For each selected item, five dose-response models (linear, Hill, exponential, 
Gauss-probit and log-Gauss-probit, see Larras et al. 2018 for their definition) 
are fitted by non linear regression. 
The best one is chosen as the one giving the lowest AICc value.

The former use of the AIC (Akaike criterion)
was replaced by the use of the AICc (second-order Akaike criterion)
in order to prevent the overfitting that may occur with dose-response designs
with a small number of data points, as recommended
and now classically done in regression 
(Hurvich and Tsai, 1989; Burnham and Anderson DR, 2004).

Items with the best AICc value not lower than the AICc value of the null model
(constant model) minus 2 are eliminated. 
Items with the best fit showing a global significant quadratic trend of the 
residuals as a function of the dose (in rank-scale) are also eliminated (the 
best fit is considered as not reliable in such cases). 

Each retained item is also classified in four classes by its global trend,
which can be used to roughly describe the shape of each dose-response curve:
  - inc for increasing curves,
  - dec for decreasing curves ,
  - U for U-shape curves,
  - bell for bell-shape curves.

Some curves fitted by a Gauss-probit model can be classified as increasing or decreasing when the 
dose value at which their extremum is reached is at zero or if their simplified version with f = 0
is retained (corresponding to the probit model).
Some curves fitted by a log-Gauss-probit model can be classified as increasing or decreasing 
if their simplified version with f = 0
is retained (corresponding to the log-probit model).

Each retained item is thus classified in a 16 class typology depending of the
chosen model and of its parameter values :

- "H.inc" for increasing Hill curves
- "H.dec" for decreasing Hill curves
- "L.inc" for increasing linear curves
- "L.dec" for decreasing linear curves
- "E.inc.convex" for increasing convex exponential curves
- "E.dec.concave" for decreasing concave exponential curves
- "E.inc.concave" for increasing concave exponential curves
- "E.dec.convex" for decreasing convex exponential curves
- "GP.U" for U-shape Gauss-probit curves
- "GP.bell" for bell-shape Gauss-probit curves
- "GP.inc" for increasing Gauss-probit curves
- "GP.dec" for decreasing Gauss-probit curves
- "lGP.U" for U-shape log-Gauss-probit curves
- "lGP.bell" for bell-shape log-Gauss-probit curves
- "lGP.inc" for increasing log-Gauss-probit curves
- "lGP.decreasing" for decreasing log-Gauss-probit curves


-- REFERENCES --

Burnham, KP, Anderson DR (2004). Multimodel inference: understanding AIC and 
BIC in model selection. Sociological methods & research, 33(2), 261-304.

Hurvich, CM, Tsai, CL (1989). Regression and time series model selection in 
small samples. Biometrika, 76(2), 297-307.

Larras F, Billoir E, Baillard V, Siberchicot A, Scholz S, Wubet T, Tarkka M,
Schmitt-Jansen M and Delignette-Muller ML (2018). DRomics: a turnkey tool to 
support the use of the dose-response framework for omics data in ecological 
risk assessment. Environmental science & technology.