method |
a character string specifying the method used for
combining the markers.
Notations:
Before getting into these methods,
let us first introduce some notations that will be used
throughout this vignette. Let
Di,i=1,2,…,n1
be the marker values of ith individual in diseased group, where
Di=(Di1,Di2) and
Hj,j=1,2,…,n2
be the marker values of jth individual in healthy group, where
Hj=Hj1,Hj2 .
Let
xi1=c(Di1,Hj1) be the values of the first marker, and
xi2=c(Di2,Hj2) be values of the second marker for the ith
individual i=1,2,…,n . Let
Di,min=min(Di1,Di2),Di,max=max(Di1,Di2),Hj,min=min(Hj1,Hj2),Hj,max=max(Hj1,Hj2) and
ci be be the resulting combination score for the ith individual.
The available methods are:
-
Logistic Regression (logistic) : Combination score obtained
by fitting a logistic regression modelis as follows:
ci=(1+eβ0+β1xi1+β2xi2eβ0+β1xi1+β2xi2)
A combination score obtained by fitting a logistic regression model typically refers
to the predicted probability or score assigned to each observation
in a dataset based on the logistic regression model’s
fitted values
-
Scoring based on Logistic Regression (scoring) : Combination score is obtained using the
slope values of the relevant logistic regression model, slope values are rounded to the number of
digits taken from the user.
ci=β1xi1+β2xi2
-
Pepe & Thompson’s method (PT) : The Pepe and Thompson combination score,
developed using their optimal linear combination technique, aims to maximize
the Mann-Whitney statistic in the same way that the Min-max method does. Unlike
the Min-max method, the Pepe and Thomson method takes into account all marker
values instead of just the lowest and maximum values.
maximizeU(α)=(n1,n21)∑i=1n1∑j=1n2I(Di1+αDi2>=Hj1+αHj2)
ci=xi1+αxi2
-
Pepe, Cai & Langton’s method (PCL) : Pepe, Cai and Langton combination score
obtained by using AUC as the parameter of a logistic regression model.
maximizeU(α)=(n1,n21)∑i=1n1∑j=1n2I(Di1+αDi2>
Hj1+αHj2)+(21)I(Di1+αDi2=Hj1+αHj2)
-
Min-Max method (minmax) : This method linearly combines the minimum
and maximum values of the markers by finding a parameter,α , that
maximizes the Mann-Whitney statistic, an empirical estimate of the ROC area.
maximizeU(α)=(n1,n21)∑i=1n1∑j=1n2I(Di,max+αDi,min>Hj,max+αHj,min)
ci=xi,max+αxi,min
where xi,max=max(xi1,xi2) and xi,min=min(xi1,xi2)
-
Su & Liu’s method (SL) : The Su and Liu combination score is computed through
Fisher’s discriminant coefficients, which assumes that the underlying
data follow a multivariate normal distribution, and the covariance matrices across
different classes are assumed to be proportional.Assuming that
D∼N(μD,∑D)
and
H∼N(μH,∑H) represent
the multivariate normal distributions for the diseased and non-diseased groups,
respectively. The Fisher’s coefficients are as follows:
(α,β)=(∑D+∑H)−1μ
whereμ=μD−μH.The combination score in this case is:
ci=αxi1+βxi2
-
Minimax approach (minimax) : Combination score obtained with the Minimax procedure;
t parameter is chosen as the value that gives the maximum AUC from the
combination score. Suppose that D follows a multivariate normal distribution
D∼N(μD,∑D) , representing diseased group and H follows
a multivariate normal distribution H∼N(μH,∑H) , representing the non-diseased group.
Then Fisher’s coefficients are as follows:
(α,β)=[t∑D+(1−t)∑H]−1(μD−μH)
ci=b1x1+b2x2
-
Todor & Saplacan’s method (TS) :Combination score obtained by using
the trigonometric functions of the Θ value that optimizes the corresponding AUC.
ci=sin(θ)xi1+cos(θ)xi2
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