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I have a question about the Logistic growth model, I am applying the equation $$ y = \frac{\epsilon_1}{1 + e^{-(\epsilon_2+\epsilon_3x)}} $$ in a non linear regression.

I am following a book that first tried to convert this into a linear model, so this is my attempt:

$$y \approx \frac{\epsilon_1}{1 + e^{-(\epsilon_2+\epsilon_3x)}} \Rightarrow \frac{y}{\epsilon_1} \approx \frac{1}{1 + e^{-(\epsilon_2+\epsilon_3x)}} \Rightarrow \ ?¿$$

Here I am stuck because I do not know how to continue, I tried to apply the logit transformation $log \left[ \frac{r}{1-r} \right]$ with $r = \frac{y}{\epsilon_1}$. But I couldn't follow how, I am a biologist so it is a little bit complicated for me heheh

Could someone give me a hint to continue this, I should get that $$log\left[ \frac{\frac{y}{\epsilon_1}}{1-\frac{y}{\epsilon_1}}\right] \approx \epsilon_2 + \epsilon_3 x$$

Thank you in advance for your time and help.

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$$y \approx \frac{\epsilon_1}{1 + e^{-(\epsilon_2+\epsilon_3x)}} \\ \Rightarrow \frac{y}{\epsilon_1} \approx \frac{1}{1 + e^{-(\epsilon_2+\epsilon_3x)}} \\ \Rightarrow 1-\frac{y}{\epsilon_1} \approx \frac{e^{-(\epsilon_2+\epsilon_3x)}}{1 + e^{-(\epsilon_2+\epsilon_3x)}} \\ \Rightarrow \frac{\frac{y}{\epsilon_1}}{1-\frac{y}{\epsilon_1}} \approx \frac{1}{e^{-(\epsilon_2+\epsilon_3x)}} = e^{(\epsilon_2+\epsilon_3x)} \\ \Rightarrow \log_e\left[\frac{\frac{y}{\epsilon_1}}{1-\frac{y}{\epsilon_1}}\right] \approx {\epsilon_2+\epsilon_3x}$$

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  • $\begingroup$ excuse me, ig I want to do the same with the Gompertzian model: $$y \approx \theta_1 e^{-\theta_2e^{-\theta_3x}} \Rightarrow log(\frac{y}{\theta_1}) \approx -\theta_2e^{-\theta_3x}$$ So, here should I apply $log$ again ? $\endgroup$ Sep 23, 2019 at 9:24
  • $\begingroup$ Because now this $\approx -\theta_2e^{-\theta_3x}$ is negative, and I can not take log of negative numbers rights? $\endgroup$ Sep 23, 2019 at 9:33
  • $\begingroup$ If you know it is negative, perhaps you could consider $\log\left(-\log\left(\frac{y}{\theta_1}\right)\right) \approx \log(\theta_2) -\theta_3x$ $\endgroup$
    – Henry
    Sep 23, 2019 at 9:43
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    $\begingroup$ There is a difference between $\log(Size/1800)$ which will be negative if $0 < Size < 1800$ and $\text{logit}(Size/1800)$ which may be negative if $0 < Size < 900$ or positive if $900 < Size < 1800$ $\endgroup$
    – Henry
    Sep 23, 2019 at 10:43
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    $\begingroup$ You did not mention "logit" before - a word economists pronounce as if it has no relationship to logic or logarithm. It might be better if people said "log-odds" instead $\endgroup$
    – Henry
    Sep 23, 2019 at 10:45

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