  Statistical anisotropy in the nanohertz-frequency gravitational-wave
background (GWB) is expected to be detected by pulsar timing arrays (PTAs) in
the near future. By developing a frequentist statistical framework that
intrinsically restricts the GWB power to be positive, we establish scaling
relations for multipole-dependent anisotropy decision thresholds that are a
function of the noise properties, timing baselines, and cadences of the pulsars
in a PTA. We verify that $(i)$ a larger number of pulsars, and $(ii)$ factors
that lead to lower uncertainty on the cross-correlation measurements between
pulsars, lead to a higher overall GWB signal-to-noise ratio, and lower
anisotropy decision thresholds with which to reject the null hypothesis of
isotropy. Using conservative simulations of realistic NANOGrav datasets, we
predict that an anisotropic GWB with angular power $C_{l=1} > 0.3\,C_{l=0}$ may
be sufficient to produce tension with isotropy at the $p = 3\times10^{-3}$
($\sim3\sigma$) level in near-future NANOGrav data with a $20$~yr baseline. We
present ready-to-use scaling relationships that can map these thresholds to any
number of pulsars, configuration of pulsar noise properties, and sky coverage.
We discuss how PTAs can improve the detection prospects for anisotropy, as well
as how our methods can be adapted for more versatile searches.
