curveFit | R Documentation |

Thirteen monotonic(sigmoidal) models ("Hill", "Hill_two", "Hill_three", "Hill_four",
"Weibull", "Weibull_three", "Weibull_four", "Logit", "Logit_three", "Logit_four",
"BCW(Box-Cox-Weibull)", "BCL(Box-Cox-Logit)", "GL(Generalized Logit)") and four
non-monotonic(J-shaped) models ("Brain_Consens", "BCV", "Biphasic", "Hill_five")
are provided to fit dose-response data. The goodness of fit of a model
is evaluated by the following statistics: coefficient of
determination (*R^2*), adjusted coefficient of determination (*R_{adj}^2*),
root mean squared error (RMSE), mean absolute error (MAE), Akaike information criterion (AIC),
bias-corrected Akaike information criterion(AICc), and Bayesian information criterion (BIC).

curveFit(x, rspn, eq , param, effv, rtype = 'quantal', sigLev = 0.05, sav = FALSE, ...)

`x` |
a numeric vector of experimental concentration. |

`rspn` |
a numeric matrix of experimental responses with one or more replicates. |

`eq` |
equation used for curve fitting: "Hill", "Hill_two", "Hill_three", "Hill_four", "Weibull", "Weibull_three", "Weibull_four", "Logit", "Logit_three", "Logit_four", "BCW", "BCL", "GL", "Brain_Consens", "BCV", "Biphasic", "Hill_five". |

`param` |
a vector of starting parameters. Use tuneFit to get the starting values. |

`effv` |
a numeric vector of responses for the calculation of effect concentrations. Minus values(e.g., -5%) are permited only in the condition of 'hormesis' dose-responses. Relative values(e.g., 5%, 10%) in the condition of 'continuous' dose-responses. |

`rtype` |
three dose-response types: 'quantal', 'continuous', 'hormesis'. Default is 'quantal'. 'quantal': dose-responses with lower limit fixed at 0 and higher limit at 1 (100%). 'continuous': dose-responses with no fixed lower or higher limits. 'hormesis': non-monotonic J or U-shaped dose-responses with lower limit fixed at 0 and higher limit at 1 (100%). |

`sigLev` |
the significant level for confidence intervals and Dunnett\'s test. Default is 0.05. |

`sav` |
TRUE: save output to a default file; FALSE: output will not be saved; a custom file directory: save output to the custom file directory. |

`...` |
other arguments passed to nlsLM in minpack.lm. |

Curve fitting is dependent on the package minpack.lm
(http://cran.r-project.org/web/packages/minpack.lm/index.html).

Monotonic(sigmoidal) equations are as follows:

Hill:

*E = 1/≤ft( {1 + {{≤ft( {α /c} \right)}^β }} \right)*

Hill_two:

*E = β c/≤ft( {α + c} \right)*

Hill_three:

*E = γ /≤ft( {1 + {{≤ft( {α /c} \right)}^β }} \right)*

Hill_four:

*E = δ + ≤ft( {γ - δ } \right)/≤ft( {1 +
{{≤ft( {α /c} \right)}^β }} \right)*

where *α* = EC50, *β* = H (Hill coefficient), *γ* = Top,
and *δ* = Bottom

Weibull:

*E = 1 - \exp ( - \exp (α + β \log (c)))*

Weibull_three:

*E = γ ≤ft( {1 - \exp ≤ft( { - \exp ≤ft( {α +
β \log ≤ft( c \right)} \right)} \right)} \right)*

Weibull_four:

*E = γ + ≤ft( {δ - γ } \right)\exp ≤ft(
{ - \exp ≤ft({α + β \log ≤ft( c \right)} \right)} \right)*

Logit:

*E = {(1 + \exp ( - α - β \log (c)))^{ - 1}}*

Logit_three:

*E = γ /≤ft( {1 + \exp ≤ft( {≤ft( { - α } \right)
- β \log ≤ft( c \right)} \right)} \right)*

Logit_four:

*E = δ + ≤ft( {γ - δ } \right)/≤ft( {1 + \exp ≤ft
( {≤ft( { - α } \right) - β \log ≤ft( c \right)} \right)} \right)*

where *α* is the location parameter and *β* slope parameter.
*γ* = Top, and *δ* = Bottom

BCW:

*E = 1 - \exp ≤ft( { - \exp ≤ft( {α + β ≤ft(
{\frac{{{c^γ } - 1}}{γ }} \right)} \right)} \right)*

BCL:

*E = {(1 + \exp ( - α - β (({c^γ } - 1)/γ )))^{ - 1}}*

GL:

*E = 1/{(1 + \exp ( - α - β \log (c)))^γ }*

Non-monotonic(J-shaped) models:

Hill_five:

*E = 1 - ≤ft( {1 + ≤ft( {γ - 1} \right)/≤ft( {1 + {{≤ft( {α /c}
\right)}^β }} \right)} \right)≤ft( {1 - 1/≤ft( {1 + {{≤ft( {δ /c} \right)}
^\varepsilon }} \right)} \right)*

Brain_Consens:

*E = 1 - ≤ft( {1 + α c} \right)/≤ft( {1 + \exp ≤ft(
{β γ } \right){c^β }} \right)*

where *α* is the initial rate of increase at low concentration, *β*
the way in which
response decreases with concentration, and *γ* no simple interpretation.

BCV:

*E = 1 - α ≤ft( {1 + β c} \right)/≤ft( {1 + ≤ft( {1 + 2β
γ } \right){{≤ft( {c/γ } \right)}^δ }} \right)*

where *α* is untreated control, *β* the initial rate of
increase at low concentration, *γ* the concentration cause 50% inhibition,
and *δ* no simple interpretation.

Cedergreen:

*E = 1 - ≤ft( {1 + α \exp ≤ft( { - 1/≤ft( {{c^β }}
\right)} \right)} \right)/≤ft( {1 + \exp ≤ft( {γ ≤ft({\ln
≤ft( c \right) - \ln ≤ft( δ \right)} \right)} \right)} \right)*

where *α* the initial rate of increase at low concentration, *β*
the rate of hormetic effect manifests itself,
*γ* the steepness of the curve after
maximum hormetic effect, and *δ* the lower bound on the EC50 level.

Beckon:

*E = ≤ft( {α + ≤ft( {1 - α } \right)/≤ft( {1 + {{≤ft(
{β /c} \right)}^γ }} \right)} \right)/≤ft( {1 + {{≤ft(
{c/δ } \right)}^\varepsilon }} \right)*

where *α* is the minimum effect that would be approached by
the downslope in the absence
of the upslope, *β* the concentration at the midpoint of the falling slope,
*γ* the steepness of the rising(positive) slope, *δ* the concentration
at the midpoint of the rising slope, and *ε* the steepness of the
falling(negative) slope.

Biphasic:

*E = α - α /≤ft( {1 + {{10}^{≤ft( {≤ft( {c - β }
\right)γ } \right)}}} \right) + ≤ft( {1 - α } \right)/≤ft
( {1 + {{10}^{≤ft( {≤ft( {δ - c} \right)\varepsilon } \right)}}}
\right)*

where *α* is the minimum effect that would be approached by the
downslope in the absence
of the upslope, *β* the concentration at the midpoint of the falling slope,
*γ* the steepness of the rising(positive) slope, *δ* the concentration
at the midpoint of the
rising slope, and *ε* the steepness of the falling(negative) slope.

In all, *E* represents effect and *c* represents concentration.

`fitInfo ` |
curve fitting information. |

`eq ` |
equation used in curve fitting. |

`p ` |
fitted parameters. |

`res ` |
residual. |

`sta ` |
goodness of fit. |

`crcInfo ` |
a numeric matrix with the experimental concentration (x), predicted and experimental responses, experimental responses, lower and upper bounds of (non-simultaneous) prediction intervals (PI.low and PI.up), and lower and upper bounds of (non-simultaneous) confidence intervals (CI.low and CI.up). |

`ecx` |
effect concentrations only if effv is provided. |

`effvAbs` |
Absolute effects corresponding to effv only in the condition of 'continuous' dose-responses. |

`rtype ` |
dose-response type. |

`rspnRange` |
response range. The lower limit is the response at extremely low dose. The higher limit is the response at infinite high dose. |

`minx ` |
concentration to induce the maximum stimulation for 'continuous' dose-response |

`miny ` |
the maximum stimulation for 'continuous' data. |

tuneFit is recommended to find the starting values.

Scholze, M. et al. 2001. A General Best-Fit Method for dose-response Curves and the
Estimation of Low-Effect Concentrations. Environmental Toxicology and Chemistry
20(2):448-457.

Zhu X-W, et.al. 2013. Modeling non-monotonic dose-response relationships: Model evaluation
and hormetic quantities exploration. Ecotoxicol. Environ. Saf. 89:130-136.

Howard GJ, Webster TF. 2009. Generalized concentration addition: A method for examining mixtures
containing partial agonists. J. Theor. Biol. 259:469-477.

Spiess, A.-N., Neumeyer, N., 2010. An evaluation of R2 as an inadequate measure for nonlinear
models in pharmacological and biochemical research: A Monte Carlo approach. BMC Pharmacol.
10, 11.

Huet, S., Bouvier, A., Poursat, M.-A., Jolivet, E., 2004. Statistical tools for nonlinear
regression: a practical guide with S-PLUS and R examples. Springer Science & Business Media.

Gryze, S. De, Langhans, I., Vandebroek, M., 2007. Using the correct intervals for prediction: A
tutorial on tolerance intervals for ordinary least-squares regression. Chemom. Intell. Lab.
Syst. 87, 147-154.

## example 1 # Fit hormesis dose-response data. # Calculate the concentrations that cause 5% of 50% inhibition. x <- hormesis$OmimCl$x rspn <- hormesis$OmimCl$y curveFit(x, rspn, eq = 'Biphasic', param = c(-0.34, 0.001, 884, 0.01, 128), effv = 0.5, rtype = 'hormesis') x <- hormesis$HmimCl$x rspn <- hormesis$HmimCl$y curveFit(x, rspn, eq = 'Biphasic', param = c(-0.59, 0.001, 160,0.05, 19), effv = c(0.05, 0.5), rtype = 'hormesis') x <- hormesis$ACN$x rspn <- hormesis$ACN$y curveFit(x, rspn, eq = 'Brain_Consens', param = c(2.5, 2.8, 0.6, 2.44), effv = c(0.05, 0.5), rtype = 'hormesis') x <- hormesis$Acetone$x rspn <- hormesis$Acetone$y curveFit(x, rspn, eq = 'BCV', param = c(1.0, 3.8, 0.6, 2.44), effv = c(0.05, 0.5), rtype = 'hormesis') ## example 2 # Fit quantal dose-responses: the inhibition of heavy metal Ni(2+) on the growth of MCF-7 cells. # Calculate the concentrations that cause 5% and 50% inhibition. x <- cytotox$Ni$x rspn <- cytotox$Ni$y curveFit(x, rspn, eq = 'Logit', param = c(12, 3), effv = c(0.05, 0.5), rtype = 'quantal') ## example 3 # Fit quantal dose-responses: the inhibition effect of Paromomycin Sulfate (PAR) on photobacteria. # Calculate the concentrations that cause 5% and 50% inhibition. x <- antibiotox$PAR$x rspn <- antibiotox$PAR$y curveFit(x, rspn, eq = 'Logit', param = c(26, 4), effv = c(0.05, 0.5))

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