Answers to Practice Problems - Appendix

A.1 Bayesian approaches can incorporate prior information (e.g., data from a previous study). This can be important in conservation studies because management decisions must often be made quickly with limited new data, in which case the use of prior data or other studies can help. Similarly, combining different kinds of data (e.g., genetic and non-genetic) can help improve inference. Bayesian approaches also yield a probability distribution (posterior distribution) whose shape allows easy visual interpretation of the probability space (e.g., tails of the distribution) and whether a hypothesis is true.

Whereas a frequentist approach typically yields only a point estimate and confidence interval with an upper and lower confidence limit but no probability distribution of curve. Bayesian approaches also more directly address a hypothesis of interest, e.g., “population x is declining”, than does a frequentist approach that typically rejects (or “fails to reject”) a null hypothesis as to whether the event (significant decline evidence) is likely occur by chance alone if we conduct thousands independent experiments.

A.2 P-values can overstate the strength of evidence because hypothesis testing based on P-values often does not fully account the effects of the multiple factors (or the uncertainty) that influence the final result (e.g., differences in the environment, climate, etc.). Bayesian approaches, on the other hand, can more easily incorporate the uncertainty caused by various factors, which often reduces the support for the hypothesis (or for rejection of the null hypothesis in a frequentist approach). Accounting for multiple effects is difficult in typical ecological studies (and frequentist approaches) that rely on observational data, e.g., without controlled experiments. See Gaggiotti (2010) and Malakoff (1999) who explain that statistically significant effects inferred from frequentist analyses (P- values) can disappear upon a Bayesian reexamination.

A.3 Point estimates are often useless without interval estimates (e.g., CIs) because point estimates alone give us no understanding of the certainty of our point estimate. For example, if a point estimate of Ne is 50 but the confidence interval around this point estimate ranges from 20 to 100,000, then we actually do not know if Ne is near 50 or is actually near say 5,000. In this case, without the CI we should not believe Ne is near 50. Similarly if FST is estimated to equal 0.05 but the CI includes zero (e.g., FST = 0.00 to 0.10) we cannot be sure FST is greater than zero and thus if genetic differentiation exists; Without a CI we should not make inferences, but someone could make erroneous inferences that populations are differentiated.

A.4 The difference between the standard deviation and standard error is that the SE is computed from the mean of many sets of observations (e.g., many samples or groups of individuals), whereas the SD is computed from each of many individual (single) observations. Not surprisingly, the means from each of many sets of observations have less variation (around the true mean value) and smaller CIs than does the SD because the SD is computed from individual observations (not sets of observations). See the text and Figure A3. A.5 See definitions of the coalescent in the Appendix and Glossary. It important to use many loci when using coalescent approaches because random genealogical processes (drift or lineage sorting, and mutation) lead to many possible random genealogies for different loci from the same population demographic or evolutionary history (Figure A12). The number of loci needed depends on the question (application), the effect size (e.g., severity of population decline, magnitude of FST etc.), and the required precision or power to detect an effect. The number of loci might be relatively low (e.g., 5-10) if the number of individuals samples is very large for certain applications – for example the estimation of a small Ne (where the effect size 2 of signal from drift is strong) or a large FST, a large r , etc.