Why does -2log-likelihood decrease when you add new variables? Suppose I'm conducting a study with follow-up survey after N years. Now I want to determine potential risk factors for mortality. I use a model selection procedure and get a table in which the maximized value of -2log-likelihood monotonically decreases when I add more variables (all risk factors and some interactions). Is there some explanation why maximized value of -2log-likelihood progressively decreases, or is that coincidence?
 A: It's not coincidence.
If you fit some model and then fit the same model plus additional predictors, the model has additional ability to fit the data. Since you're fitting by maximum likelihood, the bigger model can match the likelihood of the first model by constraining the parameters associated with the additional variables to 0 -- then, starting from that constrained point it can improve the likelihood still further by "freeing up" those parameters.
Looked at another way, consider the simple case of two models, one with two parameters both free ($\theta_1,\theta_2$) and one where one of the parameters is set to 0 ($\theta_1$ free, $\theta_2=0$). Clearly the second, smaller model can't in general maximize the likelihood of the larger model, so imposing the constraint would mean moving down the likelihood surface from $(\hat{\theta}_1,\hat{\theta}_2)$ to a point on the $\theta_2=0$ line -- from the blue point (where $\theta_2$ was "free") to the reddish-brown one ($\theta_2=0$, the constrained optimum, at $(\hat{\theta}^c_1,0)\,$).
[Here we plot contours (level-sets) of the likelihood surface - the smaller contours represent higher likelihood. I also simplify the discussion by considering an example where the likelihood surface has a single optimum - both local and global.]

Since adding the parameter associated with $\theta_2$ to a model with just $\theta_1$ is the same as allowing $\theta_2$ to go from being fixed at $0$ to being free to be nonzero, that moves you back "up the hill" to the overall maximum, increasing the likelihood from a constrained optimum to the unconstrained optimum. The same kind of argument applies each time you add more parameters to a given model.
Clearly also higher likelihood $\mathcal{L}$ means lower $-2\log\mathcal{L}$, so adding parameters to an existing model will always* reduce that.
* not quite always. With a continuous $\theta_2$ parameter, it's "almost always" and if you have a discrete parameter then it might actually have its unconstrained optimum at 0.
A: This answer is intended to supplement that of @Glen_b not replace it.
If the OP is interested in the closely related question of 'Why when I add more variables do my standard errors go up?' then the canonical reference is

@ARTICLE{altham84,
  author = {Altham, P M E},
  year = 1984,
  title = {Improving the precision of estimation by fitting a model},
  journal = {Journal of the Royal Statistical Society --- Series B},
  volume = 46,
  pages = {118--119}
}

The article treats the question from an algebraic point of view witou the rather neat geometrical perspective.
