Analysis of variance in soil research: Examining the assumptions. (17th May 2019) - Webster, Richard; Lark, Richard M.
Abstract : Three assumptions underly safe inference from an analysis of variance of soil data from designed experiments and surveys. They are (a) that the residuals are normally distributed, (b) that the within‐groups variances are equal and (c) that effects of treatments and blocks are additive. Small departures from these assumptions are unlikely to affect conclusions because the analysis is robust. Large departures could have serious consequences, however. The most common departures in the real world of the soil are strongly positively skewed distributions of the observed values, many of which are approximately lognormal. In those situations, transformation of the data to logarithms should lead to analyses from which inferences are sound and to confidence limits on estimates. Mean values can be transformed back to the original scales to aid understanding. Measurements of nitrogen in the soil of the Broadbalk wheat experiment show how transformations of the strongly skewed data to logarithms are sufficient to satisfy the assumptions underlying the analysis. Those of nitrogen in the soil of Cashmore Field at Silsoe required the somewhat more complex Box–Cox transformation to satisfy the assumptions and lead to sensible conclusions. The statistical distributions in pH, already a logarithm, in the soil in sandy drift and chalky boulder clay at Broom's Barn Farm differed so strongly that no transformation would satisfy the assumptions. Non‐parametric tests are available, but they should be used only if no suitable transformation can be found. Highlights: Safe inference from the analysis of variance is based on three conditions. The conditions are normal distribution of residuals, homogeneity of variances and additivity of effects. Most departures from the conditions in the real world of the soil are caused by positively skewed distributions. Transformation to logarithms will often deal with departures from these conditions.