Regression analysis in the context of designed experiments: Neglect not thy opportunity to test for position and parallelism

Applied biologists often look for relationships between response variables of primary interest and quantitative explanatory variables. For example, yield measures are considered in terms of concentration of nutrients, disease measures in terms of rate of application of pathogenic chemicals, and expression of genes in terms of levels of stress perhaps measured in days from withholding a vital necessity for growth such as water. In the context of a designed experiment, the levels of the explanatory variable will have been specifically chosen with replicates (multiple experimental units) at each level to consider variability in response over the levels. In other studies, collected data comprising pairs of observations will be investigated to consider the relationship between them without necessarily having replication at each level. Consequently, publications in Annals of Applied Biology (AAB) and other biological journals often include scatter plots depicting such relationships complete with lines, or curves, to show how the observed trend in the data can be explained in terms of a mathematical relationship, fitted using the statistical technique of linear, or nonlinear, least squares regression. In doing this however, it is important that the statistical quality of the proposed relationship is assessed. Firstly, in fact, it should be asked whether a fitted (e.g., linear) relationship is actually required. It can be sufficient for the purposes of the study simply to present the scatter-plot, Pearson correlation coefficient, r, number of pairs of values (n) and P-value for the F-test of the correlation coefficient. In other words, you do not need a model of the relationship if the aims of the study do not include estimating and assessing the rate of change in a response variable with respect to a unit change in an explanatory variable. Next, if modelling is deemed to be required, is a simple linear relationship adequate to explain variation in the response, or should the statistical significance of the addition of a nonlinear (quadratic) term be tested (using the F-test)? And then, if there is nonlinearity, should an empirical nonlinear model (Ratkowsky, 1990) be fitted which could provide a better biological and statistical explanation involving estimated parameters that have meaningful interpretation, as compared to the coefficients of a polynomial model? Finally, are there different treatments (e.g., varieties, species, etc.), perhaps in a factorial treatment structure, that should be considered in the modelling: does the proposed relationship between the response and the explanatory variable vary depending on the treatment? This last question is the main focus of this editorial although answers to the other questions will also be encompassed. An example for discrete response data is the comparison of five wheat cultivars for their resistance to soil-borne cereal mosaic virus in three different soils (a five by three factorial set), given increasing dilution of the soil inocula using sand (Lyons et al., 2009). Four serial dilutions were used to give the explanatory variable and four seedlings were planted per pot, providing a binomial response for infection. Three replicate pots per cultivar by soil by dilution combination were used. The experimental unit was the pot. These response data required generalised linear modelling (GLM) invoking a logit link function (McCullagh & Nelder, 1989). Here, we need to ask whether the relationship between the binomial response and the explanatory variable (dilution) varied depending on the treatment (cultivar). An example for continuous response data is the comparison of eight Arabidopsis genotypes (mutants of the GA 20-OXIDASE gibberellin biosynthesis gene family and Col-0 control) for lengths of floral organs (petals, sepals, pistils and stamens) with and without gibberellin (an eight by two factorial set), given flower position (number, 1–10) along the primary inflorescence as the explanatory variable (Plackett et al., 2018). Four replicate plants were used per treatment combination. The experimental unit was the flower; these being statistically independent along the inflorescence—see Powers & Kozak (2019) on issues of independence. For each floral organ separately, these response data required linear regression modelling in the context of designed experiments (e.g., Draper & Smith, 1998), checking for evidence of any nonlinearity by addition of a quadratic term. Again here, we need to ask whether the relationship between the response (length) and the explanatory variable (flower position) varied depending on the treatment (genotype). As a statistical editor for Annals of Applied Biology (AAB), I have noticed that opportunities for comprehensive regression analysis in the context of designed experiments have sometimes been neglected. And the implication here is that more biological information could have been extracted, given more comprehensive statistical analysis. Alternatively, simple linear regression may have been done, but with no interpretation of the modelling, begging the question as to whether it was actually required, or whether a correlation analysis would have been sufficient. Moreover, when a relationship between a response and an explanatory variable has been observed, it can be that different treatments will have a different form of the same relationship. In the simplest case of an observed linear relationship and a single treatment factor with two levels, there is a need to test the differences between the two treatments as a whole, that is, the position of the two lines on the 2D Cartesian plot, and then whether the rates of change for the two treatments differ with respect to the explanatory variable, that is, Received: 2 June 2021 Accepted: 5 June 2021