The meta-analysis of response ratios in experimental ecology - Meta-Analysis in Ecology

Ecology, June, 1999 by Larry V. Hedges, Jessica Gurevitch, Peter S. Curtis

INTRODUCTION

The increasingly large body of studies in experimental ecology has led to interest in methods for summarizing this evidence to reach general conclusions. Meta-analysis, the formal application of quantitative methods to summarize evidence across studies, has recently been introduced to ecology (Gurevitch et al. 1992, Arnqvist and Wooster 1995). In a meta-analysis, the result of each independent experiment is usually expressed as an index of effect; these effect estimates are then combined across studies to produce a summary of the findings. Subgroupings of experiments may be examined separately to determine whether experimental results differ across biologically meaningful groupings of experiments (e.g., whether the effects of a manipulation such as elevated atmospheric C[O.sub.2] on total biomass differ across plant taxa).

Meta-analyses can only provide meaningful summaries if the effect-size index used is a meaningful summary of any one experiment. Several recent meta-analyses in ecology have used the standardized difference between means (the difference between the mean of the treated group and a control group, divided by the within-group standard deviation), also called a "d index," as an index of effect. Although its statistical properties are well understood, and a substantial set of meta-analytic procedures using the d index are available (see Hedges and Olkin 1985), it is not always a meaningful way to summarize experiments in ecology (Osenberg et al. 1997).

The response ratio, the ratio of some measured quantity in experimental and control groups, is commonly used as a measure of experimental effect because it quantifies the proportionate change that results from an experimental manipulation. Examples of such ratios include relative competition intensity, relative yield, and relative crowding coefficient. The response ratio is also closely related to metrics that quantify the effects of treatments on per capita rates of change (Osenberg et al. 1997, 1999, Laska and Wootton 1998, Downing et al. 1999). The response ratio has been used informally in reviews to describe the effects of predation in streams (Cooper et al. 1990, Englund et al. 1999), the effects of grazers on algal biomass (Sarnelle 1992), and the response of plant biomass to increased C[O.sub.2] levels (Kimball 1983, Poorter 1993, Gunderson and Wullschleger 1994). To take full advantage of the statistical techniques available for statistical inference in meta-analysis, it is necessary to know the sampling properties of the effect-size index used. To fully utilize meta-analytic techniques, however, it is necessary to understand the sampling properties of the response ratio. The purpose of this article is to provide the information necessary to carry out meta-analytic procedures using response ratios and demonstrate their use with an important set of ecological data. We use data from recent experiments on the effects of increased ambient C[O.sub.2] on growth of woody plants (Curtis and Wang 1998) to illustrate the use of response ratios in meta-analysis.

THE RESPONSE RATIO AS AN EFFECT INDEX

Consider a set of experiments, each of which compares dry mass gain in a replicated control group in which plants are grown with ambient levels of C[O.sub.2] with an experimental group in which plants are grown with approximately double the ambient level of C[O.sub.2]. We denote the mean and standard deviation of the outcome in the experimental group by [Mathematical Expression Omitted] and S[D.sub.E], the sample size (number of replicates) by [n.sub.E], and the mean, standard deviation and the sample size of the outcome in the control group by [Mathematical Expression Omitted], [S.sub.DC] and [n.sub.C], respectively. Then, the (sample) response ratio, [Mathematical Expression Omitted] is an estimate of the ratio, [Rho], of the population means. However, it is desirable to perform statistical analyses in the metric of the natural logarithm of the response ratio, [Mathematical Expression Omitted], for two reasons. The first is that the logarithm linearizes the metric, treating deviations in the numerator the same as deviations in the denominator. That is, while the ratio is affected more by changes in the denominator (especially when the denominator is small), the log ratio is affected equally by changes in either numerator or denominator. The second reason is that the sampling distribution of R is skewed, and the distribution of L is much more normal in small samples than that of R (see Appendix A).

If [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are normally distributed and [Mathematical Expression Omitted] is unlikely to be negative, then L is approximately normally distributed with mean approximately equal to the true log response ratio and variance, v, approximately equal to

[Mathematical Expression Omitted]. (1)

If the variances in the treatment and control groups are approximately equal then the variance can be computed using the pooled within-group standard deviation [SD.sub.P] as