Deming Regression

Kevin O’Brien

Deming Regression

• Orthogonal regression (also known as “Deming regression”) examines the linear relationship between two continuous variables.

• Unlike simple linear regression, both the response and predictor in orthogonal regression contain measurement error. In simple regression, only the response variable contains measurement error.

• It’s often used to test whether two instruments or methods are measuring the same thing, and is most commonly used in clinical sciences to test the equivalence of measurement instruments. This is an extremely common use case (see Bland-Altman plot)

Create Some Data

library(MethComp)

# 'True' values
M <- runif(100,0,5)

# Measurements - with generated error terms
x <- M + rnorm(100)
y <- 2 + 3 * M + rnorm(100,sd=2)

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Specifying Variance

Deming regression with equal variances

Deming(x,y)

## Intercept     Slope   sigma.x   sigma.y 
##  1.753048  3.279962  1.257293  1.257293

Specifying the Variance Ratio as 2

Deming(x,y,vr=2)

## Intercept     Slope   sigma.x   sigma.y 
##  2.157498  3.112469  1.204702  1.703706

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OLS Model Estimates

# Comparing classical regression and "Deming extreme"
summary(lm(y~x))

## 
## Call:
## lm(formula = y ~ x)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.8089  -2.0615   0.4823   2.0092   9.1973 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   4.8934     0.5433   9.006 1.73e-14 ***
## x             1.9795     0.1741  11.369  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.442 on 98 degrees of freedom
## Multiple R-squared:  0.5688, Adjusted R-squared:  0.5644 
## F-statistic: 129.3 on 1 and 98 DF,  p-value: < 2.2e-16

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Plotting the Different Model Fits

• Blue: OLS Regression
• Red : Deming Regression

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Bootstrap Estimates

Deming(x,y,boot=TRUE)

## y  = alpha + beta* x

##           Estimate S.e.(boot)      50%       2.5%    97.5%
## Intercept 1.753048 1.00191990 1.749247 -0.3744253 3.430094
## Slope     3.279962 0.33099820 3.286663  2.7739982 4.050519
## sigma.x   1.257293 0.09518457 1.245123  1.0655869 1.428868
## sigma.y   1.257293 0.09518457 1.245123  1.0655869 1.428868

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