curveFit: 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_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 = 'log[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 experiment repetition.

models 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_six".

param

starting values for curve fitting.

effv

numeric values [0, 1].

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
resresidual
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 sigmoidal models
minxconcentration to induce the largest stimulation for J-shaped models
minythe largest stimulation, only for J-shaped models
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) models 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, $\beta$ = H (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) models: 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 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, $E$ 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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