Inferring Ecological Relationships from the Edges of Scatter Diagrams: Comparison of Regression Techniques

Abstract

Scatter diagrams have historically proved useful in the study of associative relationships in ecology. Several important ecological questions involve correlations between variables resulting in polygonal shapes. Two examples that have received considerable attention are patterns between prey size and predator size in animal populations and the relationship between animal abundance and body size. Each is typically illustrated using scatter diagrams with upper and lower boundaries of response variables often changing at different rates with changes in the independent variables. Despite recent statistical contributions that have stimulated an interest in characterizing the limits of a variable, a consensus on an appropriate methodology to quantify the boundaries of scatter diagrams has not yet been achieved. We tested regression techniques based on least squares and least absolute values models using several independent data sets on prey length and predator length for piscivorous fishes and compared estimated slopes for consistency. Our results indicated that least squares regression techniques were particularly sensitive to outlying y values and irregularities in the distribution of observations, and that they frequently produced inconsistent estimates of slope for upper and lower bounds. In contrast, quantile regression techniques based on least absolute values models appeared robust to outlying y values and sparseness within data sets, while providing consistent estimates of upper and lower bound slopes. Moreover, the use of quantile regression eliminated the need for an excess of arbitrary decision-making on the part of the investigator. We recommend quantile regression as an improvement to currently available techniques used to examine potential ecological relationships dependent upon quantitative information on the boundaries of polygonal relationships.

Links between plant species’ spatial and temporal responses to a warming climate

“…Thus, we adopted quantile regression, which is an effective approach for evaluating the magnitude of boundaries of the relationship between two variables (i.e. a correlation between two variables at the upper or lower boundaries of scatter plots) [35]. The choice of quantile to best represent the edge of a scatter plot can be subjective [35]; thus the 75th, 85th and 95th percentiles were used in this study.…”

Is the timing of spawning in sparid fishes a response to sea temperature regimes?

“…The reduction in deviation is signiWcant. Quantile regression analysis (Scharf et al 1998) indicated signiWcant reductions in both the upper (80% quantile: y = 9.8 ¡ 0.30x, P < 0.002, n = 88) and lower (20% quantile: y = ¡8.0 + 2.50x, P = 0.022, n = 88) boundaries of the data with increasing SST (Fig. 2c).…”

“…This relationship was plotted both for the family (i.e., all data records ignoring genus or species identities), and the genus level for Acanthopagrus (20 reports) and Pagrus (11 reports), the genera with the most reports of spawning seasons (i.e., all data records for the genus disregarding species identities). Quantile regression, a technique for characterising the boundaries of scatter diagrams that have polygonal shapes (Scharf et al 1998), was used to estimate the slopes of boundaries of the clouds of points in the family level scatter plot. The 80th and 20th quantiles were used as the estimates of the upper and lower boundaries, respectively.…”

“…Because quantile regression involves minimising the sums of absolute residuals, it is resistant to outliers and sparseness in data sets. The approach used was similar to that of Scharf et al (1998) except that more (1,000) bootstrap samples were used to estimate standard errors and signiWcance levels. At the genus level the data sets were too small for informative quantile regression.…”

Diet composition and feeding habits of the eyespot skate, Atlantoraja cyclophora (Elasmobranchii: Arhynchobatidae), off Uruguay and northern Argentina

“…Regressions on the 5, 50 and 95% quantiles were fitted in order to test an increase in minimum, medium and maximum prey size with increasing TL of A. cyplophora, respectively (Scharf et al, 1998). All statistical analyses were performed using the R statistical software, version 3.1.0.…”