  We critically examine the dependence of spectral index on luminosity in
optically selected AGN samples. An analysis of optically selected high-redshift
quasars showed an anti-correlation of $\alpha_{OX}$, the spectral index between
the rest-frame 2500 A and 2 keV, with optical luminosity (Miyaji et al. 2006).
We examine this relationship by means of Monte Carlo simulations and conclude
that a constant spectral index independent of optical luminosity is still
consistent with this high-z sample. We further find that that contributions of
large dispersions and narrow range of optical luminosity are most important for
the apparent, yet artificial, $\alpha_{OX} - l_o$ correlation reported. We also
examine another, but more complete low-z optical selected AGN sub-sample from
Steffen et al. (2006), and our analysis shows that a constant spectral index
independent of optical luminosity is also consistent with the data. By
comparing X-ray and optical luminosity functions, we find that a luminosity
independent spectral index is in fact more preferred than the luminosity
dependent spectral index model. We also discuss the selection effects caused by
flux limits, which might systematically bias the $l_x - l_o$ relation and cause
discrepancy in optically selected and X-ray selected AGN samples. To correctly
establish a dependence of spectral index of AGNs on their luminosity, a larger
and more complete sample is needed and consequences of luminosity dispersions
and selection effects in flux limited samples must be taken into account
properly.
