  It is known that the class of luminous starburst galaxies tends to have
higher $R =^{12}CO(J=1--0)/^{13}CO(J=1--0)$ integrated line intensity ratios
($R>20$) than normal spiral galaxies ($R \sim 10$). Since most previous studies
investigated only $R$, it remains uncertain whether the luminous starburst
galaxies are overabundant in $^{12}$CO or underabundant in $^{13}$CO. Here we
propose a new observational test to examine this problem. Our new test is to
compare far-infrared luminosities [$L$(FIR)] with those of $^{12}$CO and
$^{13}CO [L(^{12}CO)$ and $L(^{13}CO)$, respectively]. It is shown that there
is a very tight correlation between $L(^{12}CO)$ and L(FIR), as found in many
previous studies. However, we find that the $^{13}$CO luminosities of the
high-R galaxies are lower by a factor of three on the average than those
expected from the correlation for the remaining galaxies with ordinary $R$
values. Therefore, we conclude that the observed high $R$ values for the
luminous starburst galaxies are attributed to their lower $^{13}$CO line
intensities.
