| curveFit {mixtox} | R Documentation |
Curve Fitting
Description
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).
Usage
curveFit(x, rspn, eq , param, effv, rtype = 'quantal', sigLev = 0.05, sav = FALSE, ...)Arguments
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. |
Details
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/\left( {1 + {{\left( {\alpha /c} \right)}^\beta }} \right)
Hill_two:
E = \beta c/\left( {\alpha + c} \right)
Hill_three:
E = \gamma /\left( {1 + {{\left( {\alpha /c} \right)}^\beta }} \right)
Hill_four:
E = \delta + \left( {\gamma - \delta } \right)/\left( {1 +
{{\left( {\alpha /c} \right)}^\beta }} \right)
where \alpha = EC50, \beta = H (Hill coefficient), \gamma = Top,
and \delta = Bottom
Weibull:
E = 1 - \exp ( - \exp (\alpha + \beta \log (c)))
Weibull_three:
E = \gamma \left( {1 - \exp \left( { - \exp \left( {\alpha +
\beta \log \left( c \right)} \right)} \right)} \right)
Weibull_four:
E = \gamma + \left( {\delta - \gamma } \right)\exp \left(
{ - \exp \left({\alpha + \beta \log \left( c \right)} \right)} \right)
Logit:
E = {(1 + \exp ( - \alpha - \beta \log (c)))^{ - 1}}
Logit_three:
E = \gamma /\left( {1 + \exp \left( {\left( { - \alpha } \right)
- \beta \log \left( c \right)} \right)} \right)
Logit_four:
E = \delta + \left( {\gamma - \delta } \right)/\left( {1 + \exp \left
( {\left( { - \alpha } \right) - \beta \log \left( c \right)} \right)} \right)
where \alpha is the location parameter and \beta slope parameter.
\gamma = Top, and \delta = Bottom
BCW:
E = 1 - \exp \left( { - \exp \left( {\alpha + \beta \left(
{\frac{{{c^\gamma } - 1}}{\gamma }} \right)} \right)} \right)
BCL:
E = {(1 + \exp ( - \alpha - \beta (({c^\gamma } - 1)/\gamma )))^{ - 1}}
GL:
E = 1/{(1 + \exp ( - \alpha - \beta \log (c)))^\gamma }
Non-monotonic(J-shaped) models:
Hill_five:
E = 1 - \left( {1 + \left( {\gamma - 1} \right)/\left( {1 + {{\left( {\alpha /c}
\right)}^\beta }} \right)} \right)\left( {1 - 1/\left( {1 + {{\left( {\delta /c} \right)}
^\varepsilon }} \right)} \right)
Brain_Consens:
E = 1 - \left( {1 + \alpha c} \right)/\left( {1 + \exp \left(
{\beta \gamma } \right){c^\beta }} \right)
where \alpha is the initial rate of increase at low concentration, \beta
the way in which
response decreases with concentration, and \gamma no simple interpretation.
BCV:
E = 1 - \alpha \left( {1 + \beta c} \right)/\left( {1 + \left( {1 + 2\beta
\gamma } \right){{\left( {c/\gamma } \right)}^\delta }} \right)
where \alpha is untreated control, \beta the initial rate of
increase at low concentration, \gamma the concentration cause 50% inhibition,
and \delta no simple interpretation.
Cedergreen:
E = 1 - \left( {1 + \alpha \exp \left( { - 1/\left( {{c^\beta }}
\right)} \right)} \right)/\left( {1 + \exp \left( {\gamma \left({\ln
\left( c \right) - \ln \left( \delta \right)} \right)} \right)} \right)
where \alpha the initial rate of increase at low concentration, \beta
the rate of hormetic effect manifests itself,
\gamma the steepness of the curve after
maximum hormetic effect, and \delta the lower bound on the EC50 level.
Beckon:
E = \left( {\alpha + \left( {1 - \alpha } \right)/\left( {1 + {{\left(
{\beta /c} \right)}^\gamma }} \right)} \right)/\left( {1 + {{\left(
{c/\delta } \right)}^\varepsilon }} \right)
where \alpha is the minimum effect that would be approached by
the downslope in the absence
of the upslope, \beta the concentration at the midpoint of the falling slope,
\gamma the steepness of the rising(positive) slope, \delta the concentration
at the midpoint of the rising slope, and \epsilon the steepness of the
falling(negative) slope.
Biphasic:
E = \alpha - \alpha /\left( {1 + {{10}^{\left( {\left( {c - \beta }
\right)\gamma } \right)}}} \right) + \left( {1 - \alpha } \right)/\left
( {1 + {{10}^{\left( {\left( {\delta - c} \right)\varepsilon } \right)}}}
\right)
where \alpha is the minimum effect that would be approached by the
downslope in the absence
of the upslope, \beta the concentration at the midpoint of the falling slope,
\gamma the steepness of the rising(positive) slope, \delta the concentration
at the midpoint of the
rising slope, and \epsilon the steepness of the falling(negative) slope.
In all, E represents effect and c represents concentration.
Value
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. |
Note
tuneFit is recommended to find the starting values.
References
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.
Examples
## 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))