Bootstrap confidence intervals of mean

Description

Bootstrap confidence intervals of mean

Usage

boot.CI(x, alpha = 0.05, CI.type = "all")

Arguments

Value

boot.CI return list of confidence intervals of mean (CI.percent: percentile, CI.BC: bias-corrected and CI.BCa: bias-corrected and accelerated).

Examples

set.seed(12)
boot.CI(rnorm(1000, mean=0, sd=1), alpha=0.05, CI.type="per") # example of wrong input for type
boot.CI(rnorm(1000, mean=0, sd=1), alpha=0.05, CI.type="all") # require all type

Linear plateau model estimation by bootstrapping residuals

Description

boot.resid.linear.plateau is the core function to implement bootstrapping residuals on linear plateau models, which assumes y ~ a + b * (x - c) * (x <= c). Note that this function may take minutes up to days. Parallel computing may be necessary. We suggest users start with a smaller B and moderate n.start to see if the bootstrap models can converge. In general, increasing n.start and plus_minus may help with ease of convergence. For rigorous statistical inference, B should be on the order of a thousand.

Usage

boot.resid.linear.plateau(
  mod,
  data,
  x.range = data.frame(x = seq(0, 280, by = 40)),
  B = 100 - 1,
  plus_minus = 100,
  n.start = 1000,
  print.progress = TRUE
)

Arguments

Value

boot.resid.linear.plateau returns a list of two elements: result: matrix with B rows and columns containing bootstrap sample for parameter (a,b,c), optimal N and yield (max_x, max_y), log-likelihood (logLik) and N values of interest; x.range: range of x considered for prediction interval (same as x.range in vector form)

Examples

set.seed(1)
x <- rep(1:300, each=4)
a <- 8; b <- 0.05; c <- 100
y <- a + b * (x - c) * (x <= c) +
    rnorm(length(x), sd=1)
d <- cbind(x,y)

# a converged example:
ans <- f.linear.plateau(d, start=list(a = 7, b = 0.1, c = 150),
    plus_minus=10, n.start=10, msg=FALSE)


ans.boot <- boot.resid.linear.plateau(ans, d, x.range=seq(0,280,by=40),
    B=9, plus_minus = 1e2, n.start=1000, print.progress=TRUE) # use larger B for inference

Fitting quadratic model using multiple initial values

Description

boot.resid.linear.plateau is the core function to implement bootstrapping residuals on quadratic models, which assumes y ~ a+b*x+c*x^2. Note that this function may take minutes up to days. Parallel computing may be necessary. We suggest users start with a smaller B and moderate n.start to see if the bootstrap models can converge. In general, increasing n.start and plus_minus may help with ease of convergence. For rigorous statistical inference, B should be on the order of a thousand.

Usage

boot.resid.quad(
  mod,
  data,
  x.range = data.frame(x = seq(0, 280, by = 40)),
  B = 100 - 1,
  plus_minus = 10,
  n.start = 20,
  print.progress = TRUE
)

Arguments

Value

boot.resid.quad.plateau returns a list of two elements: result: matrix with B rows and columns containing bootstrap sample for parameter (a,b,c), optimal N and yield (max_x, max_y), log-likelihood (logLik) and N values of interest; x.range: range of x considered for prediction interval (same as x.range in vector form)

Examples

set.seed(1)
x <- rep(1:300, each=5)
a <- 8; b <- 0.05; c <- -1e-3
y <- (a + b * x + c *x^2) + rnorm(length(x), sd=0.1)
d <- cbind(x,y)

ans <- f.quad(d, start=list(a = 7, b = 0.02, c = 1e-5),
    plus_minus=10, n.start=10, msg=FALSE)


ans.boot <- boot.resid.quad(ans, d, x.range=seq(0,280,by=40),
    B=1e1-1, plus_minus = 1e1, n.start=20, print.progress=TRUE) # use larger B for inference

Quadratic plateau model estimation by bootstrapping residuals

Description

boot.resid.quad.plateau is the core function to implement bootstrapping residuals on quadratic plateau models, which assumes y = (a + b * x + c *x^2) * (x <= -0.5*b/c) + (a + -b^2/(4 * c)) * (x > -0.5 * b/c). Note that this function may take minutes up to days. Parallel computing may be necessary. We suggest users start with a smaller B and moderate n.start to see if the bootstrap models can converge. In general, increasing n.start and plus_minus may help with ease of convergence. For rigorous statistical inference, B should be on the order of a thousand.

Usage

boot.resid.quad.plateau(
  mod,
  data,
  x.range = data.frame(x = seq(0, 280, by = 40)),
  B = 100 - 1,
  plus_minus = 100,
  n.start = 5000,
  print.progress = TRUE
)

Arguments

Value

boot.resid.quad.plateau returns a list of two elements: result: matrix with B rows and columns containing bootstrap sample for parameter (a,b,c), optimal N and yield (max_x, max_y), log-likelihood (logLik) and N values of interest; x.range: range of x considered for prediction interval (same as x.range in vector form)

Examples

set.seed(1)
x <- rep(1:300, each=5)
a <- 8; b <- 0.05; c <- -1e-4
y <- (a + b * x + c *x^2) * (x <= -0.5*b/c) + (a + -b^2/(4 * c)) * (x > -0.5 * b/c) +
    rnorm(length(x), sd=0.1)
d <- cbind(x,y)

ans <- f.quad.plateau(d, start=list(a = 7, b = 0.02, c = 1e-5),
    plus_minus=10, n.start=10, msg=FALSE)


boot.resid.quad.plateau(ans, d, x.range=seq(0,280,by=40),
    B=1e1-1, plus_minus = 1e2, n.start=1000, print.progress=TRUE) # use larger B for inference

Two sample bootstrap test for comparing different in sample1 and sample2, not necessary with same sample size

Description

Two sample bootstrap test for comparing different in sample1 and sample2, not necessary with same sample size

Usage

compare.two.sample(sample1, sample2, fun = mean, R = 1000)

Arguments

Value

compare.two.sample return a list with two components, namely, p.value: two tailed p-value for the bootstrap test object: a "simpleboot" object allowing further analysis using other R packages, such as boot)

Examples

set.seed(1203)
# compare median of two expontential r.v.
compare.two.sample(rexp(100, rate=1), rexp(100, rate=2), fun=median, R=1e3)$p.value

f.Q1 <- function(x) quantile(x, probs=0.25)
compare.two.sample(rnorm(100, mean=0), rnorm(200, mean=0.5), fun=f.Q1, R=1e3)$p.value

Fitting linear plateau model using multiple initial values

Description

f.linear.plateau fits linear plateau model using multiple initial values. The multiple initial values are randomly sampled in a "cube" of parameter space. More precisely, linear plateau model assumes y ~ a + b * (x - c) * (x <= c).

Usage

f.linear.plateau(
  d,
  start = list(a = 1, b = 1, c = 1),
  plus_minus = 100,
  n.start = 1000,
  msg = FALSE
)

Arguments

Value

f.linear.plateau returns a list of two components (if converged): nls.summary: summary of the fitted model; nls.model: nls object

Examples

set.seed(4)
x <- rep(1:300, each=4)
a <- 8; b <- 0.05; c <- 100
y <- a + b * (x - c) * (x <= c) +
    rnorm(length(x), sd=0.1)
d <- cbind(x,y)

# a converged example:
ans <- f.linear.plateau(d, start=list(a = 7, b = 0.1, c = 150),
    plus_minus=10, n.start=10, msg=FALSE)

summary(ans$nls.model)

Fitting quadratic model using multiple initial values

Description

f.quad fits quadratic model using multiple initial values. The multiple initial values are randomly sampled in a "cube" of parameter space. More precisely, quadratic model assumes y ~ a+b*x+c*x^2,

Usage

f.quad(
  d,
  start = list(a = 1, b = 1, c = 1),
  plus_minus = 1,
  n.start = 10,
  msg = FALSE
)

Arguments

Value

f.quad returns a list of two components (if converged): nls.summary: summary of the fitted model; nls.model: nls object

Examples

set.seed(1)
x <- rep(1:300, each=2)
a <- 8; b <- 0.05; c <- -1e-3
y <- a + b*x + c*x^2 + rnorm(length(x), sd=0.1)
d <- cbind(x,y)

# a converged example:
ans <- f.quad(d, start=list(a = 7, b = 0.02, c = 1e-5),
    plus_minus=10, n.start=10, msg=FALSE)

summary(ans$nls.model)

Fitting quadratic plateau model using multiple initial values

Description

f.quad.plateau fits quadratic plateau model using multiple initial values. The multiple initial values are randomly sampled in a "cube" of parameter space. More precisely, quadratic plateau model assumes y ~ (a + b * x + c *x^2) * (x <= -0.5*b/c) + (a + -b^2/(4 * c)) * (x > -0.5 * b/c).

Usage

f.quad.plateau(
  d,
  start = list(a = 1, b = 1, c = 1),
  plus_minus = 100,
  n.start = 1000,
  msg = FALSE
)

Arguments

Value

f.quad.plateau returns a list of two components (if converged): nls.summary: summary of the fitted model; nls.model: nls object

Examples

set.seed(3)
x <- rep(1:300, each=4)
a <- 8; b <- 0.05; c <- -1e-4
y <- (a + b * x + c *x^2) * (x <= -0.5*b/c) + (a + -b^2/(4 * c)) * (x > -0.5 * b/c) +
    rnorm(length(x), sd=0.1)
d <- cbind(x,y)

# a converged example:
ans <- f.quad.plateau(d, start=list(a = 7, b = 0.02, c = 1e-5),
    plus_minus=10, n.start=10, msg=FALSE)

summary(ans$nls.model)