  For any $k\geq 1$, this paper studies the number of polynomials having $k$
irreducible factors (counted with or without multiplicities) in
$\mathbf{F}_q[t]$ among different arithmetic progressions. We obtain asymptotic
formulas for the difference of counting functions uniformly for $k$ in a
certain range. In the generic case, the bias dissipates as the degree of the
modulus or $k$ gets large, but there are cases when the bias is extreme. In
contrast to the case of products of $k$ prime numbers, we show the existence of
complete biases in the function field setting, that is the difference function
may have constant sign. Several examples illustrate this new phenomenon.
