curveFit: Curve Fitting

Description

Thirteen monotonic(sigmoidal) equations ("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 six non-monotonic(J-shaped) equations ("Brain_Consens", "BCV", "Beckon", "Biphasic", "Hill_six") are provided to fit concentration-response data. The statistics for goodness of fit 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, expr, eq , param, effv, fig = TRUE, ylimit, 
         xlabel = 'lg[concentration, mol/L]', ylabel = 'Inhibition [%]', 
		 sigLev = 0.05, noec = TRUE, algo = "default")

Arguments

a numeric vector of experimental concentrations

expr

a numeric matrix with one or more columns. each column represents one experimental repetition.

equations to be called: "Hill", "Hill_two", "Hill_three", "Hill_four", "Weibull", "Weibull_three", "Weibull_four", "Logit", "Logit_three", "Logit_four", "BCW", "BCL", "GL", "Brain_Consens", "BCV", "Beckon", "Biphasic", "Hill_six"

param

starting values for curve fitting.

effv

numeric vector with single or multiple effect values.

fig

a logical value (TRUE of FALSE). Whether to show the concentration-response curve.

ylimit

a two value vector defines the lower and upper limits of the figure.

xlabel

define x-axis label.

ylabel

define y-axis label.

sigLev

The significant level for confidence intervals and Dunnett's test. The default is 0.05.

noec

a logical value (TRUE of FALSE) to determine the calculation of NOEC and LOEC.

algo

algorithm used in the non-linear least squares fitting. if package 'nls2' is installed on your R platform. The following choices are available : brute-force"(alternately called "grid-search"), "random-search", "plinear-brute" and "plinear-random"

Value

fitInfocurve fitting information including the formula used to fit the concentration- response data. The fitted coefficients with standard errors, t test value, and p value. Residual standard error and the degree of freedom are also provided.
pfitted coefficients of the formula
staStatistics about the goodness of fit ($R^2$, $R_{adj}^2$, MAE, RMSE, AIC, AICc, and BIC)
crcInfoa numeric matrix with the experimental concentration (x), fitted response (yhat), experimental responses, lower and upper bounds of observation-based confidence intervals (OCI.low and OCI.up), and lower and upper bounds of function-based confidence intervals (FCI.low and FCI.up)
eciconfidence intervals of effect concentration at the response of effv
effvciconfidence intervals of the response effv for monotonic equations
minxconcentration to induce the largest stimulation for non-monotonic equations
minythe largest stimulation, only for non-monotonic equations
noecInfo$mata matrix of experimental concentrations, Student's t-statistic, F distribution at the sigLev,and sign (-1 or 1)
noecInfo$nonon-observed effect concentration (NOEC)
noecInfo$loleast-observed effect concentration (LOEC)

Details

Curve fitting is dependent on the package 'nls2' (http://cran.r-project.org/web/packages/nls2/index.html). Monotonic(sigmoidal) equations are listed 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, $beta$ = m(Hill coefficient), $gamma$ = Top, and $delta$ = Bottom Weibull: $$E = 1 - \exp ( - \exp (\alpha + \beta \lg (c)))$$ Weibull_three: $$E = \gamma \left( {1 - \exp \left( { - \exp \left( {\alpha + \beta \lg \left( c \right)} \right)} \right)} \right)$$ Weibull_four: $$E = \gamma + \left( {\delta - \gamma } \right)\exp \left( { - \exp \left({\alpha + \beta \lg \left( c \right)} \right)} \right)$$ Logit: $$E = {(1 + \exp ( - \alpha - \beta \lg (c)))^{ - 1}}$$ Logit_three: $$E = \gamma /\left( {1 + \exp \left( {\left( { - \alpha } \right) - \beta \lg \left( c \right)} \right)} \right)$$ Logit_four: $$E = \delta + \left( {\gamma - \delta } \right)/\left( {1 + \exp \left ( {\left( { - \alpha } \right) - \beta \lg \left( c \right)} \right)} \right)$$ 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: $$E = 1/{(1 + \exp ( - \alpha - \beta \lg (c)))^\gamma }$$ Non-monotonic(J-shaped) equations: Hill_six: $$E = \left( {\gamma /{{\left( {1 + \alpha /c} \right)}^\beta }} \right)/ (gamma\_one/{\left( {1 + alpha\_one/c} \right)^{beta\_one}}$$ 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 rat of the hormetic effect manifests itself, gamma the steepness of the curve after the 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 minim 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, $E$ is effect and $c$ is the concentration.

References

Scholze, M. et al. 2001. A General Best-Fit Method for Concentration-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. Di Veroli GY, Fornari C, Goldlust I, Mills G, Koh SB, Bramhall JL, et al. 2015. An automated fitting procedure and software for dose-response curves with multiphasic features. Scitific Report 5: 14701. 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

Run this code

## example 1
# Fit the non-monotonic concentration-response data

x <- hormesis$OmimCl$x
expr <- hormesis$OmimCl$y
curveFit(x, expr, eq = 'Biphasic', param = c(-0.34, 0.001, 884, 0.01, 128), effv = 0.5)

x <- hormesis$HmimCl$x
expr <- hormesis$HmimCl$y
curveFit(x, expr, eq = 'Biphasic', param = c(-0.59, 0.001, 160,0.05, 19),  effv = c(0.05, 0.5))

x <- hormesis$ACN$x
expr <- hormesis$ACN$y
curveFit(x, expr, eq = 'Brain_Consens', param = c(2.5, 2.8, 0.6, 2.44),  effv = c(0.05, 0.5))

x <- hormesis$Acetone$x
expr <- hormesis$Acetone$y
curveFit(x, expr, eq = 'BCV', param = c(1.0, 3.8, 0.6, 2.44),  effv = c(0.05, 0.5))

## example 2
# Fit the concentration-response data of heavy metal Ni(2+) on MCF-7 cells.
# Calculate the concentrations that cause 5\% and 50\% inhibition of the growth of MCF-7 and
# corresponding confidence intervals.

x <- cytotox$Ni$x
expr <- cytotox$Ni$y
curveFit(x, expr, eq = 'Logit', param = c(12, 3), effv = c(0.05, 0.5))

## example 3
# Fit the concentration-response data of Paromomycin Sulfate (PAR) on photobacteria.
# Calculate the concentrations that cause 5\% and 50\% inhibition of the growth of photobacteria 
# and corresponding confidence intervals.

x <- antibiotox$PAR$x
expr <- antibiotox$PAR$y
curveFit(x, expr, eq = 'Logit', param = c(26, 4), effv = c(0.05, 0.5))

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