In conducting likelihood ratio tests, why do statisticians normally neglect to write out the constant cutoff point, $\lambda(x) < \text{constant}$? Suppose that we are doing a likelihood ratio test. We normally wish to find for a family of pdf's $f_\theta(x)$, for the test of hypothesis $H_0: \theta \in \Theta_0$ and $H_1: \theta \in \Theta$: $$ \lambda(x) = \frac{\sup_{\theta\in\Theta_0}L(\theta\mid X)}{\sup_{\theta\in\Theta}L(\theta\mid X)} $$ In most books I have seen, they usually find this, then say that a Uniformly Most Powerful test is of the form: $$ \lambda(x) < \text{constant} $$ They normally neglect to display that constant. I am wondering why this is so. Doesn't the cutoff point depend on that constant?

The cutoff point _is_ the "constant". Normally one finds a monotone function $g$ for which the distribution of $g(\lambda(X))$ is understood and tabulated (and in this era "tabulated" should be taken to mean on-the-shelf software deals with it). Thus if, for example, if $g$ is a decreasing function and $g(\lambda(X))\sim t_n,$ and one is testing at level $\alpha$, then one finds for which value of $c$ one has $\Pr(t_n >c) = \alpha,$ and then one rejects the null hypothesis if $g(\lambda(X))>c.$ The value of $c$ of course depends on $\alpha$.

I suspect I could say more if you gave a concrete example.