statisticalrethinkin..

1.2. WRECKING PRAGUE 19model P 1B , it’s not at all clear what we are to make of either rejecting or accepting the null.e null model is not unique to any process model nor hypothesis. If we reject the null, wecan’t really conclude that selection matters, because there are other neutral models that predictdifferent distributions of alleles. And if we accept the null, we can’t really conclude thatevolution in neutral, because some selection models expect the same frequency distribution.is is a huge bother. Once we have the diagram in FIGURE 1.2, it’s easy to see the problem.But few of us are so lucky. While population genetics has recognized this issue, scholarsin other disciplines continue to test frequency distributions against power law expectations,arguing even that there is only one neutral model. 10 Even if there were only one neutralmodel, there are so many non-neutral models that mimic the predictions of neutrality, thatneither rejecting nor failing to reject the null model carries much inferential power.So what can be done? Well, if you have multiple process models, a lot can be done. Ifit turns out that all of the process models of interest make very similar predictions, thenyou know to search for a different description of the evidence, a description under whichthe processes look different. For example, while P 0A and P 1B make very similar power lawpredictions for the frequency distribution of alleles, they make very dissimilar predictions forthe distribution of changes in allele frequency over time. In other words, explicitly comparepredictions of more than one model, and you can save yourself from some ordinary kinds offolly.And even when the data do discriminate among recognized models, we have to imaginethat there might be other process models that would correspond to the same predictions.is recognition lies behind many common practices in statistical inference, such as considerationof unobserved variables and sampling bias.Rethinking: Entropy and model identification One reason that statistical models routinely correspondto many different detailed process models is because they rely upon distributions like the normal,binomial, Poisson, and others. ese distributions are members of a family, the EXPONENTIALFAMILY. Nature loves the members of this family. Nature loves them, because nature loves entropy,and all of the exponential family distributions are MAXIMUM ENTROPY distributions. Taking the naturalpersonification out of that explanation will take time. e practical implication is that one canno more infer evolutionary process from a power law than we can infer developmental process fromthe fact that height is normally distributed. is fact should make us humble about what typical regressionmodels—the meat of this book—can teach us about mechanistic process. On the other hand,the maximum entropy nature of these distributions means we can use them to do useful statisticalwork, even when we can’t identify the underlying error process. Not only can we not identify it, butwe don’t have to.1.2.2. Measurement matters. e logic of falsification is very simple. We have a hypothesisH, and we show that it entails some observation D. en when look for D. If we don’t findit, we must conclude that H is false. Logicians call this kind of reasoning MODUS TOLLENS,which is Latin shorthand for “the method of destruction.” In contrast, finding D tells usnothing necessarily about H, because other hypotheses might also predict D.A compelling, but misleading, scientific fable that uses modus tollens concerns the colorof swans. Before Europeans voyaged to Australia, all swans that any European had ever seenor read about had white feathers. is lead to the belief that all swans are white. Let’s callthis a formal hypothesis:H 0 : All swans are white.