In addition to the methods mentioned by Bill, researchers can also assess selection for particular genes by examining the dN/dS ratio (also referred to as the Ka/Ks ratio):
https://en.wikipedia.org/wiki/Ka/Ks_ratio
Essentially, the method looks at the frequency of non-synonymous mutations (mutations that change one amino acid to another) versus synonymous mutations (mutations that do not affect the amino acid sequence of the resulting protein). The synonymous mutations are assumed to be neutral, whereas at least some of the non-synonymous mutations could have a fitness effect. An excess of non-synonymous mutations over synonymous mutations suggests that some of those mutations are undergoing positive selection (the mutations confer a fitness benefit, causing the mutations to rapidly spread throughout the population), whereas a low rate of non-synonymous mutations would suggest purifying selection (mutation to the gene is deleterious to fitness, so mutations are strongly selected against).
As with most things in science, there is no one definitive test of whether a particular trait evolved under selection or neutral drift. Rather, one must build a case by looking at a variety of different sources of evidence, not just those based on DNA sequence analysis, but also experimental data testing the effects of particular genes and mutations (e.g. see this nice paper that uses experimental studies in mice to build the case for positive selection of a mutation among Asian populations:
https://www.cell.com/cell/fulltext/S0092-8674(13)00067-6). The review you cite from Pardis Sabeti would be a good source to consult, as she is one of the experts at the forefront of this field.
Evo said:
No, we cannot test for these things. We just know that there has not been the technology before to selectively do selective genetic modification on a worldwide scale. Any other questioning would wander into conspiracy theory.
There are certainly mechanisms that contribute to evolution beside natural selection. Some believe that these mechanisms, such as genetic drift, may contribute to many more of the genetic changes we see throughout evolution than natural selection: http://discovermagazine.com/2014/march/12-mutation-not-natural-selection-drives-evolution As discussed in Bill's and my post, scientists have devised (and are continuing to work on) methods to test whether certain genes/mutations are under selection.
Distinguishing between mutations that spread via positive selection versus neutral drift mechanisms is not only important in evolutionary biology, but also in cancer biology. Tumors accumulate mutations over time and some of these mutations (called as driver mutations) contribute to carcinogenesis. At the same time, other mutations are also occurring in the tumor cells (called passenger mutations) that do not appreciably affect carcinogenesis. When studying the mutations found in tumors, researchers have put a lot of effort into disentangling the driver mutations from the passenger mutations in order to figure out what genes and biological pathways are contributing to the growth and proliferation of particular cancer types.