Let {$X_{ni}|1;{le};i;{le};n$, $n;{ge};1$} be an array of rowwise negatively associated random variables and let $alpha$ > 1/2, 0 < p < 2 ${alpha}p;{ge};1$. In this paper we discuss $n^{{alpha}p-2}h(n)$ max $_{1;{le};k{le}n};|;{sum}^k_{i=1};X_{ni}|/n^{alpha};{ o};0$ completely as $n;{ o};{infty}$ under not necessarily identically distributed with a suitable conditions and h(x) > 0 is a slowly varying function as $x;{ o};{infty}$. In addition, we obtained that $n^{{alpha}p-2}h(n)$ max $_{1;{le};k{le}n};|;{sum}^k_{i=1};X_{ni}|/n^{alpha};{ o};0$ completely as $n;{ o};{infty}$ if and only if $E|X_{11}|^ph(|X_{11}|^{1/alpha});<;{infty}$ and $EX_{11};=;0$ under identically distributed case and some corollaries are obtained.