The Brain-Cousens hormesis models

'BC.4' and 'BC.5' provide the Brain-Cousens modified log-logistic models for describing u-shaped hormesis.

BC.5(fixed = c(NA, NA, NA, NA, NA), names = c("b", "c", "d", "e", "f"), ...)

  BC.4(fixed = c(NA, NA, NA, NA), names = c("b", "d", "e", "f"), ...)

Arguments

fixed	numeric vector specifying which parameters are fixed and at which values they are fixed. NAs designate parameters that are not fixed.
names	a vector of character strings giving the names of the parameters.
...	additional arguments to be passed from the convenience functions.

Details

The model function for the Brain-Cousens model (Brain and Cousens, 1989) is

$$ f(x, b,c,d,e,f) = c + \frac{d-c+fx}{1+\exp(b(\log(x)-\log(e)))}$$,

and it is a five-parameter model, obtained by extending the four-parameter log-logistic model (LL.4 to take into account inverse u-shaped hormesis effects.

The parameters have the following interpretations

$b$: Not direct interpretation
$c$: Lower horizontal asymptote
$d$: Upper horizontal asymptote
$e$: Not direct interpretation
$f$: Size of the hormesis effect: the larger the value the larger is the hormesis effect. $f=0$ corresponds to no hormesis effect and the resulting model is the four-parameter log-logistic model. This parameter should be positive in order for the model to make sense.

Fixing the lower limit at 0 yields the four-parameter model

$$ f(x) = 0 + \frac{d-0+fx}{1+\exp(b(\log(x)-\log(e)))}$$

used by van Ewijk and Hoekstra (1993).

Value

See braincousens.

References

Brain, P. and Cousens, R. (1989) An equation to describe dose responses where there is stimulation of growth at low doses, Weed Research, 29, 93--96.

van Ewijk, P. H. and Hoekstra, J. A. (1993) Calculation of the EC50 and its Confidence Interval When Subtoxic Stimulus Is Present, Ecotoxicology and Environmental Safety, 25, 25--32.

Note

This function is for use with the function drm.

Examples


## Fitting the data in van Ewijk and Hoekstra (1993)
lettuce.bcm1 <- drm(weight ~ conc, data = lettuce, fct=BC.5())
modelFit(lettuce.bcm1)
#> Lack-of-fit test
#> 
#>           ModelDf      RSS Df F value p value
#> ANOVA           7 0.088237                   
#> DRC model       9 0.118842  2  1.2140  0.3527
plot(lettuce.bcm1)

lettuce.bcm2 <- drm(weight ~conc, data = lettuce, fct=BC.4())
summary(lettuce.bcm2)
#> 
#> Model fitted: Brain-Cousens (hormesis) with lower limit fixed at 0 (4 parms)
#> 
#> Parameter estimates:
#> 
#>               Estimate Std. Error t-value   p-value    
#> b:(Intercept) 1.282812   0.049346 25.9964 1.632e-10 ***
#> d:(Intercept) 0.967302   0.077123 12.5423 1.926e-07 ***
#> e:(Intercept) 0.847633   0.436093  1.9437   0.08059 .  
#> f:(Intercept) 1.620703   0.979711  1.6543   0.12908    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error:
#> 
#>  0.1117922 (10 degrees of freedom)
ED(lettuce.bcm2, c(50))
#> 
#> Estimated effective doses
#> 
#>        Estimate Std. Error
#> e:1:50   35.023     15.427
# compare the parameter estimate and 
# its estimated standard error 
# to the values in the paper by 
# van Ewijk and Hoekstra (1993)


## Brain-Cousens model with the constraint b>3
ryegrass.bcm1 <- drm(rootl ~conc, data = ryegrass, fct = BC.5(),
lower = c(3, -Inf, -Inf, -Inf, -Inf), control = drmc(constr=TRUE))

summary(ryegrass.bcm1)
#> 
#> Model fitted: Brain-Cousens (hormesis) (5 parms)
#> 
#> Parameter estimates:
#> 
#>               Estimate Std. Error t-value   p-value    
#> b:(Intercept)  3.00000    0.72020  4.1655  0.000525 ***
#> c:(Intercept)  0.48364    0.25420  1.9026  0.072357 .  
#> d:(Intercept)  7.74462    0.21500 36.0210 < 2.2e-16 ***
#> e:(Intercept)  2.92416    0.56297  5.1942 5.162e-05 ***
#> f:(Intercept)  0.15886    0.67583  0.2351  0.816675    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error:
#> 
#>  0.5311668 (19 degrees of freedom)

## Brain-Cousens model with the constraint f>0 
## (no effect as the estimate of f is positive anyway)
ryegrass.bcm2 <- drm(rootl ~conc, data = ryegrass, fct = BC.5(),
lower = c(-Inf, -Inf, -Inf, -Inf, 0), control = drmc(constr=TRUE))

summary(ryegrass.bcm2)
#> 
#> Model fitted: Brain-Cousens (hormesis) (5 parms)
#> 
#> Parameter estimates:
#> 
#>               Estimate Std. Error t-value   p-value    
#> b:(Intercept)  2.75688    0.52461  5.2551 4.511e-05 ***
#> c:(Intercept)  0.41515    0.26309  1.5780    0.1311    
#> d:(Intercept)  7.74173    0.21415 36.1510 < 2.2e-16 ***
#> e:(Intercept)  2.77687    0.50938  5.4514 2.930e-05 ***
#> f:(Intercept)  0.35672    0.68827  0.5183    0.6102    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error:
#> 
#>  0.528689 (19 degrees of freedom)