Simple Deming Regression

We can start by creating some fake data to work with.

library(SimplyAgree)

dat = data.frame(
  x = c(7, 8.3, 10.5, 9, 5.1, 8.2, 10.2, 10.3, 7.1, 5.9),
  y = c(7.9, 8.2, 9.6, 9, 6.5, 7.3, 10.2, 10.6, 6.3, 5.2)
)

Also, we will assume, based on historical data, that the measurement error ratio is equal to 4.

The data can be run through the dem_reg function and the results printed.

dem1 = dem_reg(x = "x",
               y = "y",
               data = dat,
               error.ratio = 4,
               weighted = FALSE)
dem1
#> Deming Regression with 95% C.I.
#>               coef      bias     se df lower.ci upper.ci        t p.value
#> Intercept -0.08974 -0.044938 1.7220  8  -4.0607    3.881 -0.05212  0.9597
#> Slope      1.00119  0.003529 0.1872  8   0.5696    1.433  0.00638  0.9951

The resulting regression line can then be plotted.

plot(dem1)

The assumptions of the Deming regression model, primarily normality and homogeneity of variance, can then be check with the check method for Deming regression results. Both plots appear to be fine with regards to the assumptions.

check(dem1)

Weighted Deming Regression

For this example, I will rely upon the “ferritin” data from the deming R package.

library(deming)
data('ferritin')

head(ferritin)
#>   id period old.lot new.lot
#> 1  1      1       1       1
#> 2  2      1       3       3
#> 3  3      1      10       9
#> 4  4      1      13      11
#> 5  5      1      13      12
#> 6  6      1      15      13

Let me demonstrate the problem with using simple Deming regression when the weights are helpful. When we look at the two plots below, we can see there is severe problem with using the “un-weighted” model.

dem2 = dem_reg(
  x = "new.lot",
  y = "old.lot",
  data = ferritin,
  weighted = FALSE
)
dem2
#> Deming Regression with 95% C.I.
#>             coef      bias      se  df lower.ci upper.ci      t p.value
#> Intercept 5.2157 -0.235818 2.18603 160   0.8985    9.533  2.386 0.01821
#> Slope     0.9637  0.002597 0.02505 160   0.9143    1.013 -1.448 0.14949

check(dem2)

Now, let us see what happens when weighted is set to TRUE.

dem2 = dem_reg(
  x = "new.lot",
  y = "old.lot",
  data = ferritin,
  weighted = TRUE
)
dem2
#> Weighted Deming Regression with 95% C.I.
#>               coef       bias       se  df lower.ci upper.ci       t   p.value
#> Intercept -0.02616  0.0065148 0.033219 160 -0.09176  0.03945 -0.7874 4.322e-01
#> Slope      1.03052 -0.0001929 0.006262 160  1.01815  1.04288  4.8729 2.626e-06

plot(dem2)

check(dem2)

Measurement Error

A Deming regression model also assumes the measurement error (σ²) ratio is constant.

$$ \lambda = \frac{\sigma^2_\epsilon}{\sigma^2_\delta} $$ In SimplyAgree, the error ratio can be set with the error.ratio argument. It defaults to 1, but can be changed by the user. If replicate measures are taken, then the user can use the id argument to indicate which measures belong to which subject/participant. The measurement error, and the error ratio, will then be estimated from the data itself.

If the data was not measured in replicate then the error ratio (λ) can be estimated from the coefficient of variation (if that data is available) and the mean of x and y ($\bar x, \space \bar y$).

$$ \lambda = \frac{(CV_y \cdot \bar y)^2}{(CV_x \cdot \bar x)^2} $$

Weights

In some cases the variance of X and Y may increase proportional to the true value of the measure. In these cases, it may be prudent to use “weighted” Deming regression models. The weights used in SimplyAgree are the same as those suggested by Linnet (1993).

$$ \hat w_i = \frac{1}{ [ \frac{x_i + \lambda \cdot y_i}{1 + \lambda}]^2} $$

Weights can also be provided through the weights argument. If weighted Deming regression is not selected (weighted = FALSE), the weights for each observation is equal to 1.

The estimated mean of X and Y are then estimated as the following.

$$ \bar x_w = \frac{\Sigma^{N}_{i=1} \hat w_i \cdot x_i}{\Sigma^{N}_{i=1} \hat w_i} $$

$$ \bar y_w = \frac{\Sigma^{N}_{i=1} \hat w_i \cdot y_i}{\Sigma^{N}_{i=1} \hat w_i} $$

Estimating the Slope and Interept

First, there are 3 components ($v_x, \space v_y, \space cov_{xy}$)

$$ v_x = \Sigma_{i=1}^N \space \hat w_i \cdot (x_i- \bar x_w)^2 $$ $$ v_y = \Sigma_{i=1}^N \space \hat w_i \cdot (y_i- \bar y_w)^2 $$ $$ cov_{xy} = \Sigma_{i=1}^N \space \hat w_i \cdot (x_i- \bar x_w) \cdot (y_i- \bar y_w) $$

The slope (b₁) can then be estimated with the following equation.

$$ b_1 = \frac{(\lambda \cdot v_y - v_x) + \sqrt{(v_x-\lambda \cdot v_y)^2 + 4 \cdot \lambda \cdot cov_{xy}^2}}{2 \cdot \lambda \cdot cov_{xy}} $$ The intercept (b₀) can then be estimated with the following equation.

b₀ = ȳ_w − b₁ ⋅ x̄_w The standard errors of b1 and b0 are both estimated using a jackknife method (detailed by Linnet (1990)).