Nonlinear combinations of model parameters in regression

Published at January 9, 2020 ·  11 min read

Nonlinear regression plays an important role in my research and teaching activities. While I often use the ‘drm()’ function in the ‘drc’ package for my research work, I tend to prefer the ‘nls()’ function for teaching purposes, mainly because, in my opinion, the transition from linear models to nonlinear models is smoother, for beginners. One problem with ‘nls()’ is that, in contrast to ‘drm()’, it is not specifically tailored to the needs of biologists or students in biology....

Fitting 'complex' mixed models with 'nlme': Example #4

Published at September 13, 2019 ·  11 min read

Testing for interactions in nonlinear regression Factorial experiments are very common in agriculture and they are usually laid down to test for the significance of interactions between experimental factors. For example, genotype assessments may be performed at two different nitrogen fertilisation levels (e.g. high and low) to understand whether the ranking of genotypes depends on nutrient availability. For those of you who are not very much into agriculture, I will only say that such an assessment is relevant, because we need to know whether we can recommend the same genotypes, e....

Fitting 'complex' mixed models with 'nlme': Example #2

Published at September 13, 2019 ·  9 min read

A repeated split-plot experiment with heteroscedastic errors Let’s imagine a field experiment, where different genotypes of khorasan wheat are to be compared under different nitrogen (N) fertilisation systems. Genotypes require bigger plots, with respect to fertilisation treatments and, therefore, the most convenient choice would be to lay-out the experiment as a split-plot, in a randomised complete block design. Genotypes would be randomly allocated to main plots, while fertilisation systems would be randomly allocated to sub-plots....

Fitting 'complex' mixed models with 'nlme'. Example #1

Published at August 20, 2019 ·  9 min read

The environmental variance model Fitting mixed models has become very common in biology and recent developments involve the manipulation of the variance-covariance matrix for random effects and residuals. To the best of my knowledge, within the frame of frequentist methods, the only freeware solution in R should be based on the ‘nlme’ package, as the ‘lmer’ package does not easily permit such manipulations. The ‘nlme’ package is fully described in Pinheiro and Bates (2000)....

Germination data and time-to-event methods: comparing germination curves

Published at July 20, 2019 ·  11 min read

Very often, seed scientists need to compare the germination behaviour of different seed populations, e.g., different plant species, or one single plant species submitted to different temperatures, light conditions, priming treatments and so on. How should such a comparison be performed? Let’s take a practical approach and start from an appropriate example: a few years ago, some collegues studied the germination behaviour for seeds of a plant species (Verbascum arcturus, BTW…), in different conditions....

Survival analysis and germination data: an overlooked connection

Published at July 2, 2019 ·  16 min read

The background Seed germination data describe the time until an event of interest occurs. In this sense, they are very similar to survival data, apart from the fact that we deal with a different (and less sad) event: germination instead of death. But, seed germination data are also similar to failure-time data, phenological data, time-to-remission data… the first point is: germination data are time-to-event data. You may wonder: what’s the matter with time-to-event data?...

Stabilising transformations: how do I present my results?

Published at June 15, 2019 ·  5 min read

ANOVA is routinely used in applied biology for data analyses, although, in some instances, the basic assumptions of normality and homoscedasticity of residuals do not hold. In those instances, most biologists would be inclined to adopt some sort of stabilising transformations (logarithm, square root, arcsin square root…), prior to ANOVA. Yes, there might be more advanced and elegant solutions, but stabilising transformations are suggested in most traditional biometry books, they are very straightforward to apply and they do not require any specific statistical software....

Genotype experiments: fitting a stability variance model with R

Published at June 6, 2019 ·  8 min read

Yield stability is a fundamental aspect for the selection of crop genotypes. The definition of stability is rather complex (see, for example, Annichiarico, 2002); in simple terms, the yield is stable when it does not change much from one environment to the other. It is an important trait, that helps farmers to maintain a good income in most years. Agronomists and plant breeders are continuosly concerned with the assessment of genotype stability; this is accomplished by planning genotype experiments, where a number of genotypes is compared on randomised complete block designs, with three to five replicates....

How do we combine errors, in biology? The delta method

Published at May 25, 2019 ·  7 min read

In a recent post I have shown that we can build linear combinations of model parameters (see here ). For example, if we have two parameter estimates, say Q and W, with standard errors respectively equal to \(\sigma_Q\) and \(\sigma_W\), we can build a linear combination as follows: \[Z = AQ + BW + C\] where A, B and C are three coefficients. The standard error for this combination can be obtained as:...

Dealing with correlation in designed field experiments: part II

Published at May 10, 2019 ·  16 min read

With field experiments, studying the correlation between the observed traits may not be an easy task. Indeed, in these experiments, subjects are not independent, but they are grouped by treatment factors (e.g., genotypes or weed control methods) or by blocking factors (e.g., blocks, plots, main-plots). I have dealt with this problem in a previous post and I gave a solution based on traditional methods of data analyses. In a recent paper, Piepho (2018) proposed a more advanced solution based on mixed models....