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CPS1848 J.A. Roldán-Nofuentes et al.
which is distributed asymptotically according to Hotelling’s T-squared
distribution with a dimension 2 and n  n degrees of freedom, where 2 is the
1 2
2
ˆ
dimension of the vector Aω . When n  n is large, the statistic Q is
1 2
distributed according to a central chi-square distribution with 2 degrees of
freedom when the null hypothesis is true. To be able to calculate the global
T
test statistic, Q  2 ˆ T T   ˆ ˆ ω A T  1  Aω , it is necessary for the matrix A ˆ ˆ ω A
ˆ ω A A
to be non-singular.
Individual hypothesis tests
The hypothesis test to individually compare the two PPVs (NPVs) is
H : PV  PV vs H : PV  PV ,
0 1 2 0 1 2
where PV is PPV or NPV. Based on the asymptotic normality of the estimators,
the statistic for this hypothesis test is

PV  PV
z  1 2 ,

Var PV  ˆ   Var PV  ˆ   2Cov PV PV 2 
,
2
1
1
which is distributed asymptotically according to a normal standard
distribution, and where the variances-covariances is obtained from the
equation
Alternative methods to the global test
The global hypothesis test simultaneously compares the PPVs and the
NPVs of the two BDTs. Some alternative methods to this global hypothesis
test, based on the individual hypothesis tests, are: 1) Solving the tests

H : PPV  PPV and H : NPV  NPV , each one to an error  ; 2) Solving
0 1 2 0 1 2
the individual tests, H 0 : PPV  PPV and H 0 : NPV  NPV , and applying a
1
2
1
2
multiple comparison method such as Bonferroni’s method (1936) or Holm’s
method (1979), which are methods that are very easy to apply based on the
p-values. Bonferroni’s method consists of solving each individual hypothesis
test to an error  2 ; and Holm’s method is a step-down method which is
based on Bonferroni’s method but is more conservative.
Simulation experiments were carried out to study the type I errors and the
powers of the four methods proposed to solve the global hypothesis test: the
hypothesis test based on the chi-square, the individual hypothesis tests each
one to an error  , and the individual hypothesis tests applying Bonferroni’s
method and Holm’s method. From the results obtained in these experiments,
we propose the following method to compare the PVs of two BDTs subject to
a case-control design: 1) Applying the hypothesis test based on the chi-square
distribution to an error  , 2) If the global hypothesis test is not significant,
the equality hypothesis of the PVs is not rejected; if the global hypothesis test
is significant to an error  , the investigation of the causes of the significance

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