paired t-test advantages - literature reference I am having trouble finding any web-based resource or (even better) publication which details (or even names) the kind of problems which one avoids by running a paired t-test as opposed to an unpaired test when testing the same population twice.
I am surprised resources like wikipedia also do not detail this. Will an unpaired test overestimate or underestimate my p? Or does this depend on other considerations?
Could you help me out?
 A: In a paired t-test, each observation is of the form:  $Z =  X - Y$. That means that the variance of each observation is 
$$Var(Z) = Var(X-Y) = Var(X) + Var(Y) -2Cov(X,Y)$$
Now, suppose that X,Y and are positively related (for instance before and after scores on a test). The paired t-test will then reduce the $Var(Z)$ by the $-2Cov(X,Y)$ term. 
Thus in the equation for the paired t-test (see here) $$ t= \frac{\bar{X_D} - \mu_d}{\sigma_d /\sqrt{n}}$$
you would get the $\sigma_d$ is smaller and be more likely to reject. If for some reason, the paired scores had a negative covariance, then you would get $\sigma_d$ to be larger and be less likely to reject. Most often (and logically so), paired samples have a positive covariance and thus $Var(Z)$ is reduced. This yields to higher power. 
If you look at the wikipedia page for statistical power (here), you will see that for a one sided paired t-test the power (as a function of how large the paired difference is under the alternative -- denoted by $\tau$ is:
$$\pi(\tau)\approx 1-\Phi(1.64-\tau\sqrt{n}/\hat{\sigma}_D). $$
As you can see, for larger values of $\sigma_d$, $\Phi(1.64-\tau\sqrt{n}/\hat{\sigma}_D)$ would decrease and $\pi(\tau)$ would increase.
Thus, if the data is paired, if you don't use the paired test you are losing power and are less likely to reject. However, if the unusual (but perhaps possible) situation where paired data are more variable, then you are more likely to reject and can very well be concluding incorrectly.
It is worth noting that if the data is not paired and you use the paired test, then you also lose power. For instance, if you are using a one-sided t-test on a sample size of $2n$ observations but erroneously concluded that observation $1$ is pared with observation $n+1$, observation $2$ is paired with observation $n+2$, etc..., then in a paired t-test your sample size is $n$. By reducing your sample size, you once again lose power.
Hope that helps!
