Environmental and Ecological Statistics 11, 55-71, 2004 Corrected version of manuscript, prepared by the authors. Random denominators and the analysis of ratio data MARTIN LIERMANN,1* ASHLEY STEEL, 1 MICHAEL ROSING2 and PETER GUTTORP3 1Watershed Program, NW Fisheries Science Center, 2725 Montlake Blvd. East, Seattle, WA 98112 E-mail: [email protected] 2Greenland Institute of Natural Resources 3National Research Center for Statistics and the Environment, Box 354323, University of Washington, Seattle, WA 98195 Ratio data, observations in which one random value is divided by another random value, present unique analytical challenges. The best statistical technique varies depending on the unit on which the inference is based. We present three environmental case studies where ratios are used to compare two groups, and we provide three parametric models from which to simulate ratio data. The models describe situations in which (1) the numerator variance and mean are proportional to the denominator, (2) the numerator mean is proportional to the denominator but its variance is proportional to a quadratic function of the denominator and (3) the numerator and denominator are independent. We compared standard approaches for drawing inference about differences between two distributions of ratios: t-tests, t-tests with transformations, permutation tests, the Wilcoxon rank test, and ANCOVA-based tests. Comparisons between tests were based both on achieving the specified alpha-level and on statistical power. The tests performed comparably with a few notable exceptions. We developed simple guidelines for choosing a test based on the unit of inference and relationship between the numerator and denominator. Keywords: ANCOVA, catch per-unit effort, fish density, indices, per-capita, per-unit, randomization tests, t-tests, waste composition 1352-8505 © 2004 Kluwer Academic Publishers 1. Introduction Ratios are used extensively in fields such as environmental science, social science, economics, and zoology. Quantities that are otherwise not comparable can be presented together in a concise and easy to interpret way. Examples of the use of ratios include catch per-unit effort, morphometric indices, per-capita economic indices, and nutrient concen- trations. If we define ratio data as any data in which the quantity of interest is one random variable divided by a second random number, the examples are endless. Analyzing data as ratios can lead to a wide variety of problems including the violation of standard statistical * Corresponding author. 2 Liermann et al. assumptions and, potentially, misinterpretation of results. Yet, ratios are a convenient and intuitive way of summarizing information. They are and will continue to be used in many disciplines. The purpose of this paper is to evaluate the effectiveness of common statistical techniques when used for comparing two distributions of ratios. Ratios are often used to standardize samples. Catch per unit effort is an example of such a situation. To compare the number of animals under two different conditions, it is necessary to control for the area searched, the time searched, or the intensity of the search effort. If the samples are drawn from a population in which the animals follow a uniform distribution in space or time, then any one observation can be thought of as contributing a group of smaller, equally-sized observations. We might think of an observation such as 12 animals in 200 m2 as being equivalent to 200 1 m2 observations, each with an expected value of 12/200 animals. In this situation it is appropriate to combine the observations in each treatment with a weight proportional to the denominator (area, time, or intensity). We call this “per-unit inference”. Per-unit inference is appropriate where the denominator can be broken into discrete units that are similar across observations. We define the other extreme as “whole object inference”. In this case, the researcher does not consider the denominator to be composed of multiple, smaller, identical units. Each observation would then deserve equal weight, no matter the size of the denominator. Whole object inference is more common in applications such as morphometric indices where each ratio reflects the value for one individual. For example, in fish ecology, the ratio of body weight to length is often used as an indicator of fish condition. In many research examples, the scale of inference is clear. In other cases, the scale of inference is a subtle issue. The research question could be framed in either way. The most appropriate analytical methods will depend on which question the researcher wants to answer. The first data example below is an example of such a situation. A second analytical concern with ratio data is whether the expected relationship between the numerator and the denominator is isometric (linear through the origin). This is often true; one would not expect to observe animals in 0 km2. In these cases, there is often an implied dependence and one variable is clearly the denominator. In other cases, the ratio itself has ecological meaning and the relationship between the numerator and denominator may not go through the origin. For example, a pond may have to be greater than a certain size for fish to be present. In this case the relationship between the number of fish and pond area will not go through the origin. These types of ratio models have been less frequently considered in the statistical literature. Investigation of these situations requires graphical analysis in addition to statistical testing because of the multiple ways in which two populations may differ if not constrained to go through the origin. We hypothesize that the best analytical approach will differ between situations that reflect per-unit versus whole object inference and will also depend on the presumed relationship between the numerator and the denominator. Two problems with ratio data have been well treated in the statistical literature already and are not the subject of our analyses. First, several authors have pointed out that interpreting results of analyses based on ratio data can be non-intuitive, potentially leading to unintended inference and incorrect conclusions (Atchley, et al., 1976; Beaupre and Dunham, 1995; Jasienski and Fakhri, 1999; Raubenheimer and Simpson, 1992). They advocate reformulating the hypothesis in terms of the numerator, using the denominator as an explanatory variable. While we agree that inference based on the comparison of ratios requires caution, we argue that ratio-based inference allows for simple yet robust statistical techniques, while use and interpretation of analysis of covariance (ANCOVA) requires a greater degree of statistical sophistication and is vulnerable to more violations of assumptions (Good and Hardin, 2003). In this paper, we focus on ratio-based approaches Random denominators and the analysis of ratio data 3 but include simple ANCOVA methods for comparison. The other well-documented statistical issue is that of spurious correlation (Atchley et al., 1976; Kronmal, 1993; Pearson, 1897). Pearson drew attention to the problem of spurious correlation in 1897 demonstrating that two uncorrelated variables may appear highly correlated when divided by the same covariate. We restrict our discussion in this paper to those situations in which the researcher is simply interested in comparing two populations on the basis of the distributions of X1/Y1 and X2/Y2. In this paper we differentiate between types of ratio problems and consider which of the common methods of inference might be best in each case. We emphasize those techniques that can be easily applied by both statisticians and non-statisticians alike and do not explore multi-step procedures, the role of diagnostics, nor techniques such as likelihood ratio tests that are not available in standard software packages. We apply the approaches to a diverse set of empirical and simulated data and compare the performance of statistical approaches based on both observed versus expected alpha levels and statistical power. We begin by presenting the empirical data sets and the models from which the simulated data are generated. A series of statistical approaches to analyzing ratios are then described. Finally, the empirical and simulated data are used to compare statistical approaches across data sets. We conclude by providing suggestions for appropriate analysis of ratio data. 2. Data We motivate this paper with three examples in which inference about ratio data is desired. The first problem investigates the effectiveness of stream restoration techniques (Roni and Quinn, 2001). Stream restoration is a common tool in the effort to recover endangered salmonids in the Pacific Northwest. The data we use in this paper come from a series of restoration projects (the placement of logs to create pool habitat and cover in streams) in western Oregon and Washington states. The question of interest is whether fish density is different between restored and control streams. Fish numbers are collected in the field by electro-fishing an entire habitat unit (either a pool or a riffle), which might range in size from a few m2 to over 100 m2 (Fig. 1). Logistically, it is impossible to control for pool size. Fish density, calculated as the number of fish per m2, is therefore a ratio with a random denominator. Whether per-unit or whole object inference is most appropriate is a matter of biological importance. Per-unit inference would assume that all m2 are similar. Statistical analysis would consider differences in fish per m2 between treated and control streams. Whole object inference, on the other hand, should be used if all m2 of pool were not equivalent. Inference would then consider differences between densities calculated at the pool scale. How the analyses for these two types of inference might differ is described below. The second example concerns municipal solid waste. Researchers want to answer questions about the distribution of various materials in the wastestream. For example, they might be interested in whether hazardous materials compose a larger fraction of the material in the dry versus the wet season.