Getting the Absolute/Relative Growth Rate from growth curves

Published at November 27, 2025 ·  7 min read

Yesterday, a colleague of mine pointed me to the article “How to fit nonlinear plant growth models and calculate growth rates: an update for ecologists” (Paine et al., 2012). It addresses a relevant topic: many plant scientists are involved in growth analyses and need to determine Absolute Growth Rates (AGRs) and Relative Growth Rates (RGRs).

The main point made by Paine et al. is that we can use the observed data to fit a growth model via nonlinear regression and then calculate model-derived growth rates together with their standard errors. In principle, the process is straightforward: we select a suitable growth model to predict biomass at any given time \(t\); the AGR at time \(t\) is the derivative of the selected growth function with respect to time, and the RGR is simply the AGR at time \(t\) divided by the biomass at that same time point.

Why are derivatives important in life? A case-study with nonlinear regression

Published at November 26, 2025 ·  7 min read

In general, undergraduate students in biology/ecology courses tend to consider the derivatives as a very abstract entity, with no real usefulness in the everyday life. In my work as a teacher, I have often tried to fight against such an attitude, by providing convincing examples on how we can use the derivatives to get a better understanding about the changes on a given system.

In this post I’ll tell you about a recent situation where I was involved with derivatives. A few weeks ago, a colleague of mine wrote me to ask the following question (I’m changing it a little, to make it, hopefully, more interesting). He asked: “I am using a power curve to model how the size of the sampling area affects species richness. How can I quantify my knowledge gain?”. This is an interesting question, indeed, although I feel I should provide you with some background information.

Field Research methods in Agriculture

Published at November 25, 2025 ·  2 min read

Hi everybody, I have exciting news!

After a few months of silence, I’m thrilled to finally share the reason why: I’ve been working intensely on a new book project, which has taken up most of my spare time. And now… the book is out!

This book is titled ‘Field Research Methods in Agriculture’ and it is published by Springer Nature. It offers a clear, accessible, and practical introduction to experimental design and basic data analysis for field experiments in agriculture and related disciplines. It’s specifically designed for students, researchers, and practitioners who want to strengthen their methodological skills without getting lost in heavy mathematics.

Dealing with correlation in designed field experiments: part II

Published at February 10, 2025 ·  11 min read

With field experiments, studying the correlation between the observed traits may not be an easy task. For example, we can consider a genotype experiment, laid out in randomised complete blocks, with 27 wheat genotypes and three replicates, where several traits were recorded, including yield (Yield) and weight of thousand kernels (TKW). We might be interested in studying the correlation between those two traits, but we would need to face two fundamental problems:

A trip from variance-covariance to correlation and back

Published at January 24, 2025 ·  6 min read

The variance-covariance and the correlation matrices are two entities that describe the association between the columns of a two-way data matrix. They are very much used, e.g., in agriculture, biology and ecology and they can be easily calculated with base R, as shown in the box below.

data(mtcars)
matr <- mtcars[,1:4]

# Covariances
Sigma <- cov(matr)

# Correlations
R <- cor(matr)

Sigma
##              mpg        cyl       disp        hp
## mpg    36.324103  -9.172379  -633.0972 -320.7321
## cyl    -9.172379   3.189516   199.6603  101.9315
## disp -633.097208 199.660282 15360.7998 6721.1587
## hp   -320.732056 101.931452  6721.1587 4700.8669
R
##             mpg        cyl       disp         hp
## mpg   1.0000000 -0.8521620 -0.8475514 -0.7761684
## cyl  -0.8521620  1.0000000  0.9020329  0.8324475
## disp -0.8475514  0.9020329  1.0000000  0.7909486
## hp   -0.7761684  0.8324475  0.7909486  1.0000000

It is useful to be able to go back and forth from variance-covariance to correlation, without going back to the original data matrix. Let’s consider that the variance-covariance of the two variables X and Y is:

How do we combine errors? The linear case

Published at November 22, 2024 ·  7 min read

In our research work, we usually fit models to experimental data. Our aim is to estimate some biologically relevant parameters, together with their standard errors. Very often, these parameters are interesting in themselves, as they represent means, differences, rates or other important descriptors. In other cases, we use those estimates to derive further indices, by way of some appropriate calculations. For example, think that we have two parameter estimates, say Q and W, with standard errors respectively equal to \(\sigma_Q\) and \(\sigma_W\): it might be relevant to calculate the amount:

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

Published at November 22, 2024 ·  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:

Plotting weather data with ggplot()

Published at June 6, 2024 ·  7 min read

Very often, we agronomists have to deal with weather data, e.g., to evaluate and explain the behaviour of genotypes in different environments. We are very much used to representing temperature and rainfall data in one single graph with two y-axis, which gives a good immediate insight on the weather pattern at a certain location. Unfortunately, I had to discover that doing such graphs with ggplot() is not a straightforward task.

Here is why I don't care about the Levene's test

Published at March 15, 2024 ·  5 min read

During my stat courses, I never give my students any information about the Levene’s test (Levene and Howard, 1960), or other similar tests for homoscedasticity, unless I am specifically prompted to do so. It is not that I intend to underrate the tremendous importance of checking for the basic assumptions of linear model! On the contrary, I always show my students several methods for the graphical inspection of model residuals, but I do not share the same aching desire for a P-value, that most of my colleagues seem to possess.

Pairwise comparisons in nonlinear regression

Published at February 23, 2024 ·  8 min read

Pairwise comparisons are one of the most debated topic in agricultural research: they are very often used and, sometimes, abused, in literature. I have nothing against the appropriate use of this very useful technique and, for those who are interested, some colleagues and I have given a bunch of (hopefully) useful suggestions in a paper, a few years ago (follow this link here).

According to the emails I often receive, there might be some interest in making pairwise comparisons in linear/nonlinear regression models. In particular, whenever we have grouped data and we have fitted the same model to each group, we might like to compare the groups, to state whether the regression lines/curves are significantly different from each other. To this aim, we could consider two approaches: