# Non-linear growth curves with Stan

I suppose the go to tool for fitting non-linear models in R is nls of the stats package. In this post I will show an alternative approach with Stan/RStan, as illustrated in the example, Dugongs: “nonlinear growth curve”, that is part of Stan’s documentation.

The original example itself is taken from OpenBUGS. The data describes the length and age measurements for 27 captured dugongs (sea cows). Carlin and Gelfand (1991) model the data using a nonlinear growth curve with no inflection point and an asymptote as $x_i$ tends to infinity:
$$Y_i \sim \mathcal{N}(\mu_i, \sigma^2),\; i = 1,\dots,27\\\ \mu_i = \alpha - \beta \lambda^{x_i},\; \alpha,\, \beta > 0;\, 0 < \lambda < 1$$ Fitting the curve with nls gives the following results:

Writing the model in Stan requires a few more lines, but gives me also the opportunity to generate output from the posterior distributions.

The predicted parameters and errors are very much the same as in the least square output of nls, but with the Stan output I can also review the 90% credible intervals.

### Session Info

R version 3.2.2 (2015-08-14)Platform: x86_64-apple-darwin13.4.0 (64-bit)Running under: OS X 10.11.1 (El Capitan)locale:[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8attached base packages:[1] stats     graphics  grDevices utils     datasets [6] methods   base     other attached packages:[1] rstan_2.8.0        ggplot2_1.0.1.9003 Rcpp_0.12.1       loaded via a namespace (and not attached): [1] colorspace_1.2-6 scales_0.3.0     plyr_1.8.3       [4] parallel_3.2.2   tools_3.2.2      inline_0.3.14    [7] gtable_0.1.2     gridExtra_2.0.0  codetools_0.2-14[10] grid_3.2.2       stats4_3.2.2     munsell_0.4.2