# Skewness Calculation Function Calculates the skewness in the estimates of the parameters of a given model. ```r skewIPEC( expr, theta, x, y, tol = 1e-30, 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 - `theta`: A vector of parameters of the model - `x`: A vector or matrix of observations of independent variable(s) - `y`: A vector of observations of response variable - `tol`: The tolerance for detecting linear dependencies in the columns of a matrix for calculating its inverse. See the input argument of `tol` of the `ginv` function in package `MASS` - `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 The defined model should have two input arguments: a parameter vector and an independent variable vector or matrix, e.g. `myfun <- function(P, x){...}`, where `P` represents the parameter vector and `x` represents the independent variable vector or matrix. Let $|g[1i]|$ be a measure of the skewness of the estimate of the **i**-th parameter. If $|g[1i]| <$ 0.1 , the estimator $thetahat[i]$ of parameter $theta[i]$ is very close-to-linear in behavior; if 0.1 $<=$ $|g[1i]| <$ 0.25 , the estimator is reasonably close-to-linear; if $|g[1i]|$ $>=$ 0.25 , the skewness is very apparent; if $|g[1i]| >$ 1 , the estimator is considerably nonlinear in behavior (Pages 27-28 in Ratkowsky 1990). ## Returns - **skewness**: The skewness ## References Hougaard, P. (1985) The appropriateness of the asymptotic distribution in a nonlinear regression model in relation to curvature. **J. R. Statist. Soc., Ser. B** 47, 103$-$114. 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 The current function can be applicable to nonlinear models with multiple independent variables. ## See Also `derivIPEC`, `hessian` in package `numDeriv`, `jacobian` in package `numDeriv` ## Examples ```r #### Example 1 ################################################################################# # The velocity of the reaction (counts/min^2) under different substrate concentrations # in parts per million (ppm) (Page 269 of Bates and Watts 1988) x1 <- c(0.02, 0.02, 0.06, 0.06, 0.11, 0.11, 0.22, 0.22, 0.56, 0.56, 1.10, 1.10) y1 <- c(76, 47, 97, 107, 123, 139, 159, 152, 191, 201, 207, 200) # Define the Michaelis-Menten (MM) model MM <- function(theta, x){ theta[1]*x / ( theta[2] + x ) } par1 <- c(212.68490865, 0.06412421) res5 <- skewIPEC( MM, theta=par1, x=x1, y=y1, tol= 1e-20 ) res5 ################################################################################################# #### Example 2 ################################################################################## # Development data of female pupae of cotton bollworm (Wu et al. 2009) # References: # Ratkowsky, D.A. and Reddy, G.V.P. (2017) Empirical model with excellent statistical # properties for describing temperature-dependent developmental rates of insects # and mites. Ann. Entomol. Soc. Am. 110, 302-309. # Wu, K., Gong, P. and Ruan, Y. (2009) Estimating developmental rates of # Helicoverpa armigera (Lepidoptera: Noctuidae) pupae at constant and # alternating temperature by nonlinear models. Acta Entomol. Sin. 52, 640-650. # 'x2' is the vector of temperature (in degrees Celsius) # 'D2' is the vector of developmental duration (in d) # 'y2' is the vector of the square root of developmental rate (in 1/d) x2 <- seq(15, 37, by=1) D2 <- c(41.24,37.16,32.47,26.22,22.71,19.01,16.79,15.63,14.27,12.48, 11.3,10.56,9.69,9.14,8.24,8.02,7.43,7.27,7.35,7.49,7.63,7.9,10.03) y2 <- 1/D2 y2 <- sqrt( y2 ) # Define the square root function of the Lobry-Rosso-Flandrois (LRF) model sqrt.LRF <- function(P, x){ ropt <- P[1] Topt <- P[2] Tmin <- P[3] Tmax <- P[4] fun0 <- function(z){ z[z < Tmin] <- Tmin z[z > Tmax] <- Tmax return(z) } x <- fun0(x) if (Tmin >= Tmax | ropt <= 0 | Topt <= Tmin | Topt >= Tmax) temp <- Inf if (Tmax > Tmin & ropt > 0 & Topt > Tmin & Topt < Tmax){ temp <- sqrt( ropt*(x-Tmax)*(x-Tmin)^2/((Topt-Tmin)*((Topt-Tmin )*(x-Topt)-(Topt-Tmax)*(Topt+Tmin-2*x))) ) } return( temp ) } myfun <- sqrt.LRF par2 <- c(0.1382926, 33.4575663, 5.5841244, 38.8282021) # To calculate bias resu5 <- skewIPEC( myfun, theta=par2, x=x2, y=y2, tol= 1e-20 ) resu5 ################################################################################################# #### Example 3 ################################################################################## # 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(P1, x){ P1[3] + P1[2]*exp(P1[1]*x) } # Define the second case of Mitscherlich equation MitB <- function(P2, x){ log( P2[3] ) + exp(P2[2] + P2[1]*x) } # 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) r0 <- fitIPEC( MitA, x=x4, y=y4, ini.val=ini.val3, xlim=NULL, ylim=NULL, fig.opt=TRUE, control=list(trace=FALSE, reltol=1e-20, maxit=50000) ) parA <- r0$par parA r5 <- skewIPEC(MitA, theta=parA, x=x4, y=y4, tol=1e-20) r5 ini.val4 <- c(exp(-0.1), log(2.5), 1) R0 <- fitIPEC( MitB, x=x4, y=y4, ini.val=ini.val3, xlim=NULL, ylim=NULL, fig.opt=TRUE, control=list(trace=FALSE, reltol=1e-20, maxit=50000) ) parB <- R0$par parB R5 <- skewIPEC( MitB, theta=parB, x=x4, y=y4, tol=1e-20 ) R5 ini.val6 <- c(-0.15, 2.52, 1.09) RES0 <- fitIPEC( MitC, x=x4, y=y4, ini.val=ini.val6, xlim=NULL, ylim=NULL, fig.opt=TRUE, control=list(trace=FALSE, reltol=1e-20, maxit=50000) ) parC <- RES0$par parC RES5 <- skewIPEC( MitC, theta=parC, x=x4, y=y4, tol=1e-20 ) RES5 ################################################################################################# #### Example 4 ################################################################################### # Data on biochemical oxygen demand (BOD; Marske 1967) # References # Pages 56, 255 and 271 in Bates and Watts (1988) # Carr, N.L. (1960) Kinetics of catalytic isomerization of n-pentane. Ind. Eng. Chem. # 52, 391-396. data(isom) Y <- isom[,1] X <- isom[,2:4] # There are three independent variables saved in matrix 'X' and one response variable (Y) # The first column of 'X' is the vector of partial pressure of hydrogen # The second column of 'X' is the vector of partial pressure of n-pentane # The third column of 'X' is the vector of partial pressure of isopentane # Y is the vector of experimental reaction rate (in 1/hr) isom.fun <- function(theta, x){ x1 <- x[,1] x2 <- x[,2] x3 <- x[,3] theta1 <- theta[1] theta2 <- theta[2] theta3 <- theta[3] theta4 <- theta[4] theta1*theta3*(x2-x3/1.632) / ( 1 + theta2*x1 + theta3*x2 + theta4*x3 ) } par8 <- c(35.92831619, 0.07084811, 0.03772270, 0.16718384) cons5 <- skewIPEC( isom.fun, theta=par8, x=X, y=Y, tol= 1e-20 ) cons5 ################################################################################################# ```

skewIPEC function