Statistics for correlated data: Phylogenies, space, and time

Here we give an introduction to the growing number of statistical techniques for analyzing data that are not independent realizations of the same sampling process-in other words, correlated data. We focus on regression problems, in which the value of a given variable depends linearly on the value of another variable. To illustrate different types of processes leading to correlated data, we analyze four simulated examples representing diverse problems arising in ecological studies. The first example is a comparison among species to determine the relationship between home-range area and body size; because species are phylogenetically related, they do not represent independent samples. The second example addresses spatial variation in net primary production and how this might be affected by soil nitrogen; because nearby locations are likely to have similar net primary productivity for reasons other than soil nitrogen, spatial correlation is likely. In the third example, we consider a time-series model to ask whether the decrease in density of a butterfly species is the result of decreases in its host-plant density; because the population density of a species in one generation is likely to affect the density in the following generation, time-series data are often correlated. The fourth example combines both spatial and temporal correlation in an experiment in which prey densities are manipulated to determine the response of predators to their food supply. For each of these examples, we use a different statistical approach for analyzing models of correlated data. Our goal is to give an overview of conceptual issues surrounding correlated data, rather than a detailed tutorial in how to apply different statistical techniques. By dispelling some of the mystery behind correlated data, we hope to encourage ecologists to learn about statistics that could be useful in their own work. Although at first encounter these techniques might seem complicated, they have the power to simplify ecological research by making more types of data and experimental designs open to statistical evaluation.