F-tests can be **two-tailed** (to test that $s_1^2 ne s_2^2$) or **one-tailed** (to test that $s_1^2 > s_2^2$).

How can I modify Levene/Brown-Forsythe to be “one-tailed”, that is, to test $s_1^2 > s_2^2$ instead of $s_1^2 ne s_2^2$?

Here is a demo:

The image shows normally distributed training data (n=1000) and a model. An F-test is used to compare the variance of one point’s residuals (n=2) to the variance of *all* of the residuals (n=2000), so the point is an outlier if its residual variance is “too large.” The points are colored by p-value, where light points fit the model and dark points are outliers, and you can see that the two-tailed Brown-Forsythe rejects points that are *too close* to the model as well as too far.

*Note: A different non-parametric, one-sided variance test would be fine as well.*

Glen_b gave the information I needed, but I thought I would leave some implementation details (using scipy).

```
#basic F-test
F = var(a) / var(b)
Fp = stats.f.sf(F, df1, df2)
#Brown Forsythe
BF, BFp = stats.levene(a, b, center='median')
#two tailed t-test on transformed data
za = abs(a-median(a))
zb = abs(b-median(b))
t, tp_two_tailed = stats.ttest_ind(za, zb)
#the two tailed t test recapitulates the BF test
assert(t**2 == BF)
assert(p_BF == p_two_tailed)
#one tailed t test p value
tp = stats.t.sf(t, df)
```

Above shows scatter plots of the p values from the one-tailed $F$-test and two-tailed BF-test (left), and the one-tailed $t$ tests (right). Red points are “too close” ($s_1^2 < s_2^2$).