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

In MethComp: Analysis of Method Comparison studies

Description

The function makes a regression of y on x, assuming that both x and y are measured with error. This problem only has an analytical solution if the ratio of the variances is known, hence this is required as an input parameter.

Usage

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

Arguments

x	numerical variable.
y	numerical 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?

Details

The formal model underlying the procedure is based on a so called functional relationship:

x_i=k_i + e_1i, y_i=alpha + beta k_i + e_2i

with var(e_1i)=s, var(e_2i)=VR*s, where VR is the known variance ratio.

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 k_i's.

This is not in Peter Sprent's book (see references).

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.

Author(s)

Bendix Carstensen, Steno Diabetes Center, [e-mail], http://BendixCarstensen.com.

References

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

WE Deming: Statistical adjustment of data, New York: Wiley, 1943. [This is a reference taken from a reference list — I never saw the book myself].

See Also

MCmcmc

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,  resp. 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)

Example output

Loading required package: nlme
Intercept     Slope   sigma.x   sigma.y 
0.8790873 3.3424617 1.1140694 1.1140694 
Intercept     Slope   sigma.x   sigma.y 
 1.252795  3.206296  1.069370  1.512318 
y  = alpha + beta* x 
           Estimate S.e.(boot)       50%       2.5%    97.5%
Intercept 0.8790873 0.78192861 0.8513765 -0.7698304 2.250194
Slope     3.3424617 0.24615506 3.3430884  2.9218856 3.870070
sigma.x   1.1140694 0.07038687 1.1008308  0.9691499 1.243525
sigma.y   1.1140694 0.07038687 1.1008308  0.9691499 1.243525
           Estimate S.e.(boot)       50%       2.5%    97.5%
Intercept 0.8790873 0.79163147 0.8520398 -0.8987695 2.294932
Slope     3.3424617 0.26103801 3.3399717  2.9293248 3.963580
sigma.x   1.1140694 0.07109705 1.0971917  0.9598462 1.236436
sigma.y   1.1140694 0.07109705 1.0971917  0.9598462 1.236436
 num [1:1000, 1:4] 1.8767 0.7007 1.0333 1.021 -0.0699 ...
 - attr(*, "dimnames")=List of 2
  ..$ : NULL
  ..$ : chr [1:4] "Intercept" "Slope" "sigma.x" "sigma.y"

Call:
lm(formula = y ~ x)

Residuals:
    Min      1Q  Median      3Q     Max 
-7.1383 -2.2684 -0.1284  2.0472  6.8795 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   4.0960     0.5700   7.186 1.32e-10 ***
x             2.1703     0.1719  12.628  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3.201 on 98 degrees of freedom
Multiple R-squared:  0.6194,	Adjusted R-squared:  0.6155 
F-statistic: 159.5 on 1 and 98 DF,  p-value: < 2.2e-16

  Intercept       Slope     sigma.x     sigma.y 
4.095971994 2.170344488 0.003200742 3.200741562 

MethComp documentation built on May 2, 2019, 5:06 p.m.