# Wald Confidence Curves and the Likelihood Confidence Curves Calculates the Wald confidence curves and the likelihood confidence curves of model parameters. ```r confcurves( expr, x, y, ini.val, weights = NULL, control=list(), fig.opt = TRUE, fold = 5, np = 20, alpha = seq(1, 0.001, by=-0.001), show.CI = NULL, method = "Richardson", method.args = list(eps = 1e-04, d = 0.11, zero.tol = sqrt(.Machine$double.eps/7e-07), r = 6, v = 2, show.details = FALSE), side = NULL ) ``` ## Arguments - `expr`: A given parametric model - `x`: A vector or matrix of observations of independent variable(s) - `y`: A vector of observations of response variable - `ini.val`: A vector or list of initial values of model parameters - `weights`: An optional vector of weights to be used in the fitting process. `weights` should be NULL or a numeric vector. If non-NULL, weighted least squares is used with weights `weights`; otherwise ordinary least squares is used. - `control`: A list of control parameters for using the `optim` function in package `stats` - `fig.opt`: An option to determine whether to draw the confidence curves of each parameter - `fold`: The fold of $ \mbox{SE}\left(\hat{\theta_{i}}\right) $ for controlling the width of the confidence interval of $\hat{\theta_{i}}$ that represents the estimate of the $i$th parameter - `np`: The number of data points for forming the lower or upper bounds of a likelihood confidence interval of $ \hat{\theta_{i}} $, which controlls the step size (i.e., $\delta$) in the $y$ coordinates of the likelihood confidence curves - `alpha`: The significance level(s) for calculating the $x$ coordinate(s) of the $(1-\alpha)100\%$ Wald confidence curves, which equals to $ t_{\alpha/2}\left(n-p\right) $ - `show.CI`: The $ t_{\alpha/2}\left(n-p\right) $ value(s) associated with the confidence level(s) of each parameter to be showed, i.e., `c(0.80, 0.90, 0.95, 0.99)` - `method`: It is the same as the input argument of `method` of the `hessian` function in package `numDeriv` - `method.args`: It is the same as the input argument of `method.args` of the `hessian` function in package `numDeriv` - `side`: It is the same as the input argument of `side` of the `jacobian` function in package `numDeriv` ## Details For the $(1-\alpha)100\%$ Wald confidence curves, the corresponding $x$ and $y$ coordinates are: $$ x = t_{\alpha/2}\left(n-p\right), $$ and $$ y = \hat{\theta_{i}} \pm t_{\alpha/2}\left(n-p\right)\,\mbox{SE}\left(\hat{\theta_{i}}\right), $$ where $n$ denotes the number of the observations, $p$ denotes the number of model parameters, and $\mbox{SE}\left(\hat{\theta_{i}}\right)$ denotes the standard error of the $i$th model parameter's estimate. $\quad$ For the likelihood confidence curves (Cook and Weisberg, 1990), the corresponding $x$ and $y$ coordinates are: $$ x = \sqrt{\frac{\mbox{RSS}\left(\hat{\theta}^{\,\left(-i\right)}\right)-\mbox{RSS}\left(\hat{\theta}\right)}{\mbox{RSS}\left(\hat{\theta}\right)/(n-p)}}, $$ where $\mbox{RSS}\left(\hat{\theta}\right)$ represents the residual sum of squares for fitting the model with all model parameters; $\mbox{RSS}\left(\hat{\theta}^{\,\left(-i\right)}\right)$ represents the residual sum of squares for fitting the model with the $i$th model parameter being fixed to be $\hat{\theta_{i}} \pm k\,\delta$. Here, $k$ denotes the $k$th iteration, and $\delta$ denotes the step size, which equals $$ \delta = \frac{\hat{\theta_{i}} \pm \mbox{fold}\,\mbox{SE}\left(\hat{\theta_{i}}\right)}{\mbox{np}}. $$ $$ y = \hat{\theta_{i}} \pm k\,\delta. $$ Here, `fold` and `np` are defined by the user in the arguments. $\quad$ For other arguments, please see the `fitIPEC` and `parinfo` functions for details. ## Returns - **partab**: The estimates, standard errors and confidence intervals of model parameters; also see the `parinfo` function - **parlist**: A list for the estimate, Wald interval curves and likelihood interval curves of each model parameter. ## References Cook, R.D. and Weisberg, S. (1990) Confidence curves in nonlinear regression. **J. Am. Statist. Assoc.** 82, 221$-$230. tools:::Rd_expr_doi("10.1080/01621459.1990.10476233") Nelder, J.A. and Mead, R. (1965) A simplex method for function minimization. **Comput. J.** 7, 308$-$313. tools:::Rd_expr_doi("10.1093/comjnl/7.4.308") Ratkowsky, D.A. (1990) **Handbook of Nonlinear Regression Models**, Marcel Dekker, New York. ## Author(s) Peijian Shi pjshi@njfu.edu.cn , Peter M. Ridland p.ridland@unimelb.edu.au , David A. Ratkowsky d.ratkowsky@utas.edu.au , Yang Li yangli@fau.edu . ## Note In the value of `parlist`, there are the estimate (`pari`), the Wald interval curves (`WaldCI`), and the likelihood interval curves (`lhCI`) of the $i$th model parameter. In `WaldCI`, there are three columns. The first column, `tc`, represents $ t_{\alpha/2}\left(n-p\right) $, the second and third columns, `LCI` and `UCI`, represent the lower and upper bounds of the $(1-\alpha)100\%$ Wald confidence intervals, respectively. In `lhCI`, there are six columns. The first and second columns, `x.lower` and `lhLCI`, represent the lower bounds of the likelihood confidence intervals the and corresponding $x$ values, respectively; the third and fourth columns, `x.upper` and `lhUCI`, represent the upper bounds of the likelihood confidence intervals and the corresponding $x$ values, respectively; the fifth and sixth columns, `RSS.lower` and `RSS.upper`, represent the values of the residual sum of squares of the lower bounds and those of the upper bounds, respectively. Please see Cook and Weisberg (1990) for details. ## See Also `parinfo`, `fitIPEC`, `optim` in package `stats` ## Examples ```r #### Example 1 ################################################################################### # Weight of cut grass data (Pattinson 1981) # References: # Clarke, G.P.Y. (1987) Approximate confidence limits for a parameter function in nonlinear # regression. J. Am. Stat. Assoc. 82, 221-230. # Gebremariam, B. (2014) Is nonlinear regression throwing you a curve? # New diagnostic and inference tools in the NLIN Procedure. Paper SAS384-2014. # http://support.sas.com/resources/papers/proceedings14/SAS384-2014.pdf # Pattinson, N.B. (1981) Dry Matter Intake: An Estimate of the Animal # Response to Herbage on Offer. unpublished M.Sc. thesis, University # of Natal, Pietermaritzburg, South Africa, Department of Grassland Science. # 'x4' is the vector of weeks after commencement of grazing in a pasture # 'y4' is the vector of weight of cut grass from 10 randomly sited quadrants x4 <- 1:13 y4 <- c(3.183, 3.059, 2.871, 2.622, 2.541, 2.184, 2.110, 2.075, 2.018, 1.903, 1.770, 1.762, 1.550) # Define the first case of Mitscherlich equation MitA <- function(P, x){ P[3] + P[2]*exp(P[1]*x) } # Define the second case of Mitscherlich equation MitB <- function(P2, x){ if(P2[3] <= 0) temp <- mean(y4) if(P2[3] > 0) temp <- log(P2[3]) + exp(P2[2] + P2[1]*x) return( temp ) } # Define the third case of Mitscherlich equation MitC <- function(P3, x, x1=1, x2=13){ theta1 <- P3[1] beta2 <- P3[2] beta3 <- P3[3] theta2 <- (beta3 - beta2)/(exp(theta1*x2)-exp(theta1*x1)) theta3 <- beta2/(1-exp(theta1*(x1-x2))) - beta3/(exp(theta1*(x2-x1))-1) theta3 + theta2*exp(theta1*x) } ini.val3 <- c(-0.1, 2.5, 1) RESU1 <- confcurves( MitA, x=x4, y=y4, ini.val=ini.val3, fig.opt = TRUE, fold=5, np=20, alpha=seq(1, 0.001, by=-0.001), show.CI=c(0.8, 0.9, 0.95, 0.99) ) ini.val4 <- c(-0.10, 0.90, 2.5) RESU2 <- confcurves( MitB, x=x4, y=y4, ini.val=ini.val4, fig.opt = TRUE, fold=5, np=200, alpha=seq(1, 0.001, by=-0.001), show.CI=c(0.8, 0.9, 0.95, 0.99) ) ini.val6 <- c(-0.15, 2.5, 1) RESU3 <- confcurves( MitC, x=x4, y=y4, ini.val=ini.val6, fig.opt = TRUE, fold=5, np=20, alpha=seq(1, 0.001, by=-0.001), show.CI=c(0.8, 0.9, 0.95, 0.99) ) ################################################################################################## graphics.off() ```

confcurves function