FindFit giving weird values on a logistic equation

Question

I was trying to fit a set of data onto a logistic equation, however it ended up telling me that the initial value when $t=0$ is greater than the maximum limit, thus I am led to believe that it isn't working.

My approach:

Here are the data points with the top row being day and the second row being height (cm)

$\begin{matrix}7&14&21&28&35&42&49&56&63&70&77&84\\17.93&36.36&67.76&98.1&131&169.5&205.5&228.3&247.1&250.5&253.8&254.5\end{matrix}$

I put them into mathematica by the following:

points={{7,17.93},{14,36.36},{21,67.76},{28,98.1},{35,131},{42,169.5},{49,205.5},{56,228.3},{63,247.1},{70,250.5},{77,253.8},{84,254.5}};

I need to fit this data into this equation

$\frac{C}{1+\frac{C-N_0}{N_0}e^{-rt}}$, where $C$ is defined as the maximum possible height, $N_0$ is the height when $t=0$, $r$ is some constant, $e$ is Euler's Constant and $t$ is time

Since $C$ and $N$ are already used by mathematica, i used $c$ and $n$

FindFit[points,c/(1+((c-n)/n)*E^(-r*t)),{c,n,r},t]

The output was:

{c->163.363, n->653566, r->236.408}

What have I done wrong to get these values? Also I am positive I have the correct equation as this is what the assignment tells us is the equation and all the other questions work. (Although they are a different style and we are given $N_0$ and $C$ in those ones)

My expected output for $C$ and $N_0$ I believe is around

{c->256, n->9}

However I am unsure about $r$ but believe it to be under $1$ and positive

Answer 1

This is quite common a problem when doing nonlinear fit. As far as I know, the most general and effective solution for it is to give the fitted parameters a good enough start value, which you've already known for your specific problem:

exp = c/(1 + ((c - n)/n)*E^(-r*t));
FindFit[points, exp, {{c, 256}, {n, 9}, r}, t]
Show[Plot[exp /. %, {t, 7, 84}], ListPlot@points]

{c -> 261.04, n -> 12.3092, r -> 0.0877072}

Answer 2

If "diagnostics" or standard errors are required you can use NonlinearModelFit with the starting values:

fun = c n/(n + (c - n) Exp[-r t]);
nlm = NonlinearModelFit[points, fun, {{c, 256}, {n, 9}, r}, t]

Visualizing fit:

Show[ListPlot[points], Plot[nlm[t], {t, 0, 90}]]

You can get model:

nlm//Normal

yielding

3213.19/(12.3092 + 248.73 E^(-0.0877072 t))

You can look at model properties:

nlm["Properties"]

For example:

nlm["AdjustedRSquared"]

yields 0.999586.

Or

nlm["ParameterTable"]