For a MOSFET in deep triode region, we can approximate it as a resistor with the following:
$$ \frac{1}{{\mu}_n C_{ox}(W/L)(V_{GS} - V_{Th})} $$
However, in small-signal for a MOSFET in saturation, I know that \$R_{out} = \frac{1}{\lambda \cdot I_d}\$, where lambda is channel-length modulation factor and that is proportional to inverse of length. So increasing length increases \$R_{out}\$. But why isn't it dependent on width? Surely, if I make the transistor very wide, resistance should drop.