Step 1: Getting started with ‘Quarto’

First, I used the RStudio Quick Start (File|New project) to create a Quarto blog in a separate project folder from my original blogdown website folder. This was very easy!

Step 2: Where do I put my posts?

This was also easy and logic: blog posts, as ‘.qmd’ sorce files need to be put in the ‘posts’ folder.

Step 3: Where do I put my non-post pages?

My blog also contains static pages, such as articles, tutorials, a project page, links and other stuff. I put the source code for these pages, as .qmd’ files in the root project folder so that they could be rendered as ‘.html’ files in the root folder of the website. Tutorials, books or other projects rendered as self-contained website folders, were also put in the root project folder. All these pages which are not under the ‘posts’ folder are not listed together with the other post in the home page of the blog, but they are easy to link, considering that Quarto directly translates the source folder structure into the website URL structure. However, static folders need to be listed as ‘resources’ in the ‘_quarto.yml’ configuration file, as shown below.

project:
  type: website
  resources:
    - folder1
    - folder2
    - folder3

In this vein, I have also added figures and other materials, which I need to link in webpages and posts, to the ‘Figure’ folder, within the root project folder, by using the markdown syntax that follows.

![caption](../Figures/logo.jpg)

Alternatively, the include_graphics() function in the knitr package can be used with increased flexibility.

knitr::include_graphics("../Figures/logo.jpg")

Step 4: Support for literature citations

Whenever I need to cite literature references in a post or other page, I simply need to refer to the literature file (‘.bib’) and the the ‘.csl’ file in the YAML header of the whole project (_quarto.yml), e.g., :

bibliography: myFile.bib
csl: myStyle.csl

Lately, I can cite by using pandoc-citeproc.

Step 5: Latex equations

While this was initially a problem with ‘blogdown’, latex equations are natively supported and included in ‘quarto’, as shown below.

$$Y = a \, \exp{\left(-b \, t\right)}$$

Step 6: Creating entries for the navigation bar

This was very simple, as the navigation bar is managed within the ‘_quarto.yml’ configuration file. For example, for this blog, the navigation bar is defined as follows.

website:
  navbar:
    left:
      - aboutThis.qmd
      - about.qmd
      - text: "Posts"
        href: index.html
      - tutorials.qmd
      - Rpackages.qmd
      - links.qmd
    right:  
      - icon: github
        href: https://github.com/OnofriAndreaPG
      - icon: bluesky
        href: https://bsky.app/profile/statforbiology.bsky.social

Step 7: Adding an RSS Feed

This was super-easy! An RSS is produced by default, although I wanted to have a second feed containing only posts in one specific category (‘R’). What I had to do was to modify the YAML matter on the ‘index.qmd’ file, which was already available whitin the root project directory, as shown below.

listing:
  contents: posts
  categories: true
  sort: "date desc"
  type: default
  sort-ui: false
  filter-ui: false
  feed:
    categories:
      - R

My blog is listed on R Bloggers, which receives the feed from ‘/tags/r/index.xml’, while my feed for the R category is now produced as ‘/index-r.xml’. I did not want to change my subscription to R bloggers, and thus I used the automated script execution to have R copy the RSS feed from ‘/index-r.xml’ to ‘/tags/r/index.xml’, so that the second file is an exact copy of the first one. To do so I had to add the following line to the ‘_quarto.yml’ configuration file:

project:
  type: website
  post-render:
    - Rscript copy-feed.R

Finally I crated the R script file ‘copy-feed.R’ in the root project folder, containing the following functions:

dir.create("_site/tags/r", recursive = TRUE, showWarnings = FALSE)

file.copy(
  "_site/index-R.xml",
  "_site/tags/r/index.xml",
  overwrite = TRUE
)

Step 9 - Google Analytics (6/2/2019)

Also for the new blog, I would like to track the visits I receive. I went to Google Analytics to recover my tracking ID and I added such an information to the ’_quarto.yml’ configuration file.

website:
  google-analytics: "G-myID"

Curve shapes

Functions are often classified according to the shape they exhibit when plotted on an x–y graph. This approach is adopted, for example, in Ratkowsky (1990), and the classification presented below is largely based on that work.

• Polynomials
  1. Straight line function
  2. Quadratic polynomial function
• Concave/Convex curves (no inflection)
  1. Exponential function
  2. Asymptotic function / Negative exponential function
  3. Power curve function
  4. Logarithmic function
  5. Rectangular hyperbola
• Sigmoidal curves
  1. Logistic function
  2. Gompertz function
  3. Modified Gompertz function
  4. Log-logistic function
  5. Weibull (type 1) function
  6. Weibull (type 2) function
• Curves with maxima/minima
  1. Bragg function
  2. Lorentz function
  3. Beta function

have chosen to include a relatively large number of functions, so this post is necessarily rather long. To make navigation easier, you can first inspect Figure 1 to identify the type of curve of interest and then use the links above to jump directly to the corresponding section.

Throughout this post, denotes the predictor, the response, and the letters of the Roman alphabet (, , , …) the model parameters, which determine the shape of the function.

Polynomials

Polynomials form a class of their own because their flexible shapes allow them to approximate a wide variety of biological processes, at least over a restricted range of the predictor. The mathematical tractation is very simple, but their main limitation is that they cannot describe asymptotic processes, which are extremely common in biology. Moreover, as the polynomial degree increases, their shapes become increasingly complex and often lack a meaningful biological interpretation.

D(expr = expression(a + b*X + c*X^2), name = "X")

b + c * (2 * X)

Concave/Convex curves (no inflection points)

D(expression(a * exp(k * X)), "X")

a * (exp(k * X) * k)

D(expression(a - (a - b) * exp (- m * X)), "X")

(a - b) * (exp(-m * X) * m)

D(expression(a * X^b), "X")

a * (X^(b - 1) * b)

Logarithmic function

This function is linear in :

Because of the logarithmic transformation, the function is only defined for . The parameter determines the shape of the curve. When , the response increases with and the curve is concave down. Conversely, when , decreases as increases and the curve is concave up, as shown in Figure 1.

The interpretation of the parameters is fairly straightforward. The parameter is the value of the response when . Changing while keeping constant simply shifts the curve vertically without altering its shape, so that the difference between any two curves remains constant for every value of (Fig. 2).

The parameter controls the slope of the curve and, more specifically, it is equal to the slope at , as can be seen from the expression for the first derivative:

D(expression(a + b*log(X)), "X")

b * (1/X)

Changing while keeping constant produces curves that all intersect at (Fig. 2).

In biology, logarithmic functions are used, for example, to describe species–area relationships in ecology, enzyme kinetics in biochemistry, and sensory perception in neurobiology.

D(expression( (a*X) / (b + X) ), "X")

a/(b + X) - (a * X)/(b + X)^2

Sigmoidal curves

Sigmoidal curves are S-shaped and they may be increasing, decreasing, symmetric or non-symmetric around the inflection point. They are parameterised in countless ways, which may often be confusing. I will show some widespread parameterisations, that are very useful for bioassays or growth studies. All curves can be turned from increasing to decreasing (and vice-versa) by reversing the sign of the parameter.

Curves with maxima/minima

It is sometimes necessary to describe phenomena where the variable reaches a maximum value at a certain level of the variable, and drops afterwords. For example, growth or germination rates are higher at optimal temperature levels and lower at supra-optimal or sub-optimal temperature levels. Another example relates to bioassays: in some cases, low doses of toxic substances induce a stimulation of growth (hormesis), which needs to be described by an appropriate model. The second order (and higher order) polynomial funcion we have seen earlier is capable of accounting for maxima/minima, but there are a few other interesting functions that may turn out useful in some circumstances.

Conclusions

Here we are; I have discussed more almost 30 functions, which are commonly used to model biological processes. These functions can be found in several different parameterisations and flavours; they can also be modified and combined to suit the needs of several disciplines in biology, chemistry and so on. However, this will require another post.

Thanks for reading! And … don’t forget to check out my new book!

Prof. Andrea Onofri
Department of Agricultural, Food and Environmental Sciences
University of Perugia (Italy)
Send comments to: andrea.onofri@unipg.it

Further readings

1. Miguez, F., Archontoulis, S., Dokoohaki, H., Glaz, B., Yeater, K.M., 2018. Chapter 15: Nonlinear Regression Models and Applications, in: ACSESS Publications. American Society of Agronomy, Crop Science Society of America, and Soil Science Society of America, Inc.
2. Ratkowsky, D.A., 1990. Handbook of nonlinear regression models. Marcel Dekker Inc., New York, USA.
3. Ritz, C., Jensen, S. M., Gerhard, D., Streibig, J. C. (2019) Dose-Response Analysis Using R. CRC Press

Step 1 - The MathJax problem

I added a post to the ‘content’ directory. This was a trial post, with an equation in latex code. I immediately noted that there was no support to MathJax. Surfing the net, I found a solution for this. These were my steps:

1. The folder ‘themes/hugo-blog-jeffprod/layouts/partials’ contains the template file ‘header.html’. The blogdown documentation suggests not to directly modify this file. Thus I created a copy on the ‘layouts/partials/’ folder in my root directory. This folder did not exist at the beginning, so I created it.

2. I opened the newly created file ‘layouts/partials/header.html’ and added the following lines within the

   …

   section:

<script type="text/javascript"
  src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>

Equations seem to render ok, now.

Step 2 - Change the blog name and author

I modified the ‘config.toml’ file in the root directory and added name and author of the site. An easy step…

Step 3 - Clean the example files

There were a lot of example posts in the ‘content’ directory. I removed them all, with no problems. I also had to remove the content of the ‘static’ folder. These files were not needed, because they belonged to the example posts.

Step 4 - Where do I put my post?

New posts need to be put in the ‘content/post’ directory. In order to add a post, I am using the function ‘blogdown::new_post_addin()’ (I like working in console…), which creates a correct skeleton. Blog posts are listed under the home page of my blog, sorted by date and by tags.

Step 5 - Where do I put my non-post pages?

I did not intend to create a pure blog. I wanted also to add static pages, such as articles, tutorials, a project page, links and other stuff. I noted that, when .md or .Rmd files are left in the ‘content’ folder, a folder with the same name is created in ‘public’ and the post is rendered within that folder. Therefore, I put a ‘links.md’ file in the ‘content’ folder, containing a few links to other web pages of interest. This file is not rendered as a post and it is not visible in my home page. However, a folder named ‘link’ is created in the public directory, containing the ‘links.md’ file, rendered as ‘index.html’ file. Therefore, I can link this file by using the ‘/link/’ address.

Step 6 - How do I create further entries for the navigation bar?

I just created the ‘/link/index.html’ file in the ‘public’ directory. I would like the reader to access this web page by clicking on an appropriate link at the top of the home-page, next to the ‘Home’ and ‘About me’ links. In order to add this link, I had to edit again the ‘layouts/partials/header.html’ file. In the ‘nav’ section, there was the line:

<a class="navbar-item" href="{{ "about" | absURL }}">About me</a>

This line creates the ‘About me’ link on the navigation bar. Below this line, I added another line:

<a class="navbar-item" href="{{ "links" | absURL }}">Links</a>

I guess I can use this approach to redirect readers to other web pages of interest.

Step 7 - I need to put some order in the ‘content’ folder’ (4/1/2019)!

After adding a few non-post pages to my content folder I wanted to avoid too much confusion. I discovered that I can add folders to the ‘content’ folder. They are rendered as such in the public directory. For example, I created an ‘article’ folder and put a ‘page1.md’ file inside it. After rendering, a folder ‘article/page1/’ is created in the ‘public’ folder, containing the ‘index.html’ page, which is the rendered version of the ‘page1.md’ file.

I think I start feeling slightly more confident with blogdown!

Step 8 - Literature support (7/1/2019)!

This problem took me a while to solve. Lately, I discovered that, in order to be able to insert citations, the post file needs to be ‘.Rmd’ and not ‘.md.’ Furthermore, the literature file (‘.bib’) and the ‘.csl’ file need to be in the ‘/content/’ folder. A reference to both files need to be put in the YAML header, e.g., :

bibliography: myFile.bib
csl: myStyle.csl

Step 9 - Google Analytics (6/2/2019)

I would like to track the visits I receive. To do so I enabled Google Analytics and got my tracking id (see here). I also worked on my ‘config.toml’ file and added the following statement.

GoogleAnalytics = "UA-my_ID"

In contrast to what I expected, Google Analytics did not work and no hits were recorded for my blog. Lately, I worked on the ‘layouts/partials/header.html’ and added the following code, right after the ’

’ tag (just replaced ‘UA-my_ID’ with my tracking id).

<!-- Global Site Tag (gtag.js) - Google Analytics -->
<script async src="https://www.googletagmanager.com/gtag/js?id=UA-my_ID"></script>
<script>
  window.dataLayer = window.dataLayer || [];
  function gtag(){dataLayer.push(arguments);}
  gtag('js', new Date());

  gtag('config', 'UA-my_ID');
</script>

Now Google Analytics works ok!

Step 10 - Add an RSS Template

I wanted my RSS feed to contain the whole post and not only a description. I did the following:

1. Took the default template from this site.
2. Replaced ‘{{ .Summary | html }}’ with ‘{{ .Content | html }}’
3. Saved the file as ‘RSS.xml’under the ’layouts’ directory.

Step 11 - Further info on Google Analytics (10/12/2020)

A very kind reader (Emma) emailed me that there have been some relevant updates for Google analytics (v4). She pointed me to the following guide: https://www.websiteplanet.com/blog/ultimate-beginners-guide-google-analytics/. This is useful, to have a better comprehension of the whole Google Analytics platform, just in case someone is interested.

Step 12 - Adding an image to a blog post (19/10/2021)

It took me awhile to figure out how to include a figure in a blog post. Lately I also discovered that there is a blogdown add-in to help with this job (see this post here), but I wanted to master the whole process. I solved the problem like this:

1. created an ‘img’ folder within the ‘static’ folder in my blogdown project directory.
2. Put all the images I wanted to include in blog posts in that folder
3. Use the markdown code:

![caption](/img/figFileName.png)

Be careful: the ‘slash’ character at the beginning of file path is fundamental, otherwise the code does not work. Of course, you can use html tags to center your figure:

<p style="text-align:center">
![](/_Figures/Stat_drcte_3-method-data.png){width=90%}
</p>

Step 13 - Changing the title of the home page (14/12/2023)

It may seem stupid, but it took me awhile to figure out how I could possibly change the title for the homepage. Lately, in the cointent folder I discovered a file named ‘_index.Rmarkdown’, containing only the YAML header with

---
title: Homepage title
---

This is where the homepage title is defined. As far as I could see, all other test in the body of this file is neglected.

Which formula should we use?

The answer is easy to remember.

1. If we know the entire population, the variance is the average squared deviation from the population mean, so we divide by .
2. If we only have a random sample and our goal is to estimate the population variance, we divide by . Otherwise, our estimate will be systematically too small.

Thanks for reading! And … don’t forget to check out my new book!

Prof. Andrea Onofri
Department of Agricultural, Food and Environmental Sciences
University of Perugia (Italy)
Send comments to: andrea.onofri@unipg.it

The formal proof (look how clever I am!)

We have a population with mean equal to and variance equal to . Let’s take a sample with its mean and . For the population, we can write:

We already see that the sum of squared residuals is higher if we take instead of , unless , i.e. . From the above, we derive

If we consider the first equation above (variance for the population) we have

We see that the rightmost term of this equation ( is the sum of squared distances for the sampling distribution of the mean. That is times the squared standard error. Therefore:

and

Here we go! Indeed: