The author proves strong-type factorization theorems for positive multilinear operators with range in \(L_ q(X,\mu)\), \(q\geq 0\), with (X,\(\mu)\) a \(\sigma\)-finite measure space. If \(1\leq p\leq \infty\), a Banach lattice E is said to be p-convex if there exists a constant \(M<\infty\) such that \(\| (\sum^{n}_{i=1}| x_ i|^ p)^{1/p}\| \leq M(\sum^{n}_{i=1}\| x_ i\|^ p)^{1/p}\) if \(1\leq p<\infty\) or \(\| \bigvee^{n}_{i=1}| x_ i| \| \leq M\max_{1\leq i\leq n}\| x_ i\|\) if \(p=\infty\), for every choice of finitely many vectors \(\{x_ i\}^ n_{i=1}\) in E. The principal result of the paper can be formulated as follows. If \(E_ k\), \(1\leq k\leq n\), is a \(p_ k\)-convex Banach lattice and if \(B:E_ 1\times...\times E_ n\to L_ q(q\geq 0)\) is a positive n-linear map and if \(r\geq 1\) is such that \(1/r=\sum^{n}_{k=1}1/p_ k\) and \(r\geq q\), then there exists \(0\leq \phi \in L_ s\), where \(1/s=1/q-1/r\), such that \((1/\phi)B(E_ 1\times...\times E_ n)\subseteq L_ r\). The methods of the paper are based on the positive projective tensor product of Banach lattices combined with some ideas and results of E. M. Nikishin [Usp. Mat. Nauk 25, No.6(156), 129-191 (1970; Zbl 0222.47024)] and B. Maurey [Astérisque 11, 1-163 (1974; Zbl 0278.46028)]. Several examples are given to indicate the scope of the results.