Can balanced accuracy be higher than accuracy?

Question

I have classification tree where the balanced accuracy of the test set is higher than the normal accuracy. I thought balanced accuracy can only have at his maximum the same value as the accuracy not higher. Can anyone explain in which situation the balanced accuracy can be higher then accuracy?

The test set is unbalacned 30 % class 0 and 70 % class 1. The balacned accuracy is 73,5 % and the accuracy is 70,38 %. The true positive rate is 65,61 and the TNR is 81,38 — MasterStudent1992, May 22 '19 at 10:44
No, I meant how "balanced accuracy of the test set" is *defined*. I have never seen the term. — Stephan Kolassa, May 22 '19 at 10:45
balanced accuracy = (TP/(TP+FN)+(TN/(FP+TN)))/2 , accruacy = (TP+TN)/(TP+FN+FP+TN) — MasterStudent1992, May 22 '19 at 10:47
You've asked the same question about a week ago: [How can the balanced accurcay be bigger than the normal accuracy in unbalanced test data?](https://stats.stackexchange.com/questions/408496/how-can-the-balanced-accurcay-be-bigger-than-the-normal-accuracy-in-unbalanced-t) — Scholar, May 22 '19 at 13:53
@bi_scholar in the other question iam talking about resampling methods here its not depending on resampling methods... — MasterStudent1992, May 22 '19 at 15:13
Whether the classifications come from resampling methods or not is irrelevant. — Stephan Kolassa, May 22 '19 at 15:34

score 5 · Accepted Answer · answered May 22 '19 at 15:34

Let

$$ a:=TP,\quad b:= TN,\quad c:=TP+FN,\quad d:=TN+FP. $$

Then accuracy and balanced accuracy are

$$ Acc=\frac{a+b}{c+d},\quad BAcc=\frac{a}{2c}+\frac{b}{2d}, $$

or

$$ Acc=\frac{acd+bcd}{cd(c+d)},\quad BAcc=\frac{\frac{1}{2}ad(c+d)+\frac{1}{2}bc(c+d)}{cd(c+d)}. $$

Therefore,

$$ Acc<BAcc $$

is equivalent to

$$ acd+bcd < \frac{1}{2}ad(c+d)+\frac{1}{2}bc(c+d), $$

which in turn is equivalent to

$$ acd+bcd < ad^2+bc^2. $$

Taking a look at all possible combinations of $a,b,c,d$ (the only restriction being that $a\leq c$ and $b\leq d$), we find that this is indeed very often the case:

maximum <- 5

for ( aa in 1:maximum ) {
    for ( bb in 1:maximum ) {
        for ( cc in aa:maximum ) {
            for ( dd in bb:maximum ) {
                if ( aa*cc*dd+bb*cc*dd < aa*dd^2+bb*cc^2 ) {
                    cat("aa=",aa,", bb=",bb,", cc=",cc,", dd=",dd,
                       " ==> Acc=",(aa+bb)/(cc+dd)," < ",
                       aa/(2*cc)+bb/(2*dd),"=BAcc\n",sep="")
                }
            }
        }
    }
}

yields

aa=1, bb=1, cc=1, dd=2 ==> Acc=0.6666667 < 0.75=BAcc
aa=1, bb=1, cc=1, dd=3 ==> Acc=0.5 < 0.6666667=BAcc
aa=1, bb=1, cc=1, dd=4 ==> Acc=0.4 < 0.625=BAcc
aa=1, bb=1, cc=1, dd=5 ==> Acc=0.3333333 < 0.6=BAcc
aa=1, bb=1, cc=2, dd=1 ==> Acc=0.6666667 < 0.75=BAcc
aa=1, bb=1, cc=2, dd=3 ==> Acc=0.4 < 0.4166667=BAcc
aa=1, bb=1, cc=2, dd=4 ==> Acc=0.3333333 < 0.375=BAcc
aa=1, bb=1, cc=2, dd=5 ==> Acc=0.2857143 < 0.35=BAcc
aa=1, bb=1, cc=3, dd=1 ==> Acc=0.5 < 0.6666667=BAcc
aa=1, bb=1, cc=3, dd=2 ==> Acc=0.4 < 0.4166667=BAcc
aa=1, bb=1, cc=3, dd=4 ==> Acc=0.2857143 < 0.2916667=BAcc
aa=1, bb=1, cc=3, dd=5 ==> Acc=0.25 < 0.2666667=BAcc
aa=1, bb=1, cc=4, dd=1 ==> Acc=0.4 < 0.625=BAcc
(...)

And neither accuracy nor balanced accuracy is a good measure for assessing classification models: Why is accuracy not the best measure for assessing classification models?

Can balanced accuracy be higher than accuracy?

1 Answers1