ion is alpha diversity, as in LBPa, rather than community composition, as in TR. In fact, alpha diversity is also level-1 data in the TR scheme. This is because alpha diversity can be presented in a sites 3 alpha diversity table, which is an observation units 3 descriptors table, which in turn is the basic format of level-1 data (Fig. 1). Alpha diversity can be considered secondary raw data because it summarizes the species abundance information from the primary raw data (sites 3 species table) through the computation of an alpha diversity index, but it is still level-1 data. Both alpha diversity and beta diversity can be derived from the sites 3 species table, albeit via different computation routes (Fig. 1). When we discussed level-1 data in TR, we chose to concentrate on the sites 3 species table, rather than the sites 3 alpha diversity table, for two reasons. First, clarifying the relationships between the levels of abstraction is easier when each higher level consists ofion is easier when each higher level consists of the variation in the level below it. This made it possible to use an analogy from physics, where level-1 data consist of position, level-2 data of velocity, and level-3 data of acceleration. In mathematical terms, the level-2 data correspond to the first derivative and level-3 data to the second derivative of the original level-1 variable. Second, our main interest was in beta diversity, and the sites 3 species table can be used to calculate beta diversity, whereas the sites 3 alpha diversity table cannot. Computing beta diversity requires knowledge on species identities, and this information is lost when the alpha diversity index is computed, so the sites 3 alpha diversity table cannot be used as a starting point when one is interested in questions along the community composition path (Fig. 1). Just like one can derive level-2 data and level-3 data from the sites 3 species table and the individuals 3 species table, one can do the same with the sites3 alpha diversity table (see ‘‘The alpha diversity path’’ in Fig. 1). Alpha diversity is at the same level of abstraction as community composition (level 1), and variation in alpha diversity is, hence, at the same level of abstraction as variation in community composition (level 2). Consequently, analyzing alpha diversity is a level-2 question by our terminology, which contrasts with the view of PCD, LBPa, and LBPb, who call it a level-1 analysis. Definitions of beta diversity Over the years, beta diversity has been defined in different ways. Whittaker (1960) measured beta diversity with dissimilarity indices computed between site pairs, and later noted that the average of all such values for a data set can be considered an expression of beta differentiation (Whittaker 1972). Therefore, the individual cell values in the level-2 dissimilarity matrix along the community composition path can be termed ‘‘pairwise beta diversity,’’ and the average of all off-diagonal values in the dissimilarity matrix is one expression of ‘‘regional beta diversity.’’ Whittaker (1960) also defined the terms gamma diversity and alpha diversity, and related the three kinds of diversity to each other in a multiplicative way, i.e., c 1⁄4 a 3 b, from which follows that b 1⁄4 c/a (multiplicative beta diversity). The three diversity measures have since then also been related in an additive way, i.e., c1⁄4aþb, from which follows that b 1⁄4 c a (additive beta diversity; e.g., Lande 1996, Veech et al. 2002, PCD). In both the multiplicative and the additive approach, alpha and gamma diversities can be measured as the number of species in a site and a larger region, respectively. Multiplicative beta diversity then measures how many times more species there are in an entire region than in an average site within that region. Additive beta diversity measures how many more species the entire region has than an average site within that region. Both are variants of ‘‘regional beta diversity.’’ For pairwise beta diversity, one can choose a dissimilarity metric from a wide variety of available indices, each one of which emphasizes different aspects of the raw data. Some of these indices are ratios whereas others are not. The classical beta diversity measures are ratios and therefore conform with the concept of multiplicative beta diversity. Examples include measures based on the Jaccard, Sorensen, or Bray-Curtis indices, which are often obtained by first subtracting two values and then dividing the result by a third value (to indicate, for example, what proportion of species found in two sites are not shared between the two sites). When b1⁄4c/a is computed for site pairs, it equals 2 minus the Sorensen index. In contrast, dissimilarity indices such as the Euclidean distance and the Manhattan distance are not ratios, but have the same unit as the input data (e.g., number of species) and therefore conform with the concept of additive beta diversity. The Manhattan distance, when computed using presence–absence data, indicates how many species are found in one of the sites to be compared but not both. This is obtained by subtraction, and the value thus obtained is used without relating it to the number of species that were found in both sites COMMENTS 3246 Ecology, Vol. 89, No. 11