Fit the Asymptotic regression model to SAR data.

sar_asymp(data, start = NULL, grid_start = NULL, normaTest = 'lillie', homoTest = 'cor.fitted')

data | A dataset in the form of a dataframe with two columns: the first with island/site areas, and the second with the species richness of each island/site. |
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start | NULL or custom parameter start values for the optimisation algorithm. |

grid_start | NULL or the number of points sampled in the model parameter space or FALSE to prevent any grid start after a fail in initial optimization to run a grid search. |

normaTest | The test used to test the normality of the residuals of the model. Can be any of 'lillie' (Lilliefors Kolmogorov-Smirnov test; the default), 'shapiro' (Shapiro-Wilk test of normality), 'kolmo' (Kolmogorov-Smirnov test), or 'none' (no residuals normality test is undertaken). |

homoTest | The test used to check for homogeneity of the residuals of the model. Can be any of 'cor.fitted' (a correlation of the residuals with the model fitted values; the default), 'cor.area' (a correlation of the residuals with the area values), or 'none' (no residuals homogeneity test is undertaken). |

A list of class 'sars' with the following components:

par The model parameters

value Residual sum of squares

counts The number of iterations for the convergence of the fitting algorithm

convergence Numeric code indicating model convergence (0 = converged)

message Any message from the model fit algorithm

hessian A symmetric matrix giving an estimate of the Hessian at the solution found

verge Logical code indicating model convergence

startValues The start values for the model parameters used in the optimisation

data Observed data

model A list of model information (e.g. the model name and formula)

calculated The fitted values of the model

residuals The model residuals

AIC The AIC value of the model

AICc The AICc value of the model

BIC The BIC value of the model

R2 The R2 value of the model

R2a The adjusted R2 value of the model

sigConf The model coefficients table

normaTest The results of the residuals normality test

homoTest The results of the residuals homogeneity test

observed_shape The observed shape of the model fit

asymptote A logical value indicating whether the observed fit is asymptotic

neg_check A logical value indicating whether negative fitted values have been returned

The `summary.sars`

function returns a more useful summary of
the model fit results, and the `plot.sars`

plots the model fit.

The model is fitted using non-linear regression. The model parameters are estimated
by minimizing the residual sum of squares with an unconstrained Nelder-Mead optimization algorithm
and the `optim`

function. To avoid numerical problems and speed up the convergence process,
the starting values used to run the optimization algorithm are carefully chosen, or custom values can be provided
using the argument `start`

. The fitting process also determines the observed shape of the model fit,
and whether or not the observed fit is asymptotic (see Triantis et al. 2012 for further details).
Model validation is undertaken by assessing the normality (`normaTest`

) and homogeneity (`homoTest`

)
of the residuals and a warning is provided in `summary.sars`

if either test is failed.
A selection of information criteria (e.g. AIC, BIC) are returned and can be used to compare models
(see also `sar_average`

)

Triantis, K.A., Guilhaumon, F. & Whittaker, R.J. (2012) The island species-area relationship: biology and statistics. Journal of Biogeography, 39, 215-231.

#> #> Model: #> Asymptotic regression #> #> Call: #> S == d - c * z^A #> #> Did the model converge: TRUE #> #> Residuals: #> 0% 25% 50% 75% 100% #> -137.800 -22.475 2.600 21.650 129.000 #> #> Parameters: #> Estimate Std. Error t value Pr(>|t|) 2.5% 97.5% #> d 2.0912e+02 3.4968e+01 5.9803e+00 4.5909e-05 1.3918e+02 279.0532 #> c 1.8490e+02 4.3863e+01 4.2153e+00 1.0103e-03 9.7170e+01 272.6204 #> z 9.8865e-01 8.4606e-03 1.1685e+02 4.9588e-21 9.7173e-01 1.0056 #> #> R-squared: 0.58, Adjusted R-squared: 0.47 #> AIC: 186.04, AICc: 189.67, BIC: 189.13 #> Observed shape: convex up, Asymptote: TRUE #> #> #> Warning: The normality test selected indicated the model residuals are not normally distributed (i.e. P < 0.05)plot(fit)