Fit the Power model to SAR data.

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

Arguments

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.

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).

Value

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.

Details

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)

References

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

Examples

data(galap) fit <- sar_power(galap) summary(fit)
#> #> Model: #> Power #> #> Call: #> S == c * A^z #> #> Did the model converge: TRUE #> #> Residuals: #> 0% 25% 50% 75% 100% #> -177.900 -22.125 14.700 38.525 126.300 #> #> Parameters: #> Estimate Std. Error t value Pr(>|t|) 2.5% 97.5% #> c 33.179155 19.241710 1.724335 0.106646 -5.304264 71.6626 #> z 0.283187 0.099363 2.850028 0.012849 0.084461 0.4819 #> #> R-squared: 0.49, Adjusted R-squared: 0.41 #> AIC: 187.03, AICc: 189.03, BIC: 189.35 #> Observed shape: convex up, Asymptote: FALSE #>
plot(fit)