I have a data set that represents exponential decay. I would like to fit an exponential function $y = Be^{ax}$ to this data. I've tried log transforming the response variable and then using least squares to fit a line; using a generalized linear model with a log link function and a gamma distribution around the response variable; and using nonlinear least squares. I get a different answer for my two coefficients with each method, although they are all similar. Where I have confusion is I'm not sure which method is the best to use and why. Can someone please compare and contrast these methods? Thank you.

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#### Best Answer

The difference is basically the difference in the assumed distribution of the random component, and how the random component interacts with the underlying mean relationship.

Using nonlinear least squares effectively assumes the noise is additive, with constant variance (and least squares is maximum likelihood for normal errors).

The other two assume that the noise is multiplicative, and that the variance is proportional to the square of the mean. Taking logs and fitting a least squares line is maximum likelihood for the lognormal, while the GLM you fitted is maximum likelihood (at least for its mean) for the Gamma (unsurprisingly). Those two will be quite similar, but the Gamma will put less weight on very low values, while the lognormal one will put relatively less weight on the highest values.

(Note that to properly compare the parameter estimates for those two, you need to deal with the difference between expectation on the log scale and expectation on the original scale. The mean of a transformed variable is not the transformed mean in general.)

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