The statistical analysis of ecophysiological response curves obtained from experiments involving repeated measures

Physiological ecologists often analyze the responses of physiological or biochemical traits to environmental factors such as temperature, irradiance, water potential, or the concentrations of CO2, O2, and inorganic nutrients. The data for such a response curve typically are gathered by sequential sampling of the same plant or animal, and their analysis should explicitly allow for this repeated-measures design. Unfortunately, the statistical analysis of response curves in ecology generally has either been ignored or incorrectly done. In an effort to encourage rigorous analysis of response data, we address statistical treatment of response curves and illustrate the correct alternatives that are available. Four different statistical methods for analyzing response curves are considered: analysis of variance with repeated measures (ANOVAR), multivariate analysis of variance with repeated measures (MANOVAR), a nonparametric split-plot analysis (NP split-plot) and parametric comparison of models fitted to the data by nonlinear regression. Analyses of the CO"2 dependence of photosynthesis in the C4 grass Echinochloa crus-galli following chilling are used as examples of these different methods. ANOVAR, potentially the most powerful analysis, makes stringent assumptions about the variance-covariance structure of the data. Within limits these assumptions can be relaxed and a corrected significance level used. When the variance-covariance structure badly violates the ANOVAR assumptions, MANOVAR or NP split-plot are viable alternatives. In physiological ecology, however, the use of MANOVAR frequently is limited by small sample sizes and the tendency for the number of levels of the treatment factor to exceed the sample size. Greater attention to experimental design can avoid this problem. The NP split-plot is based on simple assumptions and could be widely used. The comparison of curves fitted by nonlinear regression is also distribution free and provides an interesting alternative when the responses are amenable to fitting. For any of these analyses to be viable the thoughtful choice of experimental protocols and design is essential.