Why is C statistic the same as area under ROC curve?
Example:
4 leaf tree, 100 1’s 100 0’s
30 1’s
20 0’s
20 1’s
20 0’s
10 1’s
50 0’s
40 1’s
10 0’s
Areas: (Number of 1’s)x(Number of 0’s)
40 1’s
30 1’s
10 0’s
20 1’s
20 0’s
10 1’s
50 0’s
More 1’s ------------------------------------------------ Less 1’s
40x90 Concordant
Pairs
IF cut is after 1st leaf
20+20+50 = 90 0’s
CUT
40 1’s
30 1’s
10 0’s
20 1’s
20 0’s
10 1’s
50 0’s
More 1’s ------------------------------------------------ Less 1’s
30x70 more
Concordant Pairs
IF cut is after 2nd
20+50 = 70 0’s
90x40 Concordant
Pairs
IF cut is after 1st leaf
40 1’s
30 1’s
10 0’s
20 1’s
20 0’s
10 1’s
50 0’s
More 1’s ------------------------------------------------ Less 1’s
30x70 more
Concordant Pairs
IF cut is after 2nd
CUT
40 1’s
30 1’s
10 0’s
20 1’s
20 0’s
10 1’s
50 0’s
More 1’s ------------------------------------------------ Less 1’s
In terms of PROPORTIONS:
ROC
curve
0.10 1’s
0.20 1’s
0.30 1’s
0.40 1’s
0.10 prop.
concordant
0.21 proportion
concordant
0.36 proportion
concordant
Left of cut point => decide 1
Diagonal line coordinates at ends
are (proportion 0’s declared 1’s,
proportion 1’s declared 1’s) =
(sensitivity , 1-specificity) !!!
0.10 0.20 0.20
0.50
proportions of 0’s = blue
box widths
Blue boxes are ties, proportions .4x.1=.04, .3x.2=.06, .2x.2=.04, .1x.5=.05
Proportion ties = .04+.06+.04+.05. Half of that = sum of blue triangles below
diagonals = .02+.03+.02+.025 = .095. Area under ROC curve (black diagonal
lines) is 0.36+0.21+0.10+0.095 = C = 0.765