Page 132 - Contributed Paper Session (CPS) - Volume 6

CPS1848 J.A. Roldán-Nofuentes et al.
ˆ
ˆ
ˆ
ˆ

and the estimators of their variances are Var Se    Se 1  1 Se 1  n ,
1
1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
  


  
Var Se 2 Se 2  1 Se 2  n 1 , Var Sp 1 Sp 1  1 Sp 1  n 2 and
ˆ
ˆ
ˆ
  
Var Sp 2 Sp 2  1 Sp ˆ 2  n . Therefore, the sensitivities and the specificities are
2
estimated as proportions of marginal totals. In this way, in the case sample we
are interested in the marginal frequencies n and n , as these frequencies
11 1 1
are the product of a type I bivariate binomial distribution (Kocherlakota and
Kocherlakota, 1992). In an analogous way, the marginal frequencies n and
20
n 2 0 of the control sample are the product of a type I bivariate binomial
distribution. In the case of individuals with the disease, the type I bivariate
binomial distribution is characterized (Kocherlakota and Kocherlakota, 1992)

by the two sensitivities ( Se and Se ) and by the correlation coefficient (  )
1 2
between T and T . In an analogous way, in the case of individuals who do
1 2
not have the disease the type I bivariate binomial distribution is characterized

by Sp , Sp and the correlation coefficient (  ) between T and T . In the
1 2 1 2
case of the individuals with the disease (cases), the correlation coefficient
between the two BDTs is
  Se Se 
   111 1 2  1 ,






Se 1 1 Se Se 2 1 Se 2  Se 1 1 Se Se 2 1 Se 2 
1
1
and in the case of the individuals who do not have the disease (controls), the
correlation coefficient between the two BDTs is
  Sp Sp 
   200 1 2  0 ,






Sp 1 1 Sp Sp 2 1 Sp 2  Sp 1 1 Sp Sp 2 1 Sp 2 
1
1
with
n n  n n n n  n n
ˆ 1 111 11 1 1 and   ˆ   2 200 20 2 0 ,
1 2 0 2
n n
1 2
ˆ ˆ
ˆ
ˆ
ˆ
Cov  Se Se   ˆ  Se Se 2  n   ˆ n
,
1
1
1
1
1
  111
2
and
ˆ
ˆ
ˆ ˆ
ˆ
ˆ


,
Cov  Sp Sp 2   200  Sp Sp 2  n   ˆ n .
0
2
1
1
2
All of the other covariances are zero, since the two samples are independent.


The estimators of  and  are
n n  n n n n  n n


ˆ   1 111 11 1 1 and ˆ   2 200 20 2 0 .
n 11 n  1 n 11 n 1 1 n  1 n 1 1  n 20 n  2 n 20 n 2 0 n  2 n 2 0 
Assuming that there is an estimation p of the disease prevalence, the
estimators of the predictive values are
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