Best test to use Suppose a group of people are exposed to some chemical for $20$ minutes. Their heart rate is measured before being exposed. The exposure to the chemical seems to cause their heart rate to speed up. What test can be used to precisely find the point where the heart rate differs significantly from baseline? A paired t-test?
 A: Depending on what you want to know, a paired $t$-test may or may not be a reasonable choice. 
Suppose your experiment works like this:


*

*Measure subjects' heart rates.

*Expose them to substance X for 20 minutes.

*Measure their heart rates again.


A paired $t$-test would be an ideal choice for asking whether subjects' heart rates differed between Steps 1 and 3. This is a simple within-subjects design and using a paired $t$-test allows you to control for fact that subjects may have very different resting heart rates, depending on age, sex, fitness, etc. 
On the other hand, suppose you running a slightly different experiment, like this:


*

*Hook subjects up to a heart-rate monitor

*Expose them to substance X for up to 20 minutes.

*Continuously monitor their heart rates.


and you are asking whether substance X has an effect at all, and, if so, the effect's latency (i.e., how long it takes). This is a somewhat thornier problem. The naive solution might be to bin your data into, say,  1 minute intervals and then whether the heart rate at time=1 minute is different from baseline, time=2 minutes vs. baseline, and so on, each of these via a paired $t$-test. 
That's not totally unreasonable, but you need to control for multiple comparisons; if you perform 20 $t$-tests with a threshold of $p<0.05$, you should expect to see one significant difference by chance alone. There are various ways of doing this (search for "multiple comparisons correction") depending on exactly what you want to control. I have seen people argue for accepting two successive time points, each of which is individually significant at $p<0.05$, as indicating a significant deviation. The logic here is that, assuming the observations are independent, the chance of seeing two consecutive significant-but-spurious results is (0.05)^2, which is probably acceptably small. Seems reasonable enough, but I'd love others' opinions on that.
Another alternative would be to fit a model to your data. If you know that the drug leads to a 20% increase vs. baseline, maybe you could fit a sigmoid with a delay parameter to each of your observations.
There are lots of other approaches--this has been extensively studied under the name "change-point analysis", and many of the popular tests are available in an R package. However, be careful and read the documentation carefully; as someone else said here recently "The large print giveth and the small print taketh away" and the devil really is in the details here, so be careful.
