MethComp (version 1.30.0)

Deming: Regression with errors in both variables (Deming regression)

Description

The formal model underlying the procedure is based on a so called functional relationship: $$x_i=\xi_i + e_{1i}, \qquad y_i=\alpha + \beta \xi_i + e_{2i}$$ with \(\mathrm{var}(e_{1i})=\sigma\), \(\mathrm{var}(e_{2i})=\lambda\sigma\), where \(\lambda\) is the known variance ratio.

Usage

Deming(
  x,
  y,
  vr = sdr^2,
  sdr = sqrt(vr),
  boot = FALSE,
  keep.boot = FALSE,
  alpha = 0.05
)

Value

If boot==FALSE a named vector with components

Intercept, Slope, sigma.x, sigma.y, where x

and y are substituted by the variable names.

If boot==TRUE a matrix with rows Intercept,

Slope, sigma.x, sigma.y, and colums giving the estimates, the bootstrap standard error and the bootstrap estimate and c.i. as the 0.5,

\(\alpha/2\) and \(1-\alpha/2\) quantiles of the sample.

If keep.boot==TRUE this summary is printed, but a matrix with columns

Intercept,

Slope, sigma.x, sigma.y and boot rows is returned.

Arguments

x

a numeric variable

y

a numeric variable

vr

The assumed known ratio of the (residual) variance of the ys relative to that of the xs. Defaults to 1.

sdr

do. for standard deviations. Defaults to 1. vr takes precedence if both are given.

boot

Should bootstrap estimates of standard errors of parameters be done? If boot==TRUE, 1000 bootstrap samples are done, if boot is numeric, boot samples are made.

keep.boot

Should the 4-column matrix of bootstrap samples be returned? If TRUE, the summary is printed, but the matrix is returned invisibly. Ignored if boot=FALSE

alpha

What significance level should be used when displaying confidence intervals?

Author

Bendix Carstensen, Steno Diabetes Center, , http://BendixCarstensen.com

Details

The estimates of the residual variance is based on a weighting of the sum of squared deviations in both directions, divided by \(n-2\). The ML estimate would use \(2n\) instead, but in the model we actually estimate \(n+2\) parameters --- \(\alpha, \beta\) and the \(n\) \(\xi s\). This is not in Peter Sprent's book (see references).

References

Peter Sprent: Models in Regression, Methuen & Co., London 1969, ch.3.4.

WE Deming: Statistical adjustment of data, New York: Wiley, 1943.

Examples

# 'True' values 
M <- runif(100,0,5)
# Measurements:
x <- M + rnorm(100)
y <- 2 + 3 * M + rnorm(100,sd=2)
# Deming regression with equal variances, variance ratio 2.
Deming(x,y)
Deming(x,y,vr=2)
Deming(x,y,boot=TRUE)
bb <- Deming(x,y,boot=TRUE,keep.boot=TRUE)
str(bb)
# Plot data with the two classical regression lines
plot(x,y)
abline(lm(y~x))
ir <- coef(lm(x~y))
abline(-ir[1]/ir[2],1/ir[2])
abline(Deming(x,y,sdr=2)[1:2],col="red")
abline(Deming(x,y,sdr=10)[1:2],col="blue")
# Comparing classical regression and "Deming extreme"
summary(lm(y~x))
Deming(x,y,vr=1000000)