Table 4.

Mahalanobis distances for bivariate combinations with CA 125

Pelvic disease group
Menopause
Pre
Post
Post
Pre/post
Pre
Post
Stage
III/IV
III/IV
I/II
I/II/III/IV
Benign
Benign
HistologyNonmucinousNonmucinousNonmucinousMucinousEndometriosisOvarian serous
Z value CA 1256.811.07.23.42.64.0
Marker pair
Square-root Mahalanobis distance
    CA 125 and CA 15.38.011.97.33.42.64.0
    CA 125 and CA 19.96.811.07.65.42.64.0
    CA 125 and CEA6.811.07.24.22.64.0
    CA 125 and SMRP8.411.87.23.42.64.0
  • NOTE: Calculation of Mahalanobis distances: denote by x1 = CA 125 value in pool, by x2 = CA 19.9/CA 15.3/CEA value in pool, and let X = [x1x2]. Denote by y1 = vector of CA 125 values from appropriate control group, and y2 = vector of CA 19.9/CA 15.3/CEA values from appropriate control group, and bivariately by Y = [y1y2]. Then, the mean on the logarithmic scale is given by m1 = mean(log(y1)) and m2 = mean(log(y2)), bivariately denoted by M = [m1m2]. The difference between the measurement in the pooled sample and the mean of the individual samples, on the logarithmic scale, is D = log(X) − M. The variance-covariance matrix is calculated from the individual observations for each biomarker, Σ = variance-covariance matrix of log(Y). The variance-covariance matrix controls for any correlation between the biomarkers; thus, for example, when two biomarkers are elevated in the presence of ovarian cancer, a negative correlation induces greater complementarity than either a zero or positive correlation, the latter inducing a degree of redundancy. The Mahalanobis distance is given by DTΣ−1D, and the square root of the Mahalanobis distance is analogous to the z value in one dimension.