TITLE: Estimate parameters through transformation and linear regression (statistics)
QUESTION [0 upvotes]: A model for a chemical process is
$$y_i = \frac{V_m}{k + x_i},$$
where $x_i$ and $y_i$, the predictor (independent) and response (dependent) variable, respectively, are obtained from data. $V_m$ and $k$ are the two parameters to be estimated.
If you wish to use a simple linear regression to estimate the two parameters (assuming that the model assumptions are satisfied), show how to achieve this by transformation, indicating what your transformed predictor and response variables are.
Also, show how to estimate $V_m$ and $k$ from B0 and B1. You may assume that $V_m \neq 0$.

I'm not really sure where to start on this, any help is appreciated.

REPLY [0 votes]: So, you have $n$ data points $(x_i,y_i)$ and you want to fit the model
$$y=\frac {V_m}{k+x}$$ which is nonlinear with respect to the parameters $V_m$ and $k$ that you want to adjust to get the best fit of the data. Because the model is nonlinear, you need estimates and this is what you are asked to do.
In a first step, rewrite
$$z=\frac 1 y=\frac {k+x}{V_m}=\frac {k}{V_m}+\frac {x}{V_m}=\alpha+\beta x$$ and you see that a linear regression based on $(x_i,z_i=\frac 1 {y_i})$ will give $\alpha,\beta$. So, your estimates $V_m=\frac 1 \beta$ and $k=\frac \alpha \beta$ are easy to get.
Now, you must continue with nonlinear regression since what is measured is $y$ and not $\frac 1 y$ but this will converge quite fast.
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