  In this paper, we define some weighted sums of the alternating multiple
$T$-values (AMTVs), and study several duality formulas for them by using the
tools developed in our previous papers. Then we introduce the alternating
version of the convoluted $T$-values and Kaneko-Tsumura $\psi$-function, which
are proved to be closely related to the AMTVs. At the end of the paper, we
study the $\Q$-vector space generated by the AMTVs of any fixed weight $w$ and
provide some evidence for the conjecture that their dimensions $\{d_w\}_{w\ge
1}$ form the tribonacci sequence 1, 2, 4, 7, 13, ....
