# Curve Fitting 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). ```r 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. ## Returns - **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. ## 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. ## Note tuneFit is recommended to find the starting values. ## Examples ```r ## 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)) ```

curveFit function