We -- my friend -- had asked a question [here](https://mathematica.stackexchange.com/questions/253282/any-suggestions-for-improving-my-fitting), which was not completely related to Mathematica, however, in a hope that someone familiar with the topic could help. There are still some aspects which we would like to know.

We have a dataset as:

```
data = {{0, 0.046}, {40, 0.111}, {80, 0.291}, {120, 0.808}, {160, 1.742}, {200, 3.319}, {240, 5.017}, {280, 5.503}, {320, 5.897}};

```

where the first components are time; and, with the corresponding standard deviations as:

```
sd = {0.003, 0.012, 0.023, 0.056, 0.083, 0.216, 0.526, 0.366, 0.313};

```

We would like to obtain the growth rate, which is the slope of line in the exponential phase of the growth.

To this end, our first approach is to employ nonlinear regression, that is, to fit the log of logistic function to the log of data, which in this case, the logistic rate approximates the slope of line in the exponential phase of growth. Thus, we have:

```
data2 = data;
data2[[All, 2]] = Log[data2[[All, 2]]];
nlm2 = NonlinearModelFit[data2, Log[a/(1 + E^(-k (t - t0)))], {{a, 1}, {t0, 50}, {k, 0.1}}, t];
nlm2["ParameterConfidenceIntervalTable"]

```

which $k$, i.e., the logistic rate, reads as: $k = 0.025$, with the confidence interval of $0.024 < k < 0.026$.

Now, we would like to see how to approximate the growth rate using linear regression. In this [paper](https://www.pnas.org/content/116/29/14698), it suggests that, first to fit a polynomial to the growth curve data, in order to obtain the maximum growth rate $t\_{\text{max}}$, by differentiating the fitted polynomial ($t\_{\text{max}}$ is where the maximum of the derivative occurs). Then, to fit $a t + b$ to the log of the growth curve data in the neighborhood of $t\_{\text{max}}$.

Now, our question is: it seems Mathematica does not return sensible result if we want to fit a polynomial, let's say, degree $6$ to the data:

```
nlm3 = NonlinearModelFit[data, 
a t^6 + b t^5 + c t^4 + d t^3 + e t^2 + f t + g, {{a, 1}, {b, 1}, {c, 1}, {d, 1}, {e, 1}, {f, 1}, {g, 1}}, t];
nlm3["ParameterConfidenceIntervalTable"]

```

which $a$, $b$, $c$, $d$, and $e$ are effectively $0$.

Any help is appreciated how to move forward from here!