 if zero==1replace n010=n010+0.5 if zero==1replace n001=n001+0.5 if zero==1replace n000=n000+0.5 if zero==1gen V13=(n111/((n111+n110)*(n111+n101)))-(n110/((n111+n110)*(n110+n100)))-(n101/((n101+n100)*(n111+n101)))+(n100/((n101+n100)*(n110+n100)))gen V24=(n000/((n001+n000)*(n010+n000)))-(n001/((n001+n000)*(n011+n001)))-(n010/((n011+n010)*(n010+n000)))+(n011/((n011+n010)*(n011+n001)))gen V14=0gen V23=0gen V12=0gen V34=0***Bivariate model fitting***mvmeta b V, vars(b1 b2)mvmeta b V, vars(b3 b4)***Model fitting of Eq. 10***mvmeta b V, vars(b1 b2 b3 b4)test b1=b3test b2=b4 ***Calculation of Di, Si***gen D1=b1-b2gen D2=b3-b4gen S1=b1+b2gen S2=b3+b4sum D1local D1_mean=r(mean)sum D2local D2_mean=r(mean)sum S1local S1_mean=r(mean)sum S2local S2_mean=r(mean)***Calculation of the slope of the regression of Di on Si***gen b1_reg=((`e(Sigma11)')-(`e(Sigma22)'))/((`e(Sigma11)')+(`e(Sigma22)')+2*(`e(Sigma12)'))gen b2_reg=((`e(Sigma33)')-(`e(Sigma44)'))/((`e(Sigma33)')+(`e(Sigma44)')+2*(`e(Sigma34)'))***Calculation of the intercept of the regression of Di on Si***gen a1_reg=(_b[b1]-_b[b2])-(_b[b1]+_b[b2])*((`e(Sigma11)')-(`e(Sigma22)'))/((`e(Sigma11)')+(`e(Sigma22)')+2*(`e(Sigma12)'))gen a2_reg=(_b[b3]-_b[b4])-(_b[b3]+_b[b4])*((`e(Sigma33)')-(`e(Sigma44)'))/((`e(Sigma33)')+(`e(Sigma44)')+2*(`e(Sigma34)'))***Construction of the plot of Di on Si***gen D1_pred=a1_reg+b1_reg*S1gen D2_pred=a2_reg+b2_reg*S2twoway (sc D1 S1 [fw=total1],msize(small) msymbol(Oh)) (sc D1_pred S1,c(line) ms(i) lwidth(thin) lpattern(dot) ) (sc D2 S2 [fw=total2],msymbol(Th) msize(small) ) (sc D2_pred S2,c(line) ms(i) lwidth(thin))***Construction of the SROC curve***gen tse1=exp(a1_reg/(1-b1_reg))*(fpr1/(1-fpr1))^((1+b1_reg)/(1-b1_reg))/(1+(exp(a1_reg/(1-b1_reg))*(fpr1/(1-fpr1))^((1+b1_reg)/(1-b1_reg))))gen tse2=exp(a2_reg/(1-b2_reg))*(fpr2/(1-fpr2))^((1+b2_reg)/(1-b2_reg))/(1+(exp(a2_reg/(1-b2_reg))*(fpr2/(1-fpr2))^((1+b2_reg)/(1-b2_reg))))***Adding values (0,0) and (1,1) in our plotlocal new = _N + 1set obs `new'replace tse1=0 in `new'replace fpr1=0 in `new'replace tse2=0 in `new'replace fpr2=0 in `new'replace total1=1 in `new'replace total2=1 in `new'local new = _N + 1set obs `new'replace tse1=1 in `new'replace fpr1=1 in `new'replace tse2=1 in `new'replace fpr2=1 in `new'replace total1=1 in `new'replace total2=1 in `new'gr7 tpr1 tse1 fpr1 [fw=total1], ysize(6) xsize(6) noaxis xline(0(.1)1) yline(0(.1)1) /// tlab(0(.1)1) xlab(0(.1)1) ylab(0(.1)1) s(Oi) c(.s) /// l1(Sensitivity) b2(1-Specificity) ti(Summary ROC Curve) key1(" ") key2(" ")gr7 tpr2 tse2 fpr2 [fw=total2], ysize(6) xsize(6) noaxis xline(0(.1)1) yline(0(.1)1) /// tlab(0(.1)1) xlab(0(.1)1) ylab(0(.1)1) s(Ti) c(.s[-]) /// l1(Sensitivity) b2(1-Specificity) ti(Summary ROC Curve) key1(" ") key2(" ")***Calculation of the AUC1,AUC2***sum a1_regsum a2_regsum b1_regsum b2_reglocal a1_reg=a1_reglocal a2_reg=a2_reglocal b1_reg=b1_reglocal b2_reg=b2_regpreserve clearset obs 10000local t1=0 local t2=1local t=`t2' - `t1'gen x= `t1'+`t'*_n/10000gen y=exp(`a1_reg'/(1-`b1_reg'))*(x/(1-x))^((1+`b1_reg')/(1-`b1_reg'))/(1+(exp(`a1_reg'/(1-`b1_reg'))*(x/(1-x))^((1+`b1_reg')/(1-`b1_reg'))))integ y x local auc1=r(integral)dis `auc1'restorepreserveclear set obs 10000local t1=0 local t2=1local t=`t2' - `t1'gen x= `t1'+`t'*_n/10000gen y=exp(`a2_reg'/(1-`b2_reg'))*(x/(1-x))^((1+`b2_reg')/(1-`b2_reg'))/(1+(exp(`a2_reg'/(1-`b2_reg'))*(x/(1-x))^((1+`b2_reg')/(1-`b2_reg'))))integ y x local auc2=r(integral)dis `auc2'restore***Calculation of the Variance of AUC1-AUC2******Calculation of Var(b1)***nlcom ((`e(Sigma11)')-(`e(Sigma22)'))/((`e(Sigma11)')+(`e(Sigma22)')+2*(`e(Sigma12)'))matrix V=r(V)local var_b1=V[1,1]dis `var_b1'***Calculation of Var(b2)***nlcom ((`e(Sigma33)')-(`e(Sigma44)'))/((`e(Sigma33)')+(`e(Sigma44)')+2*(`e(Sigma34)'))matrix V=r(V)local var_b2=V[1,1]dis `var_b2'***Calculation of Var(a1)***nlcom (_b[b1]-_b[b2])-(_b[b1]+_b[b2])*((`e(Sigma11)')-(`e(Sigma22)'))/((`e(Sigma11)')+(`e(Sigma22)')+2*(`e(Sigma12)'))matrix V=r(V)local var_a1=V[1,1]dis `var_a1'***Calculation of Var(a2)***nlcom (_b[b3]-_b[b4])-(_b[b3]+_b[b4])*((`e(Sigma33)')-(`e(Sigma44)'))/((`e(Sigma33)')+(`e(Sigma44)')+2*(`e(Sigma34)'))matrix V=r(V)local var_a2=V[1,1]dis `var_a2'***Calculation of Cov(b1,b2)***nlcom (((`e(Sigma11)')-(`e(Sigma22)'))/((`e(Sigma11)')+(`e(Sigma22)')+2*(`e(Sigma12)')))+ ///(((`e(Sigma33)')-(`e(Sigma44)'))/((`e(Sigma33)')+(`e(Sigma44)')+2*(`e(Sigma34)')))matrix V=r(V)local var_b1_b2=V[1,1]dis `var_b1_b2'local cov_b1_b2=((`var_b1_b2')-(`var_b1')-(`var_b2'))/2dis `cov_b1_b2'***Calculation of Cov(a1,a2)***nlcom ((_b[b1]-_b[b2])-(_b[b1]+_b[b2])*((`e(Sigma11)')-(`e(Sigma22)'))/((`e(Sigma11)')+(`e(Sigma22)')+2*(`e(Sigma12)')))+ ///((_b[b3]-_b[b4])-(_b[b3]+_b[b4])*((`e(Sigma33)')-(`e(Sigma44)'))/((`e(Sigma33)')+(`e(Sigma44)')+2*(`e(Sigma34)')))matrix V=r(V)local var_a1_a2=V[1,1]dis `var_a1_a2'local cov_a1_a2=((`var_a1_a2')-(`var_a1')-(`var_a2'))/2dis `cov_a1_a2'***Calculation of Cov(b1,a1)***nlcom (((`e(Sigma11)')-(`e(Sigma22)'))/((`e(Sigma11)')+(`e(Sigma22)')+2*(`e(Sigma12)')))+ ///((_b[b1]-_b[b2])-(_b[b1]+_b[b2])*((`e(Sigma11)')-(`e(Sigma22)'))/((`e(Sigma11)')+(`e(Sigma22)')+2*(`e(Sigma12)')))matrix V=r(V)local var_b1_a1=V[1,1]dis `var_b1_a1'local cov_b1_a1=((`var_b1_a1')-(`var_b1')-(`var_a1'))/2dis `cov_b1_a1'***Calculation of Cov(b1,a2)***nlcom (((`e(Sigma11)')-(`e(Sigma22)'))/((`e(Sigma11)')+(`e(Sigma22)')+2*(`e(Sigma12)')))+ ///((_b[b3]-_b[b4])-(_b[b3]+_b[b4])*((`e(Sigma33)')-(`e(Sigma44)'))/((`e(Sigma33)')+(`e(Sigma44)')+2*(`e(Sigma34)')))matrix V=r(V)local var_b1_a2=V[1,1]dis `var_b1_a2'local cov_b1_a2=((`var_b1_a2')-(`var_b1')-(`var_a2'))/2dis `cov_b1_a2'***Calculation of Cov(b2,a1)***nlcom (((`e(Sigma33)')-(`e(Sigma44)'))/((`e(Sigma33)')+(`e(Sigma44)')+2*(`e(Sigma34)')))+ ///((_b[b1]-_b[b2])-(_b[b1]+_b[b2])*((`e(Sigma11)')-(`e(Sigma22)'))/((`e(Sigma11)')+(`e(Sigma22)')+2*(`e(Sigma12)')))matrix V=r(V)local var_b2_a1=V[1,1]dis `var_b2_a1'local cov_b2_a1=((`var_b2_a1')-(`var_b2')-(`var_a1'))/2dis `cov_b2_a1'***Calculation of Cov(b2,a2)***nlcom (((`e(Sigma33)')-(`e(Sigma44)'))/((`e(Sigma33)')+(`e(Sigma44)')+2*(`e(Sigma34)')))+ ///((_b[b3]-_b[b4])-(_b[b3]+_b[b4])*((`e(Sigma33)')-(`e(Sigma44)'))/((`e(Sigma33)')+(`e(Sigma44)')+2*(`e(Sigma34)')))matrix V=r(V)local var_b2_a2=V[1,1]dis `var_b2_a2'local cov_b2_a2=((`var_b2_a2')-(`var_b2')-(`var_a2'))/2dis `cov_b2_a2'***Calculation of partial derivative a1***preserve clearset obs 10000local t1=0 local t2=1local t=`t2' - `t1'gen x= `t1'+`t'*_n/10000gen y=((x/(1-x))^((1+`b1_reg')/(1-`b1_reg')))/((1+((x/(1-x))^((1+`b1_reg')/(1-`b1_reg')))*exp(`a1_reg'/(1-`b1_reg')))^2)integ y x local integ_a1=r(integral)restorelocal deriv_a1=(1/(1-(`b1_reg')))*exp((`a1_reg')/(1-(`b1_reg')))*(`integ_a1')dis `deriv_a1'***Calculation of partial derivative a2****preserveclear set obs 10000local t1=0 local t2=1local t=`t2' - `t1'gen x= `t1'+`t'*_n/10000gen y=((x/(1-x))^((1+`b2_reg')/(1-`b2_reg')))/((1+((x/(1-x))^((1+`b2_reg')/(1-`b2_reg')))*exp(`a2_reg'/(1-`b2_reg')))^2)integ y x local integ_a2=r(integral)restorelocal deriv_a2=-(1/(1-(`b2_reg')))*exp((`a2_reg')/(1-(`b2_reg')))*(`integ_a2')dis `deriv_a2'***Calculation of partial derivative b1****preserveclear set obs 10000local t1=0 local t2=1local t=`t2' - `t1'gen x= `t1'+`t'*_n/10000gen y=(((x/(1-x))^((1+`b1_reg')/(1-`b1_reg')))*(`a1_reg'+2*log(x/(1-x))))/((1+((x/(1-x))^((1+`b1_reg')/(1-`b1_reg')))*exp(`a1_reg'/(1-`b1_reg')))^2)integ y x local integ_b1=r(integral)restorelocal deriv_b1=((1/(1-(`b1_reg')))^2)*exp((`a1_reg')/(1-(`b1_reg')))*(`integ_b1')dis `deriv_b1'***Calculation of partial derivative b2****preserveclear set obs 10000local t1=0 local t2=1local t=`t2' - `t1'gen x= `t1'+`t'*_n/10000gen y=(((x/(1-x))^((1+`b2_reg')/(1-`b2_reg')))*(`a2_reg'+2*log(x/(1-x))))/((1+((x/(1-x))^((1+`b2_reg')/(1-`b2_reg')))*exp(`a2_reg'/(1-`b2_reg')))^2)integ y x local integ_b2=r(integral)restorelocal deriv_b2=-((1/(1-(`b2_reg')))^2)*exp((`a2_reg')/(1-(`b2_reg')))*(`integ_b2')dis `deriv_b2'***Retrieve Var(AUC1-AUC2)***matrix G = (`deriv_a1',`deriv_b1',`deriv_a2',`deriv_b2')matrix G1=G'matrix V=((`var_a1'),(`cov_b1_a1'),(`cov_a1_a2'), (`cov_b2_a1')\(`cov_b1_a1'),(`var_b1'),(`cov_b1_a2'),(`cov_b1_b2')\(`cov_a1_a2'),(`cov_b1_a2'),(`var_a2') ,(`cov_b2_a2')\(`cov_b2_a1'),(`cov_b1_b2'),(`cov_b2_a2'),(`var_b2'))mat v1=G*V*G1svmat v1, names(var_auc1_auc2)gen se_auc1_auc2=sqrt(var_auc1_auc2)***Calculation of the test for the equality of the AUC for the two tests*** gen z=(`auc1'-`auc2')/se_auc1_auc2gen p_value=2*(1-normal(abs(z))) |
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