Calibration for the simple linear regression model

The function calibrate computes the maximum likelihood estimate and a condfidence interval for the unknown predictor value that corresponds to an observed value of the response (or vector thereof) or specified value of the mean response. See the reference listed below for more details.

Usage

calibrate(object, ...)

# Default S3 method
calibrate(
  object,
  y0,
  interval = c("inversion", "Wald", "none"),
  level = 0.95,
  mean.response = FALSE,
  adjust = c("none", "Bonferroni", "Scheffe"),
  k,
  ...
)

# S3 method for class 'formula'
calibrate(formula, data = NULL, ..., subset, na.action = stats::na.fail)

# S3 method for class 'lm'
calibrate(
  object,
  y0,
  interval = c("inversion", "Wald", "none"),
  level = 0.95,
  mean.response = FALSE,
  adjust = c("none", "Bonferroni", "Scheffe"),
  k,
  ...
)

Arguments

object: A matrix, list, data frame, or object that inherits from class stats::lm().
...: Additional optional arguments. At present, no optional arguments are used.
y0: The value of the observed response(s) or specified value of the mean response.
interval: The method to use for forming a confidence interval.
level: A numeric scalar between 0 and 1 giving the confidence level for the interval to be calculated.
mean.response: Logicial indicating whether confidence intervals should correspond to an observed response(s) (FALSE) or a specified value of the mean response (TRUE). Default is FALSE.
adjust: A logical value indicating if an adjustment should be made to the critical value used in constructing the confidence interval. This useful when the calibration curve is to be used k > 0 times.
k: The number of times the calibration curve is to be used for computing a confidence interval. Only needed when adjust = TRUE.
formula: A formula of the form y ~ x.
data: an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which stats::lm() was called.
subset: An optional vector specifying a subset of observations to be used in the fitting process.
na.action: a function which indicates what should happen when the data contain NAs.

Value

An object of class "invest" containing the following components:

estimate The estimate of x0.
lwr The lower confidence limit for x0.
upr The upper confidence limit for x0.
se An estimate of the standard error (Wald interval only).
interval The method used for calculating lower and upper (only used by print method).

Note

The invest() function is more general, but is based on numerical techniques to find the solution. When the underlying model is that of the simple linear regression model with normal errors, closed-form expressions exist which are utilized by the calibrate function.

References

Graybill, F. A., and Iyer, H. K. (1994) Regression analysis: Concepts and Applications. Duxbury Press.

Miller, R. G. (1981) Simultaneous Statistical Inference. Springer-Verlag.

Examples

#
# Arsenic example (simple linear regression with replication)
#

# Inverting a prediction interval for an individual response
arsenic.lm <- stats::lm(measured ~ actual, data = arsenic)
plotFit(arsenic.lm, interval = "prediction", shade = TRUE, 
        col.pred = "lightblue")
(cal <- calibrate(arsenic.lm, y0 = 3, interval = "inversion"))
#> estimate    lower    upper 
#> 2.931449 2.536740 3.325140 
abline(h = 3)
segments(cal$estimate, 3, cal$estimate, par()$usr[3])
arrows(cal$lower, 3, cal$lower, par()$usr[3])
arrows(cal$upper, 3, cal$upper, par()$usr[3])


#
# Crystal weight example (simple linear regression)
#

# Inverting a confidence interval for the mean response
crystal.lm <- stats::lm(weight ~ time, data = crystal)
plotFit(crystal.lm, interval = "confidence", shade = TRUE,
        col.conf = "lightblue")
(cal <- calibrate(crystal.lm, y0 = 8, interval = "inversion", 
                  mean.response = TRUE))
#> estimate    lower    upper 
#> 15.88820 14.65896 17.15963 
abline(h = 8)
segments(cal$estimate, 8, cal$estimate, par()$usr[3])
arrows(cal$lower, 8, cal$lower, par()$usr[3])
arrows(cal$upper, 8, cal$upper, par()$usr[3])


# Wald interval and approximate standard error based on the delta method
calibrate(crystal.lm, y0 = 8, interval = "Wald", mean.response = TRUE)
#>   estimate      lower      upper         se 
#> 15.8881952 14.6526267 17.1237638  0.5670834