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

Published at December 12, 2025 ·  8 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 \(X\) and \(Z\), with standard errors respectively equal to \(\sigma_X\) and \(\sigma_Z\) and a covariance of \(\sigma_{XZ}\) we can build a linear combination as follows:

\[ Y = f(X,Z) = aX + bZ + c\]

where \(a\), \(b\) and \(c\) are three numeric constants. The standard error for this combination can be obtained as:

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:

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. Therefore, now and then, I have to build some helper functions, to perform some specific tasks; I usually share these functions within the ‘aomisc’ package, that is available on github (see this link).