# Nonlinear Regression Tolerance Bounds, Version 2 Provides 1-sided or 2-sided nonlinear regression tolerance bounds. ```r nlregtol.int(formula, xy.data = data.frame(), x.new = NULL, side = 1, alpha = 0.05, P = 0.99, maxiter = 50, new = FALSE, ...) ``` ## Arguments - `formula`: A nonlinear model formula including variables and parameters. - `xy.data`: A data frame in which to evaluate the formulas in `formula`. The first column of `xy.data` must be the response variable. - `x.new`: Any new levels of the predictor(s) for which to report the tolerance bounds. The number of columns must be 1 less than the number of columns for `xy.data`. - `side`: Whether a 1-sided or 2-sided tolerance bound is required (determined by `side = 1` or `side = 2`, respectively). - `alpha`: The level chosen such that `1-alpha` is the confidence level. - `P`: The proportion of the population to be covered by the tolerance bound(s). - `maxiter`: A positive integer specifying the maximum number of iterations that the nonlinear least squares routine (`nls`) should run. - `new`: When `new = TRUE`, the function shows updated version of outcomes. - `...`: Optional arguments passed to `nls` when estimating the nonlinear regression equation. ## Returns `npregtol.int2` returns a list with items: - **tol**: Data frame of original response varible `y`, fitted values `y.hat`, corresponding tolerance limits. The data frame is ordered by `X` values. If there are new data for prediction, predicted values are attached in the end. - **alpha.P.side**: Model specifications of critical level, content level and side. - **reg.type**: Type of regression model. - **model**: The linear regression model fitted. - **newdata**: X values of new data for prediction. - **xy.data.original**: Original data frame ## Details It is highly recommended that the user specify starting values for the `nls` routine. ## References Wallis, W. A. (1951), Tolerance Intervals for Linear Regression, in **Second Berkeley Symposium on Mathematical Statistics and Probability**, ed. J. Neyman, Berkeley: University of CA Press, 43--51. Young, D. S. (2013), Regression Tolerance Intervals, **Communications in Statistics - Simulation and Computation**, 42 , 2040--2055. ## See Also `nls`, `nlregtol.int` ## Examples ```r ## 95%/95% 2-sided nonlinear regression tolerance bounds ## for a sample of size 50. set.seed(100) x <- runif(50, 5, 45) f1 <- function(x, b1, b2) b1 + (0.49 - b1)*exp(-b2*(x - 8)) + rnorm(50, sd = 0.01) y <- f1(x, 0.39, 0.11) formula <- as.formula(y ~ b1 + (0.49 - b1)*exp(-b2*(x - 8))) out1 <- nlregtol.int(formula = formula, xy.data = data.frame(cbind(y, x)), x.new=c(10,20), side = 2, alpha = 0.05, P = 0.95 , new = TRUE) out1 ######### set.seed(100) x1 <- runif(50, 5, 45) x2 <- rnorm(50, 0, 10) f1 <- function(x1, x2, b1, b2) {(0.49 - b1)*exp(-b2*(x1 + x2 - 8)) + rnorm(50, sd = 0.01)} y <- f1(x1 , x2 , 0.25 , 0.39) formula <- as.formula(y ~ (0.49 - b1)*exp(-b2*(x1 + x2 - 8))) out2 <- nlregtol.int(formula = formula, xy.data = data.frame(cbind(y, x1 , x2)), x.new=cbind(c(10,20) , c(47 , 53)), side = 2, alpha = 0.05, P = 0.95 , new = TRUE) out2 ```

nonlinregtolint function