Page 133 - Contributed Paper Session (CPS) - Volume 6

CPS1848 J.A. Roldán-Nofuentes et al.

PPV  ˆ 1 pn n  pn n n  n  and NPV  ˆ 1 pn n  qn n  qn n
2 11
1 20

n
qn
2 11
for Test 1, and 1 2 20 2 1 11 1 20
PPV  ˆ pn n and NPV  ˆ qn n
2 1 1
1 2 0
2

pn n  qn 1 n  n 2 0  2 pn 2 n  n 1 1  qn n
1 2 0
1
2
2 1 1
for Test 2, when q  1 p . Let the variance-covariance matrixes be defined as

 

 
 Var Se Cov Se ,Se    Var Sp Cov Sp ,Sp  
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
 ˆ Se    1 1 2   and  ˆ Sp    1 1 2   .


 
 
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
 Cov Se ,Se  Var Se   Cov Sp ,Sp  Var Sp 
 1 2 2   1 2 2 
T
,
,
,
Let θ Se Se Sp Sp 2  be a vector whose components are the sensitivities
1
2
1
T
,
,
and the specificities, and let ω  PPV PPV NPV NPV 2  be a vector whose
,
1
2
1
components are the predictive values. The variance-covariance matrix of θ is
ˆ
 1 0  0 0
        ,
ˆ Se 
ˆ θ
 0 0   0 1  ˆ Sp
where  is the product of Kronecker. Applying the delta method, the matrix
of variances- covariances of ˆ ω is
   ω     ˆ ω  ω     T .

ˆ
 θ  θ  θ   
Then, we study the joint comparison and the individual comparison of the
PVs of the two BDTs. In both cases, and as has been explained in Section 1, it
is assumed that there is an estimation of the disease prevalence based on a
health survey or other studies.
Global hypothesis test
The global hypothesis test to simultaneously compare the PVs of the two
BDTs is
H
H 0  : PPV  1 PPV 2 and NPV  1 NPV 2  vs :at least one equality is not true,
1
which is equivalent to the hypothesis test
H : Aω  vs H Aω  0 0,
:
0 1
where A is a complete range design matrix and a dimension 2 4 , i.e.
 1 0
 1
A     1  .
 0 1 
As the vector ˆ ω is distributed asymptotically according to a multivariate
normal distribution i.e. n  n 2  ˆ ω ω   N  ,0 Σ ω  , then the statistic
1 n n 
1
2
for the global hypothesis test is
Q  2 ˆ T T   ˆ ˆ ω A T  1  Aω ,
ˆ ω A A
122 | I S I W S C 2 0 1 9