### More than semantics

As Gordon Smyth mentions in the comments, it is more than just semantics. Due to the parameter $\phi_1$ the logistic growth model is not exactly equal to the logistic function used in logistic regression.

Compare

$$\phi_1/\lbrace 1+\text{exp}[-(t-\phi_2)/\phi_3]\rbrace$$

with

$$1/\lbrace 1+\text{exp}[-(t-\phi_2)/\phi_3]\rbrace$$

### Logistic regression and fractions

[Logistic regression](https://en.m.wikipedia.org/wiki/Logistic_regression#Logistic_model) deals with $\phi_1=1$. This relates to *fractions* between zero and one. The example from the book is a growth model and a different setting.

### Not all fractions are logistic regression

Sidenote: I am saying that logistic regression relates to fractions, but not all fractions relate to logistic regression.

When we deal with fractions then we still might have $\phi_1 \neq 1$ for instance the fraction of coronavirus variants relates to growth models, and has been modeled by some with logistic curves that wrongly assume $\phi_1 = 1$.

An example is the forecasting of variants by the Dutch 'RijksInstituut Voor Veiligheid (RIVM)'. Besides problems with the exponential growth of the two different strains (leading to large discrepancies in the odds far beyond the prediction intervals) the model wrongly assumed $\phi_1 = 1$. In the picture below we see that this has the effect of unrealistic predictions for April, with statements that the fraction of infections with the variant is gonna be close to 1 with a very tiny error margin (the reality is that it is currently somewhere in the range 0.8 to 0.9).

![bad logistic curve](https://uploads.disquscdn.com/images/213fbaba117070583bfd7ad813b5caf8d81c8bae8c04a702a3e3ba48f3802c09.png)

For the growth model we can come up with ideas why the $\phi_1$ should not be equal to 1. Also for other models that deal with fractions it might be wrong as well. This is in particular problematic when the curves are fitted based on little data in a small range and subsequently extrapolated. This may contribute, for instance, to the [piranha problem](https://statmodeling.stat.columbia.edu/2017/12/15/piranha-problem-social-psychology-behavioral-economics-button-pushing-model-science-eats/). Assuming linearity of effects on the odds can be problematic.